CAREY, CD

GODWIN, DP

MIDWINTER, JE

SELVIAH, DR

SONG, SH

English

UCL Discovery

15 PERFECT SHUFFLE INTERCONNECTIONS USING FRESNEL COMPUTER GENERATED HOLOGRAMS IN A PLANAR-OPTIC CONFIGURATION D.P.Godwin,C.D.Carey,S.H.  S0ngl.D.R. Selviah and J.E. Midwinter Department of Electronic  and Electrical  Engineering, University College London, UK INTRODUCTION enable multistage routing and sorting networks to be realised in a compact and lugged phar  optic form. Several  standard  interconnection  patterns  exist  but perhaps the most useful is the perfect shuffle (PS) since multiple identical copies can be cascaded in multistage interconnection networks for use in sorting of  optical telecommunication  signals, for example. This is not the case for the crossover interconnection network where each stage differs. Perfect shuffles also can be used in algorithms  to  perform  fast  Fourier  transforms, polynomial  evaluation  and  matrix  transposition (Stone(1)). The optical implementation of  a  perfect shuffle has numerous benefits (Lohmann (2)). not least that data may be arranged in  a 2-D  fashion and the whole array shuffled in one operation (Lin et al (3)). This operation may be carried out between two arrays of multi quantum well (MQW) modulators (Parry et  al (4)), with  the modulators and the optical interconnect element  all  arranged  in  the  same plane  (so called planar-optics,  Jahns et al  (5)).  Crossover networks have recently been demonstrated optically in  planar- optic configurations (Song et a1  (6))  The planar-optic configuration  has the advantage that the optical paths are folded into a compact and rugged structure and the various  components  are  kept  in  good  alignment. Moreover,  the  alignment  can  be  performed lithographically so high accuracy and reproducibility can be achieved. This makes the planar optic type of interconnect ideal for interconnecting VLSI or  WSI circuits.  Interconnects performed  electronically  are beginning  to  reach  their  limits  as  the  chips  are progressively  increasing  in size and speed. One problem is that of distribution  of the same clock signal to several points on the wafer surface at the same instant. This has been  addressed  by  the development of  novel, self- focusing  Fresnel-Dammann  gratings,  and  Fresnel computer generated  hologram  fanout  and  focusing elements which have been demonstrated in  a  slanted axis,  multiple reflection,  planar optic configuration (Godwin et al (7)). In this paper we concentrate on the task of  transferring data from one part of the chip to another in  an optical bus  i.e.  transfer of  an  image. Moreover, we  require  the optical interconnection to interchange  the pixels between the input and the output in order to perform a perfect shuffle  operation. This will lETRI, Korea THE 2-D PERFECT SHUFFLE The perfect shuffle was originally defined to act upon 1-D  data (1).  such that the data are divided into two, and the two halves perfectly interleaved, in analogy with shuffling a pack of  cards (fig.1).  If  the data are given addresses,  numbered sequentially  in binary, it can be seen from fig.1 that the perfect shuffle operation corresponds to a barrel roll of  the address bits (ie the data at address abcd goes to address bcda, where a, b, c, d are the address  bits, and can be 1 or 0).  In order to use the advantages of  optical  interconnection,  however, these  1-D  data  sets  must  be  mapped  into  two dimensions (2).  (In an electronic circuit this could be a hard-wired  mapping.)  This is not a unique operation, and there are many possible mapping schemes.  The implementation of a perfect shuffle in 2-D  may also be defined in many different ways, and the choice of 2-D shuffle must be made compatible with  the mapping scheme chosen.  In this paper we consider two  such mapping schemes, and their corresponding 2-D  perfect shuffle operations. The first mapping scheme we consider was proposed by Stirk et a1 (8), who termed the corresponding shuffle a folded perfect shuffle. The mapping scheme is a simple one, and allows all of  the algorithms designed for 1-D data to be used in 2-D.  The 1-D  data is Mitten along rows of a 2-D  matrix, a new row being started when the previous one is full (see fig.2).  This corresponds to the mapping of the I-D  address, [abcd], to the 2-D  address, [ab,cd] [row address, column address].  The required 2-D  perfect shuffle must therefore move the  data  at address [ab,cd] to  address m,da], and this operation may also be Seen in fig.2. The second mapping scheme was described by  Taylor (9).  and takes [abcd] into [ac,bd] (extending to larger addresses such that [abcdef  ...I  -+  [ace  ...,  bdf ...  1).  This mapping  scheme  may  be  seen  in  fig.3.  The corresponding 2-D  perfect shuffle comprises perfect 16 therefore refer to this type of 2-D  shuffle as a double perfect shuffle): [abcd] +  I-D IO 2-0 mapping + [ac,bd] +  double PS + [ca,db] +  2-0 IO I-D  mapping +  [cdab] [abcd] +  1-D  PS + [bcda] +  I-D PS +  [cdab] This has the advantage that, in  routing networks for instance (provided that &port  switching modules are used), only half the number of  interconnection stages are required. The effect of  a double perfect shuffle operation could be  performed optically by  cascading two sets of  optical hardware for  performing  a  I-D perfect shuffle. However, if  this can be done using a single stage of  special optical hardware it will  lead to more compact and more efficient systems. It is worth noting that the reverse is not possible; special double perfect shuffle optical  hardware  cannot  be  used  to perform a 1-D perfect shuffle. For such a shuffle it is necessary to develop special compact optical hardware to perform a folded perfect shuffle since this has the effect of a single 1-D perfect shuffle.  It  can be  seen, therefore,  that  there  is  a  requirement  for  the implementation of  different 2-D  perfect shuffles in optical hardware. HOLOGRAPHIC IMPLEMENTATION If  we have a regular. nxn, square array of  small pixels (figA), on a pitch of  d, we can get a perfect shuffled arrangement of these signals by  copying the array four times and superimposing the copies.  The first step is to copy and superimpose the array such  that  the  four quadrants of  the array overlap perfectly (fig.5).  This happens when the centres of the copies are placed at the comers of a square, having the same orientation  as the array, and of  side ndn.  The second step is to shift the centres of each of  the copies by  d/4 in both the x- and y-directions (ie, diagonally by  d/242).  The sense of these final shifts, as applied  to  each  of  the copies, determines  the type of  perfect shuffle.  A maximum of 24 different shuffles may  be  formed by  shifting the centres in different  ways (there  are four centres,  each of which may be  shifted in one of  four directions, but a given shift direction may only be used once -this  gives a total of 4 x 3 x 2 x 1 = 24 possible combinations). For instance, the folded perfect shuffle (fig.6a) is achieved by  moving the centres of  the top right and bottom left arrays apart along the diagonal joining them, and  the centres of the top left and bottom right together.  The double perfect shuffle (fig.6b) is formed by  pushing all of the centres together along the diagonals  (note that the centres move by d/4 parallel to the square edge, and so the centre spacing  decreases by d/2. From the above considerations  it is possible to see that a perfect shuffle may be formed by  using a  1:4 fanout hologram (Carey et a1 (IO)).  This is a lenslike element, but with four foci instead of one.  If it is used to image an  array of  spots, four copy arrays will appear, the centres of which will coincide with the hologram's four focal spots.  If  the distance  between  these  spots is chosen according to steps one and two above, a perfect shuffle results.  Two holograms were chosen for this work.  One produced a rhombic array of four spots, and can be used to perform a folded perfect shuffle, whilst the other gave a square array of four spots and could be used to perform a double perfect shuffle. GENERATION OF THE HOLOGRAMS The holograms were designed to operate at 30'.  in the planar optic configurations shown in fig.7.  They were generated  by  calculating  the  Fresnel  interference pattern, at the hologram plane, between waves diverging from  a  point  source  as  a  reference,  and  waves converging to an array of four spots (with the spacing calculated to produce the desired perfect shuffle).  The interference pattern can be represented by  the following equation (Kawai and Kohga (1  1)): where &$ is the phase difference between reference and image waves, x and y are coordinates in  the hologram plane, and all 0s represent phases in the hologram plane; is the phase of the reference beam, Q0 is the phase of the  image array taken as a whole, and the Qi are the phases of  the individual light spots, i, in  the image  (i runs from 1-4 in this case). The hologram is stored as a binary pattern such that, when cos &$  2 0,  1 is stored, and when cos &$ < 0.0 is stored. The hologram can be manufactured as a phase grating such that. on replay, a 1 represents a K phase delay, and 0 a 0 phase delay. Part of  the computer generated hologram pattern that creates a rhombic array can be seen in fig.8a. and that for a square array can be  seen in  fig.8b.  The arrays generated by  these holograms when  illuminated by  a white light source (in the configuration of  fig.7b)  can be seen in fig.9.  A perfect shuffle that has been generated holographically can be seen in fig.10. CONCLUSIONS We  have  described  the  design  and  experimental performance of  two binary Fresnel holograms which. when  illuminated by a point source, generate a square and a rhombic array of  light spots respectively.  The 17 square array hologram  can be  used to double  perfect shuffle  an  input  array, and  the  rhombic  hologram performs a folded perfect shuffle.  Depending  upon the relative spacings of  the holographic array and the array to be shuffled, 24 different shuffles can be performed. Their utility would depend upon  the  mapping of  data from 1-D  into 2-D,  and the function that the particular perfect shuffle operation is designed to perform. The holograms  were demonstrated  operating at 30 degree oblique  incidence in  a  planar  optic  configuration suitable for integration with VLSI or WSI circuits. Fresnel lens for planar optical perfect shuffle”, Optical computing topical meeting. Palm  Springs, California, March  16-19. 1993 Technical  digest series, volume 7, pp 291-294 11. Appl. Phys.. 30, L2101-L2103 Kawai.  S. and Kohga, Y..  1991, Japanese J. of ACKNOWLEDGEMENTS This work was funded by the SERC through the UCL Optoelectronic Rolling Grant.  Many  thanks to Mark Abbott for computer hardware  aid and  to Rutherford Electron  Beam  Lithographic  Facility  for  mask fabrication. REFERENCES 1.  Stone. S.H.,  1971, IEEE Trans. Comput..  C-20. 153-16 1 2.  Lohmann,A.W., 1986, Appl. Opt.. 25. 1543-IS49 3.  Lin, S-H.,  Krile, T.F.  and Walkup. J.F.,  1987. SPIE, 152,209-216 4.  Parry. G., Whitehead, M., Stevens, P..  Rivers, A., Barnes.  P., Atkinson,  D.,  Roberts,  J.S.,  Button,  C., Woodbridge, K. and Roberts, C., 1991, Phvsica SC~~DQ, E,  210-214 S.  Jahns, J., Lee, Y.H.,  Burrus, CA.  and Jewel], J.L.. 1992, Appl. Opt., 31,592-591 6.  Song, S. H., Lee, E. H., Carey, C. D., Selviah, D. R. and Midwinter, J. E., 1992, Optics Letters, U, No.  18. 1253-1255 7.  Godwin.  D.  P.,  Selviah,  D.  R..  Carey,  C. and Midwinter, J.  E.,  1993, “The self-focusing Fresnel- Dammann grating and Fresnel binary CGH for compact 2-D light spot array generation”.  IEE conference  on Holographic  Systems Components and  Applications, Neuchatel 8.  Stirk, C.W.,  Athale, R.A. and Haney. M.W..  1988. Appl. Opt., 27,202-203 9.  Taylor, M.G.,  1990, Ph.D. Thesis. “Devices  and Networks  for  Optical  Switching”.  University  of London, UK 10.  Carey, C.D., Selviah, D.R., Midwinter, J.E.. Song. S.H. and  Lee.  E.H.,  1993. “Four-foci  slanted  axis oo00  A-Am ooO1  B  I  oO01 0010  c  B  0010 0011  D  J  0011 0100  E  c  0100 0101  F  K  0101 0110  G  D  0110 0111  H  L  0111 lo00  I  E  1oO0 1001  J  M 1001 1010  K  F  1010 1011  L  N  1011 1100  M  G  1100 1101  N  0  1101 1110  0  H  1110 1111  P -  P  1111 Figure 1:  1-D  perfect shuffle 0000A o001  B 0010  c 0011  D 0100  E 0101  F 0110  G 0111  H lo00  I 1001  J 1-D  to 2-D mapping 00  01  10  11 Double  t  ps B  0010 J  0011 c  0100 0101  F  mapping  K  0101 0010  c 0011  D- E  1-Dto2-D .. 0110  G 0111  H lo00  I 1001  J 1010  K 1011  L 1100  M 1101  N 1110  0 1111  P D  0110 L  0111 E  lo00  t  ps  M  1001 Folded 00  01  10  11  F  1010 N  1011 G  1100 '0 1101 H  1110 P  1111 Figure 2:  1-D  to 2-D  mapping and shuffle for folded perfect shuffle A0000 E  OOO1 I  0010 M 0011 B  0100 F  0101 J  0110 N  0111 c  lo00 G  1001 K  1010  1010  K  00  01  10  11 1011  L 1100  M 1101  N 1110  0 1111  P 2-D  10  1-D L  1110 P  1111 Figure 3:  1-D  to 2-D  mapping and shuffle for double perfect shuffle, 0  0  0  0 0  0  0  0 0  0  0  0 Figure 4:  4x4 array of signals OOl!Yl!YAA OOElElAA x  x no#b@stPstP ood04bP OOOOc3c3 0000aa -M x  -  Centre of each array Figure 5: Four copies of a 4x4 array, the centres placed such that their four quadrants overlap perfectl: 19 AAAA AAAA 0000 0 000 000 0000 0000 X-  Centre of each array Q989 Figure 6a:  Shifts required for a folded perfect shuffle 7  OAOAOA A A A A 9 9 9 0  OClO1309  c X-  Centre of each array Figure 6b:  Shifts required for a double perfect shuffle camera Microscope objective ;d-  13.86" 46.76'  : I  I  6-  :\  /\[I:\  /  \  /  BLOCK1 I1 Figure 7a: Planar-optic configuration for performing  a perfect shuffle on an input array at 30' camera Microscope objective It  GLASS  I BLOCK  I Figure 7b  Planar-optic configuration for  testing a holographic fanout element at 30" 20 Figure 8:  Part of the hologram pattems for the 2x2 array generators a) rhombic b) square a)  b) Figure 9:  Output of 2x2 array generators a) rhombic b) square Figure 10:  Holographically generated perfect shuffle 

spot array generation”.

The self-focusing FresnelDammann grating and Fresnel binary CGH for compact

PERFECT SHUFFLE INTERCONNECTIONS USING FRESNEL COMPUTER-GENERATED HOLOGRAMS IN A PLANAR-OPTIC CONFIGURATION

PERFECT SHUFFLE INTERCONNECTIONS USING FRESNEL COMPUTER-GENERATED HOLOGRAMS IN A PLANAR-OPTIC CONFIGURATION

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