Design and fabrication of binary slanted surface-relief
gratings for a planar optical interconnection
J. Michael Miller, Nicole de Beaucoudrey, Pierre Chavel, Jari Turunen,
Edmond Cambril

To cite this version:
J. Michael Miller, Nicole de Beaucoudrey, Pierre Chavel, Jari Turunen, Edmond Cambril. Design
and fabrication of binary slanted surface-relief gratings for a planar optical interconnection. Applied
optics, 1997, 36 (23), pp.5717-5727. �10.1364/AO.36.005717�. �hal-00878957�

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Design and fabrication of binary slanted
surface-relief gratings for a planar opticalinterconnection
J. Michael Miller, Nicole de Beaucoudrey, Pierre Chavel, Jari Turunen,and Edmond Cambril

We propose the use of binary slanted surface-relief gratings with parallel-groove walls as input andoutput couplers in a planar optical interconnect. Parametric optimization of cascaded output couplers
is employed to design an interconnect consisting of N output couplers producing a uniform intensity
distribution with a high efficiency that may be realized in one lithographic etching step. The sensitivityof a N 5 4 interconnect to various fabrication errors is analyzed. We demonstrate the operation of aslanted surface-relief grating manufactured with electron-beam lithography and reactive-ion etching for
an operating wavelength of l 5 0.633 mm. © 1997 Optical Society of America
Key words: Subwavelength gratings, grating couplers, slanted surface-relief profiles, planar optics.

1. Introduction

Planar-integrated optical systems1 are of considerable interest in optical interconnection since the optical components in these miniature systems are
inherently aligned.2 The constituent optical elements in planar optical systems are typically surfacerelief diffractive components.1 Microlithographic
fabrication techniques are employed to generatethese structures, which may have a binary, multilevel, or continuous profile. The synthesis of these
diffractive elements is often based on the paraxial
approximation, which leads to profiles with transverse features of dimensions much larger than the
wavelength of the incident light l. If nonparaxial
deflection angles are required, as is the case in planar
optics, the diffractive profile will contain featureswith dimensions of the order of the optical wavelength l. Consequently, the diffraction characterisJ. M. Miller, N. de Beaucoudrey, and P. Chavel are with theInstitut d’Optique Théorique et Appliquée, Centre National de laRecherche Scientifique, B.P. 147, 91403 Orsay, France. J. Turunen is with the Department of Physics, University of Joensuu,
P.O. Box 111, FIN-80101 Joensuu, Finland. E. Cambrill is with
the Laboratoire de Microstructures et de Microélectronique, Centre National de la Recherche Scientifique, 196 avenue Henri
Ravera, B.P. 107, 92225 Bagneaux, France.
Received 12 November 1996; revised manuscript received 27
March 1997.
0003-6935y97y235717-11$10.00y0© 1997 Optical Society of America

tics must be analyzed by use of electromagnetic
theory.3 Furthermore, the use of resonance-domain
diffractive elements4 in planar optics may alleviate
some of the reported disadvantages of paraxial designs.5 This has only recently become possible withthe rapid advances in submicrometer lithography
and mathematical synthesis techniques in diffractive
optics.6
In this paper we concentrate on the design of binary slanted surface-relief diffractive elements withsubwavelength features for use in a planar optical
backplane. The operation is based on input and output grating couplers, and the interconnection is established by multiple total internal reflections inside
the backplane substrate, as illustrated in Fig. 1. In
this ideal configuration the input coupler is designed
to deflect normally incident light efficiently into the
backplane substrate at angles larger than the criticalangle, whereas the cascaded output couplers ensure a
uniform intensity distribution with a high efficiency
among the different processors. The main design
problem of obtaining a large deflection for a normally
incident wave can be mitigated by use of highfrequency binary lamellar gratings only with great
difficulty.7,8 Likewise, multilevel gratings9 proveequally troublesome, since the blaze effect of triangular profiles is not valid in the resonance domain.
Slanted volume holographic Bragg gratings10,11
would provide the ideal solution, but ultimately replication techniques will be used to mass-produce a
10 August 1997 y Vol. 36, No. 23 y APPLIED OPTICS

5717

Fig. 1. ~a! General configuration for a planar optical backplane interconnection.
M 5 0.

planar optical back-plane, which clearly favors theuse of some form of surface-relief diffractive element.
With the exception of a proposed backplane bus
based on volume holographic gratings12 and holographically recorded surface-relief gratings,13 thedemonstrator systems manufactured to date have
employed identical binary lamellar surface-relief
output-coupling gratings.14,15 If such a system is
employed to distribute a clock signal in a multiprocessor system, one drawback is that the intensity ateach processor diminishes monotonically with increased distance from the input coupler. This nonuniformity dephases the individual processor clocks,
i.e., it introduces a clock skew. We can avoid thisclock skew by designing individually the successive
output-coupling gratings so that, in a cascaded formation, they produce a uniform intensity distribution.
In considering the multiple output couplers of Fig.
1, the groove-slant angle of a binary grating can be
employed as a new degree of freedom in the design
process. The remaining grating parameters arethen equal for each output coupler. A continuous
choice of the diffraction-order efficiency is provided ateach output coupler by variation of the slant angle.
Some preliminary results with a variation of theslant angle were presented in Refs. 16 and 17. Although such a combination of gratings provides anelegant solution to the problem of obtaining a uniform intensity distribution, the disadvantage is that
one lithographic etching step is required to generate
the surface-relief profile of each single output-coupler
grating.
In what follows, we further elaborate the design ofthese slanted surface-relief gratings such that only asingle etching step is needed to produce N output
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APPLIED OPTICS y Vol. 36, No. 23 y 10 August 1997

~b! Ideal T out,nand Rout,nfor a system with N 5 4 and10

couplers, which yield a uniform intensity distribution
and a high efficiency. By utilizing parametric optimization, we design a series of N 5 4 and N 5 8 suchcouplers with different fill factors. The sensitivity of
the N 5 4 solution is analyzed in the presence of themost prevalent fabrication errors. Finally, we demonstrate the operation of a slanted surface-relief grating coupler at l 5 0.633 mm. This component was
realized in photoresist by use of direct electron-beamwriting and reactive-ion etching ~RIE!. To ourknowledge this is the first time that a slanted
surface-relief grating has been used in a planar optic
configuration.
2. Design of the Grating Couplers

We begin by determining the goal performance characteristics for input and output couplers in a perfect
backplane interconnection. This is followed by a description of binary slanted surface-relief gratings andthe optimization technique that we employ to implement the matching of the ideal grating-coupler attributes with such profiles.
A.

Optical Backplane

The general configuration for a planar optical interconnection is illustrated in Fig. 1~a!. Here the incident light is coupled into the backplane by means of
the l 5 61 transmitted diffraction orders of the input
grating coupler, making an angle u61 to the substratenormal. On either side of this grating reside M andN output couplers, respectively. We require the input coupler to have a splitting ratio that ensures anequal distribution of the incident light between the
M 1 N output couplers. Concentrating now on one
side of the input coupler, the kth reflected and the lthtransmitted diffraction-order efficiencies for the nth

Setting uinc 5 2uB 5 245° and nsub 5 1.5, we obtain
Lxyl 5 0.943. This result implies that the k 5 1
reflected order is also perpendicular to the substrate,
whereas the k 5 2 reflected order becomes collinear to
the incident direction. Similarly, for an input coupler with Lxyl 5 0.943 @see Fig. 2~b!# and that isilluminated at normal incidence, the l 5 61 transmitted orders have diffraction angles of 6uB.
The design problem for the output couplers consists of searching for the optimal values of the useful diffraction-order efficiencies @given by Eqs. ~1!#
while minimizing the diffraction efficiencies of theunwanted propagating orders, i.e., the k 5 1 andk 5 2 reflected orders. Since the light intensity
reaching each grating depends on the response of
the previous gratings, we must consider a global
synthesis problem with respect to the characteristics of the backplane, i.e., the N cascaded output
couplers. Accordingly, we employ the following
figures of merit to quantify the backplane performance:
1. The total efficiency
N

Fig. 2. Grating geometry for an optical backplane: propagatingorders for ~a! an output and ~b! an input coupler.

Rout,nk

T out,n.
l

output coupler are denoted byandMultiple or single connections at the nth output may
be envisaged. The useful diffraction orders then correspond either to several or to a lone propagatingtransmitted order. Here, only one connection ateach output is considered. This corresponds to tailoring the diffraction-order efficiencies T out,nand
1
Rout,n.
Toachieveauniform
distribution
of
the
in0
tensities emanating from each output coupler ~i.e., In
5 I1 for n 5 2, . . . , N!, combined with a high efficiency, the diffraction-order efficiencies for the N output couplers must follow the relations18
T out,n5 ~N 2 n 1 1!21,
1

Rout,n5 1 2 T out,n
,
0
1

(1)

out, n5 0 and n 5 1, . . . , N. The
where T out,nlÞ1 5 RkÞ0diffraction-order efficiencies T out,nand Rout,nare
1
0

shown in Fig. 1~b! for an ideal system with N 5 4
output couplers.
Turning our attention to the backplane characteristics, we see that the bounce angle uB inside thesubstrate must be larger than the critical angle ucrit 5
arcsin~nairynsub!, where nsub represents the refractive
index of the substrate. We set uB 5 45°, which isappropriate as soon as nsub . 1.41. For the component output couplers of the backplane @Fig. 2~a!#, we
fix the angle of the first transmitted diffraction orderto uout
5 0°. The continental European form of the1
grating equation19 implies the following relation between the angle of incidence uinc, the wavelength l,
the refractive index of the substrate nsub, and thespatial period of the grating Lx:l.
Lx 5 2nsub sin uinc

(2)

h5

( ~R

out,n210

3 Rout,n22
3 · · · 3 Rout,0!T out,n
,
0
0
1

(3)

n51

where we define Rout,0
as the input efficiency onto the0
first output coupler. In the synthesis problem werequire at least h . 90% when Rout,05 100%.
0
2. The uniformity error U, defined in terms of theintensities as
U5

I max
2 I min
n
n
min ,
I max
1
I
n
n

(4)

where Imax
and Imindenote the maximum and mininnmum intensities, respectively, among the N output
beams. Typically, this value should be no higher
than, say, U 5 10%, to avoid clock skew.
3. The overall efficiencies lost into the k 5 1 andk 5 2 unwanted reflected orders, given byN

h1,lost 5( ~R

3 Rout,n22
3 · · · 3 Rout,0! Rout,n
,
0
0
1

(5)

(~R

3 Rout,n22
3 · · · 3 Rout,0! Rout,n
,
0
0
2

(6)

out,n210n51Nh2,lost 5

out,n210n51which must be as low as possible in order to avoid
cross talk in the system.
It should be mentioned that surface-relief gratings
with a sinusoidal profile and varying relief depths cangive a uniform intensity distribution.13 In spite of
the relatively straightforward nature of generating
sinusoidal surface-relief profiles, a backplane basedon these coupling gratings will have a low efficiency
since large amounts of light are diffracted into thek 5 2 reflected order of the output couplers.
B.

Slanted Binary Surface-Relief Gratings

Our selection of a surface-relief profile for the output
couplers depends first on whether N cascaded ele10 August 1997 y Vol. 36, No. 23 y APPLIED OPTICS

5719

Fig. 3. Configuration for a slanted surface-relief grating.

ments can match the desired performance requirements ~see Subsection 2.A! and second, whether we
can realistically conceive of manufacturing such amultiple-grating system, i.e., the surface-relief profiles must be compatible with state-of-the-art microlithographic fabrication techniques. This latter
requisite can be met if we employ gratings with single
grooves of similar dimensions ~thereby facilitating
the control of proximity effects with electron-beamwriting! and equal relief depths. The degrees of
freedom in the design process then become the groove
width and the relief depth, and, furthermore, we liberate the slant angle of the groove so that we are notconfined to binary lamellar profiles.
In Fig. 3 is illustrated such a binary slanted
surface-relief grating.16,17 A plane electromagnetic
wave is incident upon the permittivity profile, which
has the form
n2~x9! 5

H

n1 2
,
n2 2

when 0 # x9 , c9,when c9 # x9 , L(7)when 0 # z , h. Here L is the grating period in x9
direction, h is the relief depth, a denotes the slant
angle, and K is the grating vector, which makes anangle F 5 py2 1 a with the z axis. The grating
profile is characterized in the ~x, z! coordinate system
by a single groove of width cx 5 c9ycos a, a period of
Lx 5 Lycos a, and a fill factor of ff 5 cxyLx. Regions
0 and 3 consist of isotropic materials with permittivities n20 and n23.
Table 1. High-Efficiency Output-Coupler Grating Parameters andDiffraction Efficiencies in TE and TM Polarizations with a Constant Fill
Factor ffn

Parameters andDiffracted Orders

TE Polarization

TM Polarization

hyl
a
ff
T out,4~%!
1
Rout,4~%!
0Rout,4~%!
1
Rout,4~%!
2

1.058
230.0°
0.5
97.2
0.1
0.4
2.3

2.351
230.4°
0.5
97.6
0.6
0.9
0.8

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APPLIED OPTICS y Vol. 36, No. 23 y 10 August 1997

Fig. 4. Diffraction efficiencies of the k 5 0, 1, 2 reflected and l 5
1 transmitted orders for an output coupler in TE polarization as afunction of ~a! the relief depth, expressed as hyl, for a and ff
constant ~a 5 230° and ff 5 0.5!, ~b! the grating slant angle a, forh and ff constant ~hyl 5 1.058 and ff 5 0.5!, and ~c! the fill factor
ff, for h and a constant ~hyl 5 1.058 and a 5 230°!. The numbered points in ~b! and ~c! denote the output-grating solutions forthe backplane with N 5 4 in Fig. 1~b!.

C.

Synthesis of the Grating Couplers

We begin the design procedure by concentrating on
the last output coupler ~n 5 N!, i.e., a grating thatshould deflect 100% of the incident light into the l 5
1 transmitted diffraction order ~T out,N
5 1! and there1
fore out of the backplane. This element is a highefficiency beam-deflector grating whose slant angle
satisfies the Bragg condition. In the search for the
optimal slant angle a, an initial estimate can therefore be given byainit 5

F S
S

1
n0
arcsinsin uinc2n#
1 arcsinDn0ll
sin uinc 1
n#
n# Lx

DG

,

(8)

where we have homogenized the binary grating intoa uniform slab with an effective index n# using theTable 2. Output-Coupler Parameters and Performance Characteristics for a Backplane Composed of N 5 4 and N 5 8 Slanted Gratingsa

Parameters andPerformance Characteristics
hyl
a
ff

U ~%!
h ~%!
h1,lost ~%!
h2,lost ~%!

N54

N58

ff # 0.5

ff $ 0.5

ff # 0.5

ff $ 0.5

1.059
233.50°ff1 5 0.114
ff2 5 0.129
ff3 5 0.155
ff4 5 0.484
1.0
94.2
0.3
5.0

1.057
224.69°ff1 5 0.806
ff2 5 0.771
ff3 5 0.725
ff4 5 0.575
0.4
95.6
3.2
0.8

1.041
235.55°ff1 5 0.084, ff5 5 0.109
ff2 5 0.088, ff6 5 0.122
ff3 5 0.094, ff7 5 0.144
ff4 5 0.100, ff8 5 0.504
2.2
92.2
0.8
6.1

1.046
224.64°ff1 5 0.873, ff5 5 0.804
ff2 5 0.862, ff6 5 0.707
ff3 5 0.848, ff7 5 0.726
ff4 5 0.829, ff8 5 0.575
0.5
96.4
2.6
0.7

Fill factors for the N 5 4 and N 5 8 gratings were ffn # 0.5 and ffn 5 0.5, respectively, with constant slant angles an for n 5 1, . . . , N.

atheory of form birefringence. In TE polarization n# isdetermined from20
n# 2 5 12 ~n12 1 n22!.

(9)This lowest-order approximation is sufficient for our
purposes. From Eq. ~2! and Eq. ~8!, where l 5 1, uinc
5 245°, n1 5 n0 5 1.5, and n2 5 1.0, we find that ainit
5 228°.
The diffraction-order efficiencies were evaluatedwith a rigorous coupled-wave algorithm,21 thereby
avoiding a decomposition of the permittivity profile
into multiple slabs. We then employed parametric
optimization to maximize the diffraction-order efficiency T out,Nby tuning the relief depth, the slant
1angle, and the fill factor. The resulting grating parameters and diffraction-order efficiencies in the TE
and TM polarizations are listed in Table 1. Although the diffraction efficiencies differ only slightly,
the solution in TE polarization is preferable since therelief depth is shallower than that in TM polarization. We therefore concentrate on the design of couplers in TE polarization. In Fig. 4, the k 5 0, 1, 2
and l 5 1 diffraction-order efficiencies in TE polarization are plotted around the optimum values of thethree grating parameters. In each of these curves,
one of the parameters, namely, h, a, or ff, is varied
while the two others are held constant and equal totheir respective optimum values ~see Table 1!. The
form of the curves in Fig. 4 suggests that we canattain the remaining N 2 1 output-coupler solutions
by detuning the Nth output-coupler profile from its
optimum parameters. With a constant relief depth,hn 5 hN, there exist two options: We may either
alter an, while holding ffn 5 ffN, or conversely, we canvary ffn by setting an 5 aN, for n 5 1, . . . , N. The
numbered points in Figs. 4~b! and 4~c! denote thesuccessive output gratings ~for N 5 4! withdiffraction-order efficiencies closest to those of Fig.
1~b! for different slant angles and different fill factors,
respectively.
In the quest for the N output couplers that can be
fabricated in one etching step, we now turn our attention to the synthesis of a series of such gratings
with different fill factors ffn but constant slant angle

an 5 aN for n 5 1, . . . , N. In Fig. 4~c! there exist two
separate regions containing gratings with fill factorsthat may match the requirements of Fig. 1~b!: thosewith low fill factors ~ ffn # 0.5! and those with high fill
factors ~ ffn $ 0.5!. Starting from these initial values of the fill factor ffn, we employed a steepestdescent algorithm to minimize a quadratic merit
function, defined in terms of Eqs. ~3!–~6!, in a manner
similar to the design of grating beam splitters.22
The optimized solutions are listed in Table 2 for N 5
4 and N 5 8. For each of these solutions we obtain
efficiencies of h . 90% and uniformities of U # 1.0%,
except for N 5 8 and ffn # 0.5, for which we obtainU 5 2.2%, suggesting that this solution is not optimal

Fig. 5. Effect of a systematic error in hyl on the performance ofthe backplane: uniformity error U, efficiency h, and efficiency lost
into the k 5 1 and k 5 2 reflected order h1,lost ~solid curve! andh2,lost ~dashed curve!.
10 August 1997 y Vol. 36, No. 23 y APPLIED OPTICS

5721

since the algorithm has stagnated in a local minimum. Furthermore, we can expect extremely low
values of h2,lost for a backplane employing gratings
with ffn $ 0.5.
With our fabrication process, relief profiles withfine grooves will be more readily realizable than thosewith large grooves: Estimates for the realizable fill
factors and slant angles are the ranges 0.2 , ffn , 0.9
and 0 , uau , 45° ~see Section 4!. Hence, we therefore consider only solutions with ffn $ 0.5. During
the optimization procedure we retained 11 Rayleighorders in the calculations of the diffraction-order efficiencies. After the merit-function minimum wasfound, the uniformity and overall diffraction efficiency of the backplane solution converged to thefourth significant digit by the inclusion of 61 Rayleighorders.
Now that we have synthesized a set of N outputcoupler gratings that can, in principle, be realized in
one etching step, it remains to specify an input coupler. In exploring the possibility of also using aslanted grating, we notice that, for the specific backplane configuration with M 5 0 @Fig. 1~b!#, we require
an input coupler that operates as a high-efficiency
beam-deflector element at normal incidence. By thereciprocity theorem3 we have already specified the
profile of this coupler, which is the mirror image ~withrespect to z! of the last output-coupler profile listed inTable 1. However, the addition of this element tothe backplane implies an extra etching step over andabove the first etch required for the output couplers.
Nonetheless, for a backplane with one input coupler
and M 1 N output couplers, we have reduced thetotal number of etching steps from 1~input coupler! 1
M 1 N, as would be required for output couplers withdifferent an,16,17 to 1~input coupler! 1 1~IF M Þ 0! 1
1~IF N Þ 0! # 3, with output couplers with different
ffn.

Fig. 6. Effect of a systematic error in the fill factor ff on theperformance of the backplane.

Fig. 7. Effect of a systematic error in the slant angle a on theperformance of the backplane.

5722

APPLIED OPTICS y Vol. 36, No. 23 y 10 August 1997

3. Fabrication-Error Analysis

Direct electron-beam writing offers a realistic meansof patterning the fine grating structures that were
presented in Section 3. Despite the high resolutionof this particular plotting technique, it is inevitablethat the entire fabrication process ~initial patterning,
pattern transfer, etching, and possible replication!
will introduce some form of inaccuracy to the grating
profiles. In this section, we examine the sensitivityof the solution of N 5 4 output couplers with constantslant angle and different high fill factors ff $ 0.5 ~thegrating parameters are given in the second column ofresults in Table 2! to some of the more common manufacturing errors, i.e., deviation from the ideal relief
depth, fill factors, and slant angle and incorrectly
shaped groove walls. The analysis of the two lattertypes of error is particularly important since it can be
expected to be difficult to control the slant angle whenemploying RIE to generate slanted structures.23–25
To draw a comparison with the results that follow,
it should be noted that a backplane composed of N 5
4 identical output couplers that consist of singlegrooved binary lamellar reflection gratings,14 and one
which is manufactured perfectly, will have values of
U 5 33.8%, h 5 52.4%, and h2,lost 5 7.3%. Our
intention, therefore, is to determine the expected performance gain over these parameters for a backplanecuracies in the fill factors, whereas the overall diffraction efficiency is unaltered. This surprising
effect arises from the symmetry of the l 5 1 and k 5
0 diffraction efficiencies @see Fig. 4~c!#, which leads toan exact compensation of the four output couplers’contribution to the total efficiency h, even if they are
not equal. An error of U 5 10% occurs for Dff 5
60.009, at which point h 5 95.7%, h1,lost 5 3.0%, andh2,lost 5 1.0%. Errors in the slant angle as large as
Da 5 65° relate to a uniformity error of U 5 24%.
For these errors we find that U 5 10% when Da 5
22.25° and h 5 91.6%, h1,lost 5 5.2%, and h2,lost 5
1.8%.
If RIE or focused ion-beam milling are employed togenerate slanted surface-relief profiles,17 we can expect to obtain features with sidewalls that are notparallel but are either undercut @Fig. 8~a!# or overcut23 @Fig. 8~b!#. We denote the change in the angleof the groove walls by b. A positive value of b implies undercutting @Fig. 8~a!#, whereas a negativevalue corresponds to overcutting @Fig. 8~b!#.
We attempted to compensate for the effects ofover- and undercutting by ensuring that the fill
factor in the middle of the groove ffmid is equal to thedesign value22 @Fig. 8~c!#. The groove walls thenpivot about this center position as b varies. A
groove profile without compensation implies thatthe fill factor at the top of the grooves ffup has theideal value @Figs. 8~a! and 8~b!#. We analyzed theeffects of over- and undercutting using electromagnetic grating theory,21 in which we approximatedgratings with nonparallel groove walls by stacking
multiple slanted binary profiles with varying fill

Fig. 8. Slanted binary gratings with nonparallel groove walls:~a! undercutting ~b . 0!, ~b! overcutting ~b , 0!, ~c! overcut profile
with compensation, and ~d! approximation with stacked binary
lamellar profiles and varying fill factors with ~1! nonslanted and ~2!
slanted sections.

composed of N 5 4 different output couplers and in
the presence of manufacturing errors.
In what follows the backplane figures of merit,given by Eqs. ~3!–~6!, are presented as a function of
the magnitude of a particular fabrication error. In
Fig. 5 is shown the effect on performance of the backplane solution of a systematic error in the relief depthh, expressed as Dhyl. The uniformity error is lesssensitive to negative values of Dhyl than it is topositive values of Dhyl. The overall efficiency hdrops by some 10% with Dhyl 5 60.2 and h1,lost does
not vary noticeably in the interval, whereas h2,lost
dips at values of hyl close to the optimum. For avalue of Dhyl 5 0.05 we find that U 5 11.3%, h 5
91.1%, h1,lost 5 4.0%, and h2,lost 5 1.0%.
The consequences of errors in the fill factors andnonideal slant angles are shown in Figs. 6 and 7.
The uniformity error is extremely sensitive to inac-

Fig. 9. Effects of overcutting ~b , 0! and undercutting ~b . 0!,
without compensation, on the performance of the backplane.
10 August 1997 y Vol. 36, No. 23 y APPLIED OPTICS

5723

Fig. 10. Effect of undercutting ~b . 0! and overcutting ~b , 0!, with compensation, on the performance of the backplane.

factors @Fig. 8~d!, part ~2!#. Convergence of thediffraction-order efficiencies can then be achieved
with half as many slices compared with the approximation shown in Fig. 8~d!, part ~1!.
The effects of this type of error for noncompensatedand compensated profiles are shown in Figs. 9 and 10,
respectively, with the limits for b being determinedby intersecting groove walls. The introduction ofcompensation into the profiles indeed reduces thesensitivity of the backplane solution to this type oferror: In effect we ensure that the amount of material in the grating profile remains equal to that of theparallel-grooved grating. Our initial experimental
results ~see Section 4! suggest that we can expect anerror of b ' 1.6°, i.e., undercutting. Compensationof the groove profiles is therefore essential.
In this analysis we have assumed that only one
specific fabrication error is present in the grating
profiles. If an equal error is introduced into each of
the n 5 1, . . . , N couplers, we require an accuracy ofat least Dhyl 5 0.05, Dff 5 60.009, and uDau 5 2.25°
and compensation of the groove walls to obtain an
N 5 4 backplane with values U # 10%, h $ 90%, andh1,lost and h2,lost as low as possible.
5724

APPLIED OPTICS y Vol. 36, No. 23 y 10 August 1997

4. Experimental Verification

For an initial verification of our design process, weelected to realize the output-coupler solution in TE
polarization ~Table 1! for an operating wavelength ofl 5 0.633 mm. The slanted surface-relief grating
was realized in photoresist by use of direct electronbeam writing and RIE with a procedure similar tothat used for manufacturing high-aspect-ratio
effective-index diffractive elements26; it is as follows:
Photoresist ~with a refractive index of n1 5 1.6, which
was measured interferometrically! was spin coatedon a 220-mm-thick glass wafer ~with a refractive index of 1.52! to a depth of 1.0 mm. A 0.1-mm-thicklayer of metal was then deposited onto the surface ofthe photoresist, and the ensemble was coated with
0.15-mm poly~methyl methacrelate! ~PMMA!. The
high-frequency structure was then patterned over a 2
mm 3 2 mm area by use of a Jeol Model JBX5D2Uelectron-beam writer with an accelerating voltage of
50 kV and a beam current of 10 nA, producing a spot
diameter of 100 nm. After development of thePMMA, the metal layer was chemically etched. The
PMMA was then removed, leaving the metal layer asFig. 11. SEM micrographs of a slanted surface-relief grating in photoresist generated by use of electron-beam writing and RIE: ~a! crosssection at the nonoblique perspective of the slanted-grating profile and ~b! a close-up showing the rounded edges and slight undercut.

the mask. A standard RIE chamber was employed
to generate the slanted surface-relief profiles. Here,the substrate was oriented by approximately 230° tothe electrode plates, while ensuring that the grooves
were parallel to the edge of the holder.
A scanning electron microscope ~SEM! micrographof the resulting grating is shown in Fig. 11. We
measured from Fig. 11~a! that the slant angle had aslant angle of a ' 222°. The fill factor and the relief
depth were determined in the following manner: An
accurate scale for the SEM micrograph was established by use of the grating period as a standard
length. This was deemed possible since the resolution of the electron-beam writer is 2.5 nm, so thegrating period is Lx 5 0.598 6 0.0025 mm. From
this scale these parameters were straightforwardly
calculated to be ff ' 0.35 and h ' 1.0 mm. The
difference between the grating slant angle and that of
the substrate holder is '8°. We expect that this isdue to electric-field inhomogeneities as the substrateholder extends into the plasma dark space.
We are currently examining different techniques toobtain more accurate control of the slant angle23–25
and the fill factor to achieve the required precision as
determined in Section 3. The grating realized inphotoresist does not correspond to optimal values ofthe parameters determined in Subsection 2.C ~Table
1 and Fig. 4!, therefore we cannot expect the highest
efficiency for the resulting output coupler.
The k 5 0, 1, 2 reflected and l 5 1 transmitted
diffraction orders were recalculated with the experimental values of the parameters ~i.e., grating period, fill factor, slant angle, and refractive indices ofthe photoresist and substrate! and are plotted inFig. 12 as a function of the relief depth h. A highefficiency can be expected for relief depths in therange 0.3 mm , h , 0.6 mm. In an attempt toobtain a relief depth within this region, anothergrating was prepared ~once again with a resist witha thickness of 1 mm!, and the etch time was calibrated with the aid of the SEM micrograph shownin Fig. 11.
We characterized this grating while it operated asboth an input and an output coupler at the design
wavelength of l 5 0.633 mm. For measuring thereflected and transmitted diffraction-order efficiencies, the grating substrate was index matched onto aparallelogramic prism with a 45° angle. Opticallycementing antireflection coatings onto three faces ofthe prism ~corresponding to the input beam and theFig. 12. Calculated diffraction efficiencies for an output coupler
as a function of h ~in micrometers!. Other grating parameters are
Lx 5 0.598 mm, a 5 222°, ff ' 0.35, n0 5 1.52, n1 5 1.60, n2 5 n3
5 1, uinc 5 244.03°.
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Table 3. Experimental and Theoretical Diffraction Efficiencies for the
Fabricated Slanted Grating Acting as Output Coupler or Input Coupler

Results

Diffracted
Order ~%!

Experimental

Theoretical

Output

T out
1
Rout
0
Rout
1
Rout
2

46
30
4
7

48.6
42.0
2.0
7.4

Input

T in
21
T in
0
T in
1
Rin
0

46
33
8
3

49.0
38.1
11.4
1.5

Coupler Mode

reflected diffraction orders in the output mode!
avoided multiple reflections in the prism.
The experimentally measured diffraction-order efficiencies are given in Table 3. We have included thebest-fitting numerical values for an ideal parallelwalled slanted grating, corresponding to an estimated relief depth of h 5 0.75 mm. In this analysis
we included the 0.25-mm unetched resist layer underneath the grating structure. Some 13% and 10% of
the incident light is unaccounted for when the grating
operates as an output or an input coupler, respectively. This can be explained partially by slight absorption in the photoresist and also by the existenceof extra orders around the measured one that arecaused by stitching errors ~we step and repeat at
80-mm intervals in both directions!.
The experimental and theoretical results listed inTable 3 are in reasonable agreement, although amore accurate analysis would require a SEM micrograph, for which we would need to gold-coat andcleave the grating, rendering it useless. Nonetheless, we have clearly demonstrated with these initial
results the prospects of such elements and, furthermore, that such binary slanted surface-relief profiles
are indeed compatible with state-of-the-art microlithographic fabrication techniques.
5. Conclusion and Discussion

In conclusion, we have investigated the concept ofbinary slanted surface-relief gratings for input andoutput coupling in a planar optical backplane.
Parametric optimization was used to design a backplane composed of these gratings with a high efficiency and low uniformity error and loss into the k 5
1 and k 5 2 reflected orders. We demonstrated two
configurations of output couplers: Either the slant
angle of the individual gratings varies while the fill
factor and relief depth remain constant, or the slant
angle and relief depth are constant while the fill factor varies. It is this latter possibility that enables a
set of N output-coupler gratings to be manufactured
in truly one lithographic etching step.
In light of the fabrication-error analysis and, moreover, of our initial experimental results, it seems reasonable to assume that the realization of a planar
optical interconnect consisting of multiple binary
5726

APPLIED OPTICS y Vol. 36, No. 23 y 10 August 1997

slanted gratings is indeed feasible. Furthermore,
replication techniques such as UV embossing could
be employed to copy these slanted gratings,27 leading
to the eventual mass production of such a planar
optical interconnect. In future systems the flexibility of such interconnects may also be enhanced bymultiple connections at each output coupler. The
technique of pulse-frequency modulation28 of a highfrequency carrier grating can be employed to encode
optical functions such as fan-out and focusing,29
thereby increasing the interconnection density andreducing the complexity of the receiver optics.
J. M. Miller thanks his family, friends, and colleagues for their continued support, which has enabled the completion of this study. He is grateful to
P. Ehbets for fruitful discussions and D. Joyeux formeasuring the refractive index of the photoresist. A
Human Capital and Mobility fellowship from the EC
is acknowledged. This project was partially supported by Ultimatech, Centre National de la Recherche Scientifique, France. We thank P. Pääkkönen
for his help in the preparation of the manuscript.
We also thank the reviewers for their suggestions on
improving this manuscript.
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