Large-aperture wide-bandwidth anti-reflection-coated siliconlenses for millimeter wavelengths

R. Datta,1, ∗ C. D. Munson,1 M. D. Niemack,2, 3 J. J. McMahon,1 J. Britton,3 E. J. Wollack,4 J.Beall,3 M. J. Devlin,5 J. Fowler,3 P. Gallardo,6 J. Hubmayr,3 K. Irwin,3 L. Newburgh,7 J. P.

Nibarger,8 L. Page,7 M. A. Quijada,4 B. L. Schmitt,5 S. T. Staggs,7 R. Thornton,9 and L. Zhang7

1University of Michigan Physics Department, 450 Church St., Ann Arbor, MI 481092Cornell University Physics Department, 109 Clark Hall, Ithaca, New York 14853

3National Institute of Standards and Technology, 325 Broadway, Boulder, CO 803054NASA Goddard Space Flight Center, 8800 Greenbelt Road, Greenbelt, MD 20771

5University of Pennsylvania Department of Physics and Astronomy,

209 South 33rd St., Philadelphia, PA 191046Departamento de Astronomia y Astrofisica, Facultad de Fisica,

Pontificia Universidad Catolica, Casilla 306, Santiago 22, Chile7Princeton University Department of Physics,

Jadwin Hall, P. O. Box 708, Princeton, NJ 085448Boulder Micro-Fabrication Facility, National Institute of Standards and Technology,

325 Broadway, MS 817.03, Boulder, CO 803059West Chester University of Pennsylvania Department of Physics,

700 South High St., West Chester, PA 19383

compiled: July 19, 2013

The increasing scale of cryogenic detector arrays for sub-millimeter and millimeter wavelength astro-physics has led to the need for large aperture, high index of refraction, low loss, cryogenic refractingoptics. Silicon with n = 3.4, low loss, and relatively high thermal conductivity is a nearly optimalmaterial for these purposes, but requires an antireflection (AR) coating with broad bandwidth, lowloss, low reflectance, and a matched coefficient of thermal expansion. We present an AR coating forcurved silicon optics comprised of subwavelength features cut into the lens surface with a custom threeaxis silicon dicing saw. These features constitute a metamaterial that behaves as a simple dielectriccoating. We have fabricated and coated silicon lenses as large as 33.4 cm in diameter with coatingsoptimized for use between 125-165 GHz. Our design reduces average reflections to a few tenths ofa percent for angles of incidence up to 30 with low cross-polarization. We describe the design, tol-erance, manufacture, and measurements of these coatings and present measurements of the opticalproperties of silicon at millimeter wavelengths at cryogenic and room temperatures. This coating andlens fabrication approach is applicable from centimeter to sub-millimeter wavelengths and can be usedto fabricate coatings with greater than octave bandwidth.

OCIS codes: 000.2190, 220.0220, 110.0110, 310.1210, 080.2208, 220.3630, 120.4570, 120.4610,160.1245, 230.4170

http://dx.doi.org/10.1364/XX.99.099999

∗ Corresponding author: dattar@umich.edu

1. Introduction

The development of large format superconductingdetector arrays for millimeter and sub-millimeterastrophysics (e.g. [1–3]) has driven the need

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for high-throughput optical designs that maintaindiffraction limited performance across the arrays(see [4] for a review). Silicon is an excellent materialfor these applications due to its high index of refrac-tion (n = 3.4), low loss-tangent (tan δ < 7× 10−5),and a relatively high thermal conductivity (k ∼ 200Wm−1K−1 at 4 K [5], which is only a factor of 50lesser than high quality OFHC copper [6] and or-ders of magnitude higher than plastic). The essen-tial development required to realize optical designsusing silicon optics at millimeter wavelengths is anappropriate anti-reflection (AR) coating to mitigatethe 30% (-5 dB) reflective loss from each opticalsurface.

AR coatings consist of one or more dielectric lay-ers placed on the surface of refractive optical el-ements. The index of refraction and thickness ofthe AR layers are chosen such that the reflectionsfrom the vacuum-AR interface and from the AR-substrate interface interfere and cancel. If the op-tical path length through a single layer AR coat-ing is one quarter wavelength, the refractive indexof the coating is

√n (where n is the index of the

substrate), and the coating is free from dielectriclosses, the cancelation is perfect at one particularfrequency. Applied to silicon, such quarter-wavecoatings reduce reflections below 1% over 1.25:1bandwidth at normal incidence.

Wider bandwidths and larger ranges of angles ofincidence can be accommodated by adding addi-tional layers to the AR coating to form a multi-layer coating. These coatings require layers with anumber of different refractive indices. Cryogenicapplications require that the coefficient of ther-mal expansion of the AR layers also be sufficientlymatched to that of the lens substrate to preventdamage upon cooling. The dielectric loss and bire-fringence of the AR coating material must also becarefully controlled. Locating materials with theseproperties poses a significant challenge to imple-menting cryogenic wide band AR coatings for sili-con. One avenue to solve this problem is to engineermaterials with the required dielectric constant bycutting sub-wavelength features into the lens sur-face [7, 8]. Engineered optical materials with prop-erties determined by their detailed geometric shape,size and orientation are referred to as artificial di-electrics or metamaterials.

Simulated dielectric metamaterial AR coatingsconsist of layers of sub-wavelength holes [9], posts,or grooves [10] cut into the substrate to be coated.This approach allows the index of the AR coatingto be precisely tuned by adjusting the geometry of

the machined features, naturally solves the thermalexpansion problem, and has loss lower than thatof the substrate. At cryogenic temperatures sili-con has a dielectric loss tangent ∼ 100 times lowerthan typical plastics like Cirlex, which have previ-ously found use as single layer AR coatings [11].The advantages of silicon metamaterial AR coat-ings include precisely controlled indices of refrac-tion, inherently matched thermal expansion, andsignificantly reduced dielectric losses.

This approach has been successfully applied toplastic millimeter wave optics through direct ma-chining and silicon optics for visible wavelengthsthrough patterning of geometric structures whichperform as a AR coating similar to that found ona moth’s eye [12–14]. At millimeter wavelengthsZhang et al. [15] fabricated an alternative classof simulated dielectric AR for silicon composed ofmetal resonant structures supported by a plasticsubstrate that had higher losses than the coatingdescribed here. Schuster et al. [16] fabricated sin-gle layer silicon micro-machined artificial dielectricsand Han et al. [17] have used silicon immersiongrating technology to fabricate planar artificial di-electrics for THz radiation. Our work is the firstdemonstration of a broad-bandwidth metamaterialAR coating on silicon optics with finite curvaturefor millimeter wavelengths.

The AR coated lenses presented here have beendeveloped for the ACTPol [18] project, a polariza-tion sensitive receiver for the Atacama CosmologyTelescope (ACT, [19]). In Section 2 we present thedesign requirements for the ACTPol optics as theyapply to this work. The design of the coating isdescribed in Section 3. In Section 4 we describethe ACTPol lens design. Cryogenic measurementsof the dielectric properties of silicon samples takenfrom 45 cm diameter ingots at millimeter wave-lengths are presented in Section 5. In Section 6we describe the fabrication process of the coatingon the lenses and present reflection measurementsof a finished lens. We conclude with a brief dis-cussion of the range of applicability of this coatingtechnique.

2. Requirements

The performance requirements for the AR coatedsilicon lenses described in this work are dictated bythe scientific goals of the ACTPol project [18]. Thisinstrument is designed to detect the faint polar-ized signals from the cosmic microwave backgroundusing three detector arrays fed by independent re-imaging optics. Two of the arrays are horn cou-pled polarimeters for the 150 GHz band (passband

3

lens reimaging optics distribution of angles of incidence

2 5 8 11 14 17 20 23 26 29 32 35 38 410

0.5

1

1.5

2

2.5

3

3.5

angle of incidence (deg)

num

ber o

f ray

s (u

nits

of 1

0000

)

Alllens 1lens 2lens 3

silicon lenses

Lyot Stop

detector array

to te

lesc

ope

Fig. 1. Left: A ray diagram of one set of ACTPol re-imagingoptics, which includes three silicon lenses feeding a detectorarray. Right: A histogram of the angles of incidence of raysat the surfaces of these three silicon lenses.

from 125 to 165 GHz), while the third array willbe a multichroic array including both 150 GHz and90 GHz bands (‘90 GHz’ passband from 80 to 110GHz) [20]. Extensive design studies demonstratedhigh index of refraction (n & 3 ) lenses were neededto achieve high optical quality across the requiredfield of view.

In this paper we focus on the coatings for the150 GHz band lenses. These coatings are concep-tually similar to those which will be used for thebroader-band multichroic lenses. The bandwidthrequirement for the 150 GHz band coatings is tohave > 99% transmission between 125 and 165GHz. The left panel of Figure 1 shows the re-imaging optics for a 150 GHz array. The opticsinclude three plano-convex silicon lenses with di-ameters up to 33.4 cm. The figure shows that rayspassing through the optical system refract over awide range of angles of incidence. This is quanti-fied in the right hand panel of Figure 1 which showsthat the distribution of angles of incidence is cen-tered near 17 and that more than 96% of the rayshave angles of incidence < 30. This sets the re-quirement that the coatings must be optimized tominimize reflections for angles of incidence between0 and 30.

Since ACTPol is a polarization sensitive exper-iment, low cross-polarization is another require-ment. Studies of polarization systematics (e.g.,[21]) suggest that the CMB temperature to polar-ization leakage must be controlled to better than1% which corresponds to a requirement that differ-ences in the transmission for the two polarizationsbe 0.5%.

To reduce the thermal emission, all optics arecooled to 4K or below. Thus the AR coating mustbe able to withstand cryogenic cycling.

As described in §5 high resistivity silicon whichis available in boules up to 45 cm in diameter, of-fers a cryogenic loss tangent tan δ < 7 × 10−5, an

Fig. 2. The geometry of the two layer metamaterial ARcoating. The left side shows a cross section through the cen-ter of the pillars indicating the design parameters discussedin the text. The right side shows an isometric view of thestructure resulting from making cuts along two perpendiculardirections.

index of refraction of n = 3.4, and a relatively highthermal conductivity making it an ideal materialfor this optical design. Our approach of directlymachining metamaterial AR coatings into the lenssurfaces guarantees that the coatings have low di-electric losses and coefficients of thermal expansionthat are inherently matched to that of the siliconlenses.

3. AR Coating DesignThe AR coating is comprised of two metamateriallayers. Each layer consists of an array of squarepillars on a square grid cut using silicon dicing sawblades by making two sets of parallel trenches ro-tated 90 relative to each other. It has been shownthat any electromagnetic structure with an axis ofn-fold (n>2) rotational symmetry must have po-larization independent reflectance and zero cross-polarization when light is incident along this axis[22]. Hence the 90 rotational symmetry of thiscoating leads to zero cross-polarization at normalincidence and low birefringence at oblique incidentangles. Figure 2 shows a cross-section and isomet-ric view of this geometry which is parameterized bythe depths d1, d2, and kerf widths k1, k2 of the in-ner and outer layer respectively and a pitch p. Theouter layer is cut by using a blade of width k2 andthe inner layer is cut by making a second set of cutsto greater depth with a thinner blade of width k1.We introduce the volume fill factor vf = (1− k/p)2to facilitate comparison to published analytic calcu-lations based on the second order effective mediumtheory developed by Rytov [23] for the effective di-electric constant in the quasi-static limit (p << λ)[14, 24–26].

The metamaterial layers can be treated as a vol-ume distribution of small electromagnetic scattererscharacterized by electric and magnetic polarizabil-ity densities [7]. When the pitch is small compared

4

to shortest wavelength of interest, the fields in thelayer are homogenous [8] and one can define an ef-fective dielectric constant which can be used to pa-rameterize the propagation properties of the media.

This effective dielectric function for the layer, εeffr ,is a function of the density of scatterers in the layer[27, 28], parametrized by the volume fill factor. Asthe feature size of a composite media approaches asignificant fraction of the radiation wavelength theeffective dielectric function becomes frequency de-pendent [29]. As the wavelength is further reducedthe artificial dielectric structure no longer appearshomogenous and these simple quasi-static consider-ations must be augmented to adequately model itsbehavior. In going beyond the quasi-static limit res-onant effects, diffraction and scattering can occurin the artificial dielectric structure [9, 30–32]. Thecoatings described here, which operate between thequasi-static and diffractive regimes, rely on numeri-cal simulations to estimate the relation between thevolume fill factor and the effective refractive indexof the metamaterial layer.

The values of d1, d2, k1, k2, and p were cho-sen using a three step design process. A prelim-inary design was carried out using a classical an-alytic model consisting of sheets of dielectric ma-terial each with a constant thickness and index ofrefraction [33]. These were varied to minimize thereflectance across the band yielding targets for theelectrical thickness and index of refraction of ourtwo layer coating.

The second step was to translate the index andthickness from the simple model into the pitch, kerfsand depths for our coating geometry. This step re-quired knowledge of the relation between the effec-tive index of refraction of an array of square pillarsand the pitch and kerf of that layer. These relationswere determined by fitting analytic models to HFSS(High Frequency Structure Simulator, [34]) simu-lations of single layer coatings with a wide rangeof volume fill factors for a few different choices ofpitch. The HFSS simulations were carried out us-ing Floquet ports and master-slave boundary condi-tions to model a box containing a single post of theAR coating as an infinite periodic two dimensionalarray of features [34].

Figure 3 shows a representative simulation of asingle layer pillar geometry with insets showing thesimulated geometry and the simple dielectric layermodeled in the fit. The fit constrained the effectiveindex neff , the effective electrical thickness teff andthe maximum frequency to which our metamaterialcoating behaves as a simple dielectric. For the pur-

Fig. 3. A comparison of the reflectance calculated from anHFSS numerical simulation of a pillar geometry to fits basedon a model of a simple dielectric layer. The geometry isspecified in the inset in the top. In the simulated geometry,p = 400 µm, tphys = 220 µm, k = 40 µm and nSi = 3.38.A comparison between fits where only the index n is freeand where both the index n and thickness t are varied isshown. The HFFS simulation and the best fit curve differby 5% at 220 GHz (∼ 1.36 mm).This corresponds to theminimum wavelength, λ5% (and a corresponding maximumfrequency f5%) for the specified pitch, p below (above) whichthe coating no longer behaves as a simple dielectric layer.

poses of our fit we define this maximum frequencyas the point where the simulated reflectance andthe analytic fit disagree at more than 5% absolutereflectance. We refer to this frequency as f5% andthe corresponding wavelength λ5%. This indicatesthe transition between specular and diffractive be-havior of the coating.

A second fit which fixed the electrical thickness tothe physical thickness of the pillars is also shown inFigure 3. This fit implies a lower f5%. The discrep-ancy between the best fit thickness teff and thephysical thickness tphys of the metamaterial layerarises since the geometric structure of the pillarsresults in fringing of fields at the junction betweenlayers causing a perturbative shift between the po-sition of the physical interface and the effective lo-cation of the junction’s electrical reference plane.An analogous effect is encountered in the designof metallic waveguide structures [31], which can beanalytically treated.

Figure 4 shows the results from fitting a numberof HFSS simulations with different pitch and fill fac-tor to the analytic model. Plots include the break-down wavelength λ5%, the effective index neff , andthe electrical thickness of a layer with index

√nSi

expressed as a ratio of the thickness of a quar-ter wave homogenous dielectric layer and a quarterwave metamaterial layer.

The recovered λ5% is in good agreement with theanalytic condition derived for a silicon optic in vac-

5

0.05 0.1 0.15 0.2 0.25 0.3

1

1.01

1.02

1.03

(tt

)

phys

/4ef

f

=

Fig. 4. A summary of results from HFSS simulations of single layers of pillars with varying geometry. Left: The minimumwavelength λ5% for which the single pillar layer is fit by an analytic model to 5% accuracy as a function of pitch p at normalincidence. Center: The effective index of refraction neff as a function of the volume fill factor vf obtained by fitting thesimulated reflectances upto the minimum wavelength λ5% = 1360 µm for a pitch, p = 400 µm, 40 µm. The line labeled“Biber” is a prediction from the analytic Biber model [24], which corresponds to the quasi-static circuit approximation for themedia. Right: The thickness teff for a slab dielectric quarter-wave coating divided by the thickness tphys of pillars formingan effective quarter wave coating as a function of p/λ. The index of the simulated dielectric coating was held fixed at

√nSi

for these simulations. A quadratic fit is plotted to guide the eye.

uum [9, 32] to prevent diffraction from a gratingarray,

p <λ

(nSi + sin θi), (1)

where p is the pitch, λ is the wavelength, nSi is theindex of silicon, and θi is the angle of incidence.This analytic expression shows that the pitch mustbe smaller than implied by the λ5% calculated atnormal incidence to minimize diffraction at obliqueangles of incidence. Consideration of manufactur-ing cost and mechanical robustness of the two-layerAR coating design favors the largest possible pitch,as this choice minimizes the number of cuts requiredto cover a given area and makes the pillars largerand therefore stronger. A 435 µm pitch leads toacceptable performance up to ∼ 175 GHz (∼ 1.7mm) which is the upper edge of our band.

Comparing this relation between vf and neff toanalytic models [14, 24] for the case where the elec-tric field of the incident wave is perpendicular to thegrooves, we found discrepancies that are reducedas the pitch decreases and we approach the quasi-static limit p/λ << 0.1. Given that our coatingdesign does not operate in this limit we find theanalytic models insufficiently accurate for our pur-poses. Thus we resort to numerical simulation tooptimize the geometry of our coating.

The variation in the electrical thickness comparedto the physical thickness (Figure 4, right) is shownfor the case of a quarter wave AR coating with index√nSi . Additional simulations show that the elec-

trical thickness depends on the index of refractionof the material on either side of the metamaterial

dielectric layer. We account for this small effect inthe final numerical optimization of the multilayercoating design.

With these relations (Figure 4) in hand we con-vert the analytic design of our two layer coating intoparameters for the pillar geometry. In the final stepwe performed a numerical optimization of the coat-ing at 15 angle of incidence using HFSS, with theconstraint that the cut geometry must match thekerf geometry of commercially available dicing saw(See Section 6). This step enables manufactura-bility, accounts for any discrepancies between theelectrical and physical thickness implied by our nu-merical simulations, and improves the performanceat larger angles of incidence. Table 1 gives theparameters for the resultant design and Figure 5shows the simulated performance for this coating asa function of frequency for several (0, 15, and 30)

Table 1. The parameters for the ACTPol AR coating.

parameter symbol dimension dimension

(µm) (units of p)

pitch p 435 1.0

kerf* k1 41 0.094

k2 190 0.437

depth d1 217 0.499

d2 349 0.802

volume fill factor* vf 1 0.820

vf 2 0.317

* nominal average values

6

200 um

Si

Frequency (GHz)

TE @ 30º incidence

TM @ 30º incidence

TM =

TM

@ 0º in

ciden

ce

TM

@ 15º incidence

target passband

120 130 140 150 160 170−50

−40

−30

−20

−10

simulated performance modeled geometry

TE @ 15º incidence

Fig. 5. Left: A simulation of the performance of the ARcoating designed for the ACTPol lenses at a range of angles ofincidence. Right: The unit cell geometry (based on measuredcut profiles) modeled using HFSS.

angles of incidence. Even at 30 incidence the bandaveraged reflections are at −26 dB and differencesbetween the two linear polarization states (TE andTM, see [33] for definition) are below 0.5% for allfrequencies and angles. At 15 incidence, averagereflections are below −31 dB. The tolerance of thedesign to various possible manufacturing errors aredescribed in detail in Appendix A.

4. ACTPol LensesThe ACTPol lenses are cylindrically symmetricplano-convex designs in which the convex surfaceis aspheric - a conic section with four perturbingterms proportional to the fourth, sixth, eighth, andtenth order of the distance from the axis. Thedesign optimization procedure for the re-imagingoptics is similar to that described in [19]; the pri-mary difference being that the ACTPol optics arerequired to be image-space telecentric to optimizethe coupling to the flat feedhorn arrays. This wasaccomplished by constraining the chief rays at eachfield point to be near normal incidence at the fo-cal plane and by allowing the tilts of lens 2, lens3, and the focal plane to vary during the optimiza-tion. The resulting design achieves Strehl ratiosgreater than 0.93 across the 150 GHz focal planeswithout accounting for the Gaussian illumination ofthe feedhorns, which effectively improves the imagequality. All three of the ACTPol optics tubes usethe same three silicon lens designs with the posi-tions and tilts adjusted to optimize the coupling tothe ACT Gregorian telescope [19].

The AR coating approach we have developed con-strains the perimeter of the lens designs to enableaccurate clamping during both machining of the ARcoatings and cryogenic cycling of the lenses. Fig-ure 6 shows a cross-section of the perimeter of an

1 mm

0.5 mm chamfer

4.9

mm

8 mm

0.65 mm

0.76 mm

Fig. 6. Cross-section of the perimeter of the ACTPol lens2, which is designed to enable machining of the AR coatingand clamping the lens without damaging the AR coating.

ACTPol lens design. Each lens includes a handlingring along its perimeter that is not AR coated andis used for mounting the lens during machining aswell as clamping to cool the lens in the cryogenicreceiver. The outer corners of the perimeter arechamfered to minimize chipping. There are stepsrising from the perimeter to the lens surface to pro-vide clearance between the perimeter clamping re-gion and the AR coating. The lens blanks weremachined by Nu-Tek Precision Optical Corporationand achieved 5 µm tolerances.

5. Properties of Silicon

Silicon manufacturing can produce different gradesof material, such as ultra high purity silicon pro-duced by the float zone process [35] that has a neg-ligible density of impurities and silicon producedby the Czochralski process [35] that has a higherlevel of impurities. As charge carriers and associ-ated states introduced by impurities are the cause ofdielectric loss, use of the highest available purity (asinferred from the room temperature resistivity) sil-icon minimizes the dielectric losses. Measurementsof the refractive index, dielectric permittivity andloss tangent of various high purity and high resis-tivity silicon sample over a range of frequencies andtemperature have been reported [36–38]. For exam-ple, ultra high purity silicon has been measured tohave a loss tangent of ∼ 1 × 10−5 at room tem-perature which would correspond to < 1% loss inthe ACTPol optical system. Unfortunately, surfacetension limits the zone-melt purification techniqueused to produce ultra high purity silicon to diame-ters below about 200 mm. Therefore, the substratesavailable for large diameter lenses considered in thiswork must be fabricated from Czochralski silicon.For our low temperature application we expect thebulk conductivity of silicon to freeze out dramati-cally reducing the dielectric losses, however, otherloss mechanisms can persist in the desired designband. We validate this general picture of the di-

7

electric loss in silicon by optically characterizingthe influence of the bulk resistivity as a functionof frequency and sample temperature.

Samples produced by the Czochralski process andreadily available in 450 mm diameter stock withbulk resistivities specified to be in the range of 1 to> 500 Ω-cm were characterized both at room tem-perature and 4K using a Bruker 125 high-resolutionFourier Transform Spectrometer (FTS) with a Ox-ford Cryostat CF continuous liquid helium flowsample chamber. The cryostat is equipped withpair of 75 µm thick polypropylene windows that en-able spectral measurement while allowing the sam-ple to be held at a regulated temperature. The sam-ples were cut to have a typical thickness of 180 µm,double side polished, and placed in a 25 mm di-ameter optical test fixture at the focus of an f/6beam. The reflections from the two surfaces of thesample form a Fabry-Perot resonator for which themodeling is relatively simple permitting measure-ment of dielectric properties. The silicon samplesare boron doped (p-type) to adjust the resistivity.The > 500 Ω-cm resistivity silicon used in the ACT-Pol lenses has the minimum dopant level.

Each sample’s transmission was measured be-tween 240 GHz (8 icm) and 18 THz (600 icm) usingdifferent combinations of sources, beam splitters,and detectors for three frequency bands between240-450GHz (8-15 icm), 450-2850 GHz (15-95 icm),and 2.85-18 THz (95-600 icm).

The spectral resolution employed, 7.5 GHz (0.25icm), fully resolves the sample’s spectral features.Sliding stages permit the sample or a reference clearaperture to be moved into the FTS beam while inthe cryostat for in situ calibration.

The transmission spectrum was modeled as a se-ries of homogeneous plane parallel dielectric lay-ers [39]. The dielectric function for the silicon wasapproximated by a classical Drude dispersion model[40]:

ε∗r(ω) = ε∗∞ −ω2

p

ω · (ω + iΓ)(2)

where ε∗r = ε′r + iε′′r is a complex function of fre-quency ω, the damping rate Γ, and the contribu-tion to the relative permittivity ε∗∞ of higher energytransitions. The plasma frequency and the damp-ing rates are related by, ω2

p = Γ/εoρ, where εo isthe permittivity of free space and ρ is the sam-ple’s bulk resistivity [41]. In this approximation,the material’s free carriers are treated as classicalpoint charges undergoing random collisions and theresulting damping is assumed to be independent of

0

0.2

0.4

0.6

0.8

1

Tra

nsm

issio

n (

−)

Silicon: 1 < ρ < 5 (Ω−cm)

0 2 4 6

0

0.2

0.4

0.6

0.8

1

Frequency (THz) 0 2 4 6 8

Frequency (THz)

Tra

nsm

issio

n (

−)

Measured

Modeled

Residual

Silicon: ρ > 500 (Ω−cm)

Tamb = 297K

Tamb = 4.5K

Fig. 7. Measured silicon transmission at room-temperature(upper panels) and 4.5 K (lower panels). The left and rightcolumns in the panel are for samples specified as 1 < ρ < 5 Ω-cm and ρ > 500 Ω-cm respectively, the sample thickness is∼ 180µm thick. The figure contains the measured FTS data(red), model (dashed blue), and residual (green).

the carrier energy. We find this representation suit-able to represent the sample’s properties over thespectral and temperature ranges of interest.

The sample thickness is known to ±0.5µm atroom temperature and corrected for thermal con-traction as a function of temperature [42]. Sincethe fringe rate is proportional to the product of therefractive index and sample thickness,uncertaintiesin the sample thickness directly limit the precisionof the determination of < (ε∗∞). Measurements ofan optically polished crystalline quartz sample withaccurately known thickness were used to measurea 1% amplitude uncertainty across the entireFTS band. This calibration uncertainty leads toa corresponding reduction in the measurement’ssensitivity to = (ε∗∞). These uncertainties wereaccounted for in fitting the FTS data. These fitsproduced root-mean-square deviations in the rangeof ∼0.005-to-0.016 between the model and theobservation spectra. Representative data for theρ > 500 and 1 < ρ < 5 Ω-cm samples are shownin Figure 7 for sample physical temperatures ofTphys=297 and 4.5 K.

The results of these fits are shown in Table 2. Asanticipated the finite plasma frequency and damp-ing rate contribute significantly to the room tem-perature losses, however, the ρ > 500 Ω-cm samplecould be approximated by a dielectric constant overthe spectral range of interest (e.g., Γ and ωp are con-sistent with zero). The influence of free carrier col-lisions at room temperature on the optical response

8

Table 2. Silicon Drude Model Fit Parameters.

ρ(Tphys) Tphys ε′∞ ε′′∞ Γ/2π ωp/2π

[Ω-cm] [K] [−] [−] [THz] [THz]

>500 297 11.7(1) 0.0015 − −4.5 11.5(5) <0.0008 − −

1-to-5 297 11.6(55) 0.0046 0.571 1.60

200 11.5(52) 0.0049 0.572 0.80

100 11.4(78) 0.0028 0.459 0.26

70 11.4(66) 0.0011 0.162 0.20

30 11.4(64) <0.0008 − −10 11.4(62) <0.0008 − −4.5 11.4(62) <0.0008 − −

becomes more pronounced in the 1 < ρ < 5 Ω-cmsample, manifesting as a reduction in transmissionat low frequencies. However, as the 1 < ρ < 5 Ω-cmsample is cooled, bulk conduction is suppressed andthis effect shifts to lower frequencies than of inter-est for millimeter wave applications. The observeddielectric parameters are a weak function of tem-perature below ∼ 30 K as anticipated given expo-nential thermal dependance of the bulk resistivity[43].

Both samples show ∼ 1% shifts in the magnitudeof the index of refraction (n ≈

√ε′∞) upon cooling.

This shift is accounted for in our optical design. Atlow temperature both the high and low resistivitysample give an upper limit for the loss tangent oftan δ ∼ ε′′∞/ε

′∞ < 7× 10−5. For the ACTPol optics

at 150 GHz, this corresponds to an absorptive lossof < 5% for our optical system comprising threelenses. The ACTPol lenses were fabricated fromρ > 500 Ω-cm silicon as it may reduce loss andfacilitates room temperature testing. Both highand low resistivity silicon would provide acceptableperformance in 4K cryogenic applications, butgiven the small cost differential high resistivitysilicon is the natural choice. Although we haveused high-resistivity silicon, our models and obser-vations suggest that the carriers will be frozen outat 4K for the range of bulk resistivities consideredhere. We note that the conductivity of siliconat room temperature, and therefore the loss, is astrong function of UV irradiance. We have notinvestigated this dependence cryogenically.

6. Coating Fabrication

The coatings were fabricated using silicon dicingsaw blades which are available in widths ranging

Lens Fabrication System

dicing saw300 nm repeatability

metrology sensor

micron accurate

gantry

prototype

lens

microscope

X

YZ

reference wa!er

(1 in each corner)

Fig. 8. The custom, micron accurate, three axis silicon dic-ing system used to fabricate coatings on the lenses. Thelabels identify the key components and axes described in theSection 6.

from 10 to hundreds of microns. Mounted on acommercial silicon dicing saw machine these bladesrepeatedly cut with micron level precision and cut-ting speeds up to several centimeters per second.Unfortunately, commercial silicon dicing machinesare not designed for curved surfaces and cannot ac-commodate the large diameter lenses required forthis project. Therefore we constructed a customthree axis silicon dicing system to fabricate thesecoatings.

The fabrication system is shown in Figure 8. Itconsists of a micron accurate three axis stage onwhich we mount an air bearing dicing spindle anda micron accurate depth gauge. The spindle andretractable depth gauge are attached to the ver-tical stage (Z-axis) which rides on the horizontal(Y-axis) stage. A lens is mounted on an aluminummounting plate on a horizontal stage (X-axis) be-low the spindle. This plate permits the lens to berotated by 0, 90, 180, and 270 and carries a ref-erence wafer that is used in setting up the bladesprior to cuts. A side looking microscope is mountedparallel to the Y-axis to characterize test cuts onthe reference wafer. Flood cooling water is sprayedon the dicing blade while cuts are made to carryaway debris. A temperature controlled water bathis used to regulate the temperature of the spindleand maintain the flood coolant and air surroundingthe dicing system within 1 C.

Our system does not show any appreciable changein cut shape or surface damage when cutting at themaximum travel speed of 50 mm/s. We conserva-tively operate at 25 mm/s for which it takes a total

9

Lens Front (Convex)

cm scale

Lens Front (Convex)

cm sccm sccm sccm sccm sccm sccm sccm sccm sccm sccm sccm sccm scalealealealealealealealealealeale

Pre cut Post cut

34

9 µ

21

7µ

206 µ 206 µ

198 µ 198 µ

178 µ 178 µ

39µ 37µ

43µ 40µ

55µ 60µ

Fig. 9. Left: Photograph of the curved front surface of anAR coated ACTPol silicon lens (lens 2) with zoomed view ofa small patch. Right: Photographs taken with side lookingmetrology microscope of a reference wafer taken prior to andafter cutting the coating on a lens.

of 12 hours of machining per lens side. Factoringin setup time it takes 6 8-hour days to fabricate asingle lens.

Figure 9 shows photographs of one of the ACTPolAR coated silicon lenses. The fabricated coating issufficiently robust to permit handling the lenses bytouching the AR coatings. The manufacture re-sulted in less than 10 out of 500,000 posts withdamage visible by eye. The shape of the cut pro-files was evaluated by making cuts on a referencesilicon wafer before cutting (pre-cut) and after cut-ting (post-cut) the lens and measuring them us-ing the side looking microscope. Comparison of thepre-cut and post-cut measurements (See Figure 9)show that the wide blade cuts repeatably with neg-ligible evolution to cut profile while the the narrowblade shows some evolution in width in the upperthird of the cut. Simulations show that this evolu-tion, leads to a few tenths of a percent increase inthe reflectance. The accuracy of the depths of thecuts is limited by a 3 micron uncertainty in the zeropoint for the blade depth and a ±7 micron quadru-ple warp in the lens. This warp is due to imper-fections in the lens mounting plate that stress thelens.

The reflectance of the completed ACTPol lenseswere measured using a scalar reflectometer. Thisreflectometer consisted of a tunable narrow bandcontinuous wave source which illuminated the flatside of lens through a 20 full width half power horn,from a distance of 5 cm, and tilted at an angle of15 relative to the lens surface normal. A receiverconsisting of an identical horn coupled to a detectordiode was placed at the mirror image of the sourcehorn relative to the plane defined by the point ofmaximum illumination of the lens and perpendicu-lar to a line joining the source and receiver. This

120 130 140 150 160 170−40

−35

−30

−25

−20

−15

Frequency (GHz)

Reflectance (dB)

Target Band

TM polarizationTE polarization measurement

sim. (two sided)

sim. (single sided)

Fig. 10. Comparison between simulations and measurementsof the fabricated lens. The two sided simulations model theAR coating on both sides of a flat silicon sample of thick-ness equal to that of the lens at its center. The single sidedsimulations model the AR coating on only one side of a flatsilicon sample.

system was calibrated by (1) placing an aluminumreflector at the same position as the lens to normal-ize the peak reflection to unity, and (2) by remov-ing the lens and calibration reflector to measure thestray reflections which were found to be negligible.

The results of this reflectance measurement areshown in Figure 10 for both the TE and TM po-larizations. Precise modeling of this measurementconfiguration requires accounting for the interfer-ence between reflections from the flat and curvedlens surface, which is beyond the scope of this work.However, the results are expected to be intermedi-ate between the reflectance from a single AR coatedsurface (single sided) and the easily simulated re-sult for two flat surfaces separated by the centralthickness of the lens (two sided). These two casesare presented in Figure 10 for both linear polar-ization states. These simulations incorporate themeasured cut profiles shown in Figure 9. We em-pirically found that moving the lens so that it iswell centered on the beam increases the interfer-ence effects (TM case) while moving the lens offcenter reduces these effects (TE case). Based onthe reasonable agreement between these simplisticsimulations and the measurement we are confidentwe are reducing reflections to a few tenths of a per-cent over the range of angles of incidence requiredfor ACTPol.

7. Conclusion

We have described a new approach for AR coatingsilicon lenses over broad bandwidths and a rangeof angles of incidence. Simulations, backed up bymeasurements of an AR coated lens, show thatthe fabricated coating of the lenses presented here

10

can reduce reflections below few tenths of a per-cent between 125 and 165 GHz for angles of in-cidence between 0 and 30 for cryogenic applica-tions. We have developed a micron-accurate 3-axissilicon dicing saw facility and are using it to man-ufacture AR coated lenses for ACTPol. We havealso shown that a range of p-type silicon dopinglevels can achieve low loss at cryogenic tempera-tures using silicon samples from boules as large as45 cm diameter. This approach for implementingwide-bandwidth AR coated silicon lenses is appli-cable for millimeter and sub-millimeter wavelengthranges and can be expanded to wider bandwidth byadding additional layers to the AR coating.

Acknowledgement

This work was supported by the U.S. NationalScience Foundation through awards AST-0965625and PHY-1214379 and NASA through the NASASpace Technology Research Fellowship traininggrant NNX12AM32H. The authors would like tothank Ki Won Yoon and Molly Dee for useful dis-cussions.

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Appendix A: TolerancesThe manufacturing tolerances for the eight typesof machining errors shown in Figure 11 were evalu-ated based on a fiducial design. These include: (1)errors in the pitch, δp; (2) errors in the centeringof the deeper groove relative to the shallow groove,δc; (3) errors in the kerf widths, δk1, δk2; (4) slopesin the walls of the grooves parameterized by δS1and δS2 which represent the distance the upper andlower groove tilt inward at the groove bottom if themean width is fixed; (5) errors in the depth of ei-ther grooves δd1, δd2, (6) differences in the depth ofgrooves in the two orthogonal directions assumingthe correct mean depth are parameterized δdLR1

for the upper and δdLR2 for the lower groove; (7)application of this coating to a surface tilted at anangle φ relative to the bottom edge of the saw and

Fig. 11. This figure shows the families of machining er-rors considered in the tolerance analysis and described in thetext. The dashed lines show the target shape for the groovesand that for the pillars while the white space in the greyrepresents the material as actually cut. Therefore differencebetween the dashed region and the white region representmachining errors. For the asymmetric depth and damagedrow errors the sketches show the geometry of the errors.

12

Ba

nd

Ave

rag

e R

efle

cta

nce

(d

B)

δp um

3dB region

( )

variation in pitch

TM @ 15º incidenceTE @ 15º incidence

TM = TE @ 0º inciden c e

0

Fig. 12. The band averaged reflectance at a 15 angle ofincidence as a function of changes in the pitch. The greyregion shows range for which the performance is within 3dBof the fiducial design.

where the depth of each groove is measured at thecenter of the groove (e.g., the effect of applying thiscoating at the edge of a curved lens); and (8) rowsof broken posts. The tolerance to variation in therefractive index of the silicon substrate, δnSi wasalso evaluated. The tolerances for these parameterswere quantified based on a fiducial model similar towhat was fabricated but with straight walled pillarsusing HFSS simulations in which the parameters ineach family were varied separately. For the major-ity of these effects the results were distilled to aband averaged reflection at 15 angles of incidence.The sensitivity to each parameter was quantified asthe displacement needed to bring about a 3 dB in-crease in reflectance. For the case of tilted surfaceand for broken posts the results of a small num-ber of simulations were evaluated and compared tothe fiducial performance. In the remainder of thissection we discuss these results.

a. Pitch: Variations in the pitch change the ef-fective index of both layers. Figure 12 shows theband averaged reflectance as function of variationin the pitch. A conservative tolerance of δp < 3µm(δp/p < 0.0075) limits the change in reflectance tobe less than 3 dB.

b. Centering: The performance was insensitiveto errors in the centering of the deeper grooveswithin the shallower grooves. This is consistentwith both layers of pillars behaving as layers witha constant effective index of refraction.

c. Kerf: The upper left panel of Figure 13presents the impact of errors in the kerf. Control-ling δk1 and δk2 to 3 µm insures a less than 3dBdegradation in performance. Varying both param-eters in the same direction (e.g., δk1 = δk2) haslittle impact on the performance. We have chosenthe fiducial design to be near the high end of themost favorable region to make the manufacture rel-

Fig. 13. This figure shows the impact of errors in the kerf,slopes in the sidewalls, overall depth errors, and asymmetricdifferences in depth between the two orthogonal cuts. Thecolor scale presents the degradation in reflection at 15 inci-dence compared to the simulations in Figure 5. The fiducialdesign is at the coordinate (0,0) in all the plots as highlightedby a “+” or arrow. The outer edge of the light blue contour(see label) represents 3dB degradation. The horizontal (ver-tical) axis represents errors in the inner (outer) layer of thecoating for the labeled parameters.

atively immune to blade wear, which could have(but did not) narrow the kerf width as machiningproceeds.

d. Sloped Walls: The upper right panel of Fig-ure 13 presents the impact of slopes in the walls leftbehind by the dicing saw. Conservative estimatesfor the 3 dB tolerance for these two parameters are6 µm for δs1 and 3 µm for δs2. Sloping both layersby a similar amount results in negligible degrada-tion.

e. Depth: The lower left panel of Figure 13presents the impact of depth errors. The 3 dB tol-erances for these parameters are 7.5 µm for δd1 and10 µm for δd2. Expressed in terms of optical pathlength the sensitivities are identical.

f. Asymmetric Depth: The lower right panel ofFigure 13 presents the impact of asymmetric depth.The 3 dB tolerances for these parameters are 15 µmfor δdLR1 and 12 µm for δdLR2 for asymmetry indirections common to both grooves. This error alsoaffects the 90 rotational symmetry of the geometryresulting in birefringence but the cross-polarizationcan be no larger than the absolute reflection.

13

surface tilted by 10o

TM @ 15º incidence

TE @ 15º incidenceTM = TE @ 0º incidence

n

3dB region

−0.3 −0.2 −0.1 0 0.1 0.2 0.3

−38

−34

−30

−26

δ Si

variation of substrate indexE-pol H-pol

0º incidence15º incidence30º incidence

target passband−60

−50

−40

−30

−20

Frequency (GHz)

120 130 140 150 160 170

Fig. 14. Left: The simulated impact of having the surfaceof the lens tilted by 10. Right: The simulated impact ofvariation in the refractive index of the silicon substrate.

g. Application to tilted surfaces: The left panelof Figure 14, left compares the performance of thiscoating applied to a surface tilted by 10 comparedwith the performance when applied to a flat surface.Fortuitously, this effect improves performance atlarge angles of incidence and has little effect at smallangles. Applying this coating by making groovesparallel to the lens symmetry axis of a curved lenswill produce acceptable performance.

h. Groups of broken posts: Assessing the im-pact of the broken posts is not straight forward be-cause the region and extent of illumination variesbetween the three lenses. The degradation due tobroken posts would thus be a function of the effec-tive aperture area of the lenses and the location ofthe defects in addition to the total area affected.Accurately quantifying this is beyond the scope ofthis work. Qualitatively, the degradation would benegligible if the fraction of total area affected issmall and more importantly, the length scale of theindividual affected areas is smaller than the wave-length of incident light. Simulating the impact ofbroken posts is computationally difficult. There-fore we resorted to a simplified model consisting ofa unit-cell containing seven posts in a row with onebroken off completely. This is equivalent to havingone seventh of the posts destroyed. This pessimistic

case produces band averaged reflections below 2%which is acceptable though it does produce non-trivial cross-polarization. We used simulation toestimate that keeping the number of broken postsbelow 1/700 would result in a negligible degrada-tion in the overall performance of the coating.

i. Index: Figure 14, right shows the bandaveraged reflectance as a function of variation inthe index. A conservative tolerance for the indexis δnSi < 0.1 (δnSi/nSi < 0.03). This leads toincreases in the reflectance of less than 3 dB overthe fiducial design. This tolerance is weaker thanthat imposed by the optical design.

The 3dB sensitivies derived in this section aresummarized in Table 3. The band averaged reflec-tivity of the fiducial design is -31 dB at 15 inci-dence which is significantly better than the require-ments for the system.

Table 3. A summary of the 3 dB sensitivities for all param-eters described in the text.

machining error parameter sensitivity tolerance

µm (units of p)

pitch (p) δp 3 7.5× 10−3

centering δc see text

kerf δk1 3 7.5× 10−3

δk2 3 7.5× 10−3

sloped walls δs1 6 1.5× 10−2

δs2 3 7.5× 10−3

depth δd1 7.5 1.9× 10−2

δd2 10 2.5× 10−2

asymmetric depth δdLR115 3.8× 10−2

δdLR212 3× 10−2

tilted surface φ see text

damaged row r ∼ 1/700 NA

index δn 0.1 NA