**UFRJ**

# Nonlinear silicon photonics

(Parte **1** de 4)

NATURE PHOTONICS | VOL 4 | AUGUST 2010 | w.nature.com/naturephotonics 535

Silicon off ers a variety of nonlinear eff ects that can be used to process optical signals at speeds of 100 Gbit s–1 and beyond1–5, detect signals at unprecedented sensitivities for novel sensing applications6,7 and enable broadband electro-optic modulation8–1. Ultimately, the nonlinear eff ects in silicon may also permit the generation of photons for lasing and amplifi cation12–15. It should be mentioned, however, that although these nonlinear eff ects are numerous, the interest in nonlinear silicon photonics is not so much due to the superior nonlinear characteristics of silicon over alternative integrated-optics materials, but rather because of its potentially lower cost and high compatibility with CMOS industry, which is interested in combining new optical functionalities with electronics on a single chip.

Th e challenges in the development of active silicon elements are: (1) the indirect minimum-energy bandgap of silicon, which makes spontaneous emission unlikely and thus impedes lasing; (2) the centrosymmetry of the silicon crystal, which prevents electrooptic eff ects and thus hinders the development of electro-optic silicon modulators; and (3) the variety of fast and slow higher-order nonlinear eff ects that result in both slow and fast recovery times when the material is exposed to high fi eld strengths or sudden temporal changes. Despite these issues, there has been impressive progress in the development of functional nonlinear silicon photonic devices. Today, silicon devices can emit light, modulate signals electro-optically and process data at speeds higher than electronic chips. Th is success is partially due to sophisticated technical solutions, and in particular has been enabled by progress in the fabrication of nanophotonic devices16,17.

Th e goal of this Review is to summarize the wealth of nonlinear eff ects found in silicon, and to highlight selected applications and technological solutions that have emerged during the past few years. First, a short summary of silicon is given, following which the various nonlinear eff ects and specifi c techniques used in the application of these techniques are reviewed. Finally, a selected number of applications are discussed in more detail.

The vision Silicon is the raw material for the silicon-on-insulator (SOI) platform, a fabrication approach in which a thin silicon layer on top of an insulator layer resides on a silicon substrate. Th e functional optical elements are situated in the thin top-silicon layer, and the insulator is typically made from SiO2. Th e refractive index of waveguiding silicon is high (nSi = 3.48) compared with air (n = 1) or the

SiO2 cladding layer (nSiO = 1.4), and hence strong optical guiding is guaranteed for all signals around the typical near-infrared wavelength of 1,550 nm.

Th e SOI platform has become the foundation of silicon photonics for many good reasons18–20. For example, silicon is widely available and is compatible with mature CMOS technology, thus off ering structure sizes down to 10 nm at low cost21,2. Th e strong

Nonlinear silicon photonics

J. Leuthold*, C. Koos and W. Freude

Silicon off ers an abundance of nonlinear optical eff ects that can be used to generate and process optical signals in low-cost ultracompact chips at speeds beyond those of today’s electronic devices. The Review discusses the nonlinear optical eff ects in silicon and highlights some of the associated key applications.

optical confi nement of silicon allows for very compact optical devices with bent waveguide radii of the order of a few micrometres, and functional waveguide elements of ten to a few hundred micrometres. Consequently, large-scale integration of many functional elements on a single chip is within reach16.

Silicon photonics also extends to highly integrated multifunctional devices that perform both optical and electrical operations on a single low-cost chip. Such devices could replace the large network interface cards that interconnect optical communications networks with computers, or could act as sensitive optical detectors with built-in processors for pre-processing detected signals. Other applications include the development of silicon photonic circuits for optical chip-to-chip communications23 and the development of silicon highest-speed signals processors for optical communications and computers24. Besides passive splitters, fi lters and multiplexers, such chips could also potentially comprise active elements such as lasers, amplifi ers, modulators, signal regenerators and wavelength converters. Slow-light photonic crystals could be used to enhance the nonlinear eff ects, and third-harmonic generation could be used for monitoring applications25.

Nonlinear eff ects in silicon Th ere are a plethora of nonlinearities in silicon26–28, and they all originate from the interactions of the optical fi eld with electrons and phonons. In fact, it is the electric fi eld of the optical signal that resonates with the electrons in the outer shells of the silicon atoms and thus causes polarization. Figure 1a illustrates how a photon oscillating at frequency ω polarizes an atom by displacing an electron orbital with respect to the nucleus. In its simplest form, and assuming an instantaneous dielectric response in an isotropic material, the relation between an induced polarization (P(t), scalar) and an electric fi eld (E(t), also scalar) is expressed by a power series in the electric fi eld26:

P(t) = ε0(χ(1)E(t) + χ(2)E2(t) + χ(3)E3(t) + | ) (1) |

where ε0 is the vacuum permittivity and χ(i) are the ith-order optical susceptibilities. Th e susceptibility terms are tensors of rank (i + 1), and describe whether the relation between the induced polarization is linear or nonlinear, whether the electric fi elds induce phaseshift s, absorb or amplify the incident fi eld, and whether waves at new frequencies are generated. Here, the nonlinear eff ects associated with each of these susceptibility coeffi cients are discussed, taking them to be scalars for simplicity.

χ(1) processes. Th e complex fi rst-order susceptibility term χ(1) concerns dipole excitations with bound and free electrons induced by a single photon. Th e real part of χ(1) is associated with the real part of the refractive index, whereas the imaginary part of χ(1) describes loss or gain. Some of the dipole excitations associated

Institute of Photonics and Quantum Electronics, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany. *e-mail: juerg.leuthold@kit.edu

536 NATURE PHOTONICS | VOL 4 | AUGUST 2010 | w.nature.com/naturephotonics with χ(1) processes are depicted in Fig. 1b. Th e left -hand energy level diagram suggests that the refractive index has its origin in atomic dipole oscillations between a bound state and a virtual level (an energy level that does not correspond to an excited energy eigenstate of the atom26). Lorentz developed a simple model for calculating the contribution of bound electrons to the susceptibility, and showed that the refractive index changes strongly near a resonance. His model gives the susceptibility of bound states belonging to a density of N dipoles (elementary charge q, eff ective mass me and oscillating charge) as:

where ω0 is the resonance frequency of the bound state and γL is the associated damping constant. ωL is the Lorentz plasma frequency, which is defi ned as:

An additional contribution to the susceptibility comes from free carriers, which semiconductors have many of at room temperature. Free carriers absorb photons and contribute to the refractive index. An absorption process with successive non-radiative recombination is depicted in the right-hand energy level diagram of Fig. 1b. Th e complex susceptibility induced by free carriers may ω20 − ω2 + iγLωχLorentz = (1) ω2 L = ε0me

Nq2 be derived from equation (2) by dropping the restoring force (that is, ω0 = 0). Th is is described by the Drude model:

− ω2 + iγDωχDrude = (1) (3) with the plasma frequency ωp being defi ned as:

ω2 p = ε0me

Nq2

Here, the plasma frequency ωp and damping constant γD have diff erent values than the corresponding quantities ωL and γL in equation (2). Any contribution from either bound- or free-electron oscillations contributes to the complex refractive index. Th us, for non-magnetic materials, one may write:

Lorentz + χ(1) Drude (4)

Equation (4) shows that the refractive index changes with both wavelength and carrier concentration N. Although equation (4) does provide physical insight, in practice when performing simulations researchers prefer to fi t their models with directly measured wavelength dependences and free-carrier contributions from electrons (concentration Ne) and holes (concentration Nh) to the refractive index. A useful empirical function for giving the refrac- tive index n as a function of wavelength (λ), Ne and Nh is given by:

,Nh) = n0(λ) + Δnf (Ne,Nh) − i | Δαf(Ne,Nh)n(λ,Neλ |

where n0(λ) is the wavelength dependence of the refractive index, Δnf is the free-carrier index (FCI) change and Δαf is the free-carrier absorption (FCA) change. n0(λ) for silicon is incorporated through the Sellmeier relation:

n20(λ) = ε + | + λ2 λ2 – λg |

A | Bλ12 |

2 (5) with ε = 1.686, A = 0.9398 μm2, B = 8.1046 × 10−3 and the bandgap wavelength of silicon λg = 1.1071 μm (refs 27,29). Th e closer the photon gets to the bandgap energy of silicon (1.12 eV), the stronger the change in refractive index becomes. Equation (5) describes what is known as the material dispersion of silicon. Further below we will see that a precise knowledge of the material dispersion and the ability to engineer the waveguide dispersion will be instrumental in optimizing the conversion effi ciency of higher-order nonlinear eff ects30.

Th e changes in FCI and FCA at λ = 1,550 nm are oft en incorporated by the following empirical expressions27,31:

Δnf = − 8.8 × 10−4 | + 8.5 ×10−18 |

cm−3

Ne 0.8 cm−3 Nh

= − 8.5 + 6.0 ×10−18 |

cm−3

Ne cm−3

Nh cm−1 Δαf where Ne and Nh have units of cm–3 and αf has units of cm–1.

χ(2) processes. Th e second-order susceptibility term χ(2) is absent in centrosymmetric crystals such as silicon. However, such nonlinear eff ects would be highly desirable as they would allow the creation of electro-optic modulators. Measures to overcome this defi ciency include breaking the crystal symmetry by depositing straining c –hω –hω y z

Figure 1 | Illustration of a dipole excitation and possible energy level diagrams. a, An electromagnetic wave with electric ﬁ eld E passing through an atom and thereby inducing a dipole oscillation P(E). b, Energy level diagrams showing possible single-photon dipole transitions with contributions to refractive index changes (left) or to free-carrier absorption (right). c, Third-order nonlinear dipole transitions, showing self-phase modulation (SPM), two-photon absorption (TPA), cross-phase modulation (XPM), third-harmonic generation (THG), partially degenerate and non-degenerate four-wave mixing (FWM) and stimulated Raman scattering (SRS).

REVIEW ARTICLES | FOCUSNATURE PHOTONICS DOI: 10.1038/NPHOTON.2010.185

NATURE PHOTONICS | VOL 4 | AUGUST 2010 | w.nature.com/naturephotonics 537 layers on top of silicon32, or using another χ(2)-nonlinear electrooptic active material as a cladding33–35. At this point it should be mentioned that electrical modulation has also been successfully demonstrated by simply injecting and removing carriers, thereby making use of the aforementioned χ(1)-related FCA and FCI eff ects8,36–39. However, χ(2) electro-optical modulators would be preferred as they do not suff er from carrier-related speed limitations.

χ(3) processes. Th ird-order nonlinearities are especially important in silicon as they exhibit a wide variety of phenomena. Th is can be easily demonstrated for an electric fi eld E comprising three fre- quency components (ωk):

E(r,t)=∑E = | ∑ (E |

(r,ω

)e iωt + c.c.) ωk where c.c. denotes the complex conjugate. By substituting equation (6) into equation (1) and expanding the frequency components of the third-order nonlinear term χ(3), one obtains a multitude of terms at new frequencies for the third-order polarization P(3):

+ | ε0χ(3)[ (E2 |

+ | ε0χ(3)[(|Eω |

+ | ε0χ(3)[(E3 |

+ | ε0χ(3)[ (E2 |

+ | ε0χ(3)[ (EωEω |

+ | ε0χ(3)[ (EωEωEω |

(7)

where the symbol | stands for all possible permutations of fre- |

quencies. Each of the terms on the right-hand side of equation (7) corresponds to a nonlinear optical excitation, of which a few are visualized in Fig. 1c. Th e energy level diagrams show how three photons induce dipole transitions to excited states, which subsequently relax back by releasing a fourth photon. If the excited states do not correspond to bound eigenstates of the crystal then the relaxation processes take place almost instantaneously. It is these fast processes that have caught the interest of the research community, and in fact many of the processes depicted in Fig. 1c are ultrafast. Among the many nonlinear processes, only those that maintain both energy and momentum conservation (known as phase-matching) result in effi cient excitation26. A particular process among the many nonlinear processes available can be selected to some degree by choosing the energy levels and phase-matching appropriately.

Th e fi rst term in equation (7) corresponds to a phenomenon called self-phase modulation (SPM), which results from dipole excitations induced by three photons (for example, all at an angular frequency of ω1). Th is process is depicted in Fig. 1c. SPM leads to an intensity-dependent refractive index change n2, which in turn modifi es the spectral composition of the very same pulse that has gener- ated it. As a consequence, SPM leads to a broadening of the pulse spectrum. Extreme levels of SPM may generate a supercontinuum, which has, for example, applications in the generation of ultrashort pulses in spectroscopy, telecommunications and metrology.

Th e photons that generate SPM can also excite an energetically higher state, as depicted in the two-photon absorption (TPA) diagram of Fig. 1c. TPA excitations are ‘absorbing’ because they correspond energetically to an excitation of an electron in the valence band to the conduction band. TPA leads to an intensity-dependent contribution α2 to the linear absorption coeffi cient α0. TPA generates free carriers in the conduction band that subsequently act as sources for FCA and FCI changes. Th e long lifetime of the free carriers in the conduction band leads to long-lasting FCA and FCI processes that ultimately slow down the speed of silicon photonic devices40,41.

Th e intensity-dependent refractive index and absorption changes associated with SPM and TPA aff ect the complex refractive index n by the relation

n = n0 + n2I – i | (α0 + α2I)λ |

where I is the intensity, and n2 (the Kerr coeffi cient) and α2 are interrelated with the real and complex part of the third-order sus- ceptibility by the equations

n2 = | Re(χ(3)) 3 |

α2 = | Im(χ(3)) 3 |

A fi gure of merit (FOM) is oft en used to compare the magnitude of the Kerr coeffi cient n2 with the strength of the TPA coeffi cient α2:

A large FOM is preferable, to avoid TPA-related speed limitations. For silicon, although the Kerr nonlinear coeffi cient is large

(n2 = (4.5 ± 1.5) × 10–18 m2 W–1 at 1.5 μm; ref. 42), the large TPA coeffi cient results in a low FOM of ~0.4 (ref. 41). Th e diffi culty in non- linear optics is therefore to fi nd a material that has both a large Kerr nonlinear coeffi cient and a small TPA coeffi cient. Th e quantities n2 and α2 vary with wavelength. Relative magnitudes of these two coeffi cients as a function of photon energy normalized to the bandgap energy have been derived for silicon43, using a method similar to that of an earlier approach for direct bandgap materials44. Th e silicon Kerr coeffi cient actually peaks at around 1.8 μm and 1.9 μm, whereas the TPA absorption coeffi cient decreases considerably beyond 2 μm; that is, beyond half the bandgap energy26. Silicon is therefore expected to show very favourable Kerr nonlinearities with a large FOM in the near-infrared, as will be discussed further below45,46.

(Parte **1** de 4)