T. Tanaka and S. Kawata Vol. 13, No. 5/May 1996/J. Opt. Soc. Am. A 935

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1 T. Tanaka and S. Kawata Vo. 3, No. 5/May 996/J. Ot. Soc. Am. A 935 Comarison of recording densities in three-dimensiona otica storage systems: mutiayered bit recording versus anguary mutiexed hoograhic recording Takuo Tanaka and Satoshi Kawata Deartment of Aied Physics, Osaka University, Suita, Osaka 565, Jaan Received May 5, 995; revised manuscrit received August 8, 995; acceted October, 995 We comare the geometrica imits of the recording density that can be attained in three-dimensiona otica memory systems that emoy the mutiayered bit-recording method and the anguary mutiexed hoograhic recording method. Both recording methods have the otentia to overcome the recording density imitations in current otica storage systems. Using the Ewad shere construction, we anayze the atera and ongitudina bandwidths for each recording method. The resective recording densities of the two methods are aso derived directy in the sace domain and comared with each other. With the bit-recording method we found that the memory density increases inversey with the f-number of the recording ens. On the other hand, the density that can be achieved using hoograhic recording first increases with the f-number vaues, attains a maximum, and then decreases at arger f-number vaues. This imies that the bit-recording method yieds arger memory densities when enses of smaer f-numbers are used in the otica system. 996 Otica Society of America. INTRODUCTION The continued deveoment of faster rocessors has necessitated the need for storage systems with arger memory caacity and higher data storage density. Because it utiizes otica icku, otica memory offers one of the highest recording densities among the various memory recording and retrieva schemes. owever, because of the wave nature of ight, the aser beam sot used to record a bit of datum on the otica disk can never be smaer than the diffraction imit given by haf of the iumination waveength; the attainabe storage density never exceeds 0 9 bits cm if the waveength is onger than 350 nm. A way to overcome this imitation is to record the bit information in three dimensions, so that the entire voume of the recording medium is utiized. We have reviousy demonstrated mutiayered threedimensiona (3-D) bit-data otica memory recording on a thick materia, where the recorded data are read using a confoca microscoe., Besides us, Rubin et a. of IBM Amaden recenty roosed and deveoed 3-D otica recording that bonds substrate disks together and reads and writes to distinct eves of a mutieve disk structure. 3 In addition, Parthenoouos and Rentzeis 4,5 of the University of Caifornia Irvine and Stricker and Webb 6 of Corne University have roosed and deveoed 3-D memory utiizing two-hoton absortion. Another we-known method of achieving 3-D otica memory storage is by means of hoograhic recording, in which various images are recorded in a common medium at different incident anges of the reference beam. This recording method has, in fact, aready found its way into disay systems. 7 In this aer we anayze the storage density for both mutiayered bit memory and anguary mutiexed hoograhic memory. 8 We cacuated and then comared the geometrica imits of the storage densities that can be attained using 3-D mutiayered otica memory recording and 3-D hoograhic memory recording. We restrict the comarison to transmission tye in both mutiayered memory and hoograhic memory.. OPTICS FOR MULTILAYERED AND ANGULARLY MULTIPLEXED TREE-DIMENSIONAL MEMORIES Figure shows the recording and readout otics of the 3-D mutiayered bit-recording otica memory device, that wi be anayzed in this aer. In recording, the aser beam is moduated using a shutter, in accordance with the ogic of the bit datum to be recorded, and is focused in a fixed oint of the recording medium by ens L. We used a hotorefractive oymer and a hotorefractive crysta (such as LiNbO 3 ), which change the refractive index ocay in resonse to confined ight distributions. A series of bit data is recorded bit by bit in the medium using a motorized 3-D transation stage. Recorded bit information is read using a transmissiontye aser scanning confoca microscoe, which is aso shown in Fig.. Since the data are recorded as variations in the refractive index, the hase-contrast method is emoyed for reading data. For hase-contrast imaging with a aser scanning system a hase-contrast objective ens is used in the iuminating (focusing) otics, and an annuar ui is used in the detection otics, which is the oosite otica arrangement to that of hase-contrast imaging with a conventiona microscoe. The otica arrangement needed to imement 3-D hoograhic memory recording is essentiay simiar to that for 3-D bit memory recording, excet that it incudes the /96/ $ Otica Society of America

2 936 J. Ot. Soc. Am. A/Vo. 3, No. 5/May 996 T. Tanaka and S. Kawata Fig.. Otica system for 3-D mutiayered bit memory recording and reading. In recording, the aser beam is moduated using a shutter and is focused in a fixed oint of the recording medium by ens L. A series of bit data is recorded bit by bit in the medium using a motorized 3-D transation stage. Recorded bit information is read using a transmission-tye aser scanning confoca microscoe. The hase-contrast method is emoyed for reading bits recorded as refractive index distributions. Fig.. Otica system for 3-D hoograhic memory recording and reading. In hoograhic recording, the signa beam B s, that is, the Fourier-transform image of the data mask, interferes with the reference beam B r and is recorded as interference fringes. To record a mutie number of hoograms, the medium is iuminated sequentiay by the mutie numbers of reference beams at distinct incident anges using a rotating mirror. In reading the recorded hoograms recorded signas B s are reconstructed by iuminating the medium with the aroriate reference beams B r. The diffracted beams B s are inverse Fourier transformed by ens L 3 and detected by the array detector. reference beam and the data mask. Figure iustrates a tyica configuration of a transmission-tye 3-D hoograhic storage system. In hoograhic recording, the ens L Fouriertransforms the data mask to generate a signa beam B s. The beam B s is constructed with ane waves. Each ane wave in B s interferes with the reference beam B r in the recording medium. The data inutted in the mask are recorded as interference fringes (hoogram) within the medium. For recording a mutie number of hoograms, the medium is iuminated sequentiay by the mutie numbers of reference beams at distinct incident anges, with use of a rotating mirror. As a resut, 3-D hoograhic memory uses x y u sace for recording data in three dimensions, in contrast with 3-D bit memory, which uses x y z sace. Different ages of signa atterns, B s, B s,..., B sn, are generated by a satia ight moduator (SLM) and the Fourier-transform ens L. To signa a age attern B si i,..., N, there corresond reference beams B ri i,..., N that iuminate the recording medium at given anges u ri i,..., N. In reading the recorded hoograms recorded signas B si i,..., N are reconstructed by iuminating the medium with the aroriate reference beams B ri i,..., N. The diffracted beams B si i,..., N are inverse Fourier transformed by ens L 3 to image the data on the ane containing the array detector. Note that the effective anguar range of u r, over which the recording medium is iuminated by reference beams, is restricted by the hysica dimension of ens L. eanue et a. roosed a method in which the reference beams are entered from a facet erendicuar to that for signa beams. With their method the reference beam enters the crysta from the to or the bottom ane in Fig.. 3 This reeases the anguar restriction for u r, whie it can be aied ony to the memory having a cubic shae (and not to the widey extended disk). To otimize the ossibe range of ange u r,l must have a reativey ong foca ength. Moreover, both L and L 3 must cover a wide fied of view because a hoograhic memory device records and reads the two-dimensiona data in arae. This means that both a ong foca ength and a arge diameter are required for enses L and L 3. Such a requirement is not imosed onto the enses for bitdata memory recording and reading. This imies that a sma-f-numbered ens such as a microscoe objective can be used for bit-recording memory. In 3-D ange sace we may mutiex the reference beam with, in addition to u [as seen in Fig. 3(a)], the other ange f [in Fig. 3(b)] for a given ange u that is arger than the ui ange of ens L. 4 Furthermore, in rincie it is ossibe to mutiex the reference beam in 3-D sace with both anges u and f [Fig. 3(c)], athough it is difficut to imement and aign such a u f scanner in the ractica otica memory device. In this aer, therefore, we discuss mainy u mutiexing and f mutiexing for 3-D hoograhic memory. 3. COMPARISON OF RECORDING DENSITY IN TREE-DIMENSIONAL SPATIAL-FREQUENCY BANDWIDT In this section we comare the recording density of the two tyes of 3-D storage memory using satia-frequency bandwidth. The recording density for bit-recording memory is determined by the size of the focused beam sot in the recording medium. The sot size is determined by the transmission bandwidth of ens L. Figure 4(a) shows the transmission bandwidth of L as a shaded region in k sace (satia-frequency domain) using the Ewad shere reresentation, where is the iumination waveength and k is the wave number. As shown in Fig. 4(a), the atera bandwidth WL B is

3 T. Tanaka and S. Kawata Vo. 3, No. 5/May 996/J. Ot. Soc. Am. A 937 Eq. () can hence be written as W B L f. (3) The ongitudina bandwidth WZ B is given by the axia comonent of grating vector K 0 formed by the wave vectors k and k 0, which gives the maximum vaue [see Fig. 4(a)]: W B Z cos u 0. (4) Combining Eqs. () and (4) yieds WZ B! 4f. (5) It is found from Fig. 4(a) and Eqs. (3) and (5) that as the f -number of the ens decreases or u 0 increases [numerica aerture (N.A.) increases], the bandwidths WL B and WZ B both increase. In other words, the size of a focused beam sot decreases in both the atera and ongitudina directions as the ens f -number decreases. Next to be discussed is the hoograhic memory device. Figure 5(a) shows again a ongitudina section of the Ewad shere for a hoograhic memory; the atera bandwidth WL of one hoogram is equa to the atera band- Fig. 3. Recording ossibiities in mutiexed hoograhy at different vaues of (a) ange u, (b) ange f, and (c) both anges u and f. given by the magnitude of the grating vector K, which reresents the interferometric fringe generated by wave vectors k and k for the maximum anguar searation of u 0 imited by the ens ui [see Fig. 4(b)]: W B L 4 sin u 0. () Because the view ange u 0 of ens L is reated to its f-number f according to sin u 0, () Fig. 4. Satia-frequency bandwidth of bit memory recording: (a) transmission bandwidth of ens L using the Ewad shere construction, (b) orientation of signa beam during recording. u 0 is the view ange of ens L. The transmission bandwidth of L is shown as the shaded region. WL B is the atera bandwidth, and WZ B is the ongitudina bandwidth.

4 938 J. Ot. Soc. Am. A/Vo. 3, No. 5/May 996 T. Tanaka and S. Kawata f -number. It can be seen in Fig. 6(a) that bit-recording memory increases both the atera and ongitudina bandwidths as the f-number decreases. They are at their maxima at f 0.5 (N.A..0. Figure 6(b), on the other hand, iustrates that in hoograhic recording ony the atera bandwidth WL increases as the f-number decreases. The anguar bandwidth Wu of the reference beam decreases as the f -number decreases. In articuar, at f 0.5, there is no mutiexity in u. In this section we have not deat with the mutiexing in f. This is because the anguar bandwidth in f is constant at, indeendent of the f-number of the ens. 4. NUMERICAL ANALYSIS FOR RECORDING DENSITY In this section a detaied numerica anaysis is made comaring the recording density characteristics for the two tyes of 3-D otica memory. Fig. 5. Satia-frequency bandwidth of hoograhic memory recording: (a) Ewad shere reresentation of transmission bandwidth of ens L, (b) orientation of signa and reference beams during recording. WL is the atera bandwidth, and WZ is the ongitudina bandwidth. The atera bandwidth WL is the same as that for bit recording, WL B. width of Fourier-transform ens L. This means that the atera bandwidth WL for hoograhic recording is the same as that for bit-data recording, i.e., W L W B L 4 sin u 0. (6) f A. Bit Memory Recording In bit memory recording, a focused aser sot forms a bit datum in the 3-D voume of a recording medium. The atera and axia intensity distributions I r, z 0 and I r 0, z of such a focused beam sot are given by J I r, z 0 I I r 0, z I 0! 3 f r, 7 (9) 5 f r # sin u 4, u u 4! z, (0) where f is the f -number of ens L, I 0 is the beam intensity at r z 0, and r, z are the oar coordinates. 5 The foca ane of the ens intersects the otica z axis at This equation indicates that a sma f-number yieds a high recording density. The anguar bandwidth Wu for the anguar scanning for the reference beam iumination is given, as shown in Fig. 5(a), by the anguar region eft by the signa band (shaded region) of the Ewad shere, i.e., W u u 0. (7) Using Eq. (), we can exress the anguar bandwidth Wu in terms of the f-number f of L as! Wu sin. (8) Equation (8) shows that a ens L with a arge f-number yieds a arge Wu, aowing a arge mutiexity in the hoogram recording. owever, it was found from Eq. (6) that recording density of each hoogram decreases with f -number. As a concusion, a trade-off exists between the anguar bandwidth Wu and the atera bandwidth WL of an individua hoogram. From Fig. 6 we can comare the transmission bandwidths between two otica systems for three vaues of the Fig. 6. Transmission bandwidth as a function of f-number in (a) bit memory recording and (b) hoograhic memory recording. Both the atera bandwidth WL B and the ongitudina bandwidth WZ B of bit-recording memory increase as the f-number decreases. In hoograhic recording, on the other hand, the atera bandwidth WL increases as the f-number decreases. The anguar bandwidth Wu of the reference beam decreases as the f-number decreases.

5 T. Tanaka and S. Kawata Vo. 3, No. 5/May 996/J. Ot. Soc. Am. A 939 r 0 and z 0. The centra sot radius Dr is given by I r, z 0 0 in Eq. (9) as Dr 3.833f. () We hereby define the atera recording density DL B by the inverse of the area of the centra sot size: DL B Dr () f The haf-width Dz of the ongitudina extent of the sot aong the otica axis from the intensity-maximum osition z 0 to the first zero is given by I r 0, z 0 in Eq. (0) as Dz 8f. (3) We define the ongitudina recording density DZ B as D B Z Dz. (4) 6f The 3-D recording density D3-D B is given by the roduct of the atera density DL B and the ongitudina density DZ B, i.e., D3-D B DB L DB Z (5) 3 f 4 Equation (5) indicates that the density D3-D B associated with 3-D bit recording is inversey roortiona to F 4. Figure 7 shows D3-D B as a function of the f-number of ens L for the aser beam of 780 nm. For f 0.7 (i.e., N.A. 0.7), a tyica vaue of a good objective ens, the attainabe recording density is bits cm 3. k r, k r,..., k rn and a comonent of a signa beam vector k si of k s, k s,..., k sn. Since the recording medium has finite thickness T, vector K i does not reresent a singe oint on the Ewad shere but is burred in the k z direction, as shown in Fig. 8(a). This gives rise to unwanted cross tak, wherein reconstructed images of different hoograms overa with each other. The beam k ri or k ri may coue with a smeared comonent of K i, resuting in the reconstruction of k si [Fig. 8(b)]. If such cross tak is to be avoided, reference beams must iuminate the medium at a certain amount of anguar interva. Let us define the uncertainty (the sread in the k z axis) in K i by the ength between the first two zeros of the burring function [a sinc function in Fig. 8(a)]. Then the minimum anguar interva Du required is Du T sin u B, where T is the thickness of the recording medium and u B is the ange for Bragg diffraction. 6 The mutiexity Nu for the hoogram with thickness T is Fig D recording density of bit recording as a function of f-number. The waveength of the aser beam is 780 nm. B. Latera Density of oograhic Recording In hoograhic recording, the atera density is imited by either the bandwidth of the ens aerture or the size of the recording medium. In this anaysis we assume that the size of the medium is arger than the ens aerture, so that the Fourier transform of the signa data is band imited by the Fourier-transform ens L. The intensity distribution of a singe bit reconstructed from the hoogram is the same as the one formed with bit writing-and-reading memory, which is given by J I r I f r f r! 3. 7 (6) 5 of hoo- As a resut, the atera recording density DL grahic recording is given by DL Dr , (7) f which is the same as Eq. (). C. Three-Dimensiona Density of oograhic Recording In mutiexed hoograhic recording, the wave vector K i that describes the interferometric fringe is given by the difference between a reference beam vector k ri of Fig. 8. Grating vector K i and the effect of diffraction: (a) the osition of K i aong the otica axis is uncertain over the extent of the sinc function, (b) occurrence of unwanted diffraction that is due to reference beam iumination at a different incident ange u ri.

6 940 J. Ot. Soc. Am. A/Vo. 3, No. 5/May 996 T. Tanaka and S. Kawata As we have seen in Fig. 6, hoograhic memory has a maximum in density at a certain vaue of f-number; it is f 0.75 for u mutiexing. In Fig. 9 the 3-D density D 3-D u,f of u f mutiexed hoograhic memory is incuded as a dashed curve, which exhibits the highest density among the 3-D memory for f. 0.59, athough the imementation of this mutiexing is the most difficut. Fig. 9. Attainabe 3-D recording density in mutiexed hoograhic recording using anguar variations in u, f, and both u and f. Aso shown is the recording density of bit memory recording. The waveength is 780 nm, and the thickness T of the recording medium is 00 mm. The recording density of mutiayered bit memory tos u mutiexing for f, 0.74, f mutiexing for f,., and u f mutiexing for f, 0.6. hence given by (see Aendix A) 8! Nu 4 < 3 : 4 sin 3 T sin 8 4 sin!# 9 = ;. (8) Now the 3-D density D 3-D u of the hoograhic recording mutiexed in ony u [Fig. 3(a)] is given by D3-D u D L Nu T < 3 : 3 T sin f T! 4 sin 8 4 sin!# 9 = ;. (9) The mutiexity Nf on the circe [in Fig. 3(b)] at a fixed incident ange u is derived in Aendix B as N f. sin (0) sin u 5. CONCLUSION AND DISCUSSION We made a comarison in 3-D recording density between mutiayered bit-recording memory, which we reviousy roosed,, and anguary mutiexed hoograhic memory. A concusion is that a bit-recording memory gives the highest density for a sma ens f -number, whie for a arge ens f-number the density with hoograhic memory becomes higher than that with bit memory. We examined ony the transmission otica memory for both hoograhic and bit recording. For refectiontye hoograhy the tota recording density wi be amost four times higher than D3-D u and two times higher than D3-D u,f. In contrast, bit-recording refection-tye memory shoud give a density two times higher than D3-D. B In this case the configuration is equivaent to that of the socaed 4-confoca microscoe. owever, the definition of density for 4-confoca otics is different from that for conventiona otics, so that the quantitative comarison in density is not made easiy. The comarison that we made in this aer may not be fair because we gave an extra anguar sace surrounding the Fourier-transform ens for reference beam iumination for hoograhic memory, resuting in the difference in the tota transmission bandwidth of the otics. There is no reason to excude such a sace for bit memory recording and reading. We can use a wide ens for bit memory recording, resuting in a higher density than that for hoograhic memory. Another robem with hoograhic memory is that the voume where a reference beam transmits through is not the same as that for a signa beam in a thick medium, i.e., the reference beam exoses a sace (shown in Fig. 0 as shaded regions) where the signa does not exist, resuting The recording density D 3-D f is hence given by D 3-D f D L N f T f T. sin sin u () D. Comarison of Mutiayered Bit Memory Versus oograhic Memory Figure 9 shows the comarison between 3-D bit memory and 3-D hoograhic memory in recording density as a function of f -number. Vaues of 780 nm and of thickness of recording medium T 00 mm are given for the cacuation. For the cacuation of recording density in f mutiexing a fixed incident ange u is given, which gives the maximum vaue. It is found that the mutiayered bit memory tos u mutiexing for f, 0.74 and f mutiexing for f,.. Fig. 0. Interference region of signa beam and reference beam in the hoograhic recording medium. Because the voume where the reference beam transmits through is not the same as that for the signa beam, reference beams exose the voume where a signa does not exist.

7 T. Tanaka and S. Kawata Vo. 3, No. 5/May 996/J. Ot. Soc. Am. A 94 in the reduction of memory caacity for the hoogram. Such a comonent of a reference beam may read the neighboring age as cross tak. This robem is evident, in articuar, when the ange of incidence for the reference beam is aart from the anguar range of the signa beam. In our cacuations we have assumed that the recording medium has a inear resonse to variations in ight intensity. In actua otica systems both the medium and the otics can be seected to exhibit a noninear resonse, and a noninear resonse in an otica system imroves resoving ower (and therefore recording density). Exames of a noninear otica system are fuorescence and refection tyes of confoca otica systems, where the resonse is roortiona to the squared resonse to ight 7 0 intensity. Using a recording medium with a noninear resonse aso imroves the density of bit memory recording. Exames are two-hoton absortion rocesses and threshod rocesses. On the other hand, an imrovement in the recording density is not gained in hoograhic recording, even with otics and recording media that exhibit noninear resonses. A deterioration in the reconstructed signa can arise even from either otica noise amification or the unwanted enhancement of higher-order diffracted ight. APPENDIX A We derive the anguar seectivity Du in hoograhic memory recording in terms of the thickness T of the recording medium. Because of the finite thickness of the medium, the wave vector K i sreads out in the k z direction, as shown in Fig. 8(a). This burring of K i determines the minimum searation Du of the neighboring reference beams necessary to avoid the cross-tak image (see Fig. 8). For simifying the cacuation we made an assumtion. It is that the signa beam and the reference beam are symmetrica with resect to the otica axis and that the incident anges of them are the same. This means that u si is equa to u ri. If we substitute u ri u si into Kogenik s wave-couing theory, 6 the vaue Du is given as Du T sin u B, (A) where u B is the ange satisfying the Bragg condition. Equation (A) imies that the anguar seectivity Du imroves as the thickness T or the ange u B for Bragg diffraction increases. In genera, the vaue u ri is not equa to u si. owever, for simifying the cacuation, we made the assumtion given above; we used Eq. (A) for cacuating the anguar seectivity Du. The aroximation u ri u si is not disadvantageous for hoograhic memory, because the anguar seectivity Du is a minimum at u ri u si. Therefore this aroximates an anguar seectivity better than the true vaue. The searation Du in anguar mutiexing is a function of the ange u B, which is formed with the reference beam in the direction of u ri and the signa beam in the direction of u si. The maximum of the ange Du searating a the signa images is given when the ange u B is at a minimum. The condition for obtaining u Bmin, which denotes the minimum of u B, is that u si equa u 0 and, at the same time, that the reference beam be in the direction of u r. u Bmin is given as u Bmin u r u 0 u r sin. (A) By substituting the minimum ange u Bmin into u B in Eq. (A), we obtain the maximum ange Du max as Du max #. (A3) u r sin T sin The mutiexity N u r is given as N u r u r Du max Du max #! u r sin T sin u r. (A4) In Eq. (A4) the numerator of the first term is the anguar range for the reference beam mutiexity excuding the first reference beam. In Eq. (A4) we sti need to determine the ange u r of the reference beam nearest to the signa beam band. If we choose a vaue cose to u 0 as u r, then Du max becomes arge, resuting in the reduction of the mutiexity N u r. On the other hand, a arge vaue for u r reduces the numerator of the first term in Eq. (A4), which is the anguar range for the reference beams excet for k r. To maximize the mutiexity, we shoud otimize u r. Ifwe differentiate Eq. (A4) with resect to u r and set the resut to zero, the maximum N u r is given as! # N u r max N sin 4! 4 sin 3 T sin 8 4 sin!#. (A5) Note that we have used the maximum ange Du max for deriving Eq. (A4). owever, since the incident ange of each reference beam is different, the anguar seectivity for each reference beam is aso different and satisfies Du.Du.Du 3..Du N, where Du i denotes the anguar seectivity for the reference beam in the direction of u ri. Therefore we can say that Eq. (A4) undercounts the mutiexity of hoograhic memory. We made a ostuation for cacuating a more accurate vaue in the foowing. From Eq. (A) the minimum of Du is given when u B is at a maximum. The condition for obtaining u B max is that u ri be equa to and that u si be equa to u 0. u B max is given as

8 94 J. Ot. Soc. Am. A/Vo. 3, No. 5/May 996 T. Tanaka and S. Kawata mutiexed hoogram. The ongitudina cuts of the Ewad shere are shown to the right of Figs. (a) and (b). A mutie number of hoograms are recorded in the same medium as the one recorded with K. The satiafrequency band for the hoogram recorded with reference beam k r, which is the nearest beam to k r in f, is shown in Fig. (b) as the oen circe above and to the eft of the shaded circe fied with the hoogram recorded with k r. The ange k r must be chosen so that the hoogram sace for K does not overa the sace for K. The ange Df satisfying the above condition is found to be Df sin r r, (B) where r k sin u, (B) r k (B3) by sime anaysis of geometry. As a resut, the mutiexity Nf aong the f direction is obtained by dividing by Df: Nf. sin (B4) sin u Fig.. oograhic mutiexity aong the f direction: (a) wave vectors of reference beam and signa beam using Ewad shere, (b) recording sace of mutiexed hoogram.! u B max sin. (A6) ence, by substituting the ange u B max into u B in Eq. (A), we obtain the minimum ange Du min as Du min #. (A7) sin T sin Since Du min is neary haf of Du max, the average ange is given as Du av Du av Du max Du min 3Du max 4. (A8) We therefore aroximate that the true mutiexity Nu of the hoograhic memory equas four-thirds of N u r max in Eq. (A5): 8 < N u 4 3 : 3 T sin APPENDIX B 4 sin! 8 4 sin!# 9 = ;. (A9) In the front view of the Ewad shere shown in Fig. (a) et us fix one reference beam vector ange k r on the k y axis; then the satia-frequency band of a hoogram recorded with the signa beam sectrum [shaded circe in Fig. (a)] is given by the stried circe in Fig. (b). Figure (b) shows the satia-frequency band for the ACKNOWLEDGMENTS The authors thank Caesar Saoma and Osamu Nakamura for their review of this manuscrit and vauabe discussions. This research was suorted by a grant in aid for scientific research from the Ministry of Education, Science, Sorts, and Cuture, Jaan, and by a grant from the CASIO Science Promotion Foundation. The authors may be reached as foows: fax, ; e-mai, t-tanaka@a.eng.osaka-u.ac.j. REFERENCES. S. Kawata, T. Tanaka, Y. ashimoto, and Y. Kawata, Three-dimensiona confoca otica memory using rotorefractive materias, in Photooymers and Aications in oograhy, Otica Data Storage, Otica Sensors and Interconnects, R. A. Lessard, ed., Proc. SPIE 04, (993).. Y. ashimoto, Y. Kawata, and S. Kawata, Three dimensiona confoca otica memory with hotorefractive materias, in Proceedings of Jaan Otics 9 (Kyoto, Jaan, 99), K. Rubin,. Rosen, T. Strand, W. Imaino, and W. Tang, Muti-ayer voumetric storage, in Otica Data Storage, Vo. 0 of 994 OSA Technica Digest Series (Otica Society of America, Washington, D.C., 994), D. A. Parthenoouos and P. M. Rentzeis, Two-hoton voume information storage in doed oymer systems, J. A. Phys. 68, (990). 5. D. A. Parthenoouos and P. M. Rentzeis, Threedimensiona otica storage memory, Science 45, (989). 6. J.. Stricker and W. W. Webb, Three-dimensiona otica data storage in refractive two-hoton oint excitation, Ot. Lett. 6, (99). 7. L. Soymar and D. J. Cooke, Voume oograhy and Voume Gratings (Academic, London, 98), L. d Auria, J. P. uignard, C. Sezak, and E. Sitz, Exerimenta hoograhic read write memory using 3-D storage, A. Ot. 3, (974). 9. F.. Mok, M. C. Tackitt, and. M. Sto, Storage of 500 high-resoution hoograms in a LiNbO 3 crysta, Ot. Lett. 6, (99).

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