Foundations of Physics

, Volume 44, Issue 8, pp 856–872 | Cite as

Large Scale Integrated Photonics for Twenty-First Century Information Technologies

A “Moore’s Law” for Optics
Article

Abstract

In this paper, we will review research done by the Large-Scale Integrated Photonics group at HP Laboratories, and in particular we will discuss applications of optical resonances in dielectric microstructures and nanostructures to future classical and quantum information technologies. Our goal is to scale photonic technologies over the next decade in much the same way as electronics over the past five, thereby establishing a Moore’s Law for optics.

Keywords

Classical optics Quantum optics Integrated photonics  Information technology 

1 Introduction and Overview

In 1965, Gordon Moore claimed in an article entitled “Cramming More Components onto Integrated Circuits” that by 1975 as many as 65,000 electronic elements would be integrated onto a silicon chip [1]. The general observation that the number of components on semiconductor dies will double every year or two is known as “Moore’s Law,” and has remained largely true over the last 50 years. In this paper, we will review research done by the Large-Scale Integrated Photonics group at HP Laboratories, and in particular we will discuss applications of optical resonances—such as that shown by the “ring of fire” in Fig. 1—in dielectric microstructures and nanostructures to future information technologies. Our goal is to scale photonic technologies over the next decade in much the same way as electronics over the past five, thereby establishing a Moore’s Law for optics.
Fig. 1

a An SEM image of a silicon microring resonator that has a diameter of only 3 \(\upmu \)m. Nevertheless, this device has quality factor for the quasi-TE mode of \(Q \approx 9000\). b Microscope image of a silicon nitride microring cavity with a diameter of 20 \(\upmu \)m fabricated for quasi-TE-mode operation. Here a laser with a wavelength of 634 nm has been tuned to a resonance of the cavity, which has a quality factor of \(Q \approx 400\)

In Sect. 2 we discuss applications of classical optics to classical computing, emphasizing photonic technologies amenable to large-scale CMOS integration for high-performance interconnects over distance scales of 1–100 m [2]. Then, in Sect. 3, we report on our research on quantum optics for classical computing, which has focused on cavity quantum electrodynamics in diamond [3]. In the spirit of the Workshop on Quantum Horizons, we ask whether quantum technologies can scale, and speculate on a framework within which we can think about applications of quantum computers. Finally, in Sect. 4, we explore “domains” of quantum mechanics, and try to make the case that much promising research remains to be done using nonlinear quantum optics for future ultra-low-power classical computers.

2 Classical Optics for Classical Computing

We can gain an appreciation for the consequences of Moore’s Law for large-scale information technology (e.g. high-performance computers or data centers) by comparing the power consumption of the input–output (IO) layer of a network switch, as shown in Table 1 [4]. We have assumed in this comparison that the port bandwidth will need to double at each new fabrication process generation to accommodate increases in processor performance. In a forced-air cooling environment with a heat sink, the total power that can be consumed by the switch is approximately 130 W. Requiring liquid cooling would allow a total device power as high as 200 W, but this would limit the field of application to extreme high end systems. If we assume that 50 % of the power budget (65 W) is available for IO, then it is apparent that power considerations alone will mandate the use of photonics as link speeds increase.
Table 1

Router IO power scaling for switches with 64, 100, and 144 ports

Node (nm)

BW

IO

Energy

64(W)

100 (W)

144 (W)

45

80

E

7,000

35.8

56.0

80.6

  

O

451

2.3

3.6

5.2

35

160

E

5,048

51.7

80.8

116.3

  

O

284

2.9

4.5

6.5

22

320

E

4,049

82.9

129.6

186.6

  

O

191

3.9

6.1

8.8

Here “Node” represents the ITRS process generation [11]; “BW” is the bandwidth of each port in units of gigabits/second; “IO” is the input/output type (either electronic or optical); “Energy” consumption is expressed in femtojoules/bit; and the final three columns are the power consumption of the switch in Watts while operating at the stated bandwidth per port

A compact approach to photonics on a silicon platform is the heterogeneous integration of III–V laser gain and a silicon optical infrastructure, which uses a laser resonator and output-coupling waveguide defined completely in the CMOS-compatible SOI layer, but relies on evanescent coupling between the laser cavity modes and a wafer-bonded InAlGaAs layer to provide gain [5]. As shown in Fig. 2, the propagating electromagnetic field mode is confined primarily in the Si waveguide, but the dimensions of the waveguide are reduced until approximately 5–10 % of the power in the mode emerges into the III–V material. These hybrid microring laser structures have diameters of only 50 \(\upmu \)m or less [6, 7, 8]—as well as very low optical and electrical losses—which allow the devices to be operated in a single transverse mode at a wavelength near 1530 nm with threshold currents as low as 2 mA and 2 mW output power. Ring diameters below 10 \(\upmu \)m should enable threshold currents below 1 mA, and the correspondingly small capacitance of the device should allow direct on-off-keying modulation rates above 10 Gb/s [5]. Several of these microring lasers could be coupled to the same waveguide bus, enabling highly integrated coarse wavelength-division multiplexed (CWDM) data transmission at low energy expenditures per bit. One significant challenge for operating these active devices in HPC and datacenter systems is the high temperatures typical of integrated circuit packages in these environments. The most common silicon photonic platform is silicon-on-insulator (SOI) [9], which incorporates a thick buried oxide layer with low thermal conductivity that significantly raises the temperature of the III–V junction above that of the ambient. A promising approach to significantly improve the thermal impedance of the low-index material below the top silicon of the wafer replaces the oxide with polysilicon diamond [10]. Silicon-on-diamond waveguides with cross sections of \(0.7 \times 2.5~\upmu \)m\(^2\) exhibited propagation losses of \(0.74\) dB/cm compared to \(0.54\) dB/cm in SOI, with numerical simulations predicting a factor of up to 11 improvement in heat dissipation.
Fig. 2

Schematic (a) and microscope image (b) of a hybrid silicon–InAlGaAs microring laser with a diameter of 50 \(\upmu \)m. This device has a threshold current of 2 mA and an output power of 2 mW

These hybrid laser systems could function very well as high-bandwidth transmitters in CWDM systems, but inevitably communication bandwidth requirements will mandate dense wavelength-division multiplexing (DWDM) with CMOS based photonics. Systems and applications requiring bandwidths approaching 1 Tb/s per optomechanical connector (e.g. a single fiber optical core affixed to a chip or board) will require DWDM and therefore resonant filters such as microrings or microdisks. The small size of these resonant devices allows them to be operated as electro-optic modulators at the low voltages and low driving power required by integrated circuit technologies beyond 22 nm CMOS. The most dramatic recent improvements in silicon electro-optical modulator speed and energy have been achieved using a lightly-doped waveguide as a p-n diode that can be operated in reverse bias to deplete carriers and thereby change the refractive index [12]. However, if carrier injection is used to change the refractive index of the silicon [12], the waveguide region containing the optical mode is limited to the broad intrinsic region of a p-i-n diode to avoid any significant optical insertion losses in the heavily-doped p-type and n-type regions. When the junction is forward-biased, carriers can be injected into the waveguide through the doped regions in contact with metal wires, requiring low bias voltages (\(<1\) V) and allowing large extinction ratios and therefore low bit-error rates at moderate laser power. For example, the 2.5 \(\upmu \)m-radius modulator shown in Fig. 3 has demonstrated an 18 dB DC extinction ratio and a 8 Gb/s modulation rate while dissipating only 45 fJ/b [13].
Fig. 3

Microscope image of a silicon microring modulator with a diameter of 10 \(\upmu \)m optimized for TM-mode operation. This device has \(Q > \hbox {10,000}\), and exhibits an 18 dB extinction ratio at 8 gigabits/s while dissipating only 45 femtojoules per bit

Although integrated active photonic devices receive most of the attention in the literature, much of the data link-level loss budget of an optical circuit depends critically on the quality of the passive devices that are used to couple and transport light from one location to another in the system. For example, one of the most critical enabling technologies for market-ready silicon photonics is the packaging of an interface between the small transverse mode of a high-index-contrast waveguide and the larger transverse mode profile of an optical fiber, with a cost that can compete with that of a typical data pin (\(\sim \)US$0.01–0.02 per pin) on a standard electronic integrated circuit package. (In DWDM systems, the optical connector/convertor cost may be divided by the total number of data channels, resulting in a potentially significant reduction in packaging costs.) Although end-coupled adiabatic tapers have been used by many researchers [14], a subwavelength grating coupler fabricated in SOI using a CMOS pilot line with 193 nm DUV photolithography has been demonstrated with a peak coupling efficiency of about 70 % [15], allowing light to be coupled onto the chip at any location that is convenient, enabling both wafer-scale testing and packaging layout optimization. We have used gratings with features smaller than an optical wavelength for a variety of applications, most recently including glasses-free 3D displays [16]. But returning to our theme of resonance effects in dielectric optical structures, we discovered that systematically varying the period and/or feature size of high-index-contrast (e.g. silicon/air) grating devices allows them to perform a variety of surprising optical functions, such as focusing like a curved lens or mirror [17], violating Snell’s Law [18], or dramatically enhancing field intensities in air or biological fluids [19]. A mirror fabricated in this way is shown in Fig. 4; it is a flat layer of amorphous silicon (about 450 nm thick) deposited on quartz. Even though the surface is not curved at all, the variation in the width of the grating grooves allows this device to perform as a mirror with a focal length of about 25 mm, even at a wavelength of 1550 nm where silicon is transparent.
Fig. 4

Optical microscope picture of a fabricated spherical SWG mirror. The groove width in various locations is shown in SEM images in the insets (Figure reproduced from [17] with permission 2010 NPG.)

3 Quantum Optics for Quantum Computing

Diamond has several properties of interest for potential information technology applications [20]. In particular, for quantum information processing, diamond has an optically active structural defect—the nitrogen-vacancy (NV) center—that is optically addressable and can exhibit electron spin coherence lifetimes exceeding \(1 \, \mathrm {ms}\) at room temperature [21]. These capabilities have enabled some remarkable demonstrations such as controlled coupling between single electronic and nuclear spins [22]. In the long term, a chip-scale quantum computer could (in principle) be built by connecting multiple NV centers together optically, allowing long-distance quantum communication through repeaters [23], or to test one-way quantum computation approaches [24], but many of these protocols require that the photons emitted by the nodes in the network be identical [23, 25, 26]. This is particularly problematic for the NV center, because 97 % of its spontaneous emission falls into spectrally broad multi-phonon bands, while only 3 % occurs within the relatively narrow zero-phonon line (ZPL). Our strategy has been to couple the emission of NVs to resonances of microcavities, thereby enhancing the fraction of spontaneous emission into the ZPL, and significantly improving photon indistinguishability [3, 27].

Historically, it has been difficult to fabricate photonic structures such as waveguides and cavities in diamond, primarily because wafers consisting of a thin layer of high-quality single-crystal diamond bonded to a lower-index insulator, analogous to silicon-on-insulator wafers, are not yet commercially available. In our laboratory, we have attempted two approaches to surmount this difficulty:
  1. 1.

    coupling diamond nanoparticles to silica microdisk structures [3, 28, 29];

     
  2. 2.

    attaching higher-index gallium phosphide nanocavities to bulk single-crystal diamond [3, 30, 31, 32, 33, 34]; and

     
  3. 3.

    fabricating photonic structures containing NV centers directly in diamond [35, 36, 37].

     
In all of these cases, we have demonstrated coupling between NV centers and whispering-gallery-type optical modes of photonic nanocavities; in the latter two cases, we have also observed Purcell enhancement of the spontaneous emission rate into the ZPL of NV-centers encapsulated within the resonators, which is proportional to the ratio \(Q/V_\mathrm {mode}\), where \(V_\mathrm {mode}\) is the volume of the coupled mode.
For example, in 2011 we were able to demonstrate coupling of the zero-phonon line of individual NV centers to modes of microring resonators—similar to the device shown in Fig. 1a—fabricated in single-crystal diamond thin films [35]. We used measurements of the radiative lifetime of the NV ZPL to show that the corresponding spontaneous emission rate had been enhanced by more than a factor of 10 by rings with quality factors \(Q \approx 4000\) and mode volumes \(V_\mathrm {mode} \approx 20 (\lambda /n)^3\) (where \(\lambda = 638\) nm is the wavelength of the ZPL transition, and \(n = 2.4\) is the index of refraction of diamond). These devices were fabricated using predominantly standard semiconductor techniques and off-the-shelf materials, and was the first published indication that integrated diamond photonics was possible in principle [37]. Within a year, we improved on this result in two significant ways [36]. First, we relied only on standard industrial semiconductor fabrication and materials, and we fabricated photonic bandgap crystal cavities such as those shown in Fig.  5 to reduce the mode volume to \(V_\mathrm {mode} < (\lambda /n)^3\). These cavities exhibited quality factors \(Q \approx 3000\), and as shown in Fig. 6 the spontaneous emission enhancement into the ZPL increased to a factor of 70. In this case, the fraction of the spontaneous emission at the wavelength of the ZPL has increased from 3 to 70 %.
Fig. 5

a Color plot showing the electric field energy density in the fundamental mode of a linear three-hole defect cavity. The lateral three holes are shifted laterally by \(d_1\), \(d_2\), and \(d_3\) to increase the quality factor. bd Scanning electron microscope images of a photonic crystal cavity fabricated in monocrystalline diamond. The black rectangles in (b) are openings etched in the membrane to facilitate etching underneath the membrane. The darker gray region around the openings indicates the extent of the etched Si under the diamond (Color figure online)

Fig. 6

a Measurement of the lifetime of the NV center when not on resonance with the cavity. The measurement is performed on photoluminescence emitted in the ZPL. b Lifetime measurement when the NV is coupled to the cavity. c Second-order correlation measured on the zero-phonon line of the coupled NV. The bin size in the histogram is 1.1 ns

Given that we appear to have assembled several of the ingredients needed for a practical scalable integrated photonic technology, it is appropriate to try to answer the simple question

Will my quantum technology scale?

This question actually goes to the heart of our use of the word “practical,” which has different meanings in academia and industry. To illustrate this, let’s restate the original question as a series of questions of industrial and commercial relevance to information technology.
  1. 1.

    Can my technology be integrated? Integration is the ability to manufacture your devices at low cost with high reliability/yield (“wafer-scale fabrication and testing”).

     
  2. 2.

    Can it be packaged? Packaging is the ability to “abstract away” the functional details of your device and present a simple, low-cost interface to the user.

     
  3. 3.

    Can it be interconnected? Interconnection is the ability of the packaged interface to connect your technology at high bandwidth and low cost to other components in the system.

     
  4. 4.

    Is it defect tolerant? Defect tolerance allows circuits and systems to function properly even if nominally similar devices are significantly different.

     
With these requirements in mind, is an integrated quantum technology based on NV-centers scalable in practice? Although it has shown great potential, at this time the answer must be “No.” First of all, single-crystal “diamond on insulator” wafers are not yet commercially available, so wafer-scale fabrication and testing is simply beyond our reach. There are other issues—tuning the optical properties of the NV centers in photonic nanostructures continues to be a significant technical challenge, which significantly reduces defect tolerance—but the inability to integrate at the wafer scale remains a serious roadblock. In this context, we see that practicality as understood by industry adds a significant layer of complexity to the common use of the term in academia. (We should recall here the quote “In theory, there is no difference between theory and practice. But, in practice, there is.”—which has been attributed to several possible originators, including Yogi Berra.)

It seems natural, then, to ask the more general question

How do I build a quantum computer?

But this is almost certainly the wrong question, and could be driving our field in a direction that could miss a promising path to commercialization of quantum technology. For example, the very first commercially significant application of the transistor certainly wasn’t the computer, and it wasn’t even the transistor radio; it was the hearing aid [38]! The first all-transistor hearing aids appeared in 1953, and were the direct result of engineers finding answers to the list of questions above. (To be fair, these first hearing aids used integrated discrete transistors—wafer-scale manufacturing was still some time off.) This advance dramatically increased the wearability and accessibility of hearing aids, and simultaneously significantly decreased the cost of packaged transistors. The prices of these devices subsequently fell within the budgets of researchers in both academic and industrial laboratories in the 1950s, ultimately leading to wonderful electronic products like transistor radios, and then to the information technologies that we take for granted today. So, a better general question would be

How do I build the “quantum hearing aid?”

4 Quantum Optics for Classical Computing

We began our discussion of applications of classical optics to classical computing with Moore’s Law. In a sense, Moore’s Law is a curse, because it has created an expectation in our customer base that computer performance will continue to scale exponentially forever. Within another few years we will reach the 12-nm ITRS technology node, and it is already clear that the performance gains that we will achieve as we move from 16-nm processors to 12-nm processors will be minuscule. However, dramatic performance improvements per unit power are now the plan of record for the U.S. Department of Energy Extreme Scale Computing Technology Roadmap [39], which states unambiguously that a machine—to be built by the year 2020—that can perform exaflop-class computations must consume no more than 20 MW of electrical power. For comparison, that is 50 GFLOPS/W, or 25 times more efficient than the most efficient supercomputer built-to-date [40]. Moore’s law is not driving that kind of improvement in energy-based performance, which corresponds roughly to an average of 10 attojoule/gate operation; although this seems to be extremely small, it can also be stated as \(1000~k_\mathrm {B} T\), where \(k_\mathrm {B}\) is Boltzmann’s constant, and \(T \approx 300\) K is the temperature of the computer. The Landauer limit [41] tells us that, in principle, if we could build a reversible computer, we could do so in an almost energy-free way, with erasures only costing us \(\ln (2) k_\mathrm {B} T\). In practice, there are bound to be other constraints, but a tantalizing open question is: “Can we use intrinsically quantum technologies to allow us to compute at lower energies?”

The semiconductor industry has grown exponentially by exploiting the nonrelativistic quantum mechanics of electronics in periodic lattices (here we think of the Heisenberg Uncertainty Principle and the Shrödinger equation), even though most integrated circuit modeling tools have abstracted and parameterized the underlying physics. While we might think that some information technology is based on the quantum coherence of light, the lasers that are used for data communications can be described (again) by quantum electronics and classical coherence theory. Many ideas for reversible computing (by Bennet et al. [41]) benefited from quantum coherence (i.e. coherent superpositions of quantum states), but most of that research was “repurposed” in the late 1980s and early 1990s as excitement over quantum computing grew. In our research, we are attempting to resurrect some of this early research to explore new approaches to low-energy computing.

We should begin by asking what properties new computing technology should have—if we intend to replace the silicon transistor with a new device, what are reasonable expectations for its performance characteristics? In a paper on the feasibility of optical transistors, Miller [42] noted that current silicon devices have a number of critically important properties, including
  • logic-level restoration;

  • input-output isolation;

  • non-critical biasing;

  • fan-out;

  • cascadability;

  • loss independence of logic levels; and

  • small device footprints.

He also noted that “...a device at the 100-photon level could still operate in a quasi-classical fashion where optical or quantum coherence is neither necessary nor even desirable.” We will take this challenge as an inspiration to look more deeply at the problem of low-energy optical information processing (which we will use as a representation of the more general problem of low-energy computing).
One strategy for low-energy classical computing (whether or not we are trying to operate reversibly) is simply to reduce the number of particles needed to represent a bit. In the extreme case of digital quantum computing, the number of particles used to represent a bit is one, but even in the case of 10–100 particles we are very likely to see quantum behavior that we will not be able to ignore. In other words, if we compute using intrinsically quantum technologies, then in quantum regimes with very low particle counts we won’t necessarily require very much energy to process information. However, as we see in the upper graph of Fig. 7, there’s a catch: quantum noise inhibits reliable computation. This plot shows a quantum trajectory simulation of the mean intracavity photon number for an optical resonator driven by a quantum coherent state that incorporates a Kerr nonlinear medium, which results in behavior known as “dispersive optical bistability” [43]. The vertical axis of the plot gives the mean number of photons in the cavity; the jumps between lower and higher photon-number states, which occur spontaneously with the drive amplitude held fixed, are symptomatic of the destabilization of optical bistability by the quantum noise associated with such small particle counts. It might seem that quantum mechanics must inevitably prevent us from building reliable classical devices at these energy scales (where 10 photons with wavelengths of 1 \(\upmu \)m have a total energy of 2 aJ), but in this case the solution to the problem of “too much quantum mechanics” is “more quantum mechanics.” The lower plot of Fig. 7 shows a quantum trajectory simulation of the mean intracavity photon number of the same system assuming nonlinear dynamic coherent-quantum-feedback control (described in detail in [43]). The “classical” performance of this quantum system has improved dramatically.
Fig. 7

Upper plot A quantum trajectory simulation of the mean intracavity photon number for a driven optical resonator including a Kerr nonlinear medium. The jumps between low and high photon-number states, which occur spontaneously with the drive amplitude held fixed, are symptomatic of quantum destabilization of dispersive optical bistability. Lower plot A quantum trajectory simulation of the mean intracavity photon number assuming nonlinear dynamic coherent-feedback control (Figure reproduced from [43]2011AIP.)

In our research, we are focusing on complex coherent nanophotonic circuits even though it is not at all clear that building an optical computer is either a feasible or useful goal. We want to study the dynamics of many coherently coupled nonlinear circuit elements in regimes where quantum effects are important, and we want to determine whether or not quantum coherent feedback control could allow us to demonstrate reliable low-energy classical information processing. At this time, it makes sense to use photonics because properly conditioned lasers are convenient sources of coherence over many orders of magnitude in mean particle number. In addition, ever-higher levels of integration creates many research opportunities in nonlinear and quantum photonics that are not really envisioned today, allowing us to explore serial and parallel interaction of many devices in ways that scientists and engineers began to explore in electronics over 40 years ago. In the case of large-scale integrated photonics, circuits formed by coupling many small-footprint devices will inevitably interact coherently, and this could expose exciting new behaviors that are qualitatively distinct from those of integrated electronics.

Let’s begin to understand how we might be able to operate complex nanophotonic circuits at low powers by building simple models of optical bistability using nonlinearities of the Kerr type. If we neglect dispersion, then the instantaneous change \(\Delta n(\mathbf {r})\) in the refractive index \(n(\mathbf {r})\) at position \(\mathbf {r}\) of a dielectric material is given by [44, 45, 46]
$$\begin{aligned} \frac{\Delta n(\mathbf {r})}{n(\mathbf {r})} = \epsilon _0\, c\, n_2(\mathbf {r})\, |\mathbf {E}(\mathbf {r})|^2 \end{aligned}$$
(1)
where \(\epsilon _0\) is the permittivity of the vacuum, \(c\) is the speed of light, \(n_2(\mathbf {r})\) is the nonlinear optical Kerr coefficient and \(\mathbf {E}(\mathbf {r})\) is the amplitude of the harmonic macroscopic electric field in the cavity defined by \(\mathcal {E}(\mathbf {r}, t) \equiv \mathfrak {R}[\mathbf {E}(\mathbf {r})\, e^{-i \omega t}]/2\). Kerr coefficients for crystalline materials compatible with large-scale integrated fabrication technologies are listed in Table  2.
Table 2

Nonlinear Kerr coefficients for crystalline materials compatible with large-scale integrated fabrication technologies

Material

\(\lambda \) (nm)

\(n\)

\(n_2\)

Si [47]

1,550

3.4

5

GaAs [46]

1,064

3.47

\(-\)33

GaAs [46]

1,550

3.38

\(-\)15

GaAs/AlAs MQW [48]

1,505

\(\sim \)3

55

GaAs/AlGaAs MQW [49]

816

\(\sim \)3

\(1.5 \times 10^{10}\)

Units are \(10^{-14}\) cm\(^2\)/W = \(10^{-6}~\upmu \)m\(^2\)/W = 1 nm\(^2\)/W. The GaAs multi-quantum-well materials have nonlinear absorptions of \(\sim \)5 cm/GW. For comparison, the last entry in the table is the effective nonlinear coefficient due to a carrier-based effect

For a resonator with mode volume \(V\), the total intracavity stored energy is given by
$$\begin{aligned} W_c = \frac{1}{2} \epsilon _0 \int _V d^3 \mathbf {r} \, n^2(\mathbf {r})\, |\mathbf {E}(\mathbf {r})|^2 \end{aligned}$$
(2)
or, if we approximate both the refractive index and harmonic field amplitude as uniform throughout the resonator, \(W_c \approx \frac{1}{2} \epsilon _0 n^2 |\mathbf {E}|^2 V\). Under the same assumption, the shift in a resonant frequency \(\omega _r\) of the cavity caused by the Kerr effect is given by Eqs. 1 and 2 as
$$\begin{aligned} \frac{\Delta \omega _r}{\omega _r} = \frac{\Delta n}{n} = \epsilon _0\, c\, n_2 |\mathbf {E}|^2 = \frac{2\, n_2\, c}{n^2 V} W_c . \end{aligned}$$
(3)
If we define the “characteristic energy” \(W_0\) of the cavity as the stored energy required to shift the resonance frequency by \(\Delta \omega _\mathrm {FWHM} = \omega _r/Q\), where \(\Delta \omega _\mathrm {FWHM}\) is the full-width at half-maximum linewidth of the resonance and \(Q\) is the optical quality factor of the resonator, then we find
$$\begin{aligned} W_0 = \frac{\lambda _r^3}{2\, n\, n_2\, c} \frac{\tilde{V}}{Q} , \end{aligned}$$
(4)
where \(\lambda _r = 2 \pi c/\omega _r\) is the vacuum resonance wavelength and \(\tilde{V} \equiv V/(\lambda _r/n)^3\) is the volume scaled by the cube of the physical wavelength in the material. Since the output power of the cavity is related to the stored energy by \(P_\mathrm {out} = (\omega _r/Q) W_c\), we can also define a “characteristic power” as
$$\begin{aligned} P_0 = \frac{\pi \lambda _r^2}{n\, n_2} \frac{\tilde{V}}{Q^2} . \end{aligned}$$
(5)
Two well-understood cavity geometries appropriate for fabrication using crystalline semiconductor materials are the microring resonator shown in Fig. 1a, and the photonic bandgap crystal cavity shown in Fig. 5. Small microring resonators coupled to waveguides have values of \(Q/\tilde{V} \approx 10^3\) [50], while those for photonic crystal cavities theoretically can be as high as \(Q/\tilde{V} \approx 10^6\) [32]. Therefore, in Fig. 8, we plot the characteristic energy \(W_0\) as a function of \(Q/\tilde{V}\) for different values of the nonlinear optical Kerr coefficient \(n_2\) over a similar range of values of \(Q/\tilde{V}\). We chose the values of the resonant wavelength and refractive index to be \(\lambda _r = 1~\mu \)m and \(n = 3.38\), respectively. (The energy of a single photon at this wavelength is approximately 200 zeptojoules.) Using Table 2, we note that the silicon microring resonator shown in Fig. 1a operated at \(\lambda _r = 1550\) nm would have \(W_0 \approx 0.4\) picojoules, and a resonator with a similar value of \(Q/\tilde{V}\) fabricated using a GaAs/AlGaAs MQW could have a characteristic energy as low as 2 zJ.
Fig. 8

Characteristic energy \(W_0\) as a function of \(Q/\tilde{V}\) for different values of the nonlinear optical Kerr coefficient \(n_2\). The resonant wavelength and refractive index have the values \(\lambda _r = 1~\upmu \)m and \(n = 3.38\), respectively. The energy of a single photon at this wavelength is approximately 200 zJ. Note that in practice large values of \(n_2\) may not be accessible at high \(Q/V\) due to nonlinear absorption

We now examine single-cavity optical bistability using a simple Lorentzian model of transmission as a function of frequency [45]. For an input power \(P_\mathrm {in}(\omega )\), the output power \(P_\mathrm {out}(\omega )\) is given by
$$\begin{aligned} \frac{P_\mathrm {out}(\omega )}{P_\mathrm {in}(\omega )} \approx \frac{\gamma ^2}{\gamma ^2 + (\omega - \omega _r - \Delta \omega _r)^2}, \end{aligned}$$
(6)
where \(\Delta \omega _r = \gamma P_\mathrm {out}/P_0\) and \(\gamma \equiv \omega _r/2 Q\). Therefore, assuming continuous-wave operation at frequency \(\omega _0\), defining \(\delta \equiv (\omega _0 - \omega _r)/\gamma \), \(y \equiv P_\mathrm {out}(\omega _0)/P_0\), \(x \equiv P_\mathrm {in}(\omega _0)/P_0\), we obtain
$$\begin{aligned} x = y \left[ 1 + (y - \delta )^2\right] . \end{aligned}$$
(7)
For given values of \(x\) and \(\delta \le \sqrt{3}\), there is only one value of \(y\) that satisfies 7. But when \(\delta > \sqrt{3}\), there are three values of \(y > 0\) for a given \(x\), as shown in Fig. 9 for the case \(\delta = 4\). The portion of the curve that lies between the inflection points at \(y = (2 \delta \pm \sqrt{\delta ^2 - 3})/3\) is unstable, since small perturbations in \(P_\mathrm {in}(\omega )\) cause \(\{x, y\}\) to decay to either the upper or lower branch.
Fig. 9

Curve of solutions to Eq. 7 with \(y \equiv P_\mathrm {out}(\omega _0)/P_0\), \(x \equiv P_\mathrm {in}(\omega _0)/P_0\), and \(\delta = 4\). The dashed portion of the curve that lies between the inflection points at \(y = (2 \delta \pm \sqrt{\delta ^2 - 3})/3\) is unstable, since small perturbations in \(P_\mathrm {in}(\omega )\) cause \(\{x, y\}\) to decay to either the upper or lower branch

As we might expect, our simple model—while illustrative—breaks down in real semiconductor material systems, particularly because nonlinear versions of the Kramers–Kronig relation between refraction and absorption tell us that any material that has a high value of \(n_2\) will also tend to have a high value of \(\alpha _2\), the two-photon absorption coefficient [51]. In this case, as the resonance frequency of our nonlinear optical cavity shifts, the width of the resonance will also broaden spectrally, reducing the contrast of our bistable switch. However, the relative magnitude of these two phenomena can be manipulated by manipulating optical nanostructures and engineering semiconductor bandgaps. For example, the Natomi group used a nearly-resonant band-filling nonlinearity in a InGaAsP photonic bandgap crystal cavity with \(Q/V\!\sim \hbox {85,000}\) to demonstrate a switch that required only 96 aJ—corresponding to about 740 photons in the cavity at a wavelength of 1550 nm. Of course, much work remains to be done to show that large numbers of these cavities can be operated at 10\(\times \) lower energies and then coupled into large, complex arrays, but early research in this field gives us hope that the technical obstacles can be overcome.

5 Conclusion

Research on large-scale integrated photonic components has been so productive and rapid over the last decade that we are encouraged to think grandly about applications of dozens, thousands, and even millions of coupled linear and nonlinear resonant optical devices to current and future information technology. But, in practice, it is not yet clear exactly how these radically new technologies could reach their intended markets. In this paper, we have tried to dimly outline what we believe are the primary practical challenges which must be met in order for photonics to play a significant role in classical data interconnects, quantum computing, and future speculative research in applications of quantum technologies to classical information processing. Micron-scale devices operating near the fundamental limit of light-matter interaction could enable future signal processing circuits and sensors operating at ultra-low power levels, as well as provide a platform for fundamental studies of few-particle interacting systems.

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© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.HP LaboratoriesPalo AltoUSA

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