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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 21 — Oct. 8, 2012
  • pp: 23144–23155
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Graphene–assisted critically–coupled optical ring modulator

Michele Midrio, Stefano Boscolo, Michele Moresco, Marco Romagnoli, Costantino De Angelis, Andrea Locatelli, and Antonio-Daniele Capobianco  »View Author Affiliations


Optics Express, Vol. 20, Issue 21, pp. 23144-23155 (2012)
http://dx.doi.org/10.1364/OE.20.023144


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Abstract

Graphene’s conductivity at optical frequencies can be varied upon injection of carriers. In the present paper, this effect is used to modulate losses of an optical wave traveling inside a ring cavity. This way an optical modulator based on the critical–coupling concept first introduced by Yariv can be realized. Through numerical simulations, we show that a modulator featuring a bandwidth as large as 100 GHz can be designed with switching energy in the order of few fJ per bit. Also, we show that operations with driving voltages below 1.2 volt could be obtained, thus making the proposed modulator compatible with requirements of low–voltage CMOS technology.

© 2012 OSA

1. Introduction

Optical modulators are key components for telecommunications and optical interconnections. They are among the first integrated optics devices that have been studied since the mid 1980s.

Earliest designs were based on ferroelectric materials or III–V compounds. However, the dominance of silicon as the semiconductor choice for electronics, and the need of integration of optical and electronic functionalities onto the same chip eventually shifted the research efforts towards structures based on silicon-on-insulator substrates. A list of contributions that cover some relevant steps toward the design of modern modulators can be found in refs. [1

1. A. Yariv, “Critical coupling and its control in optical waveguide–ring resonator systems,” IEEE Photon. Technol. Lett. 14, 483–485 (2002). [CrossRef]

14

14. A. Liu, L. Liao, D. Rubin, H. Nguyen, B. Ciftcioglu, Y. Chetrit, N. Izhaki, and M. Paniccia, “High–speed optical modulation based on carrier depletion in a silicon waveguide,” Opt. Express 15, 660–668 (2007). [CrossRef] [PubMed]

].

Most of the structures reported in the papers above essentially are straight waveguides in a Mach–Zehnder configuration. The key physical mechanism they rely on is the plasma dispersion effect: silicon losses are modified upon injection of free carriers. Through Kramers–Kronig relations, the refractive index of the material is modified as well. This may be used to produce a π–shift in one of the Mach-Zehnder arms, and finally obtain the desired intensity modulation. The major drawback of this scheme is in the relative large dimensions it requires. Indeed, as plasma–dispersion can only induce small variations of the refractive index, interaction lengths in the range of hundreds of microns to millimeters are needed.

In an attempt to reduce dimensions, and therefore allow for greater integration, ring–modulators were successively proposed. Plasma dispersion effect still is the key physical mechanism. However, light which travels inside a cavity may now experience the refractive index change for thousands of time. This way, even a very tiny variation of the propagation constant may shift the wavelength of resonance by an amount large enough to allow for efficient amplitude modulation.

Based on this idea, devices with footprints in the range of 10 squared microns have been demonstrated [15

15. Q. Xu, S. Manipatrumi, B. Schmidt, J. Shakya, and M. Lipson, “12.5 Gbit/s carrier–injection–based silicon microring silicon modulators,” Opt. Express 15, 430–436 (2007). [CrossRef] [PubMed]

17

17. P. Dong, R. Shafiiha, S. Liao, H. Liang, N.-N. Feng, D. Feng, G. Li, X. Zheng, A. V. Krishnamoorthy, and M. Ashgari, “Wavelength-tunable silicon microring modulator,” Opt. Express 18, 10941–10946 (2010). [CrossRef] [PubMed]

]. Owing to these very small dimensions, these devices also proved to have exceptional performance in terms of power consumption [18

18. W. A. Zortman, M. R. Watts, D. C. Trotter, R. W. Young, and A. L. Lentine, “Low-power high-speed silicon microdisk modulators,” in Proc. Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference (CLEO/QELS) (San Jose, Calif., 2004), paper CThJ4.

, 19

19. M. R. Watts, D. C. Trotter, R. W. Young, and A. L. Lentine, “Ultralow power silicon microdisk modulators and switches,” in Proc. of the 5th IEEE International Conference on Group IV Photonics (Cardiff, Wales, 2008).

]. The price to pay is the narrow bandwidth that resonances can offer. These are often limited to few GHz, and this finally brings to the need for stringent fabrication capabilities and thermal control.

More recently, a novel kind of modulator has appeared in the open literature [20

20. M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang. “A graphene-based broadband optical modulator,” Nature 474, 64–67 (2011). [CrossRef] [PubMed]

, 21

21. M. Liu, X. Yin, and X. Zhang, “Double–layer graphene optical modulator,” Nano Lett. 12, 1482–1485 (2012). [CrossRef] [PubMed]

]. It might represent a real break-through in the integrated–optics scenario, as it differs from plasma–dispersion–effect modulators in two respects.
  1. The first is the use of graphene, the single sheet of carbon atoms in an hexagonal lattice that, among many other exceptional properties, has a much higher electron mobility than silicon [22

    22. A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6, 183–191 (2007). [CrossRef] [PubMed]

    ].
  2. The second is the operation principle it exploits. Indeed, this modulator directly relies on loss modulation rather than on loss-induced refractive index variations. The striking point is that, essentially, this is “just” an electro–absorption modulator. As such, it may operate over a virtually endless optical band, but with much better performance than “conventional” electro–optic modulators in terms of electrical bandwidth.

Unfortunately, the differential loss that can be achieved by modulating graphene losses is relatively small. Values in the range of 0.1÷0.15 dB/μm were demonstrated in refs. [20

20. M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang. “A graphene-based broadband optical modulator,” Nature 474, 64–67 (2011). [CrossRef] [PubMed]

, 21

21. M. Liu, X. Yin, and X. Zhang, “Double–layer graphene optical modulator,” Nano Lett. 12, 1482–1485 (2012). [CrossRef] [PubMed]

]. This in turn means that if an extinction ratio of 6÷10 dB is looked for between the “ON” and “OFF” states of the modulator, a device length in the order of 50÷100 μm must be foreseen. As we shall illustrate below this could lead to rather severe impairments in terms of switching–energy consumption.

At this stage, one might observe that it looks like the potential of graphene in the framework of optical modulation were following the same path as that of the plasma–dispersion–effect. Indeed, a promising technology has become available, but dimensions (and energy consumption) yet are the main limiting factors. Just in the same way as it was done with plasma–dispersion–effect modulators, one might therefore wonder whether inserting graphene into a resonant ring structure could help to improve the performance or not. The aim of the present paper is that of investigating the point, to quantify the bandwidth and energy consumption that can be foreseen when designing a graphene–assisted optical ring modulator.

2. Principle of operation

We exploit the physical effect introduced in 2002 by A. Yariv: the critical coupling [1

1. A. Yariv, “Critical coupling and its control in optical waveguide–ring resonator systems,” IEEE Photon. Technol. Lett. 14, 483–485 (2002). [CrossRef]

] that can be obtained in ring structure as the one sketched in the inset of Fig. 1.

Fig. 1 Left: transmission curve as a function of the ring transmission α for t=0.8 at resonance. Inset: schematic diagram of the ring structure, with definition of the straight waveguide and ring single pass transmission, t and α, respectively. Right: the layout of the graphene–assisted optical ring modulator we have used in our simulations. The dashed area is the region where graphene is present.

For the sake of clarity we briefly review the main result of ref. [1

1. A. Yariv, “Critical coupling and its control in optical waveguide–ring resonator systems,” IEEE Photon. Technol. Lett. 14, 483–485 (2002). [CrossRef]

], and we use the same symbols as in the original paper. Letting t and α the single–pass transmission through the straight waveguide, and through the ring, respectively, the scattering matrix of the coupler reads as
[b1b2]=[tk*kt][a1a2],with|t|2+|k|2=1anda2=αb2eiφ.
(1)
At resonance (φ = m2π), the transmitted power is
T=|b1a1|2=(α|t|)2(1α|t|)2.
(2)

When ααcrit = t no power is transmitted; whereas, when α → 1, almost unitary transmission is obtained (at the wavelength of resonance). At this stage, we quote an observation made by Yariv in his paper: “If we can learn how to control α and/or t, we have a basis for a switching technology. If we can do it sufficiently rapid, we have the basis of a new type of an optical modulator”. As it has been shown by Liu et al. in [20

20. M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang. “A graphene-based broadband optical modulator,” Nature 474, 64–67 (2011). [CrossRef] [PubMed]

], rapid control of α is precisely what one can do with graphene, and this is the operation principle we want to investigate.

Let us indeed consider the left part of Fig. 1, where the transmission curve of Eq. (2) is shown for t = 0.8. Observe that for α > αcrit = t the transmission rapidly raises from zero towards one. The closer is αcrit to 1, the steeper is the raise of the curve on the right hand side of the critical coupling point.

If now the graphene’s conductivity is switched low, only curvature and/or scattering losses remain in the ring. If these are small enough, the overall single–pass ring transmission α → 1. This way, one moves to the right side of the critical coupling point of Fig. 1 and almost lossless transmission is obtained (at resonance).

This is, essentially, the procedure that we have followed to design the device with the help of numerical simulations we describe below.

2.1. Waveguide and ring geometry

Fig. 2 Left panel: Schematic diagram (not to scale) of the ring waveguide. Light blue is the silicon core, yellow is alumina and red is graphene, respectively. Silica cladding is not shown. Right panel: Power density and lines of force of the electric field of the fundamental TE mode.

3. On the conductivity of graphene at optical frequencies

Before entering into the details of the numerical simulations, we recall some basic facts concerning the values that graphene’s conductivity can attain at optical frequencies. To this end, we use a result derived in ref. [23

23. T. Stauber, N. M. R. Peres, and A. K. Geim, “Optical conductivity of graphene in the visible region of the spectrum,” Phys. Rev. B 78, 085432 (2008). [CrossRef]

], which gives the following dependence of the real part of the conductivity on frequency (ω), temperature (T) and chemical potential (μC):
σR(ω)σ02(tanhh¯ω+2μC4kBT+tanhh¯ω2μC4kBT).
(3)
Here h̄ and kB are the reduced Planck’s and Boltzmann constants, respectively, and
σ0=e24h¯6.0853×105Siemens.
(4)
with e the electron charge.

A plot of Eq. (3) is reported in the left panel of Fig. 3 for T=300K and chemical potential μC equal to 0.3 and 0.5 eV with solid and dashed lines, respectively. The vertical line at 0.8 eV shows the conductivity for photons having free-space wavelength in the vicinity of 1550 nm. Observe that at these wavelengths, a modest shift of the chemical potential from 0.3 to 0.5 eV allows to span the conductivity from 0.02σ0 to 0.98 σ0, i.e. over the almost entire range of values it may attain. A decrease of the chemical potential below 0.3 eV shifts the curve towards the left side, eventually yielding an almost constant value of conductivity σ = σ0 for wavelengths around 1550 nm. This in turn means that the device should be operated by applying a bias–voltage to reach the 0.3 eV chemical potential, and switching between high and low conductivities can successively be performed with a 0.2 eV increase of the chemical potential.

Fig. 3 Left panel: Conductivity of graphene for chemical potential μC = 0.3 eV (solid line) and μC = 0.5 eV (dashed line) for T=300 K. The vertical line highlights value of the conductivity for photons with free-space wavelength equal to 1550 nm. Right panel: relation between carrier density and chemical potential for an isolated sheet of graphene.

Also observe that Eq. (3) gives a bulk conductivity with the dimensions of Siemens. Whereas, in our simulations, graphene is modeled as a brick with a finite thickness (hGraphene = 0.34nm), which needs to be assigned a Siemens/meter conductivity. Therefore, following the method used in ref. [24

24. A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332, 1291–1294 (2008). [CrossRef]

], when we want to simulate highly or poorly conductive graphene, we set the values as follows:
σHigh=0.98σ0hGraphene1.75×105S/m,σLow=0.02σ0hGraphene3.58×103S/m.
With the same procedure, we also evaluated the relative dielectric constant of graphene in both the low and high conductivity states. Using the expression for the complex interband conductivity of ref. [25

25. G. W. Hanson, “Dyadic Green’s function and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. 103, 064302 (2008). [CrossRef]

], we found that for the high and low conductivity cases εHigh ≃ 10.44 and εLow ≃ 11.91, respectively. Notice that, as we will show below, given the very reduced thickness of graphene, a change in the value of the dielectric constant of graphene has a very minor effect on the propagation constant of the traveling mode.

We close the present section by also giving an estimate of the carrier densities (nS) needed to reach the above values of chemical potentials. These densities will be needed later to evaluate the energy required for switching. To this end we recall that, at least for an isolated sheet of graphene, Hanson has derived the following relation between the chemical potential and the carrier density [25

25. G. W. Hanson, “Dyadic Green’s function and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. 103, 064302 (2008). [CrossRef]

]
ns=2πh¯2vF20+ε[fd(ε)fd(ε+2μC)]dε,
(5)
where fd(ε) = (e(εμC)/kBT + 1)−1 is the Fermi–Dirac distribution and vF ≃ 9.5 × 105 m/s is the Fermi velocity. A plot of Eq. (5) is shown in the right panel of Fig. 3. It is observed that chemical potentials equal to 0.3 and 0.5 eV can be obtained with carrier densities roughly equal to ρOFF = 7.5 × 1012 cm−2 and ρON = 20.5 × 1012 cm−2, respectively. Use of subscripts OFF and ON will become clear later on.

4. Numerical simulations and results

A preliminary analysis was conducted to characterize the propagation constant of the mode in the waveguide (a straight one), either in the absence or in the presence of graphene, both in the high and low conductivity states.

It turned out that, for free space–wavelength λ0 ≃ 1550nm, the mode has the following effective indices:
Nographeneneff=2.2576Graphenewithlowconductivity(σ=σLow,ε=εLow)neff=2.2543i8.42×105Graphenewithhighconductivity(σ=σHigh,ε=εHigh)neff=2.2546i3.79×103

It is observed that graphene’s effect is mostly on the imaginary part of the effective index, i.e. on losses. Indeed the mode profiles in the three cases above were almost undistinguishable. This in turn means that no scattering from mode–mismatch is expected when the field crosses the interfaces between the ring sections with and without graphene. Indeed, both because of its extremely tiny thickness, and of relatively small conductivity, graphene acts as a sort of first–order perturbative term: mainly, it introduces an imaginary part in the propagation constant without altering the mode profile in a significant way.

A second preliminary analysis was also conducted in order to evaluate the bending losses as a function of radius of curvature. The results of this analysis are reported in Fig. 4 for rings in the absence (solid line) and presence (dashed line) of graphene, respectively. In these simulations, the conductivity of graphene, when present, is set equal to σHigh. The abscissa in the figure has to be intended as the average ring radius (i.e. the internal ring radius plus half the ring waveguide width).

Fig. 4 Left panel: a plot from CST main window, showing the structure used in the simulation performed to compute bending losses along with the mesh. The ocher, blu and red regions are the silicon core, alumina coating and graphene layers, respectively. Though barely observable, two layers of graphene are present. Right panel: bending loss in the absence and presence of the two graphene layers (solid and dashed lines, respectively). The graphene conductivity is set equal to σHigh.

In the absence of graphene, the bending losses decrease towards zero for increasing radii, as expected. Whereas, the behavior in the presence of graphene is more interesting. For large radii, the curve tends to a limiting value, in the order of 0.13 dB/μm. This value exactly corresponds to the loss of the mode in the straight waveguide we have discussed above, when we found the effective index neff = 2.2546 – i3.79 × 10−3. We conclude that the curvature loss is negligible with respect to graphene absorption for curvature radii greater than about 3 μm. For smaller radii, bending losses tend to become so large that they shadow graphene’s absorption.

Since the critical coupling is the key factor in the proposed device, and since this strongly depends on ring losses, we conclude the presentation of these preliminary results by stressing one more time the issue of scattering. We recall that we used a very fine tetrahedral mesh throughout the whole set of our simulations (see the left panel of Fig. 4 for an example of such mesh). This way, we removed the roughness of sidewalls, eventually nulling any scattering loss. One might wonder whether this is a too ideal case, so that our numerical result will not fit a real experiment. Actually, we remark that in a real device the losses due to scattering are in the order of 2×10−4 dB/μm or less [27

27. T. Barwicz and H. A. Haus, “Three-dimensional analysis of scattering losses due to sidewall roughness in microphotonic waveguides,” J. Lightwave Technol. 23, 2719–2732 (2005). [CrossRef]

29

29. M. Moresco, M. Romagnoli, S. Boscolo, and M. Midrio, “Method for Characterization of Si waveguide propagation loss,” submitted to Opt. Express (2012).

]. Therefore, if we look at the solid curve in Fig. 4 we see that scattering losses can be neglected with respect to bending losses as long as the radius of curvature is smaller than 9 μm. Since in all the simulations we performed the average ring radius was set equal to r = 5μm, we conclude that having numerically nulled the scattering losses is not a limit of our simulations indeed.

We finally come to the details of the numerical simulations we performed to design the graphene–assisted critically–coupled ring resonator. With the graphene in the high–conductivity state, we varied the gap values from 80 to 160 nm at steps of 10 nm. For each value of gap, we evaluated the ring transmission when we also varied the extension of the graphene region through the angle ψGraphene (see Fig. 1). The ring transmission swept from over to under–coupling conditions for increasing ψGraphene. The critical–coupling could be easily determined by finding the transmission minimum vs. ψGraphene. For instance, it turned out that when the gap g = 100 nm, the critical coupling was reached for ψGraphene = 116 degrees. As it is shown by the solid line in the left panel of Fig. 5, for such a case the transmission dropped to almost −45 dB at the resonance frequency of 193.66 THz (corresponding to the free–space wavelength λ0 ≃ 1550nm).

Fig. 5 Left panel: Transmission in a critically–coupled ring with gap g = 100 nm, average radius equal to 5 μm for graphene conductivity σ = σHigh (solid line) and σ = σLow (dashed curve). Right panel: numerically computed 6 dB bandwidth versus the gap distance g.

An estimate of losses in the ring is useful at this stage. Roughly, it is given by the sum of loss due to graphene absorption (in the region where the graphene is present) plus loss due to bending in the rest of the ring. This accounts to
AHigh0.13dB/μmLGraphene+1.4×103dB/μm(LRingLGraphene)1.34dB,
(6)
where we used LRing = 10π μm, LGraphene = 116/360LRing, and we took the value of bending losses for the 5μm average radius from Fig. 4. In linear scale, this value would correspond to a single–pass ring loss α = αcrit ≃ 0.73 in Eq. (2). Recall that at the critical condition, also t = αcrit.

If the graphene’s conductivity is switched to the low state, the ring loss changes due to the reduced graphene absorption. Indeed, we have seen above that the effective index of the mode which travels in a waveguide coated by graphene with σ =σLow is neff = 2.2543−i8.42×10−5, corresponding to an absorption rate of about 2.8 × 10−3 dB/μm. Therefore, the single–round–trip loss now is
ALow2.8×103dB/μmLGraphene+1.4×103dB/μm(LRingLGraphene)0.058dB
(7)
In linear scale, this would account to α = 0.987. Inserting this value into Eq. (2) with t = αcrit = 0.73 one get T = 0.84: at the frequency of resonance the transmission loss is expected to be limited to about 0.75 dB.

When we run the numerical simulation of this case we found that the above prediction was matched almost perfectly. Indeed, we observed that:
  • the frequency of resonance changed by about 10 GHz with respect to the graphene’s high–conductivity case. This in turn confirms that graphene’s conductivity has a very minor effect on the phase constant of the mode;
  • at resonance, the “dip” in the transmission curve was equal to 0.48 dB for the low–conductivity case. This curve is shown in the left panel of Fig. 5 with a dashed line. We acknowledge that neither the frequency of resonance nor the depth of the dip are easily inspected in the figure due to the large scale we used. Nevertheless, we preferred to keep both the transmission curves on the same plot as this allows to verify that the proposed device might indeed be regarded as a spectacular optical modulator. The two transmission curves are those of an almost ideal device: a large absorption in the “OFF” state, along with a very reduced insertion loss in the “ON” state, over a remarkably large bandwidth. For instance, if a 6 dB contrast ratio is looked for, a bandwidth as large as 119 GHz would be obtained.

We repeated the same analysis for all the gap values between 80 and 160 nm (at steps of 10 nm). The available 6 dB bandwidth that we computed numerically is shown in the right panel of Fig. 5 as a function of the gap distance g. In all the cases, the insertion losses we measured never exceeded the level of 0.5 dB.

5. An estimate of the energy required for switch

Graphene’s conductivity can be changed by adding (or removing) electrical charges on it. This can be done by realizing a capacitor in which one or both the armatures are formed by graphene layers. Liu et coauthors have shown experimentally both the configurations (see Fig. 1 of ref. [20

20. M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang. “A graphene-based broadband optical modulator,” Nature 474, 64–67 (2011). [CrossRef] [PubMed]

] and Fig. 1 of ref. [21

21. M. Liu, X. Yin, and X. Zhang, “Double–layer graphene optical modulator,” Nano Lett. 12, 1482–1485 (2012). [CrossRef] [PubMed]

] for the single and double layer configurations, respectively). Also, they suggested that the double–layer configuration is the best one, as this allows to avoid use of lossy doped silicon to form one of the electrical contacts. In addition, the presence of two graphene layers doubles the effects of the graphene itself.

Observe in our case that lower values of graphene’s conductivity are obtained when the charge concentration is raised (see Fig. 3). Also observe in the left panel of Fig. 5 that low graphene’s conductivity brings the modulator to its “ON” (i.e. transmitting) state. The reverse applies by lowering the charge concentration. This explains the choice of subscripts we made below Eq. (5).

The structure we have used in our simulations follows Liu’s suggestion: two sheets of graphene sandwich a layer of alumina and form the capacitor (see paragraph 2.1). Estimate of the energy required for switching is a trivial application of the capacitor paradox problem. For a pseudo–random bit–sequence this yields
EPRBS=12QON2QOFF22C
The 1/2 factor accounts for the fact that in a random sequence, switch from bit “0” to bit “1” (or viceversa) only happens one half of the times. The amount of charge required to bring the modulator to the “ON” state is
QON=ρONeS,S=ψGrapheneRAveW
with e the electron charge and S the extension of the area covered with graphene with RAve the ring average radius, and W the ring waveguide width, respectively. An analogous expression holds for the “OFF” state too. Neglecting fringing effects, the capacitance of the graphene–alumina–graphene structure is
C=ε0εrel,dielShdiel
with εrel,diel and hdiel the relative dielectric constant and thickness of the dielectric oxide between the graphene sheets respectively. The voltage required for charging is
VON,OFF=QON,OFFC=ρON,OFFeε0εrel,dielhdiel
(8)
and the energy required for switching
EPRBS=12(ρON2ρOFF2)e2hdielS2ε0εrel,diel.
(9)
Some interesting relationships in Eq. (9) are worth of mention:
  • Graphene’s surface S. The charging voltage is independent on S. Whereas, the switching energy linearly scales with it. In this respect, the present configuration might turn out to be advantageous with respect to those of refs. [20

    20. M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang. “A graphene-based broadband optical modulator,” Nature 474, 64–67 (2011). [CrossRef] [PubMed]

    , 21

    21. M. Liu, X. Yin, and X. Zhang, “Double–layer graphene optical modulator,” Nano Lett. 12, 1482–1485 (2012). [CrossRef] [PubMed]

    ] because of the reduced amount of graphene we needed to use here.
  • Oxide’s thickness hdiel. Both the voltage and the energy scale linearly with it.
  • Oxide’s dielectric constant εrel,diel. Both the voltage and the energy scale inversely with it.

A plot of Eq. (9) for oxide being constituted by a 7 nm thick layer of alumina (εrel,diel = 10 at microwave frequencies) is shown in Fig. 6. Also shown in the figure is the dependence of the energy required for switching as a function of the available bandwidth. A nice linear fit is found, with a pace of about 5 fJ for each 10 GHz increase of the bandwidth.

Fig. 6 Left panel: Estimate of the energy required for switching with a pseudo–random bit–sequence (PRBS) in a ring modulator with oxide being constituted by a 7 nm thick layer of alumina (εrel,diel = 10 at microwaves). Right panel: numerically evaluated dependence of the required switching energy versus the available bandwidth.

It is also observed that energies below the level of 100 fJ/bit are estimated for all the values of gap. In addition, from Eq. (8), it is also found that voltages equal to 0.9 and 2.6V would be needed to drive the modulator to the OFF and ON states, respectively. However, it is remarked that, according to the parameters’ list of dependencies we have listed above, these values could be further reduced if an oxide layer with either thinner dimensions or higher refractive index were used. For instance, if a 3 nm thick oxide were realized, the energy required for switching would drop to the range of 10÷30 fJ/bit, the pace of growth with bandwidth extension would be reduced to about 2 fJ for additional 10 GHz and, most important, the voltages needed to drive the modulator would fall below 1.2 V, i.e. it would become compatible with state of the art low–voltage CMOS requirements.

A rough estimate of heating effects can also be made. To this end, we first notice that graphene has been proved to be an excellent thermal conductor [30

30. A. A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao, and C. N. Lau, “Superior thermal conductivity of single-layer graphene,” Nano Lett. 8, 902–907 (2008). [CrossRef] [PubMed]

], while the silica cladding which surrounds the whole structure is a thermal insulator. As a consequence, any heating due to dissipation in the graphene layers is transferred very efficiently to the silicon core, which eventually acts as a sort of heat sink for the whole structure.

In the OFF state of the modulator, the light injected in the ring is absorbed by graphene. In order to estimate the change of temperature due to this absorption, we start from the experimental results reported by DeRose and coauthors in ref. [31

31. C. T. DeRose, M. R. Watts, D. C. Trotter, D. L. Luck, G. N. Nielson, and R. W. Young, “Silicon microring modulator with integrated heater and temperature sensor for thermal control,” in Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference 2010, paper CThJ3.

]. They have shown that the response of a silicon microring can be tuned thermally by dissipating electrical power. The rate of tuning is in the order of 4.4 μW/GHz. Hence, if the ring is fed with a 1 mW continuous–wave laser light, and since this light is absorbed by graphene only for one half of the time, a shift in the the resonant frequency in the order of 125 GHz may be expected when comparing operations in the “cold” and “heated” graphene cases. Remark that this is a thermal effect. That is, the frequency shift can not and does not follow the modulation repetition rate, and only depends on the average level of dissipated power. The frequency of operation of the modulator sets at a steady value after initial thermalization has been completed.

Physically, the 125 GHz shift of the resonant frequency is due to a change of silicon refractive index. One may assume that
ΔnSinSiΔffΔnSi0.125THz200THz122×103.
(10)
Since the thermo-optic coefficient of silicon is in the order of 2 × 10−4 K−1, a temperature change in the order of ten degrees is therefore expected in silicon (and graphene) when operating the modulator. With respect to room temperatures, this represent less that a 5% change, so that the effect on both the Fermi–Dirac distribution and the complex conductivity of graphene is expected to be rather modest. Presence of cooling devices outside the modulator might further decrease this effect.

6. Conclusion

We have proposed a novel device which exploits controllability of graphene’s optical conductivity to realize a critically–coupled ring–modulator based on the physical mechanism first introduced by Yariv in ref. [1

1. A. Yariv, “Critical coupling and its control in optical waveguide–ring resonator systems,” IEEE Photon. Technol. Lett. 14, 483–485 (2002). [CrossRef]

]. The modulator is designed to be at the critical condition for high graphene’s conductivity. In this state, no light is transmitted from the input to the output port. Whereas, when graphene’s conductivity is switched to its low state, propagation in the ring becomes almost lossless, and a dramatic increase of the transmitted power is observed.

Notice that we deliberately added losses to the propagation in the ring through insertion of graphene. This way, the overall Q–factor of the ring turned out to be rather low and no sharp resonances were formed. This is the major difference with respect to ring modulators based on the plasma–dispersion–induced effect. Indeed, in those modulators, ON–OFF switching is obtained by changing the refractive index in the ring, causing this way a detune of the frequency of resonance. However, since the change of refractive index one may reasonably obtain is small, a sharp transmission resonance is needed in order to realize a respectable contrast ratio. Typical values of bandwidth that can be obtained in this way are in the order of 10÷20 GHz. On the contrary, our design does not rely on frequency detuning. This in turn means that it may work with broad resonances too, as we proved with the help of numerical simulations. Bandwidths in the range of 100 GHz were found, though larger ones might actually be obtained if gap smaller than 100 nm could be manufactured.

Observe that a bandwidth in the order of 100 GHz is not merely a valuable figure of merit because of the transmission rates this modulator could work at. Actually, such a large bandwidth could allow to get rid of temperature control circuits that are usually needed to lock a sharp resonance to the right wavelength. Keeping in mind that the power consumption of these control circuits greatly exceeds those of the modulators themselves, the present device could eventually lead to a dramatic reduction of the energy required for switching which, in our case, could really approach the level of few fJ per bit.

References and links

1.

A. Yariv, “Critical coupling and its control in optical waveguide–ring resonator systems,” IEEE Photon. Technol. Lett. 14, 483–485 (2002). [CrossRef]

2.

R. A. Soref and B. R. Bennett, “Kramers–Kronig analysis of electro–optical switching in silicon,” Proc. SPIE 704, 32–37 (1987).

3.

J. P. Lorenzo and R. A. Soref, “1.3 μm electro–optic silicon switch,” J. Appl. Phys. 51, 6–8 (1987).

4.

L. Friedman, R. A. Soref, and J. P. Lorenzo, “Silicon double–injection electro–optic modulator with junction gate control,” J. Appl. Phys. 63, 1831–1839 (1988). [CrossRef]

5.

S. R. Giguere, L. Friedman, R. A. Soref, and J. P. Lorenzo, “Simulation studies of silicon electro–optic waveguide devices,” J. Appl. Phys. 68, 4964–4970 (1990). [CrossRef]

6.

G. V. Treyez, P. G. May, and J. M. Halbout, “Silicon optical modulators at 1.3 micrometer based on free–carrier absorption,” IEEE Electron. Dev. Lett. 12, 276–278 (1991). [CrossRef]

7.

G. V. Treyez, P. G. May, and J. M. Halbout, “Silicon Mach–Zehnder waveguide inteferometers based on the plasma dispersion effect,” Appl. Phys. Lett. 59, 771–773 (1991). [CrossRef]

8.

U. Fischer, B. Schuppert, and K. Petermann, “Integrated optical switches in silicon based on SiGe–waveguides,” IEEE Photon. Technol. Lett. 5, 785–787 (1993). [CrossRef]

9.

H. C. Huang and T. C. Lo, “Simulation and analysis of silicon electro–optic modulators utilizing the carrier–dispersion effect and impact–ionization mechanism,” J. Appl. Phys. 74, 1521–1582 (1993). [CrossRef]

10.

A. Cutolo, M. Iodice, P. Spirito, and L. Zeni, “Silicon electro–optic modulator based on a three-terminal device integrated in a low–loss single–mode SOI waveguide,” J. Lightwave Technol. 15, 505–518 (1997). [CrossRef]

11.

A. Sciuto, S. Libertino, A. Alessandria, S. Coffa, and G. Coppola, “Design, fabrication and testing of an integrated Si–based light modulator,” J. Lightwave Technol. 21, 228–235 (2003). [CrossRef]

12.

C. A. Barrios, V. R. de Almeida, and M. Lipson, “Low–power–consumption short–length and high–modulation–depth silicon electrooptic modulator,” J. Lightwave Technol. 21, 1089–1098 (2003). [CrossRef]

13.

A. Liu, R. Jones, L. Liao, D. Samara–Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high–speed silicon optical modulator based on a metal–oxide semiconductor capacitor,” Nature 427, 615–618 (2004). [CrossRef] [PubMed]

14.

A. Liu, L. Liao, D. Rubin, H. Nguyen, B. Ciftcioglu, Y. Chetrit, N. Izhaki, and M. Paniccia, “High–speed optical modulation based on carrier depletion in a silicon waveguide,” Opt. Express 15, 660–668 (2007). [CrossRef] [PubMed]

15.

Q. Xu, S. Manipatrumi, B. Schmidt, J. Shakya, and M. Lipson, “12.5 Gbit/s carrier–injection–based silicon microring silicon modulators,” Opt. Express 15, 430–436 (2007). [CrossRef] [PubMed]

16.

Q. Xu, D. Fattal, and R. G. Beausoleil, “Silicon microring resonators with 1.5-μm radius,” Opt. Express 16, 4309–4315 (2008). [CrossRef] [PubMed]

17.

P. Dong, R. Shafiiha, S. Liao, H. Liang, N.-N. Feng, D. Feng, G. Li, X. Zheng, A. V. Krishnamoorthy, and M. Ashgari, “Wavelength-tunable silicon microring modulator,” Opt. Express 18, 10941–10946 (2010). [CrossRef] [PubMed]

18.

W. A. Zortman, M. R. Watts, D. C. Trotter, R. W. Young, and A. L. Lentine, “Low-power high-speed silicon microdisk modulators,” in Proc. Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference (CLEO/QELS) (San Jose, Calif., 2004), paper CThJ4.

19.

M. R. Watts, D. C. Trotter, R. W. Young, and A. L. Lentine, “Ultralow power silicon microdisk modulators and switches,” in Proc. of the 5th IEEE International Conference on Group IV Photonics (Cardiff, Wales, 2008).

20.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang. “A graphene-based broadband optical modulator,” Nature 474, 64–67 (2011). [CrossRef] [PubMed]

21.

M. Liu, X. Yin, and X. Zhang, “Double–layer graphene optical modulator,” Nano Lett. 12, 1482–1485 (2012). [CrossRef] [PubMed]

22.

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6, 183–191 (2007). [CrossRef] [PubMed]

23.

T. Stauber, N. M. R. Peres, and A. K. Geim, “Optical conductivity of graphene in the visible region of the spectrum,” Phys. Rev. B 78, 085432 (2008). [CrossRef]

24.

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332, 1291–1294 (2008). [CrossRef]

25.

G. W. Hanson, “Dyadic Green’s function and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. 103, 064302 (2008). [CrossRef]

26.

CST Microwave Studio 2012. Darmstadt, Germany.

27.

T. Barwicz and H. A. Haus, “Three-dimensional analysis of scattering losses due to sidewall roughness in microphotonic waveguides,” J. Lightwave Technol. 23, 2719–2732 (2005). [CrossRef]

28.

K. K. Lee, D. R. Lim, L. C. Kimerling, J. Shin, and F. Cerrina, “Fabrication of ultralow-loss Si/SiO2 waveguides by roughness reduction,” Opt. Lett. 23, 1888–1890 (2001). [CrossRef]

29.

M. Moresco, M. Romagnoli, S. Boscolo, and M. Midrio, “Method for Characterization of Si waveguide propagation loss,” submitted to Opt. Express (2012).

30.

A. A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao, and C. N. Lau, “Superior thermal conductivity of single-layer graphene,” Nano Lett. 8, 902–907 (2008). [CrossRef] [PubMed]

31.

C. T. DeRose, M. R. Watts, D. C. Trotter, D. L. Luck, G. N. Nielson, and R. W. Young, “Silicon microring modulator with integrated heater and temperature sensor for thermal control,” in Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference 2010, paper CThJ3.

OCIS Codes
(060.4510) Fiber optics and optical communications : Optical communications
(160.3130) Materials : Integrated optics materials
(250.3140) Optoelectronics : Integrated optoelectronic circuits
(250.4110) Optoelectronics : Modulators

ToC Category:
Optoelectronics

History
Original Manuscript: June 1, 2012
Revised Manuscript: July 13, 2012
Manuscript Accepted: August 6, 2012
Published: September 24, 2012

Citation
Michele Midrio, Stefano Boscolo, Michele Moresco, Marco Romagnoli, Costantino De Angelis, Andrea Locatelli, and Antonio-Daniele Capobianco, "Graphene–assisted critically–coupled optical ring modulator," Opt. Express 20, 23144-23155 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-21-23144


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References

  1. A. Yariv, “Critical coupling and its control in optical waveguide–ring resonator systems,” IEEE Photon. Technol. Lett.14, 483–485 (2002). [CrossRef]
  2. R. A. Soref and B. R. Bennett, “Kramers–Kronig analysis of electro–optical switching in silicon,” Proc. SPIE704, 32–37 (1987).
  3. J. P. Lorenzo and R. A. Soref, “1.3 μm electro–optic silicon switch,” J. Appl. Phys.51, 6–8 (1987).
  4. L. Friedman, R. A. Soref, and J. P. Lorenzo, “Silicon double–injection electro–optic modulator with junction gate control,” J. Appl. Phys.63, 1831–1839 (1988). [CrossRef]
  5. S. R. Giguere, L. Friedman, R. A. Soref, and J. P. Lorenzo, “Simulation studies of silicon electro–optic waveguide devices,” J. Appl. Phys.68, 4964–4970 (1990). [CrossRef]
  6. G. V. Treyez, P. G. May, and J. M. Halbout, “Silicon optical modulators at 1.3 micrometer based on free–carrier absorption,” IEEE Electron. Dev. Lett.12, 276–278 (1991). [CrossRef]
  7. G. V. Treyez, P. G. May, and J. M. Halbout, “Silicon Mach–Zehnder waveguide inteferometers based on the plasma dispersion effect,” Appl. Phys. Lett.59, 771–773 (1991). [CrossRef]
  8. U. Fischer, B. Schuppert, and K. Petermann, “Integrated optical switches in silicon based on SiGe–waveguides,” IEEE Photon. Technol. Lett.5, 785–787 (1993). [CrossRef]
  9. H. C. Huang and T. C. Lo, “Simulation and analysis of silicon electro–optic modulators utilizing the carrier–dispersion effect and impact–ionization mechanism,” J. Appl. Phys.74, 1521–1582 (1993). [CrossRef]
  10. A. Cutolo, M. Iodice, P. Spirito, and L. Zeni, “Silicon electro–optic modulator based on a three-terminal device integrated in a low–loss single–mode SOI waveguide,” J. Lightwave Technol.15, 505–518 (1997). [CrossRef]
  11. A. Sciuto, S. Libertino, A. Alessandria, S. Coffa, and G. Coppola, “Design, fabrication and testing of an integrated Si–based light modulator,” J. Lightwave Technol.21, 228–235 (2003). [CrossRef]
  12. C. A. Barrios, V. R. de Almeida, and M. Lipson, “Low–power–consumption short–length and high–modulation–depth silicon electrooptic modulator,” J. Lightwave Technol.21, 1089–1098 (2003). [CrossRef]
  13. A. Liu, R. Jones, L. Liao, D. Samara–Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high–speed silicon optical modulator based on a metal–oxide semiconductor capacitor,” Nature427, 615–618 (2004). [CrossRef] [PubMed]
  14. A. Liu, L. Liao, D. Rubin, H. Nguyen, B. Ciftcioglu, Y. Chetrit, N. Izhaki, and M. Paniccia, “High–speed optical modulation based on carrier depletion in a silicon waveguide,” Opt. Express15, 660–668 (2007). [CrossRef] [PubMed]
  15. Q. Xu, S. Manipatrumi, B. Schmidt, J. Shakya, and M. Lipson, “12.5 Gbit/s carrier–injection–based silicon microring silicon modulators,” Opt. Express15, 430–436 (2007). [CrossRef] [PubMed]
  16. Q. Xu, D. Fattal, and R. G. Beausoleil, “Silicon microring resonators with 1.5-μm radius,” Opt. Express16, 4309–4315 (2008). [CrossRef] [PubMed]
  17. P. Dong, R. Shafiiha, S. Liao, H. Liang, N.-N. Feng, D. Feng, G. Li, X. Zheng, A. V. Krishnamoorthy, and M. Ashgari, “Wavelength-tunable silicon microring modulator,” Opt. Express18, 10941–10946 (2010). [CrossRef] [PubMed]
  18. W. A. Zortman, M. R. Watts, D. C. Trotter, R. W. Young, and A. L. Lentine, “Low-power high-speed silicon microdisk modulators,” in Proc. Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference (CLEO/QELS) (San Jose, Calif., 2004), paper CThJ4.
  19. M. R. Watts, D. C. Trotter, R. W. Young, and A. L. Lentine, “Ultralow power silicon microdisk modulators and switches,” in Proc. of the 5th IEEE International Conference on Group IV Photonics (Cardiff, Wales, 2008).
  20. M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang. “A graphene-based broadband optical modulator,” Nature474, 64–67 (2011). [CrossRef] [PubMed]
  21. M. Liu, X. Yin, and X. Zhang, “Double–layer graphene optical modulator,” Nano Lett.12, 1482–1485 (2012). [CrossRef] [PubMed]
  22. A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater.6, 183–191 (2007). [CrossRef] [PubMed]
  23. T. Stauber, N. M. R. Peres, and A. K. Geim, “Optical conductivity of graphene in the visible region of the spectrum,” Phys. Rev. B78, 085432 (2008). [CrossRef]
  24. A. Vakil and N. Engheta, “Transformation optics using graphene,” Science332, 1291–1294 (2008). [CrossRef]
  25. G. W. Hanson, “Dyadic Green’s function and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys.103, 064302 (2008). [CrossRef]
  26. CST Microwave Studio 2012. Darmstadt, Germany.
  27. T. Barwicz and H. A. Haus, “Three-dimensional analysis of scattering losses due to sidewall roughness in microphotonic waveguides,” J. Lightwave Technol.23, 2719–2732 (2005). [CrossRef]
  28. K. K. Lee, D. R. Lim, L. C. Kimerling, J. Shin, and F. Cerrina, “Fabrication of ultralow-loss Si/SiO2 waveguides by roughness reduction,” Opt. Lett.23, 1888–1890 (2001). [CrossRef]
  29. M. Moresco, M. Romagnoli, S. Boscolo, and M. Midrio, “Method for Characterization of Si waveguide propagation loss,” submitted to Opt. Express (2012).
  30. A. A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao, and C. N. Lau, “Superior thermal conductivity of single-layer graphene,” Nano Lett.8, 902–907 (2008). [CrossRef] [PubMed]
  31. C. T. DeRose, M. R. Watts, D. C. Trotter, D. L. Luck, G. N. Nielson, and R. W. Young, “Silicon microring modulator with integrated heater and temperature sensor for thermal control,” in Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference 2010, paper CThJ3.

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