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data capacity is required. However, it is expected that slow-light buffers will find applications where compact all-optical buffers are needed, with capacities of around | Mb or less. light in an atomic vapor using electromagnetically induced transparency (EIT) [12], [13], a slow-down factor of more than 31000 using population oscillation (PO) in quantum-well mate- rial [14], EIT in solid-state material [15], and a series of theoret- ical predictions in coupled resonator optical waveguides [16], and other photonic-crystal (PC)-coupled microcavities [17].

Figure 1 data capacity is required. However, it is expected that slow-light buffers will find applications where compact all-optical buffers are needed, with capacities of around | Mb or less. light in an atomic vapor using electromagnetically induced transparency (EIT) [12], [13], a slow-down factor of more than 31000 using population oscillation (PO) in quantum-well mate- rial [14], EIT in solid-state material [15], and a series of theoret- ical predictions in coupled resonator optical waveguides [16], and other photonic-crystal (PC)-coupled microcavities [17].

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Fig. 3. (a) Cascaded and (b) parallel buffers. Fig. 2. Unit optical buffers. (a) Variable delay line. (b) Delay loop. (c) Staggered delay line. Thin lines represent optical paths. Broad arrows represent control signals. L/(TsBpacket). In the following sections, we will examine the limitations on SD and Ly; for a number of device types. analyze the propagation properties of variable delay lines using simple models of the devices. Fig. 4 defines some parameters used in the analysis. The length of the delay line is L, and x represents the distance of a data bit or packet from the input of the delay line. Fig. 4 shows a packet in which the number of bits in the packet bpacket is equal to five. The physical spacing of each bit is Lpit. The capacity C’ of the delay line (i.e., the maximum number of bits that could be stored on the delay line) is C = T's Bpacket = L /Lyit. Note that in some applications, it may be desirable to operate the delay line at less than full capacity, as is the case in Fig. 4. In Section I of this paper, we highlighted the importance of miniaturization in the design of optical buffers. To achieve a small buffer, it is necessary that the physical size or spacing Lyit of the stored bits is small. Another way of looking at this is in terms of the storage density in the device, i.e., the number of stored bits per unit length of the buffer. For a delay line of length L, as shown in Fig. 4, the storage density SD of the device is SD=C/L= gee and the size Lyi, of a stored bit is Lyi =
Fig. 3. (a) Cascaded and (b) parallel buffers. Fig. 2. Unit optical buffers. (a) Variable delay line. (b) Delay loop. (c) Staggered delay line. Thin lines represent optical paths. Broad arrows represent control signals. L/(TsBpacket). In the following sections, we will examine the limitations on SD and Ly; for a number of device types. analyze the propagation properties of variable delay lines using simple models of the devices. Fig. 4 defines some parameters used in the analysis. The length of the delay line is L, and x represents the distance of a data bit or packet from the input of the delay line. Fig. 4 shows a packet in which the number of bits in the packet bpacket is equal to five. The physical spacing of each bit is Lpit. The capacity C’ of the delay line (i.e., the maximum number of bits that could be stored on the delay line) is C = T's Bpacket = L /Lyit. Note that in some applications, it may be desirable to operate the delay line at less than full capacity, as is the case in Fig. 4. In Section I of this paper, we highlighted the importance of miniaturization in the design of optical buffers. To achieve a small buffer, it is necessary that the physical size or spacing Lyit of the stored bits is small. Another way of looking at this is in terms of the storage density in the device, i.e., the number of stored bits per unit length of the buffer. For a delay line of length L, as shown in Fig. 4, the storage density SD of the device is SD=C/L= gee and the size Lyi, of a stored bit is Lyi =
Fig. 4. Variable delay line. position x by appropriate positioning of the pump beam. In the present paper, we assume that v, can be separated into a product of two terms, one dependent on time and one dependent on position. Thus, we put
Fig. 4. Variable delay line. position x by appropriate positioning of the pump beam. In the present paper, we assume that v, can be separated into a product of two terms, one dependent on time and one dependent on position. Thus, we put
Fig. 5. FIFO buffer using cascaded Class A delay lines. Fig. 6. Pulses in Class A slow-light variable delay line. change when data are in the delay line. However, this does not preclude the delay from changing when no data are present. Indeed, as pointed out earlier, a delay line whose delay cannot be changed is not an optical buffer. In Class A delay lines, the delay through the device can be changed through changes to the group velocity vg in the slow-light region and/or through changes to the length of the slow-light region (i.e., via changes to x4 and xp in Fig. 6).
Fig. 5. FIFO buffer using cascaded Class A delay lines. Fig. 6. Pulses in Class A slow-light variable delay line. change when data are in the delay line. However, this does not preclude the delay from changing when no data are present. Indeed, as pointed out earlier, a delay line whose delay cannot be changed is not an optical buffer. In Class A delay lines, the delay through the device can be changed through changes to the group velocity vg in the slow-light region and/or through changes to the length of the slow-light region (i.e., via changes to x4 and xp in Fig. 6).
Fig. 7. Pulses in Class A slow-light variable delay line with transition region. duced by material effects and slow light produced by devices in which a wave at frequency w propagating with a wave vector k “feels” that it is traveling in a homogeneous medium with an effective refractive index n(w,k) even if the medium has a nonuniform index distribution n(z,y,z) such as a Bragg grating or PC filter.
Fig. 7. Pulses in Class A slow-light variable delay line with transition region. duced by material effects and slow light produced by devices in which a wave at frequency w propagating with a wave vector k “feels” that it is traveling in a homogeneous medium with an effective refractive index n(w,k) even if the medium has a nonuniform index distribution n(z,y,z) such as a Bragg grating or PC filter.
Fig. 8. Characteristics of slow-light delay lines. where ng; =n; + w(dn;/dw) is the group index in wave- guide 7. It is easy to see from (7) that the slow-down factor can made large by ensuring that the dn2/dw term is larger than the dn,/dw term. In many of the slow-light devices, we consider here, waveguide 2 is designed to produce slow light, while waveguide | is not a slow-light waveguide (i.e., dn, /dw = 0). Thus, the slow-down factor in (7) is controlled primarily by the w(dn2/dw) term. For low-distortion slow-light propagation, the term w(dn2/dw) in (7) should be large and independent of the frequency w across the signal passband. In most practical situations, the fractional bandwidth of the signal is small, and this requirement translates into a require- ment that the effective refractive-index slope dn2/dw is large and independent of frequency across the passband, i.e., that a plot of nz versus w is a straight line with large slope.
Fig. 8. Characteristics of slow-light delay lines. where ng; =n; + w(dn;/dw) is the group index in wave- guide 7. It is easy to see from (7) that the slow-down factor can made large by ensuring that the dn2/dw term is larger than the dn,/dw term. In many of the slow-light devices, we consider here, waveguide 2 is designed to produce slow light, while waveguide | is not a slow-light waveguide (i.e., dn, /dw = 0). Thus, the slow-down factor in (7) is controlled primarily by the w(dn2/dw) term. For low-distortion slow-light propagation, the term w(dn2/dw) in (7) should be large and independent of the frequency w across the signal passband. In most practical situations, the fractional bandwidth of the signal is small, and this requirement translates into a require- ment that the effective refractive-index slope dn2/dw is large and independent of frequency across the passband, i.e., that a plot of nz versus w is a straight line with large slope.
Fig. 9. Active microresonator delay line. have the same fundamental limitations as passive PC delay lines, but active devices can approach the limitations more rapidly in practice.
Fig. 9. Active microresonator delay line. have the same fundamental limitations as passive PC delay lines, but active devices can approach the limitations more rapidly in practice.
Fig. 10. Effective refractive versus frequency for coupled microresonators. where gq is an integer. The passbands of the cavity reso- nances are centered at k,D = 2pm (i.e., at frequencies w, = 2pmc/NavgD), where p is an integer. The passbands are sepa- rated by the free spectral range [(FSR), in radians per second], where FSR = 27¢/naygD. Thus, w, = pFSR.
Fig. 10. Effective refractive versus frequency for coupled microresonators. where gq is an integer. The passbands of the cavity reso- nances are centered at k,D = 2pm (i.e., at frequencies w, = 2pmc/NavgD), where p is an integer. The passbands are sepa- rated by the free spectral range [(FSR), in radians per second], where FSR = 27¢/naygD. Thus, w, = pFSR.
THEORETICAL LIMITATIONS ON STORAGE TIME, DELAY-BANDWIDTH PRODUCT, MINIMUM BIT SIZE, MAXIMUM STORAGE DENSITY, AND HOLD-OFF TIME FOR IDEAL CLASS A DEVICES, EIT/PO DEVICES WITH A LORENTZIAN-SHAPED ATTENUATION FUNCTIONS, PC DEVICES AS DEFINED IN THE TEXT, CLASS B DEVICES, AND COMBINED A/B/A DEVICES AS SHOWN IN FIG. 16(a). TR IS THE RECONFIGURATION TIME FOR GROUP- VELOCITY CHANGE
THEORETICAL LIMITATIONS ON STORAGE TIME, DELAY-BANDWIDTH PRODUCT, MINIMUM BIT SIZE, MAXIMUM STORAGE DENSITY, AND HOLD-OFF TIME FOR IDEAL CLASS A DEVICES, EIT/PO DEVICES WITH A LORENTZIAN-SHAPED ATTENUATION FUNCTIONS, PC DEVICES AS DEFINED IN THE TEXT, CLASS B DEVICES, AND COMBINED A/B/A DEVICES AS SHOWN IN FIG. 16(a). TR IS THE RECONFIGURATION TIME FOR GROUP- VELOCITY CHANGE
Fig. 11. WDM slow-light characteristics.
Fig. 11. WDM slow-light characteristics.
Fig. 12. FIFO buffer using Class B delay lines.
Fig. 12. FIFO buffer using Class B delay lines.
We now consider wave propagation through the Class B device from time ¢ to time ¢ + dt. Using (5) and substituting (53), it is easy to show that We now analyze the behavior of a device with time- dependent refractive index. We assume that the time scale 7,, of the changes of refractive index is very much larger than the time scale of the optical carrier 7, = 1/v, ie., assume that 7, >> T.. (This assumption is equivalent to the adiabatic assumption in [29].) Under this assumption, the variables w and ¢ can be separated. From Fig. 13, the refractive index at frequency w and time ¢ can be expressed in terms of the width (Wmax — Wmin) of the region of linear refractive slope frequency w and time ¢ can be expressed in terms of the width
We now consider wave propagation through the Class B device from time ¢ to time ¢ + dt. Using (5) and substituting (53), it is easy to show that We now analyze the behavior of a device with time- dependent refractive index. We assume that the time scale 7,, of the changes of refractive index is very much larger than the time scale of the optical carrier 7, = 1/v, ie., assume that 7, >> T.. (This assumption is equivalent to the adiabatic assumption in [29].) Under this assumption, the variables w and ¢ can be separated. From Fig. 13, the refractive index at frequency w and time ¢ can be expressed in terms of the width (Wmax — Wmin) of the region of linear refractive slope frequency w and time ¢ can be expressed in terms of the width
Fig. 14. Pulses in Class B slow-light variable delay line with time-dependent uniform group-velocity distribution.
Fig. 14. Pulses in Class B slow-light variable delay line with time-dependent uniform group-velocity distribution.
Fig. 15. Pulses in Class B slow-light variable delay line with moving velocity transition.
Fig. 15. Pulses in Class B slow-light variable delay line with moving velocity transition.
Fig. 16. Mixed delay lines. (a) Increased slow-down factor. (b) Packet decompressor. bit rate. In this case, the device is a Class A device. In the other extreme, when the velocity uv, of the transition is large, t nen Bout = SBin. Note that this bandwidth increase is identical to the bandwidth increase in a device with a uniform gro up- velocity distribution that ramps in time (Section IV-A). The example considered in Fig. 15 is for a ramp increase in group velocity moving to the left. If the ramp is changed from increase to a decrease in group velocity (from v1 to vg2), t it is easy to show that Bou: = Bin/S. an nen
Fig. 16. Mixed delay lines. (a) Increased slow-down factor. (b) Packet decompressor. bit rate. In this case, the device is a Class A device. In the other extreme, when the velocity uv, of the transition is large, t nen Bout = SBin. Note that this bandwidth increase is identical to the bandwidth increase in a device with a uniform gro up- velocity distribution that ramps in time (Section IV-A). The example considered in Fig. 15 is for a ramp increase in group velocity moving to the left. If the ramp is changed from increase to a decrease in group velocity (from v1 to vg2), t it is easy to show that Bou: = Bin/S. an nen
"Fundamental limit “Example only — not a fundamental limit * For a 1-cm long device ** 40 Gb/s data *** Not including Demux/Mux EXAMPLES OF EFFECTIVE BANDWIDTH, DELAY-BANDWIDTH PRODUCT, BIT SIZE, CAPACITY, AND Aa(wo) TABLE II
"Fundamental limit “Example only — not a fundamental limit * For a 1-cm long device ** 40 Gb/s data *** Not including Demux/Mux EXAMPLES OF EFFECTIVE BANDWIDTH, DELAY-BANDWIDTH PRODUCT, BIT SIZE, CAPACITY, AND Aa(wo) TABLE II
Related topics:
OpticsMedical ImagingFiber OpticsOptical WaveguidesSlow LightElectromagnetically Induced TransparencyOptical physicsOptical FiltersInformation RateElectrical and Electronic Engineering
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