connection density, time delay, and power dissipation for three types of connection ... connection density of a space-variant free space system has a connection ...
Guided-wave and free-space optical interconnects for parallel-processing systems: a comparison Linda Jean Camp, Rohini Sharma, and Michael R. Feldman
Guided-wave and free-space optical interconnects are compared based on insertion loss, link efficiency, connection density, time delay, and power dissipation for three types of connection networks. Three types of free-space interconnect systems are analyzed that are representative of a wide variety of free-space systems: space-variant basis-set and space-invarient systems. Results indicate that the connection density of a space-variant free space system has a connection density roughly equivalent to a two level guided-wave system with a pitch of 10 pum (for a 1-pum wavelength) and a core refractive index of 2.0. It is also shown that the connection density of basis-set and space-invariant free-space systems can be several orders of magnitude higher than fundamental limits on the connection density of dual-level guided-wave interconnect systems when large-scale highly connected networks are employed.
1. Introduction
The cost per performance of many very-large-scaleintegrated computational systems can be improved by replacement of particular conventional electrical interconnects within and/or between chips with optical interconnects. Optical interconnects have the potential to increase the interconnect bandwidth while reducing the area, power dissipation, and cross talk of the system. 1-7
These improvements may be achieved with either free-space or guided-wave optical interconnects, depending on the technological issues involved in the specific implementation. Many manufacturers are currently trying to decide whether to invest resources into guided-wave or free-space optical interconnects. On the one hand, free-space interconnects, because of their three-dimensional nature, have fundamental advantages over guided-wave interconnect systems. 7 On the other hand, guided-wave systems have advantages in terms of packaging, simplicity of fabrication, and possibly, cost.4 Therefore it is important to quantify the specific performance trade-offs. In particular, in this paper, guided-wave interconnects, based on rather optimistic assumptions, are compared with realistic present-technology-based models of free-space interconnects. The point of The authors are with the Department of Electrical Engineering, University of North Carolina at Charlotte, Charlotte, North Carolina 28223. Received 15 October 1993. 0003-6935/94/266168-13$06.00/0. ©1994 Optical Society of America. 6168
APPLIED OPTICS / Vol. 33, No. 26 / 10 September 1994
this comparison is to illustrate the minimum performance advantages that can be expected from freespace optical interconnects despite potential advances in the technology of guided-wave interconnect systems. The particular free-space optical interconnect system to be considered here consists of semiconductor laser diodes, photodetectors, and holographic optical elements (Fig. 1). The holograms are used to connect the lasers and the detectors in the desired pattern. 6 Two free-space optical-interconnect systems are considered: (1) the space-variant doublepass holographic system5 and (2) the basis-set system.k' 0 These two systems are illustrated in Figs. 1 and2. In the double-pass system the hologram is divided into many subholograms. Source subholograms are placed over each laser, and detector subholograms are placed over each detector. Each source subhologram divides the incident wave front into F beams (for a fan-out of F), deflects each beam at the appropriate angle, and changes the divergence angles of the beam so that after reflection from the planar mirror, each beam is focused onto the appropriate detector subhologram. Each detector subhologram focuses the incident beam onto the detector located directly below it. In the basis-set system, described in Refs. 8 and 9 illustrated in Fig. 3, a lens is used to perform a Fourier transform of the input plane. A computergenerated hologram (CGH) consisting of M facets is placed in the Fourier plane, and a second lens is used to perform a Fourier transform between the Fourier
Fourier Transform Lens
Module
M facet Hologram
Fourier Transform
Module
IUR Hologram
,
_
w
Laser
Len
A
--
-I-, L
Detector
Fig. 1. Double-pass hologram system configuration as assumed in derivations: n, index of refraction; L, distance between the laser and the detector; R, distance between the CGH and the mirror. f
plane and the output plane. Each CGH facet in the Fourier plane implements a different connection. This requires that each connection pattern be formed by the summing together of a subset of the M connection patterns implemented by the M facets in the Fourier plane. A compact version of the basis-set system, termed the folded basis set, is illustrated in Fig. 2. The upper CGH contains the Fourier-plane subholograms. The planar mirror in Fig. 1 is replaced by a reflective CGH. The lens in Fig. 2 implements the function of both of the lenses in Fig. 3. A guided-wave optical-interconnect system consists of the same laser diodes and photodetectors used in the free-space approach and the single- or dual-level channel waveguides. A dual-level channel-waveguide system is illustrated in Fig. 4. Both holograms 6 11 and waveguides 2 12 can be constructed with existing microelectronics facilities and processing methods. Throughout this paper, holographic and guidedwave interconnects are compared in terms of their performance in implementing particular processorarray connection networks. Three connection networks are considered: mesh, hypercube, and fully connected. Properties of these connection networks are discussed in Section 2. In Section 3 the efficiencies of the two opticalinterconnect implementations are compared for a single isolated net. The dependence of the total optical link efficiency on particular variables such as fan-out and waveguide attenuation is illustrated. In Section 4 the connection densities of the two connection schemes are compared. Fundamental
Fig. 3.
f
Basis-set hologram configuration.
connection-density limitations are compared with practical limitations for particular waveguide and hologram properties and particular processor-array connection networks. It is shown that single-level guided-wave interconnects are impractical for large processor arrays with high connectivity because of cross-talk limitations. In the remainder of the paper, dual-level waveguides are considered. Expressions are derived for the dependence of module area on the number of processing elements (PE's). In Section 5, time delays for the two optical systems are compared. The expressions for area growth from Section 4 are used to determine connection lengths and interconnect latency for particular connection networks. The power dissipations of the three connection networks for the two optical systems are compared in Section 6. 2.
Connection-Network Properties
A connection network describes the pattern of connections used to interconnect PE's within a processor array. We assume that N PE's are evenly spaced over a square module in a rectangular array with dimensions N 1 /2 x N 1/2 . We use the following five parameters to characterize the processor-array conhection network: (1)the normalized longest communication link, L'X; (2) the network bisection width, B; (3) the fan-out from each processor, F; (4) the protective glass layer layer of horizontal guides
layers of low index _
Reflecting M -Faceted Hologram
(cladding) material
layer of verticle guides module via
Fourier Transform Lens
Detector and Source
Subhologram Plane Module
Fig. 2. Folded-basis-set system:
(D.
Fig. 4. Two-level waveguide configuration as assumed in Table 2. 10 September 1994 / Vol. 33, No. 26 / APPLIED OPTICS
6169
number of different connection patterns to be implemented, M; (5) the minimum number of crossovers required in a single-level guided-wave implementation. The normalized longest communication link, LmAx, is defined as LiMA
2 2 = LMAxN1/ /A1/ ,
(1)
where A is the module area and Lm 1 is the maximum distance separating any two directly connected PE's. If the longest connection link lies along a straight line parallel to a module edge, then L' 1A is equal to the maximum number of PE's separating any two directly connected PE's. The bisection width, B, is defined in terms of a partition that lies parallel to an edge of the module and divides the processor array such that N/2 nodes lie on either side of the partition. The bisection width is defined as the number of links crossing such a partition, minimized over all possible layouts. This is a slightly different definition than that given by Thompson in Ref. 10. While Thompson's definition was designed to yield lower bounds on the layout area in terms of an asymptotic dependence on N, our definition can be used to obtain estimates on the actual layout area. Values of normalized longest communication links (L' 1 ), network bisection widths, fan-out, number of crossovers, and number of different connection patterns are given in Table 1 for the three connection networks to be considered. The larger the value for each of these parameters, the more highly connected the processor array. Thus the networks are listed in order of increasing connectivity in Table 1. These three networks were chosen because of their widely varying connectivity. 3. Optical Link Efficiency
The optical system efficiency, aq, of an optical interconnect is defined as the ratio of the power incident upon the detector (PD) to the power emitted by the laser
focus here is upon single isolated nets. A net refers to a single transmitter connected to one or more receivers. A. Free-Space Optical Interconnects The diffraction efficiency of a hologram, 1h, is defined as the ratio of the power diffracted by the hologram to the desired location divided by the power incident upon the hologram. For simplicity we assume that the diffraction efficiencies of each subhologram in Figs. 1 and 2 are identical. For a free-space optical-interconnect system, if the f-number of the laser subholograms matches the divergence angles of the lasers, essentially all of the light emitted by a laser illuminates the appropriate source subhologram. In this case the optical system efficiency of a space-variant double-pass system (rqsv) is given by 'isv = shF,
where F is the fan-out. The optical link system efficiency (ibs) of a folded basis-set system is 7lbs = Nh/F.
(2)
In this section the optical link efficiency of guidedwave and free-space interconnects is compared. The
Table 1.
Parameters for Three Connection Networks that Cover a Wide Range of Connectivity
Connection Network Parameter
Mesh 2
Hypercube
Full Connected
Fan-out Crossovers
4 0
log2 N N/2
NN112 N- 1 2N'12N
Lmax
1
N / /2
2N1/2
4
2 log 2 N
4N
Bisection Width
2NI/
N
1 2
Number of Connection
Patterns (M) 6170
APPLIED OPTICS / Vol. 33, No. 26 / 10 September 1994
(4)
We assume that the diffraction efficiency of each hologram is independent of the fan-out, F. This has been shown to be approximately correct by several research groups. For example, in Ref. 13 the diffraction efficiency of holograms implementing large fanout functions remains > 75% for fan-out from 16 to 1024. B. Guided-Wave Optical Interconnects The optical link efficiency for guided-wave interconnects depends on length-dependent attenuation, r(d), source-to-waveguide coupling, K5W,and waveguide-todetector coupling, Kwd. For unity fan-out the efficiency of a waveguide of length d is given by 1 gw
(PL): 'q = PD/PL-
(3)
= KWdKSWF(d),
(5)
where r(d) = log'1(--ydd/10);
(6)
is the waveguide loss in decibels per unit length. Two types of nonunity fan-out are evaluated. Linear fan-out occurs when all destinations are distributed along a line. Linear fan-out can be implemented with a single waveguide. Remote fan-out, which occurs when the destinations lie in widely separated locations, requires a separate guide for each destination. Fan-out for linearly distributed destinations in a guided-wave system is accomplished by power splitting of a single channel. For F linearly distributed destinations addressed by a series of F - 1 power splitters, each of loss y, designed so that each detector receives the same amount of incident power (neglecting length-dependent attenuation effects), the Yd
power lost, PL, because of the splitters is given by pL
=
1
-
/[T(r 8 )k + 1]}
{Fsf
x (sumisfork = Otok =F- 2),
(7a)
where (7b)
F5 = log-(-1y5 /10).
This yields a total link efficiency for a guided-wave system with linear fan-out of 'igw = KWdKsw[(d)FS]F
(8)
'/{:[1(d)rk] + 1},
where d is the distance between destinations. For remote fan-out a separate waveguide is required for each location. Thus either F sources are required, in which case there is no splitting loss, or one source can be coupled to F waveguides with the resulting splitting loss. For remote fan-out with F sources the link efficiency for each destination is given by IgW =
KWdKSF(d)/F,
(9)
where d is the length of travel along the waveguide from source to destination. For remote fan-out with one source either a tree configuration with log 2 F couplers or a 1 x F star coupler would be required. In many cases a combination of remote and linear fan-out is needed. In this case the remote fan-out power loss would be calculated from Eq. (9), and the linear fan-out power loss would be calculated from Eq. (7). C. Comparison Figure 5 shows the dependence of optical link efficiency on source-to-detector distance as calculated from Eqs. (3)-(6). The source-to-waveguide and waveguide-to-detector coupling efficiencies are assumed to be 90% each. A hologram diffraction efficiency of 85% was assumed. Holographic efficiency
depends on design, optimization method, minimum feature size, deflection angle, and number of phase levels implemented.' 4 Several research groups have fabricated holographic elements with diffraction efficiencies between 70% and 90%.11"14"15 Linear attenuation values in the range from 1.0 to 0.1 dB/cm are considered in this figure. This reflects currently published values.3 4" 2 Generally, low attenuation values require thicker and wider channel-waveguide structures. Attenuation results from various sources including scattering, absorption and radiation losses,' 6 leakage into the substrate,' 7 and coupling between guides. Although noise and linear attenuation generally increase with connection density 3 (from increased fan-out and decreased dimension), they are held constant in these comparisons. For long connection lengths, coupling loss represents only a small fraction of the power loss, with the remaining loss caused by waveguide attenuation. It is evident from Fig. 5 that as the source-todetector distance increases, the link efficiency of an optical waveguide system decreases, while the link efficiency of a free-space system remains constant. This is analogous to comparisons between free-space and electrical interconnect systems,' 8 in which for lengths longer than a particular break-even line length it is advantageous to use optical interconnects. Figure 5 shows that, similarly, when free-space and waveguide interconnects are compared, there is a break-even distance for which free-space interconnects have superior performance. This break-even distance depends on hologram efficiency, waveguide attenuation, and coupling losses, but for current state-of-the-art waveguides and holograms this distance is 5 cm. Figure 6 shows the effect of linear fan-out on interconnect efficiencies calculated with Eqs. (3) and (8). This example assumes constant splitter loss for each guide. Low splitter loss values of 1.2 dB and 1.8 dB (Ref. 19) are used in the figure. Attenuations of 0.1 dB/cm, 3 0.3 dB/cm, 4 and 1.0 dB/cm (Ref. 12) are shown. Connection length is 5 cm. 0.8
100%
spacevariant 80% *
Link Efficiency (r)
~,
0.7
waveguide = 0.1, NIArl, =0.81
Product of Fanroutand Link Efficiency
basisset 0.6
60%
40%
20%
0
1
2
3
4
5
6
7
8
9
10
11
12
Distance (cm)
Fig. 5. Dependence of optical link efficiency on source-to-detector distance. Coupling efficiency is neglected.
Fan-out
Fig. 6. Dependence of optical link efficiency for the longest link on the number of linear fan-out destinations. Connection length is 5cm. 10 September 1994 / Vol. 33, No. 26 / APPLIED OPTICS
6171
implemented is given by
4.
Connection Density
A.
Fundamental Connection-Density Limitations
B < A1/ 2 /(S + W).
In Subsection 4.A the fundamental limits on connection-density capacities are calculated. The maximum number of waveguides that can cross a given width (w = A/ 2 ) is determined. This is compared with the number of connections that can be made with a hologram of area A. 1. Guided-Wave Optical Interconnects Guided-wave systems are limited by size and spacing considerations. 2 0 21 The minimum dimension for confinement to occur depends on the design wavelength and the difference between the refractive indices of the material of the guide, n, and of the cladding surrounding the waveguide, n,. For confinement to occur the waveguide core diameter, W, must have a minimum dimension of 20 W = X/(n2
-
n2)1/2.
(10)
The spacing between guides is limited by cross-talk considerations. The ratio of the amount of power coupled to a neighboring guide to the power remaining in an excited waveguide is referred to as the power ratio. The power ratio between two identical waveguides that are parallel for a distance L is PR = L2 C2 ,
(11)
coefficient. 2 2
The coupling where C is the coupling coefficient between two identical waveguides of width W and index ng surrounded by material of index nc and separated by a distance S is2 3 =(n~k2 -
132)(132
-
n 2k 0)
(1 + yd)k( 2- 2-)
exp[(W
-
S)],
where ko = 2r/X0, p2
=
ngk21
y = (k2nc2 -
-
2)1/2,
(Ak)/]
(14)
The number of processors that can be interconnected can then be determined from the bisection width of the connection network. Equivalently, the area of the interconnect network, A, can be determined from the bisection width, B, given in Table 1 for the three networks considered here. 2. Free-Space Optical Interconnects The fundamental limits on the density capabilities of holographic interconnects can be determined from diffraction-limited beam-divergence properties. Maximum connection density can be achieved with a space-invariant optical system. In a space-invariant system every PE employs the same connection pattern. One can achieve this by choosing a connection pattern that is a superset of all desired connection patterns and by masking off undesired connection links. This can be implemented with a basis-set optical system (Figs. 2 and 3) with the number of basis-set connections, M, set equal to 1. One may find the area of the hologram by totaling the area of each source subhologram (As) and of the detector subholograms (Ad). Setting the two areas equal optimizes total area. This yields Ah =
NAs + NFAd
=
2NFAd
(15)
for the total hologram area. The area of each detector subhologram is limited by8 4 2 2 2 Ad 2 4(1.2) X f /A cos (DF
(16)
where f is defined in Fig. 2 and CF is the deflection angle. Since the PE's are uniformly distributed, the subhologram area associated with a given PE is identical for every PE in the network. To determine the lower bound on connection density, set Ad to the minimum area of the largest subhologram. The size of each detector subhologram is given by the value of Ad for a subhologram at the edge of the network for which tan CF = A'/ 2/(2f), yielding Ad =
(1.2) 2X2 /(cos 4 CDtan2
D).
(17)
where An
=
is minimized for C = 45°, resulting in a lower bound on the hologram area of Ad
n2) ~C
(n 2
(n ng
A
The total number of connections is represented by #c. The number of connections is equal to the product of the number of elements to be connected, N, and their respective fan-out, F. The total connection density in number of connections per unit length is #c/A1/2 = 1/(S + W) = [S +
X/(n2- n2)1/ 2 ]-l.
(13)
Thus the maximum bisection width that can be 6172
APPLIED OPTICS / Vol. 33, No. 26 / 10 September 1994
2
4(1.2) 2X2 NF.
(18)
A maximum connection density is achieved for a bisection width, B, equal to NF, for which the number of connections per unit width is given by B/Al/2 = A1/ 2 /[4(l.2) 2 X2 ].
(19)
3. Comparison While the number of connections per unit width increases with area for free-space interconnects, the
number is independent of area for guided-wave systems. The maximum number of connections per millimeter of module width can be calculated with Eqs. (13) and (19) for guided-wave and free-space interconnects, respectively. The waveguide width can be set to minimum width for containment accordingto Eq. (10). For both free-space and guided-wave interconnects the connection density increases with increasing index of refraction. Index of refraction refers to the core index for the waveguide and the index between the module surface and the reflecting hologram in Fig. 2 for the free-space case. However, increasing index-of-refraction values result in slower optical propagation speed for both guided-wave and free-space systems. For this reason both optical technologies tend to employ low-index-of-refraction materials (typically 1.0 < n < 2.0). In addition, the use of high-index-of-refraction materials in the freespace case can result in additional losses because of Fresnel reflections. These can be minimized through the use of antireflection coatings (see Ref. 24, in which a silicon hologram with index of refraction of 3.5 was fabricated with a net Fresnel loss on both surfaces of less than 3%) and with planarizing techniques.
link efficiency is 'n,
=
K
log1'(_-ydd/10)log'1(-,yx/10).
(20)
Next consider the signal-to-noise ratio (SNR) for a single-level guided-wave interconnect system. In this analysis we consider only the signal and the noise inherent to waveguide systems, which are not present for free-space systems. These are the reduced signal from attenuation and the increased noise from power leakage between waveguides from crossovers. (Of course the actual SNR will be smaller than these values because of additional noise sources common to both free-space and waveguide systems, such as noise resulting from optical quantum effects, and because of noise associated with the transmit and receive electronics.) Consider the signal power remaining in a waveguide after the signal has traveled a distance d and crossed x other guides. Let Yd be the power lost per crossover. Let noisex be the power that is coupled to the waveguide under consideration from the guide perpendicular at each crossover and noise be the sum of power transferred into the waveguide during crossovers. Assume identical guides. The ratio of power remaining (signal) to noise from crossovers (noise) follows:
B. Practical Limitations 1. Guided-Wave Optical Interconnects The large number of crossovers in a single-level guided-wave system used to implement a highly connected network can reduce the connection density drastically. The number of crossovers as a function of number of processors for several interconnect configurations may be found in Table 1. Consider the system efficiency of a single-level guided-wave system. If Yd is the linear attenuation and Yxis the crossover loss (in decibels per unit length), then the 10 6
signal = log'(1yYx)log'1( dd),
(21)
noise = x log-(noisex),
(22)
SNR =
log-(Yxx)log-(Ydd)
(23)
x log-'(noisej)
Figure 7 shows the number of processors that could be connected in a hypercube according to basic, efficiency, and noise limits for a single-level guided wave as described by inequality (14) and Eqs. (20) and (23). The physical limit for guides of index 2.0 is nearly 105 possible connections for a 10-cm module diameter if 10% coupling loss between guides can be
Fundamental Limit (10-cm diameter)
10
10 5.
104.
Signal to Noise Ratio
Efficiency Limit (10%)
Number of Processors 10
SNRLimit (SNR>5) 102I
l 1 f.-
I
Fig. 7. Connection-density limitations for waveguide optical interconnects. The fundamental limit on the number of connections in a 10-cm module diameter for guide index 2.0, cladding index 1.45, and 10% cross-talk loss is shown in Fig. 5. The number of connections that can be made in a hypercube if the tolerable loss is 10% and the number of connections that can be connected if the lowest tolerable SNR is 5 are shown.
1 .408 '
_
18 2 - 10 3 Number of Processors
-
l 4 -
Fig. 8. Dependence of SNR on the number of processors connected in a hypercube and in fully connected networks. The signal-to-noise ratio corresponds to that of the longest connection in an array of a given number of processors. 10 September 1994 / Vol. 33, No. 26 / APPLIED OPTICS
6173
values for attenuation and cross talk per crossover are used in Fig. 7. Only a few PE's can be reasonably connected. The severe limitations on connection density illustrated in Figs. 7 and 8 are primarily caused by the large number of guided-wave crossovers encountered in single-level waveguide interconnect systems. If a mesh or any other crossover-free network is implemented, the connection density is limited only by Eqs. (10)-(13) and inequality (14). If single-level guided-wave interconnects are to be used in highly connected large-size processor arrays, a multiple-layer implementation is necessary. Therefore for the remainder of the paper all comparisons between guided-wave and free-space systems are based on a dual-level guided-wave system when appropriate. A two-layer system consisting of layers of waveguides in which each guide is parallel to all guides on the
tolerated. Values of 0.1-dB/cm (Ref. 3) attenuation and 0.006-dB/crossover loss4 were used for the efficiency and the SNR limits in Fig. 7. If a minimum requirement of 10% system efficiency were imposed, only a fraction (N < 4000) of the basic limit would be reached. The electrical-to-optical and optical-toelectrical conversion efficiencies are not included in our definition of optical system efficiency. This is because these efficiencies are common to free-space and guided-wave systems. Given a system that has -52-dB/crossover cross talk and that can tolerate a SNR of not less than 5, only a few processors can be connected. Figure 8 shows the number of processor elements that could be connected as a function of tolerable cross talk as determined from Eq. (12) and Table 1 for fully connected and hypercube networks. The same 103.
105
waveguide area pitch 20gm pitch 10pmn pitch = 4sm \ minimum pitch
102.
lo'.-
1t
4
waveguide area pitch =20ism pitch 0pm pitch = 41m minimum pitch
13
processor arcs .
free space space variiht
- --
--
102
10t
t
s a.t
proce
/
//
free space space variant
/
/
/
,
--
,'
/
, 1
/ /
" c
io8
so
10'
Number of Processors
10,000 (the actual number will depend on the specific PE dimensions), such optical systems can reduce the interconnect delay of the longest link in a connection network by at least one order of magnitude less than fundamental limits on dual-level guided-wave optical-interconnect systems. In discussing packing density in this paper, we have ignored several potential limiting factors, including noise that is due to a cross talk between closely spaced lasers, and for free-space interconnects, cross talk that is due to power in undesired diffraction orders. While care must be taken to separate laser sources by sufficient distances to avoid cross talk, this can be done in a manner that would not increase the system area significantly for virtually all of the cases presented in this paper. For example, for closely spaced guided-wave interconnects, lasers could be distributed along the length of the waveguides. For freespace interconnects, as the number of processors in the system grows, the center-to-center spacing between holograms also grows, resulting in increasing separations between lasers and therefore decreasing laser-to-laser proximity cross talk. Similarly it was shown in Ref. 28 that cross talk in holographic interconnect systems resulting from power contained in undesired diffraction orders decreases with increasing system size. However, this analysis was based on the use of a particular type of hologram and on the assumption that the only significant noise power was contained in the unfocused zeroth diffraction order of each hologram facet. When large fan-outs are implemented, it is possible to couple power into highly focused diffraction orders that could significantly reduce the SNR. Nevertheless, it is also possible to design a system so that these highly focused spots do not coincide with any detectors. An additional issue not raised in this paper is the yield and the reliability of these systems. It is our belief that holographic interconnects have a significant advantage in this area over channel-waveguide systems. For example, from our analysis, a spacevariant free-space system had an area comparable to that of a guided-wave system with a channel width of 5X. Yet as the area of the system grows, the probability that a defect in the fabrication process of the waveguide will cause a critical failure increases since the channel width is kept constant while the length increases. On the other hand, with the spacevariant holographic system the size of the hologram facets increases with increasing area, thereby decreasing the failure probability that is due to defects in the holograms as the system grows.
References and Notes 1. E. E. E. Frietman, W. van Nifterick, L. Dekker, and T. J. M. Jongeling, "Parallel optical interconnects: implementation of optoelectronics in multiprocessor architecture," Appl. Opt. 29, 1161-1167 (1990). 2. Y. Yamada, M. Yamada, H. Terui, and M. Kobayashi, "Optical interconnections using a silica-based waveguide on a silicon substrate," Opt. Eng. 28, 1281-1287 (1989). 3. R. Selvaraj, H. T. Lin, and J. F. McDonald, "Integrated optical waveguides in polyimide for wafer scale integration," J. Lightwave Technol. 6, 1034-1037 (1988). 4. A. Guha, J. Briston, C. Sullivan, and A. Husain, "Optical interconnects for massively parallel architectures," Appl. Opt. 29, 1077-1093 (1990). 5. M. R. Feldman and C. C. Guest, "Interconnect density capabilities of computer generated holograms for optical interconnection of very large scale integrated circuits," Appl. Opt. 28, 3134-3137 (1989). 6. M. R. Feldman and C. C. Guest, "Computer generated holograms for optical interconnection of very large scale integrated circuits," Appl. Opt. 26, 4377-4384 (1987). 7. M. R. Feldman, C. C. Guest, T. J. Drabik, and S. C. Esener, "Comparison between electrical and free-space optical interconnects for fine grain processor arrays based on interconnect density capabilities," Appl. Opt. 28, 3820-3829 (1989). 8. B. K. Jenkins, P. Chavel, R. Forchheimer, A. A. Sawchuck, and T. C. Strand, "Architectural implications of a digital optical processor," Appl. Opt. 23, 3465-3474 (1984). 9. C. W. Stirk, R. A. Athale, and M. W. Haney, "Folded perfect shuffle optical processor," Appl. Opt. 27, 202-203 (1988). 10. C. D. Thompson, "Area-time complexity for VLSI," presented at the Eleventh Annual Association for Computing Machinery Symposium on the Theory of Computing, Atlanta, Ga., 30 April 1979. 11. J. Turunen, J. Fagerholm, A. Vasara, and M. Tahizadeh, "Detour-phase kinoform interconnects: the concept and fabrication considerations," J. Opt. Soc. Am. A 7, 1202-1208 (1990). 12. F. Hickernell, "Optical waveguides on silicon," Solid State Technol. 83-87 (1988). 13. J. D. Stack and M. R. Feldman, "Recursive mean-squarederror algorithm for iterative discrete on-axis encoded holograms," Appl. Opt. 31, 4839-4846 (1992). 14. W. H. Welch and M. R. Feldman, "Iterative discrete on-axis encoding of diffractive optical elements," in Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 72. 15. J. Jahns and S. J. Walker, "Two-dimensional array of diffractive microlenses frabricated by thin film deposition," Appl. Opt. 29, 931-936 (1990). 16. S. Somileno, B. Crosignani, and P. D. Porto, GuidingDiffraction and Confinement of Optical Radiation (Academic, New York, 1986), p. 563. 17. W. Stutius and W. Streiter, "Silicon nitride films on silicon for optical waveguides," Appl. Opt. 16, 3218-3222 (1977). 18. M. R. Feldman, S. C. Esener, C. C. Guest, and S. H. Lee, "Comparison between optical and electrical interconnects based on power and speed considerations," Appl. Opt. 27, 1742-1751 (1988). 19. Y. Chung, R. Spickermann, D. B. Young, and N. Dagli, "A low-loss beam splitter with an optimized waveguide structure," IEEE Photon. Technol. Lett. 4, 1009-1011 (1992). 20. P. R. Haugen, S. Rychnovsky, A. Husain, and L. D. Hutcheson, "Optical interconnects for high speed computing," Opt. Eng. 25, 1076-1085 (1986).
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21. H. Kogelnik, "Limits of integrated optics," Proc. IEEE 69, 232-238 (1981). 22. D. Marcuse, Light Transmission Optics, 2nd ed. (Van Nostrand Reinhold, New York, 1982), p. 425. 23. Ref. 22, p. 427. 24. W. H. Welch, J. E. Morris, and M. R. Feldman, "Iterative discrete on-axis encoding of radially symmetric computergenerated holograms," J. Opt. Soc. Am. A 10, 1729-1738 (1993). 25. T. Baba and Y. Kokobun, "High efficiency light coupling from antiresonant reflecting optical waveguide to integrated photo-
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detector using an antireflecting layer," Appl. Opt. 29, 27812791 (1990). 26. T. E. V. Eck, A. J. Ticknor, R. Lytel, and G. F. Lipscomb, "A complementary optical tap fabricated in an electro-optic polymer waveguide," Appl. Phys. Lett. (to be published). 27. G. Mak, D. Bruce, and P. Jessop, "Waveguide-detector couplers for integrated optics and monolithic switching arrays," Appl. Opt. 28, 4629-4636 (1989). 28. J. Schwider, W. Stork, N. Streibl, and R. V6lkel, "Possibilities and limitations of space-variant holographic elements for
switching networks and general interconnects," Appl. Opt. 31, 7403-7410 (1992).
