Designofhigh-efficiencydielectricreflection gratings · Designofhigh-efficiencydielectricreflection gratings B.W.Shore,M.D.Perry,J.A.Britten,R.D.Boyd,M.D.Feit,H.T.Nguyen,R.Chow,

1124 J. Opt. Soc. Am. A/Vol. 14, No. 5 /May 1997 Shore et al.

Design of high-efficiency dielectric reflectiongratings

B. W. Shore, M. D. Perry, J. A. Britten, R. D. Boyd, M. D. Feit, H. T. Nguyen, R. Chow,and G. E. Loomis

Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, California 94550

Lifeng Li

Optical Sciences Center, University of Arizona, Tucson, Arizona 85721

Received February 29, 1996; revised manuscript received August 26, 1996; accepted November 11, 1996

We discuss examples of designs for all-dielectric reflection gratings that tolerate high intensity and are poten-tially capable of placing up to 99% of the incident light into a single diffraction order, such as are needed forcontemporary high-power lasers utilizing chirped-pulse amplification. The designs are based on placing a di-electric transmission grating atop a high-reflectivity (HR) multilayer dielectric stack. We comment on theconnection between transmission gratings and reflection gratings and note that the grating and the HR stackcan, to a degree, be treated independently. Because many combinations of gratings and multilayer stacksoffer high efficiency, it is possible to attain secondary objectives in the design. We describe examples of suchdesigns aimed toward improving fabrication and lowering the susceptibility to laser-induced damage. Wepresent examples of the dependence of grating efficiency on grating characteristics. We describe examples ofhigh-efficiency (95%) gratings that we have fabricated by using hafnia and silica multilayers. © 1997 OpticalSociety of America [S0740-3232(97)01105-8]

1. INTRODUCTIONContemporary developments of high-power lasers, oftenbased on chirped-pulse amplification1–14 rely extensivelyon the dispersive properties of diffraction gratings. Asnoted below, it would be desirable if these reflection grat-ings could be made entirely of dielectric, rather than me-tallic, material, in order to minimize energy absorbed andthe consequent damage. In this paper we describe someof the concerns related to the design and the fabrication ofsuch all-dielectric gratings.A reflection grating must incorporate two optical func-

tions: It must combine high reflectivity with diffraction.Conventional metallic gratings combine these two func-tions in a single conducting surface. The conductivity ofthe metal forces reflection, while periodic grooves creatediffraction. Because metallic gratings owe their reflec-tivity to conductivity, expressible as a complex-valued re-fractive index or permittivity, they have a disadvantagefor applications that subject the grating to intense radia-tion: The absorption of radiation causes heating anddamage.15 Transparent dielectric materials have muchsmaller absorption coefficients than do metals, and there-fore optical devices based on dielectric materials have po-tential for withstanding more intense radiation.16 More-over, dielectric structures, when used to create gratings,have unique properties that can offer additionalbenefits.17 The present paper describes details of the de-sign of such gratings.

2. DIELECTRIC REFLECTION GRATINGSContemporary techniques of interference lithography (orholographic grating fabrication; see Refs. 18–23) offer op-

0740-3232/97/0501124-13$10.00 ©

portunities for producing a variety of one-dimensional pe-riodically corrugated or grooved surfaces. Such a gratingcreated in a semi-infinite dielectric medium can, for suit-ably deep and closely spaced grooves, place nearly all ofthe incident light into one transmitted diffractionorder.20,21,23–28 However, a single air–dielectric interfacewill not, by itself, produce a grating that has highreflection efficiency. A second optical element is requiredto produce high reflectivity. For some purposes, andwithin appropriate wavelength regions, the reflectivitycould be provided by a metallic layer under the dielectricgrating.29,30 Here, instead, we consider some possibili-ties for creating purely dielectric devices by utilizing thehigh reflectivity of a dielectric multilayer stack.Multilayer stacks of thin (subwavelength-thickness) di-

electric films are widely used in the optics industry as an-tireflection (AR) coatings, polarizers, beam splitters, fil-ters, and highly reflecting mirrors.31–36 Rather thanrelying on conductivity or absorption to produce reflec-tion, multilayer dielectric stacks rely on interference. Asuccession of (horizontal) planar layers are fabricatedwith thicknesses such that, for light of a specified wave-length and polarization and incident from above at agiven angle, the phases of upward- and downward-traveling waves within each layer reinforce either the up-ward wave (for reflection) [a high-reflectivity (HR) coat-ing] or the downward wave [for transmission (anantireflection (AR) coating)]. Commercial optical designsoftware now permits the design of elaborate multilayerstructures, simultaneously satisfying a multitude of de-sign goals (e.g., high transmission within one bandwidthand high reflectivity within another bandwidth). How-ever, the range of possible solutions to the multilayer de-

1997 Optical Society of America

Shore et al. Vol. 14, No. 5 /May 1997/J. Opt. Soc. Am. A 1125

Fig. 1. Multilayer dielectric designs of high-index (H) and low-index (L) layers, showing relative locations of substrate, multilayers, andgroove corrugations (G). (a) Overcoated grating; dielectric layers effectively form a volume grating. (b) Undercoated grating; layersform a high-reflectivity (HR) stack under the corrugations.

sign task must be curtailed by considerations of fabrica-tion ease.A reflecting multilayer stack can be combined with a di-

electric grating in several ways. Multilayer dielectricgratings that have been designed and fabricated for useat x-ray wavelengths typically use hundreds of layerpairs, and the grating is etched through many of theselayers (so that each grating ridge comprises a multilayerstack37–42). The fabrication of devices for use at longerwavelengths, from the visible to the infrared, typically fol-lows a different approach. When the goal is a highly ef-ficient reflection grating for optical wavelengths, twotypes of design have particular interest (see Fig. 1).First, the grating may be placed onto a substrate and

then overcoated with dielectric layers.43–46 In principle,the initial grating may be created in photoresist, and theoverlying layers may be created with standard vacuumdeposition techniques. This procedure does not requirean etching step. If the layers remain conformal with theunderlying grating, then the structure effectively forms avolume grating. A particular concern with this type ofgrating is the lack of conformability as the number of lay-ers increases: The flattening of the layers may create areflecting structure rather than a diffracting structure.Alternatively, the grating may be placed on top of a

multilayer reflecting stack. This type of design allowsthe multilayer stack to be fabricated independently of thegrating. This is the method on which we have concen-trated our major efforts.

3. GRATING FABRICATIONThe procedures used to fabricate a grating have a signifi-cant effect on the theoretical design. In our work themultilayer thin-film stack is used as a substrate for thegrating, which we create by using previously describedprocedures of interference lithography.15 In brief, wefirst coat the substrate with a carefully controlled uniformlayer of photoresist. Next we expose this photosensitivesurface to the stabilized interference pattern at the inter-section of two collimated laser beams in a two-arm inter-

ferometer (see Fig. 2). Our exposure takes place in anenvironment of carefully controlled temperature and vi-brational isolation, supplemented by an active feedbackcontrol of the interference fringes.15 We develop the la-tent image to create a corrugated surface, using in situmonitoring to control the profile by terminating the devel-opment step at the optimal moment.47

Many controllable factors affect the photoresist gratingprofile: the type and the thickness of the photoresist, theintensity and the duration of the exposure, the type andthe strength of the developer, and the duration of the de-velopment step.47–49 By adjusting these variables, it ispossible to obtain groove shapes in the photoresist thatvary between nearly sinusoidal and nearly rectangular,with depths and spacings of hundreds of nanometers(comparable with a wavelength of visible light). The in-herent flexibility of the several steps and materials offersthe designer of gratings a wide range of profiles, althoughprecise connection between the controllable factors andthe profile typically involves much trial and error. Al-

Fig. 2. Schematic diagram of our holographic exposure layout,showing cw laser, beam splitter (s), turning mirrors (m), aper-tures (a), collimating lenses (L), and photoresist-coated substrate(p).


though the developed photoresist could serve as thegrating,19,20,50 it is more fragile than other materials. Tocreate a more robust structure, the photoresist can beovercoated with dielectric material, or, as in our work, thepattern can be transferred to the underlying substrate byusing lithographic etching techniques.51–56

4. GRATING MODELINGThe theoretical and computational problems associatedwith dielectric gratings16,17,21,24,26,28,39,43,57–68 are muchsimpler than those occurring with metal gratings.38,69–83

Nevertheless, because the grooves are likely to be quitedeep (depth comparable with a wavelength l) and closelyspaced (period comparable with l) for high efficiency, it isimportant to use a rigorous method based on full-vectorMaxwell equations. A variety of such methods have beendescribed in the literature.84 To provide realistic de-scriptions of actual gratings, it is essential that we be ableto model a variety of grating profiles, not just simpleshapes such as sinusoids, rectangles, and triangles.Most of our results have been based on the multilayer

modal method, with R- or S-matrix propagation, devel-oped by Li85–88 and described by him in great detail.Briefly stated, in this approach we idealize the gratingprofile as a set of discrete slices, rectangular in cross sec-tion, in each of which the material consists of two sepa-rate dielectrics. Within each of the two dielectrics ofsuch a horizontal slice, the material is uniform, and hencethere exist for the Helmholtz equation simple plane-wavesolutions (e.g., sines and cosines of possibly complex argu-ments). At the vertical interfaces between two regions,appropriate field components must be continuous. As inthe simpler cases of fields between infinite parallel plates,this continuity requirement leads to a transcendental ei-genvalue equation,76,77,89 whose infinite set of solutionsprovides the arguments of the sine and cosine functions.The constructed modal functions are exact solutions tothe Maxwell equations within a slice. Having obtainedsolutions within a slice, it is then necessary to enforce thecontinuity equations between layers. The procedure is ageneralization to multiple waves of what is required increating solutions for single waves and multiple thinfilms.34 However, the straightforward multiplication ofcharacteristic matrices, which is used for thin-film model-ing, is unsatisfactory for grating modeling, owing to theneed to retain evanescent waves. These can introducegrowing exponentials that ruin the computation, unlessspecial care is taken, as we do with our R- or S-matrixmethod.88 As a final step of the computations, the fieldabove and below all grating and dielectric structure is ex-pressed as a Rayleigh expansion, with reflection or trans-mission amplitudes obtained from the solution to alge-braic equations as just described.As in other rigorous methods, the exact Maxwell equa-

tions are transcribed into coupled algebraic equations.The computer codes provide not only efficiencies but alsofield distributions within the grating. They can be usedto model quite general shapes of grating profiles. Forpredictions of general trends and possible design targets,we use simple parameterized groove shapes (e.g., trap-ezoids and sinusoids), although it is possible to digitize

scanning electron micrograph (SEM) profiles for use as in-put data to the code.

5. EXAMPLES OF DESIGN CONCERNSDifferent grating uses pose different constraints and ob-jectives. For example, in the laser program at theLawrence Livermore National Laboratory (LLNL) ourconcerns include the following.

A. High EfficiencySome of our gratings are intended for use in chirped-pulseamplification pulse compressors. For such applicationswe require that the diffraction efficiency be as high aspossible, over a bandwidth surrounding the nominal laserwavelength (typically 1053 or 1064 nm).4 (Efficiencyhere refers, as is customary in grating theory, to the frac-tion of the normal-incidence component of the time-averaged Poynting vector that emerges into order 21.)The key to high efficiency lies in the grating equation

nb sin um 5 na sin u 1 ml/d (1)

relating angle of incidence u in a material of refractive in-dex na to the diffraction angle um of order m in a materialof refractive index nb for wavelength l and groove spacingd. To ensure highest possible efficiency, we choose thegroove spacing so that, in accord with the grating equa-tion, only two orders occur90 (specular reflection, of orderm 5 0, and backreflection, of order m 5 21) and we usethe grating in a near-Littrow mount, i.e., the angle of in-cidence is near the Littrow angle, arcsin(l /2d), at whichreflected order 21 travels back along the incidence direc-tion. (Strictly speaking, confining the grating period toallow only two orders is neither necessary nor sufficientfor high efficiency. Nevertheless, in practice, this con-straint provides excellent results.)We have used a range of groove spacings. Most re-

cently, we have concentrated on designs with 1480grooves/mm (groove spacing d 5 676 nm) to produce gold-overcoated gratings with high efficiency over the wave-length range 800–1100 nm.91 For 1053-nm light thesegratings have l/d 5 1.56.

B. Large SizeSome of our gratings, intended for use in large chirped-pulse amplification pulse compressors, are tens of centi-meters in diameter (a 100-TW laser system operating atLLNL uses gratings 40 cm in diameter, and a 1-PW laserat LLNL uses 94-cm 3 75-cm gratings). For satisfactorypulse compression, it is important that the wave frontmaintain high quality across the grating. Therefore werequire that the grating efficiency and phase be uniformover a large area.The requirement for uniform efficiency imposes a re-

quirement for uniform groove profile, implying uniformphotoresist exposure. Our holographic procedure ex-poses the entire surface simultaneously to two large in-tersecting collimated beams from a krypton-ion laser.15

(Alternative methods of exposure raster the interferencepattern across the surface.) Because there is unavoid-able spatial intensity variation across the large surface of


photoresist, it is essential to have a grating design that isrelatively insensitive to such groove variations as resultfrom the nonuniform exposure.

C. High Tolerance for Intense RadiationMany dielectric materials are available for considerationas thin films in stacks, and the consequent range of re-fractive indices is appreciable. However, our intendedusage with pulses of high peak power, at wavelengths ofapproximately 1 mm, limits the choice of dielectric mate-rials to high-band-gap oxides, such as silica (SiO2) andhafnia (HfO2). This severely restricts the range of re-fractive indices available for designs.92

Because damage is initiated by deposited radiation en-ergy and because this deposition is dependent on the localvalue of the (cycle-averaged) electric field, it may be de-sirable for the grating design to minimize high values ofthe electric field, particularly at material interfaces wheremechanical strength may be weakest.

D. Low Reflectivity of Stack during ExposureThe wavelength of our exposure laser, 413 nm, is wellmatched to the photoresist. To create the desired inter-ference pattern of 1480 lines/mm, we must expose thephotoresist to light incident at approximately 17° (the Lit-trow angle for 413 nm). It is essential that during thisexposure there be no spurious interference patterns, suchas occur with reflections from the underlying structure (orfrom nearby hardware). It is possible to meet this goalby placing an AR coating (or temporary absorbing layer)between the dielectric stack and the photoresist. Suchlayers add to the complexity of the fabrication and the de-sign. We describe (in Section 7 and beyond) a simpler,alternative procedure.

6. RELATIONSHIP BETWEENTRANSMISSION AND REFLECTIONOur gratings owe their high efficiency, in part, to thechoice of groove spacing and angle of incidence: thespacing d , 3l/2 and the near-Littrow mount allow onlyorders 0 and 21 to propagate. Under such conditionsboth the simple single-dielectric transmission grating andthe multilayer dielectric reflection grating exhibit regularmaxima and minima of diffraction efficiency as a functionof groove depth. However, conditions that produce highdiffraction efficiency for transmission will produce low dif-fraction efficiency for reflection. The reason can be un-derstood with the aid of Fig. 3 and the recognition thatwhen a transmission grating directs all downward-propagating incident light into downward-transmitted ra-diation (light incident at angle u i and emerging down-ward at angle u t , say), then the time-reversed situation(radiation from below the grating at angle u t propagatingupward out of the grating into angle u i) will also apply.Figure 3(a) illustrates a low-efficiency grating (basically atransparent layer) atop a multilayer mirror. The undif-fracted downward wave (solid arrow) returns from the re-flection (dashed arrow) and continues upward, throughthe transparent grating, to emerge as specular reflection.The opposite case of a high-efficiency transmission grat-ing is illustrated in Fig. 3(b). Here the grating directs all

of the incident light into order 21 (solid arrow). Uponreflection this light will, on passing again through thehighly efficient transmission grating, be redirected(dashed curve). This second deflection places the radia-tion again on the course of specular reflection, and the neteffect will be for the grating plus (multilayer) reflector tobehave as a mirror. Detailed computations, illustratedbelow, bear out this intuitive observation.

7. DIELECTRIC TRANSMISSION GRATINGThe design of an undercoated multilayer dielectric grat-ing [e.g., Fig. 1(b)] involves two structures: the gratingand the multilayer stack. Although the final design in-volves the joint action of each structure, it is instructiveto consider each separately: the surface-relief grating (adielectric transmission grating) and the multilayer HRstack.Reliable theoretical descriptions of high-efficiency di-

electric transmission gratings date back twodecades.21,24–26,50,57,62,93–96 The 1982 paper of Moharamand Gaylord26 presented illustrative transmission effi-ciencies for gratings whose groove spacing was equal to 1wavelength (l/d 5 1) and that were used at the Littrowangle (30°). All such gratings, whatever their groove pro-file, have only two reflected orders (specular reflection, oforder 0, and retroreflection, of order 21). If the refrac-tive index is less than 1.5, only these two orders will bepropagating orders within the material; for higher refrac-tive indices, orders 11 and 22 can propagate within thedielectric, although they will not emerge as transmittedorders into air below the grating. Figure 4 shows the dif-fraction efficiency of a simple lamellar transmission grat-ing as a function of groove depth and duty cycle (definedhere as the ratio of ridge thickness at half-height to grat-ing period).The transmission efficiency (into order 21) exhibits a

well-known series of periodic maxima and minima. For aduty cycle of 0.5, the first of these maxima occurs at adepth of approximately 1.7 wavelengths, where it reachesa value of 98%. Efficiencies exceeding 96% are availableover a range of depths and duty cycles near these values.(For wavelength 351 nm, a high-efficiency transmissiongrating with duty cycle approximately 0.5 requiresgrooves approximately 600 nm deep.)Many of the features of transmission gratings occur

also with metallic reflection gratings. Although the sur-face boundary conditions of dielectric materials differfrom those of conductors, some properties of metallic grat-

Fig. 3. Schematic diagram of arrangements that will producelow diffraction efficiency in reflection: (a) low-efficiency trans-mission grating over mirror, (b) high-efficiency transmissiongrating over mirror.


Fig. 4. Diffraction efficiency (order-1 in transmission) for alamellar-profile grating etched into silica (refractive index 1.47),for a grating whose spacing is equal to 1 wavelength and that isused at the Littrow angle (30°) for TE polarization. The plotshows the dependence of this transmission efficiency as a func-tion of groove depth (expressed in wavelengths) and duty cycle(the ridge thickness in fractions of a groove period).

Fig. 5. Same as Fig. 4, but for TM polarization.

Fig. 6. Diffraction efficiency (order-1 in transmission) for alamellar grating as a function of relative depth (in wavelengths)and refractive index. The groove spacing is 1 wavelength, theangle of incidence is the Littrow angle (30°), the groove dutycycle is 0.3, and the light is TE polarized.

ings can be expected to appear for reflecting dielectricgratings. Metallic gratings that allow only two ordersand whose profiles have Fourier sine expansions have thegeneral property that their diffraction efficiency dependsprimarily on the fundamental Fourier coefficient of thegroove profile.74,97 Thus results obtained for sinusoidalprofiles are applicable to lamellar profiles having dutycycle 0.5. It is to be expected, and is seen, that dielectricgratings show a similar insensitivity to details of thegroove profile.The response of a grating depends, often markedly, on

the polarization of the incident light. It is possible toachieve high efficiency for either TE or TM polarization.(For TE polarization the electric field is parallel to thegrooves; for TM polarization the magnetic field is parallelto the grooves.) In general, high efficiency of a dielectricgrating requires deeper grooves for TM polarization thanfor TE polarization, as can be seen by comparing the TMresults in Fig. 5 with the previously shown TE results ofFig. 4.The efficiency attainable from a simple dielectric trans-

mission grating depends not only on the groove depth butalso on the difference between refractive indices ingrooves and ridges. The phase difference between wavestraveling down through ridges and those passing downthrough valleys provides the opportunity for interference.As Fig. 6 illustrates (for TE polarization and fixed dutycycle 0.3), the situation is similar to the behavior of an in-terferometer: A given phase difference (and hence agiven diffraction efficiency) occurs for fixed values of theproduct of depth and refractive-index difference.

8. GRATING BOUNDARYCONDITIONS: TE AND TM POLARIZATIONThe practical preference for TE polarization for dielectricgratings contrasts with the case for metallic gratings, forwhich TM polarization has higher efficiency for shallowgrooves. The difference between the two types of gratingoriginates in the boundary conditions of the Maxwellequations, which force continuity across any interfaces ofthe components of electric and magnetic fields that liewithin the surface (the tangential components of Eand H).For a grating coated with high-conductivity metal, the

interface between the air and the grating is a surface overwhich the field has simple constraints: The interfacemust be a nodal surface of the electric field for TE polar-ization, and it must be an antinodal surface of the mag-netic field for TM polarization.A dielectric grating is more complicated than a metal

grating because the boundary condition involves match-ing the field above and below the surface and not just en-forcing a node or an antinode. The boundary condition atthe interface can be expressed as continuity of ratios ofelectric- and magnetic-field components (H/E is an admit-tance, E/H is an impedance). The effect of a multilayerstack need enter the computations of grating fields onlyas an admittance (or an impedance).34,98,99 Thus it ispossible, for fixed incidence wavelength and angle, totreat the multilayer stack independently of the grating;stacks that have the same admittance will impose identi-


cal boundary conditions upon the grating fields and willtherefore produce identical behavior. Our concern in thepresent paper is with high-efficiency reflection gratings,for which the multilayer stack must be highly reflective.As presented in the text by Macleod (Ref. 34, p. 43), the

reflectance at a planar interface between region a and re-gion b is

R 5 u@h~a ! 2 h~b !#/@h~a ! 1 h~b !#u2, (2)

where h (r) is the complex-valued tilted optical admit-tance for region r:

h~r ! 5n~r !

Z03 H cos ur

1/cos ur

for TEfor TM . (3)

It is possible to achieve high reflectance at an interface invarious ways. The traditional quarter-wave stack has alarge real admittance (see Macleod,46 p. 164) that domi-nates the unit admittance of air. If the stack has asmaller and complex-valued admittance, then the phaserelationship between the two admittances becomes im-portant. When the two admittances have equal magni-tudes, then complete reflectance occurs when they are 90°out of phase.

9. EXAMPLES OF MULTILAYER GRATINGDESIGNSThe basic design problem posed here is to choose thethickness of the uppermost layer and the depth and theshape of the grooves cut into this so that, when combinedwith a HR multilayer stack, the combination will place al-most all of the radiation into the reflected order 21. Theexact structure of the HR stack is not important for thegoal of high efficiency, and there exist many equally high-efficiency designs for multilayer dielectric gratings fabri-cated from given materials, differing in groove shape,depth, and multilayer thicknesses. This great variety of

choices allows us to include subsidiary constraints, suchas those listed above. Following are examples of howthese objectives can be met.

A. Quarter-Wave StackThe most common form of a HR multilayer dielectricstack is the (HL)n stack based on n pairs of high-index (H)and low-index (L) material, each of which introduces onequarter-wave of phase shift at the desired angle.31,32,34,100

As the number of layers increases, the reflectance ap-proaches closer to unity, within a wavelength band whosewidth is set by the ratio of refractive indices. With thisstack in mind, the simplest high-efficiency multilayergrating designs are based on placing a dielectric-gratingatop such a quarter-wave stack. Typically, between 6and 12 layer pairs prove satisfactory. Either an H or anL layer may be placed at the bottom; the choice may de-pend on the robustness of the coating procedure. Eitheran H or an L layer (or some third material) may be madethe top layer; in general, the groove depths for high effi-ciency are inversely proportional to the index of refractionof the material comprising the grating layer.

B. Partially Etched GroovesIt is obviously desirable to have a point design that hasrelatively little sensitivity to small variations in designparameters, such as layer thicknesses, groove depth, andduty cycle. From a purely theoretical standpoint (disre-garding fabrication considerations), robust designs can beobtained by allowing some portion of the top layer to liebeneath the grooves—the grooves are not etched entirelythrough the top layer. Figure 7 illustrates how gratingefficiency is affected by groove depth and by the thicknessof the upper layer below the groove for a given duty cycleof 0.3. (The sum of groove depth and subgroove thick-ness is the total thickness of the dielectric material.) Fig-ures 7(a) and 7(b) show clearly the periodic dependence ofefficiency on groove depth. Efficiencies exceeding 99%are predicted for a variety of parameter choices. From

Fig. 7. Efficiency (order-1 in reflection) versus depth of groove (in micrometers) and top-layer thickness below the groove (in microme-ters) for a quarter-wave stack. The grating grooves are lamellar, with 1480 grooves /mm (groove period 677 nm), and the duty cycle is0.3. The radiation is the 1053-nm wavelength at the Littrow angle (51.2°), with TE polarization. The high-index material is hafnia(index 1.9), and the low-index material is silica (index 1.46). The quarter-waves are defined for the use wavelength and angle. (a)High-index surface layer over (HL)7 stack, (b) low-index surface layer over (HL)7H stack.


Fig. 8. Efficiency versus depth (in micrometers) and duty cycle for a quarter-wave HR stack, with grooves etched completely throughthe top layer. The grating grooves are lamellar, with 1480 grooves /mm (groove period 677 nm), and the duty cycle is 0.3. The radiationis the 1053-nm wavelength at the Littrow angle (51.2°), with TE polarization. The high-index material is hafnia (index 1.9), and thelow-index material is silica (index 1.46). The quarter-waves are defined for the use wavelength and angle. (a) High-index surface layerover (HL)7 stack, (b) low-index surface layer over (HL)7H stack.

Fig. 9. Efficiency in transmission versus depth (in micrometers) and duty cycle for the grating layer (without underlying multilayerstack) of Fig. 8. (a) High index, (b) low index.

Fig. 7(a) it can be seen that when the surface layer hashigh index (H on top), one high-efficiency design would begrooves approximately 200 nm deep etched into approxi-mately 400 nm of surface layer (leaving a 200-nm sub-groove thickness). When the low-index material is up-permost, efficient performance requires much deepergrooves than this. Figure 7(b) shows that one high-efficiency design would be grooves approximately 700 nmdeep in 900 nm of surface layer (leaving a 200-nm sub-groove thickness). Whereas Fig. 7(a) shows that the ef-ficiency with a high-index grating tends to be either veryhigh or very low, Fig. 7(b) shows that when the gratingmaterial is of lower index, there occur designs with inter-mediate efficiencies.

C. Etch-through DesignsHigh-efficiency designs can be obtained in which thegrooves are either partially or completely etched throughthe top layer. Designs of the latter type are attractive forease of fabrication: The etching chemistry can be de-signed to halt at the interface, so that the groove depth is

fixed by layer thickness rather than by careful timing ofthe etching. Figure 8 shows how efficiency depends ondepth and duty cycle, for the two possible choices of upperlayer, when the grooves are etched all the way throughthe top layer.Figures 8(a) and 8(b) display several noteworthy fea-

tures. In each case gratings with duty cycle greater than0.5 exhibit a similar periodic variation of efficiency withduty cycle. To bring out this feature, the range of depthsfor the figures has been taken inversely proportional tothe refractive indices of the top layer: 1910 nm of low-index material (index 1.41) and 1410 nm of high-indexmaterial (index 1.91), each having the same optical phase.In each of these cases, the shallowest high-efficiency

design is rather insensitive to duty cycle. Efficiency ex-ceeds 90% for a range of 50 nm at approximately 330-nmdepth for H on top and for a range of 100 nm at approxi-mately 730-nm depth for L on top. Peak efficienciesreach 99% for appropriate choices of depth and duty cycle.These examples show that, in principle, high-efficiency

solutions are obtainable for either H or L as the top layer.


Designs based on etching into the higher-index materialrequire significantly shallower grooves. The final choicebetween the two possibilities involves practical consider-ations, such as the ease of etching the material and thedamage threshold (expected to be higher for low-indexmaterial).

D. Connection with TransmissionFrom the remarks above, it is to be expected that the de-signs that produce the highest reflected diffraction effi-ciency will be those for which the grating interface itselfhas neither very high nor very low efficiency for diffrac-tion in transmission. Figure 9 demonstrates this conclu-sion, by showing the transmission efficiency of the gratingalone, without underlying multilayer structure, for thedesigns presented in Fig. 8. The parameter choices thatproduce either high or low efficiency in transmission arechoices that produce low efficiency for reflection.

10. EXAMPLES OF SUPPLEMENTARYCONSTRAINTSThe illustrations presented in Section 9, like numeroussimilar results in the literature, demonstrate the widerange of parameter choices (e.g., grating shapes) that canmeet the primary design objective of high diffraction effi-ciency. The diversity of solutions offers opportunity forimposing additional requirements upon the design. Herewe mention examples of these considerations.

A. Problems with Quarter-Wave StacksThe quarter-wave stack provides one easily understoodpossibility for creating a high-efficiency reflection grating.However, in our holographic method it has one drawback.The dielectric stack, intended to be highly reflecting atthe use angle of 52° for 1053-nm light, is also highly re-flecting near the exposure angle of 17° for 413-nm light.(This light is near the reflectivity maximum associatedwith a third harmonic of the design wavelength.) Figure10 illustrates an idealized (dispersionless) model of thereflectance expected for a quarter-wave stack at the useangle of 52° and at the exposure angle of 17°. The expo-sure conditions nearly coincide with the high reflectanceof a third harmonic of the use wavelength.This undesired reflectance can be diminished by alter-

ing various layer thicknesses. Commercial optical designsoftware can provide a variety of alternative stacks, witha range of fabrication difficulty, that can minimize the re-flectance (at selected wavelengths) during exposure andmaximize the reflectance (at selected wavelengths) duringuse. However, these designs all require precise thick-ness control of all the layers.

B. HLL DesignA simple alternative to the quarter-wave design (HL)n isa (HLL)n design, comprising quarter-waves of high-indexmaterial and half-waves of low-index material. Ratherthan define the quarter-waves for the use wavelength andangle, we choose a shorter wavelength and normal inci-dence. We have chosen 830 nm for the stack-designwavelength (a choice that simplifies the monitoring of thelayer deposition). Figure 11 shows the (idealized disper-

sionless) reflectance for this stack, both for the use angleand for the exposure angle.Although details of the actual response of the

multilayer stack are affected by dispersion (and by fabri-cation irregularities in thicknesses and refractive indi-ces), the simple model shows that the HLL design canprovide a broad reflection minimum around the exposurewavelength. Such a design eliminates the need for spe-cial AR or absorbing coatings between the photoresist andthe stack and therefore greatly simplifies fabrication.As with the quarter-wave designs, grating designs

based on this stack can achieve reflected diffraction effi-ciency exceeding 99%. Such high efficiencies are avail-able for a range of groove depths and duty cycles and witheither the H or the L dielectric as the uppermost layer.

C. Fields in HLL GratingsThe efficiencies obtained with the HLL design (at the usewavelength and angle) are almost indistinguishable from

Fig. 10. Theoretical reflectance for quarter-wave stack (HL)7H.The thick curves are for TE polarization (s polarization), and thethin curves are for TM polarization (p polarization). The com-putations assume constant refractive indices, n 5 1.91 for H andn 5 1.41 for L. (a) At the use angle of 52° under air; the verticalarrow marks the use wavelength 1053 nm. (b) At the exposureangle of 17° under photoresist; the vertical arrow marks the ex-posure wavelength 413 nm.

Fig. 11. Same as Fig. 10, but for the HLL stack (HLL)7H.


Fig. 12. Calculated electric-field distribution in and above multilayer dielectric grating structures. (a) Quarter-wave stack with high-efficiency grating, showing field maxima at layer interfaces and at the corner of the grating ridge; (b) HLL stack with field maximaconcentrated in the low-index layers and in the air space between the gratings.

those obtained with the quarter-wave stack: Both de-signs have, apart from possible phase differences, thesame far-field Rayleigh expansion coefficients Rm . How-ever, the two designs differ appreciably in the near-fieldregion within the grating grooves. Because the designshave high efficiency, there is a strong standing-wave pat-tern of the electric field, with antinodal planes where theelectric field is roughly twice the value of a free-spacetraveling wave. With the quarter-wave design, theenhanced-field regions extend into the surface, wherethey may cause photoinduced breakdown and damage.With the HLL design, adjustments to the uppermost layercan, while retaining high efficiency, shift the enhanced-field regions to fall within the grooves, where they are lesslikely to induce damage.Figure 12 shows, for selected high-efficiency cases of

the two designs, the spatial distribution of electric-fieldmagnitude. Both designs have high-index material asthe top layer and nonzero thickness of this material belowthe grooves. In the quarter-wave design [Fig. 12(a)], theelectric-field maxima are situated at the interfaces in themultilayer stack and at the edges of the grating. In theHLL design [Fig. 12(b)], the maxima are shifted into thelow-index silica layers inside the stack and into the airspace between the grating ridges above the stack. There-fore this HLL design has theoretically a higher probabil-ity of laser damage resistance. (This particular HLL de-sign improved the field placement by adjusting thethickness of the H layer, thereby giving up the fabricationease of an etch-through design.)

11. FABRICATION EXAMPLESWe have fabricated a variety of small (5-cm-diameter)sample gratings on borosilicate glass substrates. Herewe present some details of multilayer dielectric gratings,intended for use at wavelengths near 1 mm and created byusing hafnia (HfO2) (index 5 1.90) and silica (SiO2)(index5 1.45) as the high- and low-index materials.

The gratings are intended to have 1480 lines/mm, forwhich the Littrow angle is 51.2° for our intended wave-length (1053 nm), and have only two propagating ordersin reflection.In principle, high-efficiency gratings can be fabricated

with either hafnia or silica as the top (corrugated) layer.The etch chemistry differs for the two materials, and con-sequently the groove profiles differ. In general, one ex-pects that lower-index material will require deepergrooves to produce high efficiency.The multilayer stacks were prepared by using local

vacuum deposition facilities. We coated these stackswith 500–700-nm-thick layers of photoresist by using aspin coater. After soft-baking the films, we exposed themto 413-nm light in the two-arm interferometer describedpreviously.15 The latent-image photoresist gratings weredeveloped by using in situ monitoring47 to ensure thatgrooves were carried through the photoresist to the sub-strate. The samples were then submitted to reactive-ionetching at remote facilities. Etches in silica were per-formed by Rochester Photonics Corp. (Rochester, N.Y.),using a reactive-ion etching system and CHF3 /Ar/O2chemistry. Etches in hafnia were performed at HughesAerospace Malibu Research Center (Malibu, Calif.), usingan ion-beam etcher with CF2Cl2 /Ar chemistry. Follow-ing the etching, the photoresist mask was stripped byplasma ashing or acetone rinse.Figure 13 shows an example of predicted diffraction ef-

ficiency (when viewed at the Littrow angle) for a HLLstack with (high-index) hafnia as the uppermost (corru-gated) layer. For these computations the top layer wastaken to have fixed thickness (350 nm), into which werecut grooves of various depths and widths. (The depth of350 nm corresponds to grooves cut completely through thehafnia.) The profiles were modeled as trapezoids, whichis a good approximation to what we produce [see Fig.14(a)]. According to these computations, we expect thatgratings with efficiency exceeding 95% should be pro-duced for a range of groove shapes and base widths; for


duty cycles of approximately 0.3, efficiency is insensitiveto depth once the grooves become 150 nm deep.We have fabricated several hafnia gratings targeting

this design. Figure 14(a) shows a SEM picture of a wit-ness sample of a grating whose efficiency, for 1.064-mmlight at TE polarization, was 95% (61%); the remaininglight was observed in zero-order reflection. This is lowerthan our target value of 99%. From measurements onthe SEM, it appeared that the silica layers were thinnerthan the intended 294 nm. Modeling this grating withsilica layers of 260 nm, and a topmost hafnia layer 365nm thick into which are etched trapezoidal grooves 165nm deep [see Fig. 14(b)], we predicted an efficiency of95%.Damage tests16,101 of this grating using 300-fs pulses

(at 1053 nm and TE polarization) show visible signs ofdamage at a fluence of 0.21 J/cm2 and massive damage(including damage to the multilayer structure) at 1.25J/cm2.

Fig. 13. Calculated efficiency (order-1 in reflection) versusgroove depth (in nanometers) and trapezoid base width (in na-nometers) for a trapezoidal-groove grating having 1480 grooves/mm, etched in hafnia atop a multilayer stack (HLL)10 of hafnia(H) and silica (L), when viewed at the Littrow angle (51.2°) forTE-polarized light of wavelength 1053 nm. The top layer,hafnia, is 350 nm thick (depth of 350 nm is etched through).

Fig. 14. (a) SEM of a multilayer dielectric grating fabricated tothe specifications of Fig. 13, showing near-trapezoidal profilesetched into hafnia; (b) idealized model used for computations.The light regions are low index (silica), and the dark regions arehigh index (hafnia). The diffraction efficiency of this gratingwas 95% (TE polarization 1.053 mm).

When silica is used as the top layer, as in the predic-tions portrayed in Fig. 15, the grooves must be muchdeeper than for hafnia to produce comparable efficiencies.The grooves tend to be rectangular, as can be seen in Fig.16(a). Although these grooves are deeper than thoseneeded for the hafnia grating, silica etches much moreeasily than does hafnia, so the greater depth does notpose a problem. Furthermore, the etching will slow con-siderably when the grooves reach the hafnia under thesilica, and this provides useful control of the depth. As inFig. 13, the computations here assume a fixed total thick-ness of silica (800 nm), into which the grooves are etched.As can be seen, for grooves 600 nm deep there occur high-efficiency solutions that are relatively insensitive to thegroove width. Alternatively, given a fixed duty cycle of0.6, there are high-efficiency solutions that are indepen-dent of the groove depth for grooves deeper than 650 nm.We have fabricated several silica gratings that targeted

this design. Figure 16(a) shows a SEM picture of a wit-ness sample of a grating whose efficiency was 94%(61%); the remaining light was observed in zero order.

Fig. 15. Calculated efficiency (order-1 in reflection) versusgroove depth (in nanometers) and groove duty cycle for alamellar-groove grating having 1480 grooves/mm, etched in silicaatop a multilayer stack (HLL)10 of hafnia (H) and silica (L), whenviewed at the Littrow angle for TE-polarized light of wavelength1053 nm. The top layer, silica, is 800 nm thick (depth of 800 nmis etched through).

Fig. 16. (a) SEM of a multilayer dielectric grating fabricated tothe specifications of Fig. 15, showing lamellar profiles etched intosilica; (b) idealized model used for computations.


In this sample the etching was incomplete and did notreach entirely through the silica.Modeling this grating with silica layers of 260 nm, and

a topmost silica layer 800 nm thick into which are cutrectangular grooves 720 nm deep [see Fig. 16(b)], we pre-dicted an efficiency of 95%. We expect that, with optimi-zation of our fabrication process, we will be able to raisethe efficiency appreciably. For comparison, the highestefficiency that we observe for gold-overcoated gratings, at1.06 mm, is 95%.

12. UNIQUE PROPERTIES OF MULTILAYERDIELECTRIC GRATINGSBecause the properties of a multilayer dielectric gratingderive from interference rather than from conductivity,such a grating will show pronounced variation of effi-ciency with wavelength. As with multilayer dielectricstacks used for other purposes, it is possible to have mul-tiple design criteria; a grating may be effective in reflect-ing a selected band of wavelengths while efficiently trans-mitting other wavelengths.The behavior of a multilayer dielectric grating, for any

given wavelength, polarization, and angle of incidence, isgoverned by the phase retardation properties of themultilayer stack and the depth and the shape of the grat-ing grooves. These grating characteristics can be ad-justed during fabrication in order to control the distribu-tion of energy among reflected, transmitted, anddiffracted beams. The wavelength discrimination inher-ent in a multilayer stack makes it possible to build grat-ings that transmit or reflect light with high efficiencywithin a narrow optical wavelength band. Diffraction ef-ficiency (for specific incident radiation) can be adjustedbetween 0.01% and 98%, so that a grating can be designedto have nearly any desired efficiency and bandwidth.This extreme optical selectivity, which is not possiblewith conventional metallic or bulk dielectric transmissiongratings, allows a narrow spectral region to be selected tothe exclusion of all others.As we have discussed in Section 5, when the groove

spacing is sufficiently small only two orders occur in theair outside a grating. Under these conditions one can de-sign a grating to control the relative energy flowing infour directions. For a near-Littrow mounting, these arespecular reflection, direct transmission, retroreflection,and Bragg transmitted diffraction. With an appropriateHR multilayer stack, the reflected diffraction efficiencyaround a selected wavelength can exceed 99%.A wide variety of designs are possible for combining di-

electric gratings with an underlying multilayer dielectricstack. Many of these designs have relatively slight sen-sitivity to variations in groove shape and depth. In par-ticular, the variety of high-efficiency designs permitschoices based on additional constraints, such as ease offabrication and resistance to damage.

ACKNOWLEDGMENTSThis work was supported under the auspices of the U.S.Department of Energy at Lawrence Livermore NationalLaboratory (LLNL) under contract W-7405-Eng-48. We

are indebted to B. Stuart of LLNL for performing damagetests on our sample gratings, to D. Raguin of RochesterPhotonics and H. Garvin of Hughes Research Center foretching services, and to J. Yoshiyama of LLNL for SEManalysis. We also acknowledge the assistance of C. Ho-aglan, E. Furst, and R. Agayan of LLNL.

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