Flat Dielectric Grating Reflectors with High Focusing Power

David Fattal,∗ Jingjing Li, Zhen Peng, Marco Fiorentino, and Raymond G. BeausoleilHewlett-Packard Laboratories, 1501 Page Mill Road, Palo Alto, CA 94304–1123

Sub-wavelength dielectric gratings (SWG) haveemerged recently as a promising alternativeto distributed-Bragg-reflection (DBR) dielectricstacks for broadband, high-reflectivity filteringapplications. A SWG structure composed of asingle dielectric layer with the appropriate pat-terning can sometimes perform as well as thirtyor forty dielectric DBR layers, while providingnew functionalities such as polarization controland near-field amplification. In this paper, we in-troduce a remarkable property of grating mirrorsthat cannot be realized by their DBR counter-part: we show that a non-periodic patterning ofthe grating surface can give full control over thephase front of reflected light while maintaininga high reflectivity. This new feature of dielec-tric gratings could have a substantial impact on anumber of applications that depend on low-cost,compact optical components, from laser cavitiesto CD/DVD read/write heads.

Resonant effects in dielectric gratings were first clearlyidentified in the early 1990’s [1] as having promising ap-plications to free-space optical filtering [2, 3] and sens-ing [4, 5]. They typically occur in sub-wavelength grat-ings (SWG), where the first-order diffracted mode cor-responds not to freely propagating light but to a guidedwave trapped in some dielectric layer. The trapped waveis scattered into the zeroth diffracted order and interfereswith the incident light to create a pronounced modula-tion of transmission and reflection. When a high-index-contrast grating is used, the guided waves are rapidlyscattered and do not propagate very far laterally. In thiscase it is appropriate to think of the grating as a coupledresonator system, where each high index groove behavesas a (lossy) cavity. In such gratings, broad transmis-sion and reflection features can be observed, and havebeen used to design novel types of highly reflective mir-rors [6, 7]. Recently, SWG mirrors have been used toreplace the top dielectric stacks in vertical-cavity surface-emitting lasers (VCSELs) [8, 9], and in novel MEMS de-vices [10, 11]. Besides being more compact and cheaperto fabricate, these structures provide new optical featuressuch as polarization control.

In this paper, we show — both numerically and ex-perimentally — how a non-periodic design of the SWGpattern allows dramatic control of the phase front of thereflected beam, without affecting the high reflectivity ofthe mirror. We provide complete design rules on how toobtain a given phase front for the reflected beam, and wedesign a set of effectively cylindrical and spherical mirrors

which can be fabricated using simple lithography followedby etching. We believe our method can considerably re-duce the cost of optical components while offering newfunctionalities for a host of applications from laser cavityreflectors to CD/DVD read/write heads. The paper willfocus on 1D gratings with TM-polarized incident light,but the design rules apply equally well to TE-polarizedand unpolarized gratings (the latter necessitating a 2Dpattern). Transmissive gratings are also possible and willbe described elsewhere.

To understand the phase front modification providedby a non-periodic SWG mirror, we start by consider-ing the complex reflection coefficient r0(λ) of a partic-ular periodic SWG mirror, as shown in Fig. 1(a). Thewavelength-dependent reflectance and phase shift of themirror can be easily obtained using either the finite ele-ment method (FEM) or rigorous coupled wave analysis(RCWA) [12]. In the particular case of Fig. 1(a), thegrating is a set of linear silicon grooves with a periodof 0.76 µm on a quartz substrate, and is illuminated atnormal incidence with a TM-polarized plane wave (i.e.,the E-field vector is perpendicular to the grooves). Dueto the strong index contrast between silicon and air,the grating has a broad spectral region of high reflec-tivity, consistent with other reports in the literature[6].A less obvious but critical observation is that the phaseof the reflected beam varies significantly across the high-reflectivity spectral region.

In order to design grating reflectors that imprint a par-ticular phase profile on an incident beam, we first exploita well-known property of Maxwell’s equations with re-spect to a uniform spatial scale transformation. If wephysically change the spatial dimensions of a periodicgrating uniformly by a factor of α, the new reflectioncoefficient profile will be identical to that of the originalgrating, but with a wavelength axis that has also beenscaled by α. Therefore, if we design a grating that has aparticular complex reflection coefficient r0 at a vacuumwavelength λ0, then we obtain a new grating with thesame reflection coefficient at wavelength λ by multiply-ing all grating dimensions by the factor α = λ/λ0, givingr(λ) = r0(λ/α) = r0(λ0).

The critical conceptual advance that enables designswith high focusing power is the realization that a non-uniform scaling of the reference grating should allow lo-cal tuning of the value of the reflected phase while main-taining a large reflectivity for a monochromatic incidentbeam. Since high-contrast gratings operate with local-ized resonances, we expect their reflection properties ata given point in space to depend only on the local geom-

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FIG. 1: Magnitude-squared and phase of the reflection co-efficient from a 1D silicon grating of 470 nm thickness ona SiO2 substrate, subject to normal incidence illuminationwith a TM-polarized plane wave. The reflection coefficient isplotted as (a) a function of wavelength for a fixed period of0.76 µm, and (b) as a function of spatial period at a fixedwavelength of 1.55 µm.

etry around that point. Suppose that we wish to imprinta phase φ(x, y) on a reflected beam at some point onthe grating with transverse coordinates (x, y). Near thatpoint, a nonuniform grating with a slowly-varying spatialscale factor α(x, y) behaves locally as though it was a pe-riodic grating with a reflection coefficient r0[λ/α(x, y)].Therefore, given a periodic grating design with a phaseφ0 at some wavelength λ0, choosing a local scale fac-tor α(x, y) = λ/λ0 will set φ(x, y) = φ0. So long asall required values of φ0 are available within the high-reflectivity spectral window, the new non-periodic grat-

ing can be designed to support a specific phase map for aspecific application. We have followed this procedure togenerate the plot of the magnitude and phase of the com-plex reflection coefficient at λ = 1.55 µm as a functionof grating period shown in Fig. 1(b). Notice that thisapproach is in general not valid for weak gratings wherethe local structure of the incident beam is spatially aver-aged laterally upon reflection [13]. However, as we showbelow, this is a very robust and reliable guiding principlefor high contrast grating designs.

Two simple phase profiles are noteworthy. First, a lin-early varying phase profile φ(x, y) = κxx + κyy (whereκx and κy are design parameters) gives rise to a de-flected reflected beam, just as a tilted mirror would.The deflection angle θ measured from normal is given

by sin θ =√κ2x + κ2y/k0, where k0 = 2π/λ0 and λ0 is the

free space wavelength. Second, a quadratic phase pro-file φ(x, y) = k0

(x2/fx + y2/fy

)/2 gives rise to elliptical

“thin lens” focusing with focal lengths fx and fy in thexz and yz planes, respectively. In that case, the gratingbehaves as a perfect parabolic mirror. Of course, morecomplicated reflection phase profiles can also be imple-mented.

Continuously scaling the height of a particular gratinggroove typically requires grey-scale lithography. Whilethis is a mature fabrication technique, it is not readilyaccessible to most research groups and is more expen-sive than binary lithography. For this reason, wheneverpossible we seek purely planar designs for our SWG re-flectors, maintaining the grating groove thickness acrossthe entire optic. In practice, it is generally possible toobtain a phase shift similar to that achievable in 3D scal-ing using purely 2D scaling, varying the grating periodand/or duty cycle (defined as the groove width divided bythe period) only. Figure 1(b) shows an example wherea phase shift close to 2π is obtained for a fixed wave-length by varying the grating period only while keep-ing the reflectance above 98%. The resulting design isreminiscent of output-coupler gratings used to extractlight from a dielectric slab into a controlled radiationmode [14–16]. However, the operating principle is verydifferent here: it is based on the excitation of localizedresonances in the high-contrast grating rather than onleaky modes in a low-contrast grating. It is also very dif-ferent from (and superior to) conventional Fresnel lenses,where prisms and wedges of well-engineered geometricfeatures are utilized to exploit refraction and total inter-nal reflection. In contrast to Fresnel lenses, SWGs haveonly sub-wavelength features that provide a smooth re-flected phase front despite the sharp discontinuities indielectric constant.

Once we have chosen a particular spatial distributionof duty cycle and period, we must determine the gratingpattern that best approximates that distribution. Oneway to think about this problem is to view the non-

3

periodic grating as a distorted version of the referenceperiodic grating. A local change in duty cycle simplycorresponds to a local scaling of the grating pattern thatleaves the underlying lattice unchanged, a procedure thatis easily implemented for 1D or 2D gratings. A localchange of period can be implemented by a coordinatetransformation that distorts the lattice itself. A moredetailed discussion of this idea can be found in the Ap-pendix section.

Our freedom in designing optical devices using purelyplanar technology is determined by the range of phasevariation we can obtain through lateral scaling of a dielec-tric grating. Many applications — including phase frontcorrection, laser cavity reflectors, and low-numerical-aperture optics — require only a relatively small phasevariation (e.g., less than π/2) achievable through the lat-eral scaling of high-contrast gratings having a single reso-nance. For example, in the case of the planar “parabolic”reflectors described below, a phase variation of 0.8π at afixed operating wavelength can be realized by varying theduty cycle alone. Larger phase differentials can be ob-tained using dielectric gratings that exhibit merged mul-tiple resonances in their spectrum, such as the 470 nm-thick silicon grating on an SiO2 substrate operating in the1.2–2 µm wavelength range shown in Fig. 1(a). Apply-ing the design principle outlined above, Fig. 1(b) demon-strates that a total phase range of 1.7π can be realizedat an operating wavelength of 1.55 µm by varying bothduty cycle and local grating period, while the reflectanceremains above 98%. Although this phase variation is al-ready large enough to support a wide range of free-spaceoptical applications, achieving a full 2π is important be-cause it enables arbitrary phase front control. A straight-forward means of reaching this significant goal is to usea few (as little as two) grating thicknesses with comple-mentary phase variations that together cover the entire2π range. This allows a reflectance in excess of 98% tobe maintained using a process compatible with planartechnology. Another solution (used next to simulate alarge-NA parabolic reflector) arises from the observationthat we can realize a 2π phase variation (and thus fullwavefront control) using a single-thickness design by ac-cepting a small decrease in the reflectance.

Figure 2 provides an example of arbitrary wavefrontcontrol with reasonably high reflectance using lateralscaling alone. Here we have used a finite element methodto simulate a 1D (cylindrical) parabolic reflector of largenumerical aperture (NA = 0.45) with a diameter of50 µm and a focal length of 50 µm . The design of thedevice requires a total phase variation of 8π from cen-ter to edge, which is obtained by modulation of the localperiod only using the reflectance curve of Fig. 1(b). Asshown in Fig. 2, this optic focuses an incident gaussianbeam with a 20 µm waist at the grating surface downto a spot with a 1.66 µm waist (i.e., within 25% of thediffraction limit), with a total reflectance of 85%. This

rational design method constitutes an excellent startingpoint for a systematic optimization of the groove widthand period, which is very likely to improve the amplitudeand beam quality of the reflected field.

We have tested the non-periodic SWG concept experi-mentally by fabricating TM reflectors with aperture radiiof 150 µm (limited by our e-beam lithography tool) anda relatively long focal length of 17.2 mm to facilitate themeasurement of the beam profile. This design requireda phase variation of up to ∼ 0.8π that was obtainedthrough spatial modulation of the duty cycle alone, asdescribed in detail in the Appendix section. We fabri-cated a flat, a cylindrical, and a spherical SWG mirrormade of a 450 nm amorphous silicon layer on a quartzsubstrate with a (uniform) grating period of 670 nm. Thefabrication procedure involving e-beam lithography is de-scribed below in the Methods section. The groove widthis uniform for the flat reflector but is spatially modu-lated for the cylindrical and spherical mirrors. An opticalmicroscope image of a fabricated spherical grating mir-ror is shown in Fig. 3, along with scanning microscopeimages of the silicon grooves at various locations. Wetested the device with a collimated laser beam havinga 100 µm waist radius. Our main experimental resultsare summarized in Fig. 4, with the schematics of thegrating patterns shown in the top row, and the measuredbeam profiles after reflection off each grating displayed inthe middle row. The reflected beam parameters can bereconstructed from the measured beam radii at variouspositions, as shown in the bottom row of Fig. 4. Usingthese parameters, we can calculate the focal lengths to be20± 3 mm for both mirrors, close to the design value of17.2 mm. The reflectance of the mirrors is in the range of80–90%, lower than the expected 98%, primarily due toproximity effects in the e-beam lithography step, as wellas the surface roughness of the silicon grooves evidentin Fig. 3. We believe that focal length and reflectancewill be much closer to the design specifications after op-timization of the fabrication procedure.

The non-periodic SWG mirrors presented here couldprovide new optical functionalities for a host of passiveor active devices. They can provide tight lateral con-finement of light in a VCSEL cavity, improving on theperiodic-SWG-based structure demonstrated in [9] andallowing high-speed, high-power, single-mode operation.They can also be used to shape directly the transversemode profile of a microlaser through the output mir-ror, replace the expensive compound lens systems in anumber of consumer electronic products (DVD players,digital cameras, etc), and provide a cheaper alternativeto micro-lens arrays used in CMOS and CCD sensorsand quantum computation implementations relying ontrapped atoms. Combined with techniques to modulatethe position, shape, or effective index of the grating layer(electro-optic effect, MEMS tuning, free carrier plasmaeffect, etc.), the approach presented here could become

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FIG. 2: Numerical simulation of a 50 µm-aperture focusing reflector with NA = 0.45, featuring an 8π differential phase shiftfrom center to edge. The grating is illuminated with a collimated beam of waist 20 µm at a wavelength of 1.55 µm. The plotshows the intensity of the reflected field as a function of position, as well as both the phase distribution 1 µm below the gratingand the beam intensity profile in the focal plane. The beam is almost perfectly gaussian at the focus with a waist of 1.66 µm.

150 μm

670 nm

FIG. 3: Optical microscope picture of a fabricated sphericalSWG mirror. The groove width in various locations is shownas SEM images in the insets.

the tool of choice for applications involving active phasefront correction, dynamic focusing, or cavity resonancetuning.

METHODS

The devices of Figures 3 and 4 were fabricated in450 nm thick amorphous silicon deposited on a quartzsubstrate at 300 ◦C by PECVD. The refractive index ofthe silicon layer was measured by ellipsometry over thewavelength range of interest, and those measurement val-ues were used in the numerical simulations. The gratingpatterns were defined by electron beam lithography us-ing the negative resist FOX-12 from Corning exposed at200 µC/cm2 and developed for 3 minutes in a solution ofMIF 300. After development, the patterns were weaklydescummed using CF4/H2 reactive ion etching to clearthe resist residue between the grating lines. The silicongrooves themselves were formed by dry etching using aHBr/O2 chemistry. At the end of the process, a 100 nmresist layer remains on top of the silicon grooves and wasincluded in the numerical simulations. Scanning electronmicroscope images of the fabricated grating are shown inFig. 3.

The measurement setup consisted of a fiber-coupledlaser that was collimated using an aspheric lens to havea waist of 100 µm at the grating position. The reflectedbeam was picked off using a cube beam-splitter. To ob-

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FIG. 4: Top row: schematic of the groove distribution for the three structures: flat (left), cylindrical (middle), and spherical(right) SWG mirrors. Middle row: measured beam profiles at the foci for the three mirrors. Bottom row: plots of the beam radii(1/e2) as a function of distance for the three mirrors. The blue and red diamonds are the measured values for the horizontaland vertical radii, respectively, and the continuous lines are best fits. The green line represents the radius of a beam reflectedfrom a reference plane mirror.

tain the reference data, a planar dielectric mirror withreflectivity >99% was used instead of the grating. Re-flectance data were also confirmed using a spectroscopicellipsometer.

APPENDIX

This appendix section provides detailed discussions ofa few technical points raised in the main text. First,we explain how to design a grating pattern to imprint aspecific phase profile on a reflected beam, in both 1D and2D. Second, we present more materials on the design ofthe fabricated reflectors of Fig. 3 and 4.

Designing a non-periodic grating pattern for aspecific reflected phase profile

In this section, we explain the procedure for designingthe pattern of a high-index-contrast non-periodic gratingto imprint a particular phase profile φ(x, y) on a reflectedbeam. For a given periodic grating, the magnitude andphase of the reflection coefficient r at a given wavelengthl will vary with the period p and the duty cycle η (de-fined as the ratio of the groove width to the period). Forcertain grating thicknesses, we can achieve a high reflec-tivity (i.e., |r|2 > 98%) with a wide variation of phaseover a relatively large range of values of {p, η}, as indi-cated by the closed solid line in Fig. 5. In general, thisresult serves as a look-up table to determine the distri-bution of duty cycle period p(x, y) and η(x, y) according

6

to the target phase distribution φ(x, y). Here we providea detailed explanation of how to determine p(x, y) andη(x, y) in practice.

Designing a 1D non-periodic grating pattern using a discretealgorithm

For a 1D non-periodic grating design, the goal is tofind the local period p(x) and duty cycle η(x) that givea particular known target phase profile φ(x) along thex axis. We begin by choosing a path (such as Path 1in Fig. 5) enclosed within the high reflectivity region,defined by a set of parametric functions of the parametert as

p = P (t) , η = I (t) , φ = Φ (t) , (1)

where P (t) and I(t) are known, single-valued functionsthat generate coordinates on the horizontal and verti-cal axes, respectively, of Fig. 5, and Φ(t) generates thephase of the reflection coefficient (encoded by color in thefigure). Given our target phase at the origin, φ(0), weselect the value of t ≡ t0 that gives Φ(t0) = φ(0), andrecord the corresponding period p0 = P (t0) and dutycycle η0 = I(t0).

Our task now is to use Fig. 5 to create a discrete rep-resentation of the target function φ(x) by choosing theposition and width of consecutive grating lines using thea straightforward recurrence procedure. (As noted in themain text, the physics of high-index-contrast subwave-length gratings results in a smoothed spatial distributionof phase with local values interpolated between adjacentgrooves.) Suppose that we have already found the pe-riod and duty cycle of grating line n to be ηn ≡ I(tn)and pn ≡ P (tn), respectively, for some value of the pa-rameter t = tn, and that the center of that line is at xn.The center position of line n+ 1 (with period pn+1) canthen be written as

xn+1 = xn +1

2(pn + pn+1) . (2)

Therefore, we must find the value of tn+1 that satisfiesΦ(tn+1) = φ(xn+1), or

Φ(tn+1) = φ

[xn +

1

2pn +

1

2P (tn+1)

], (3)

which depends on only one unknown tn+1, and can beiterated numerically. The period and duty cycle of grat-ing line n+ 1 are then computed as pn+1 = P (tn+1) andηn+1 = I(tn+1), and this process can be repeated untilthe boundary of the grating aperture is reached.

Local period modulation of a 1D non-periodic grating througha coordinate transform

As we described in the main text, the local period mod-ulation of a 1D non-periodic grating can be realized an-alytically as a coordinate transformation from the refer-ence periodic grating described by the dielectric permit-tivity distribution ε(u) = ε(u+ Λ). Consider a target pe-riod distribution p(x), and define α(x) ≡ p(x)/p0, wherep0 is the period of the reference grating. For simplic-ity, we assume that α(x) is piecewise continuous and isbounded, i.e. 0 < A ≤ α(x) ≤ B < +∞, which is al-ways the case in practice. We then define the coordinatetransformation

u(x) = u(0) +

∫ x

0

ds

α(s)(4)

By our assumptions on α(x), this map is continuouslydifferentiable everywhere in x-space with

u′(x) =1

α(x)(5)

Consider a grating in x-space defined by the dielectricfunction ε̃(x) = ε(u(x)). When α(x) varies slowly com-pared to L, we can write

u(x+ α(x) L) ≈ u(x) + α(x) Lu′(x) = u(x) + L, (6)

which leads to the relation

ε̃(x+ α(x) L) ≈ ε̃(x). (7)

That is, the grating defined in x-space by the dielectricfunction ε̃ has a local period α(x) L, so that ε̃(x) is indeedthe grating pattern we are seeking.

Designing a 2D non-periodic grating pattern

As in the 1D case, the 2D grating pattern should bedesigned by using the magnitude/phase plot of the pe-riodic grating design in Fig. 5 as a look-up table. Forexample, the discrete procedure for the 1D case can berepeated for discrete yn to determine the position of eachperiod center (xn, yn) with its duty cycle ηn,n. The cor-responding grid points of neighboring yn are then con-nected to form the non-uniform grating grooves with thelocal width determined by ηn,n. Since the period at ynand yn+1 is in general not the same, it is possible that thegrids at yn and yn+1 are not one-to-one matched, result-ing in terminated grooves at a few locations similar tothe dislocation defects in a crystal. The density of suchdefects is low under slow-varying conditions, and theirinfluence on the reflected field is localized, similar to thesituation in the large NA reflector (shown in Fig. 2 of themain text) where discontinuities at the 2mπ interfacescause no noticeable effects in the final result. The possi-ble negative consequences of these defects can be furthercompensated by local modulation of the duty cycle.

7

FIG. 5: A plot of the phase of the reflection coefficient r of a periodic SWG mirror as a function of period and duty cycle,used to design the large numerical aperture reflector shown in Fig. 2 in the main text. The area enclosed within the solid linecorresponds to |r|2 > 98%.

FIG. 6: Simulated reflected phase of a periodic SWG mirror as a function of period and duty cycle, used to design the fabricatedstructures of Fig. 3 and 4 in the main paper. The grating is fabricated in 450 nm thick silicon on a fused silica substrate, andoperates at a wavelength of 1550 nm. The dashed and solid lines mark the contours for |r|2 = 95% and 98%, respectively.

8

FIG. 7: Solid black line: Local duty cycle (i.e., the ratio ofgroove width to period) as a function of distance from thecenter of the grating, designed for a focal length of 17.2 mm.Blue solid line and red broken line: Expected (blue) and sim-ulated (red) phase front of the reflected beam. The incidentfield is a collimated beam having a waist of 100µ m focusedon the surface of the reflector.

Design of fabricated parabolic reflectors

Here we provide more detailed explanations of the de-signs of the parabolic reflectors shown in Figs. 3 and 4 ofthe main text. It is basically an application of the dis-crete algorithm introduced above. The design is based ona periodic grating of 450 nm thick silicon with a 670 nmperiod on a quartz substrate, which has the propertiesof the reflection coefficient shown in Fig. 6 (similar tothat of Fig. 5, but for a different reference period). Werealized that varying the duty cycle while keeping theperiod fixed was sufficient for the parabolic focal lengthswe wished to fabricate, corresponding to a design alonga path parallel to the duty cycle axis (i.e., the brokenwhite line in Fig. 6, with the segment inside the blacksolid lines as the counterpart of Line 1 in Fig. 5). For thecylindrical reflector, this resulted in uniform, parallel sil-icon grooves of different widths, with centers uniformeddistributed along the x direction with a period of 670 nm.A schematic of this pattern is shown in the middle of thefirst row of Fig. 4 in the main text, and the local dutycycle used in the design is shown in Fig. 7 as the solidblack line. For the spherical reflector, the result is a set

of silicon grooves whose centers were also uniformly dis-tributed along the x direction with a period of 670nm,but with each groove having a y-dependent width. Thisschematic is shown on the right of the first row of Fig. 4in the main text.

We also used used the finite element method to per-form a full-wave simulation of the designed cylindricalSWG mirror (fx = 17.2 mm, fy = +∞). The targeted(parabolic) and simulated reflection phase profiles areshown in Fig. 7 as the solid blue and broken red lines,respectively. The expected and simulated phase profileagree very well, except for a small feature near period120 where the grating reflectance of the period/duty cy-cle combination is slightly smaller (although still above98%) than the rest of the grating.

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