Charge and field driven integrated optical modulators: comparative analysis: opinion

Jacob B. Khurgin; Volker J. Sorger; Rubab Amin

doi:10.1364/OME.452872

Optical modulators are key components of any photonic system. Historically, for many years, electro-optic (i.e. based on refractive index change) modulators (EOM) made of LiNbO₃ (LN) have been devices-of-choice when it comes to data speed [1,2]. All the attractive features of LN EOM– high speed, low insertion loss, decent efficiency, wide optical (spectral) bandwidth, absence of frequency chirp, come at the expense of a major drawback – the size. This is why for about 20 years many photonic integrated circuits (PIC) have been using electro-absorption modulators (EAM) based on quantum confined Stark effects (QCSE) in III-V quantum wells [3,4]. QCSE (and similar Franz–Keldysh (FK) effect [5]) EAMs are more compact since they use thin epitaxial layers rather than diffused waveguides, while trading in the versatility of EOMs which are capable of supporting a variety of modulation formats. Thus, EOMs and EAMs have been co-existing, each in its own niche up until the advent of silicon photonics; which demands modulators to be integrated on chip platforms. This has been first achieved by integration of III-V devices on Si [6], and later by the development of thin film LN technology [7,8]. But integration on Si platform requires additional fabrication steps, and, while the size of thin film LN modulators is reduced, further size reduction is required in next generation modulators [9,10]. For this reason, the dominance of LN and QCSE modulators has been first challenged by Si modulators based on depletion/accumulation of carriers [11,12], and more recently by modulators based on emerging materials including transparent conductive oxides (TCO), i.e. indium tin oxide (ITO) [13–16] and 2D materials [17,18], such as graphene [19–21] and transition metal di-chalcogenides (TMDC) [22,23]. In all these devices the modulating action is due to injection of carriers into the active material itself, rather than simply applying an electric field across the active layer. We shall refer to them as charge driven (CD), rather than field driven (FD) modulators. Often CD modulators are used in conjunction with plasmonic techniques that allows a tighter, subwavelength optical confinement.

There is a healthy number of literature on EOM and EAM modulators [24–27], yet it appears that there exists no direct and all-encompassing comparison of the performance of different materials and techniques. In fact, modulator performance depends not only on the active material and modulation technique (CD or FD), but also on geometry, electrical circuitry, and various extrinsic enhancement techniques (e.g. microresonators, photonic crystals, etc.). Therefore, it is difficult to separate pure material factors as single stand-alone parameters, especially when comparing very different modulator setups. Building on our prior work [28,29], we develop such single metrics, for both EOMs and EAMs to perform a holistic comparison. The conclusions are thought-provoking and unequivocal; FD modulators are superior when it comes to conventional waveguide dimension with thicknesses measured in 100’s of nm, whereas in smaller (usually plasmonic-assisted waveguides), CD modulators’ performance truly shines.

Two classes of modulators are shown in Fig. 1; the CD modulator (Fig.1a) incorporates an active region and a control gate. The active region is shown schematically as a multilayer structure with N ≥1 active layers, which may be monolayers of graphene, or other 2D material, semiconductor quantum wells (QWs), or layers of quantum dots (QDs). Even if the modulating material is three-dimensional, such as ITO, one can still use the ‘layered picture’. For the sake of simplicity in this illustration we consider the two level entity, such as QDs or excitonic transition in TMDC. The absorption α and index n in the absence of injected charge is highlighted in Fig.1c. When charge Q is injected into the active region, the transition gets saturated. This leads to a reduction in both α at the peak wavelength λ₀ and n at λ₁>λ₀, appropriately chosen to avoid excessive absorption (Fig.1d). Thus, either amplitude modulation at λ₀ or phase modulation at λ₁ is achieved.

Fig. 1. Comparison of fundamental principles of (a) electrical charge (Q)-driven (CD) modulators vs (b) Field (E)-driven (FD) design setup. (c) Absorption, α (red) and index, n (blue) in the absence of charge/voltage in a two-level model. (d) Changes (Δα and Δn) imposed by injected charge in CD modulators; and (e) by applied field in FD modulator. (f) Changes imposed by charge or field as absorption edge shifts in the modulators with wide band absorption. (g)Intuitive picture of modulation – each injected carrier is represented as a perfect absorber ‘disc’ with area, σ_a or a half-wave plate ‘disc’ with area, σ_Φ – when the effective area S_eff is fully covered, full modulation is achieved. (h) Sketch of the effect of scaling down on switching energy, U_sw for CD and FD modulators on a log-log scale. The ratio of length, L to the thickness, t (i.e. capacitance and modulation speed) is kept constant. The insertion loss is also presented.

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In the FD modulator (Fig.1b) the charge appears only on two electrodes outside the active region, and the switching is performed by the field E = Q/ɛ₀ɛ_sWL where ɛ_s is static permittivity, W is the width and L is the length. A plane capacitor geometry is assumed, but in practical devices W and t should be understood as effective dimensions that take into account the fringe fields. In the absence of an applied voltage, i.e. plate charge Q, α at wavelength λ₀ (below the peak) is small, same as shown in Fig.1c. When the charge is generated the absorption peak moves towards longer wavelength – Stark or FK effect (Fig.1e), increasing α at λ₀ and n at λ₁>λ₀.

In case of semiconductors and graphene, the band-to-band absorption is continuous, but the injection/depletion of carriers or Stark effect only change the absorption in a relatively narrow spectral region near the edge (Fig.1f). Hence, the picture of Fig. 1(d, e) remains valid as long as one is interested in changes of α and n. If one considers a case when free carriers are injected (e.g. ITO or Si [30]) this picture still holds as long as we assume the resonance frequency being 0.

The key parameter that defines the modulator performance is the total charge Q_sw injected into the active layer in CD modulators or applied to the electrodes of FD modulators and required to attain switching. For amplitude-altering modulators (EAM), switching can be defined by a certain change in absorption, which in [28] was chosen to be 10 dB. For the phase (EO) modulator switching occurs when a phase shift of 180 degrees is attained [29]. Using Q_sw rather than V_π makes it independent on the capacitance, C which can be varied by altering geometry and permittivity. Then, the switching energy per bit is U_sw=½Q_sw²/C, whereas the bandwidth is determined by B∼(RC)⁻¹, where R is input impedance of the modulator matched to the impedance of the transmission line connecting the modulator to a driver (typically few tens of Ω’s). Clearly, there always exists a tradeoff between U_sw and B, but the ratio of two (or the product of U_sw and switching time) U_swτ_sw=½Q_sw²R remains constant even as C changes. Thus, switching energy-time product U_swτ_sw and by extension Q_sw are the best fundamental figures of merit for any modulator that depends only on essential material properties and the degree of optical field confinement, as explained below. Note, the dissipated power increases quadratically with the bandwidth P_dis =U_swB=½Q_sw²RB² as one is forced to reduce capacitance to increase speed, and also the fact that in CD modulator most of the power is dissipated within the active region of the device, while in FD modulator it is dissipated in the electrodes, current leads and the source output resistance.

We start with the CD EAM. As has been shown in [28] Q_sw can be evaluated as Q_sw= 2.2eS_eff/σ_a which depends only on two parameters – the effective waveguide area, S_eff and differential absorption cross-section of the material, σ_a (Fig. 1 g). The effective area can be found as S_eff= S_act/Γ, where S_act is the cross-section area of the active region, and

(1)$$\; \Gamma = \int\!\!\!\int\limits_{active} {{E^2}({x,y} )\textrm{ }dxdy/} \int\!\!\!\int\limits_{total} {{E^2}({x,y} )\textrm{ }dxdy}$$

is a confinement factor. The differential absorption cross-section of the material is the change of the absorption coefficient imposed by injection of a single electron (hole) and can be found in CD EAMs near the absorption edge, independent of geometry, as σ_a(ω)=Aπα₀ℏ/m_effγ_eff [28] Here, A is a material-dependent coefficient that is always of the order of unity, α₀ =1/137 is a fine structure constant, m_eff is the effective mass (for graphene one can introduce m_eff(ω) = 2ℏω/υ_F², where υ_F is Fermi velocity). γ_eff is the effective broadening of the transition. For the QD or excitonic TMDC media γ_eff is just the broadening of the material transition, while for the state filling (Pauli-blocking) of Fig.1f the broadening is thermal ℏγ_eff =3k_BT. For free carriers, i.e. non-resonant EAMs, the effective broadening is very large ℏγ_eff =ℏω²/γ where γ is the carrier scattering rate and this fact makes them inefficient compared to the resonant CD EAMs, such as graphene or excitonic cases [28], thus they are not considered here.(Note that, using Kramers-Kronig synergistically via absorption-coupling schemes allows for efficient devices (V_pp <<1 V), yet are not considered here [31].) The effective mass is inversely proportional to the bandgap, i.e. to the photon energy, and is pretty much fixed for a given value of ℏω. As a result, for all the aforementioned mechanisms one obtains σ_a ≈10⁻¹⁶ cm²/ℏγ_eff, where ℏγ_eff is in meV [28]. The outcome is rather striking yet logical – σ_a depends only on the value of the dipole matrix element of the transition and the broadening. For the allowed transitions at a given frequency, the value of that dipole is fixed by the oscillator sum rule and in the visible/near IR region is typically on the scale of the bond length, i.e. a few Å [31]. In the end, the strength of the absorption is not relevant. This is because the stronger the absorption (e.g. exciton vs. regular 2-level system), the more robust it is and hence more difficult to alter. These two factors balance each other out. Note, that free carrier (e.g. Si, ITO) EAMs operating far from resonance have γ_eff ∼ω²/γ i.e. rather large and they would not be competitive, but for all other CD EAMs σ_a ∼10⁻¹⁵ cm².

For CD EOM modulation the switching charge required for π phase shift can be written as Q_π=πeS_eff/σ_ϕ, where the phase shift cross-section σ_ϕ is the amount of phase shift per one injected carrier. One can estimate (in line with KK relation) as σ_a(ω)=Bπα₀ℏ/m_effΔω_eff, where B is in the order of unity, and Δω_eff is effective detuning chosen to avoid spurious changes of absorption. For the QD and TMDC modulators Δω_eff is the detuning from the absorption edge; and for ITO and Si, it is the frequency itself, ω. Note that since EOMs operate away from resonance, free carrier-based EOMs, such as ITO, are highly competitive [29]. As a consequence, and applicable for all the CD EOMs, we find σ_Φ ∼1–3×10⁻¹⁶ cm² i.e. the switching energy time product is about one order of magnitude higher than that for EAMs, which maybe a small price to pay for the EOMs advantages, especially their lower insertion loss.

Now we turn our attention to FD EAM, such as, e.g. QCSE. The shift of the absorption edge is proportional to the electric field E as Δ(ℏω_edge)∼eEΔz_cv, where Δz is the effective dipole commensurate with the width of the QW [32]. Then the change of α is proportional to the above shift and the joint density per unit energy. In the end, the differential cross section for QCSE FD EAM is

(2)$${\sigma _{a,QCSE}}\sim \textrm{ }\pi X{\alpha _0}{e^2}\Delta z{N_{QW}}/2{\varepsilon _0}{\varepsilon _s}\hbar {\gamma _{eff}}$$

σ_a,QCSE ∼ πXα₀e²ΔzN_QW/2ɛ₀ɛ_sℏγ_eff, where N_QW is the number of QWs and X is excitonic enhancement factor on the order of 1.5-2. Exemplary, assuming an InP based modulator with 5 nm QWs (Δz ∼1 nm) we obtain

(3)$${\sigma _{a,QCSE}}\sim 0.5 \times {10^{ - 16}}c{m^2} \times {N_{QW}}/\hbar {\gamma _{eff\; }}\sim {N_{QW}}{\sigma _a}.$$

σ_a,QCSE ∼ 0.5×10⁻¹⁶ cm²×N_QW/ℏγ_eff ∼N_QWσ_a. For FD modulator, σ_a is not a pure material parameter which also depends on the thickness of the active region. The efficiency of the QCSE FD EAM increases with the number of QWs and can exceed the efficiency of CD EAMs by an order of magnitude or more, as long as there is space for 10–20 QWs. Since the period is about 20 nm, such space can only be found in the relatively thick dielectric waveguides and not in the plasmonic ones, where CD modulators gain advantage. We can visualize this in Fig. 1; in FD modulators the field induced by the charge Q on electrodes penetrates the entire waveguide cross section causing the change in all available (N) layers (Fig. 1(b)), whereas in CD modulator (Fig.1a) the change occurs only in a single layer into which the same charge Q has been injected. Hence, the thicker the waveguide, the more advantageous it is for FD schemes. However, once a good overlap between the optical mode and active medium (Γ∼1) is achieved, the reduction in S_eff can only be attained by reducing S_act, which will also reduce N_QW accordingly, and Q_SW will remain the same. In CD modulators, on the other hand, reduction in S_eff always causes decrease in Q_SW which gives them advantage in sub-λ devices. This is sketched in Fig. 1 h – as one tries to scale down the modulator by reducing L and t (keeping L/t, i.e. C and modulation speed constant), U_sw stays the same in FD devices, while loss increases. For the CD devices U_sw decreases as L⁻² ∼t⁻² which is extremely attractive, even taking the higher loss in consideration. Below the critical thickness t_x ∼100 nm for both QCSE and LN (see below) devices, the CD modulators gain clear advantage.

Next, we consider LN EOM, where the key parameter is Pockels coefficient, r₃₃ =30 pm/V and static permittivity, ɛ_s= 28 and obtain σ_ϕ,LN= Kπt_aen³r₃₃/ɛ₀ɛ_sλ ≈ 4×10⁻¹⁸ cm²×t_a, where K∼1 is related to polarization and t_a is the active layer thickness in nm. Again, we find that in this FD modulator σ_ϕ increases with the thickness since each carrier placed on electrode changes refractive index in the entire active layer. For a waveguide with 500 nm active layer thickness σ_ϕ,LN ∼0.5×10⁻¹⁵ cm², i.e. about an order of magnitude higher than in any CD EOM. This result is a bit surprising considering that the LN EOM operates far away from the absorption edge, but the nature of the EO effect in LN is somewhat different from that effect in QWs. In QW the change in index is caused by the changes of the envelope wavefunction, i.e. by changes occurring in the states near the center of the Brillouin zone. The origin of the Pockels effect in LN is the very strong change of the bond lengths [33], i.e. the states throughout the entire Brillouin zone contribute to the index change which compensates for the fact that one operates far from the resonance.

In conclusion, we have set out to provide an answer on how modulator performance is fundamentally impacted between electric field-driven (FD) versus charge driven (CD) setups. In both cases, electrical charge must move – either to change the carrier concentrations directly (CD) or via loading a capacitor (FD). The required switching charge Q_sw or Q_π forms the fundamental metric for all modulators. The main difference between the FD and CD modulators is that once the charge is introduced into the electrodes, the resulting absorption and index modulation is ‘felt’ throughout the entire waveguide in the FD case, whereas it is only modulated locally in the CD case. We find that the legacy FD devices, both EAM and EOM, will hold an advantage for larger waveguides of a few hundred nm thicknesses. But if one attempts to further reduce switching charge by reducing effective area, employing ultra-thin, plasmonic, or hybrid waveguides, the advantages of FD fade and CD modulators, such as ITO and graphene among others, become a very attractive alternative for either index or absorption modulation. Another thought-provoking observation is that from the fundamental material point of view, the change of absorption (index) per unit injected charge for all CD materials are roughly comparable and there is no reason to expect a new material with dramatically improved performance to appear on the scene. So, all future improvements will, in all probability, come not as much from fundamental material breakthroughs as from further gradual progress in miniaturization while lowering insertion loss, series resistance and refining low cost fabrication techniques.

Funding

Air Force Office of Scientific Research (FA9550-16-10362, FA9550-17-1-0071, FA9550-17-1-0377.); Directorate for Engineering (1741694); Defense Advanced Research Projects Agency (HR00111720032).

Acknowledgement

The authors acknowledge discussions with Prof. D. A. B. Miller or Stanford Univ. JK is grateful to Prof. P. Noir and Dr. S. Artois of JHU for their unwavering support

Disclosures

The authors declare no conflicts of interest

Data availability

Data underlying the results presented in this paper are contained within its contents and available in Refs. [28] and [29].

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Charge and field driven integrated optical modulators: comparative analysis: opinion

Abstract

Funding

Acknowledgement

Disclosures

Data availability

References

Data availability

Cited By

Figures (1)

Equations (3)

Optical Materials Express