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Modeling nonlinear optical phenomena in silicon-nanocrystal composites and waveguides

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2014 J. Opt. 16 015207

(http://iopscience.iop.org/2040-8986/16/1/015207)

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Journal of Optics

J. Opt. 16 (2014) 015207 (11pp) doi:10.1088/2040-8978/16/1/015207

Modeling nonlinear optical phenomenain silicon-nanocrystal composites andwaveguides

Ivan D Rukhlenko

Advanced Computing and Simulation Laboratory (AχL), Department of Electrical and ComputerSystems Engineering, Monash University, Clayton, VIC 3800, AustraliaSaint Petersburg National Research University of Information Technologies, Mechanics and Optics,197101 Saint Petersburg, Russia

E-mail: ivan.rukhlenko@monash.edu

Received 25 September 2013, revised 18 November 2013Accepted for publication 19 November 2013Published 11 December 2013

AbstractA high demand in relation to simple propagation equations describing nonlinear opticalphenomena in silicon-nanocrystal composites is caused by the impracticality of solvingMaxwell’s equations while treating thousands of silicon nanocrystals in the compositesindividually. Recently, we derived such equations for the case of two optical fields using anumber of simplifying assumptions (Rukhlenko 2013 Opt. Express 21 2832–46). In particular,we neglected the effects of cross-phase modulation and cross-two-photon absorption due tothe interaction between waves of different frequencies, set equal different kinds ofmode-overlap factors, and assumed a uniform effective mode area for both fields regardless oftheir polarizations. Also, it was assumed that the fields interact inside a highly birefringentsilicon-nanocrystal waveguide, which provides tight lateral confinement of its propagatingmodes and ensures the absence of phase matching between them. Here we abandon theseapproximations and generalize the coupled-amplitude equations for the case of an arbitrarynumber of quasi-monochromatic optical fields interacting through the third-order nonlinearpolarization of silicon nanocrystals. We derive two sets of equations enabling one to studytheoretically the anisotropy of such effects as Raman amplification, four-wave mixing, andwavelength conversion—one for unbounded silicon-nanocrystal composites and the other fordifferent kinds of silicon-nanocrystal waveguides—and provide an overview of the materialparameters entering these equations.

Keywords: silicon nanocrystals, nonlinear optics, nonlinear optical materials

1. Introduction

The progress in silicon photonics and silicon-on-insulator(SOI) technology has enabled the realization of submicron-cross-section waveguides [2–5] and modulators [6–9],silicon Raman amplifiers [10–20] and lasers [21–24]. Themanipulation and generation of light in SOI-based photonicsdevices rely on a variety of relatively strong nonlinear effectsdue to the third-order susceptibility of silicon [25–32]. Inaddition to being naturally strong, these nonlinearities getfurther intensified in SOI waveguides thanks to the tightoptical confinement provided by silicon. The possibility

of integrating key silicon-based components on a singlephotonics chip and their low fabrication costs [33] have beenensuring the fast commercialization of all novel technicaldevelopments in the field of silicon photonics [34–39].

In the early 2000s, silicon nanocrystals embedded infused silica emerged as an alternative to bulk silicon,owing to their nonlinear optical phenomena being morepronounced than for bulk silicon [40–48]. Unfortunately,since the refractive index of silicon-nanocrystal compositesis quite low (typically less than 2.2), they cannot be used bythemselves to guide light in photonic nanostructures with deepsubwavelength mode confinement. The confinement problem

12040-8978/14/015207+11$33.00 c© 2014 IOP Publishing Ltd Printed in the UK

J. Opt. 16 (2014) 015207 I D Rukhlenko

can be solved by placing the composite in the gap betweentwo sufficiently thick silicon layers, which significantlyenhance optical intensity within the gap [40, 49, 50].The major difficulty in numerically modeling the non-linear optical phenomena occurring either inside suchsilicon-nanocrystal–silicon waveguides or in bulk silicon-nanocrystal composites is the large number (∼105) ofnanometer-size silicon nanocrystals contributing to thenonlinear polarization [51–54]. To overcome this difficulty,we recently developed the first theoretical treatment ofnonlinear pulse propagation in silicon-nanocrystal-dopedsilica waveguides [1]. Although our treatment was quitegeneral and based on only a few common assumptions, itstill needs to be generalized for the case of an arbitrarynumber of interacting fields characterized by different mode(or beam) profiles and different effective mode areas. Thispaper presents such a generalization to the cases of opticalpropagation through bulk silicon-nanocrystal compositesand straight silicon-nanocrystal waveguides, and provides aunified theoretical platform for the numerical modeling ofsilicon-nanocrystal-based photonics components and devices.In order to facilitate the application of our model, the setsof generalized nonlinear equations derived are supplied by adetailed discussion of their material parameters.

2. The vectorial model for pulse propagation

The starting point of our derivation of the nonlinear equationsgoverning the evolution of quasi-monochromatic optical fieldsat carrier frequencies ωµ (µ = µ1, µ2, µ3, . . .) propagatingin the z direction through silicon-nanocrystal compositeis the coupled-amplitude equations for the slowly varyingamplitudes, aµν , of propagating modes ν = ν1, ν2, ν3, . . .

[1, 55–57],

∂aµν∂z−

∞∑n=1

in+1β(n)µν

n!

∂naµν∂tn

=iωµ

4√

Nµν

(1+

iωµ

∂

∂t

)e−iβµνz

∫∫ (e∗µν · PNL

ωµ

)dr⊥, (1)

where

Nµν =14

∫∫z ·(e∗µν × hµν + eµν × h∗µν

)dr⊥ (2)

is the normalization constant, β(n)µν = ∂nβν/∂ωn|ω=ωµ and βµν

are the nth-order dispersion parameter and the propagationconstant of mode ν at frequency ωµ, and the slowlyvarying amplitude, PNL

ωµ, of the nonlinear material polarization

is the sum of Kerr (electronic), Raman, and free-carriercontributions, i.e., PNL

ωµ= PK

ωµ+ PR

ωµ+ PFC

ωµ[58, 59, 35, 36].

The integrations in equations (1) and (2) are evaluated overthe entire transverse plane xy, so dr⊥ ≡ dx dy.

Complex-valued vector functions eµν and hµν oftransverse coordinates r⊥ ≡ (x, y) form complete sets oforthogonal profiles of the propagating electric and magneticfield modes, which obey the relations(eµν · eµν′

)= e2

µνδνν′ ,(eµν · e∗µν′

)= |eµν |2δνν′ , (3a)

and ∫∫z ·(e∗µν × hµν′ + eµν × h∗µν′

)dr⊥ = 4Nµνδνν′ . (3b)

For known material polarization, equation (1) can besolved to find the complex-valued electric and magnetic fieldsinside the composite. These fields are given by

E(r, t) =∑µ

Eωµ(r, t) e−iωµt

=

∑µ

∑ν

aµν(z, t)eµν(r⊥)√

Nµνei(βµνz−ωµt) (4a)

and

H(r, t) =∑µ

Hωµ(r, t) e−iωµt

=

∑µ

∑ν

aµν(z, t)hµν(r⊥)√

Nµνei(βµνz−ωµt), (4b)

where the slowly varying amplitudes Eωµ and Hωµ of thefields are implicitly defined.

Using equations (3b) and (4), it is easy to show that thetotal power stored at field frequency ωµ is expressed throughthe slowly varying amplitudes as

Pµ(z, t) =14

∫∫z ·(Eωµ ×H∗ωµ + E∗ωµ ×Hωµ

)dr⊥

=

∑ν

|aµν(z, t)|2. (5)

2.1. Kerr polarization

An almost instantaneous response of bound electrons ofsilicon atoms to the external optical field is described by thematerial polarization [57]

PKωµ(r, t) = ε0χ

(3)K (ωµ;ωµ,−ωµ, ωµ)

× Eωµ(r, t)E∗ωµ(r, t)Eωµ(r, t)

+ 2ε0

∑µ′ 6=µ

χ(3)K (ωµ;ωµ,−ωµ′ , ωµ′)

× Eωµ(r, t)E∗ωµ′ (r, t)Eωµ′ (r, t), (6)

where the first tensor product describes the effects ofself-phase modulation (SPM) and two-photon absorption(TPA) due to the quasi-monochromatic field at frequencyωµ, whereas the terms under the sum take into account thetwo-frequency processes of cross-phase modulation (XPM)and cross-TPA.

Random orientation of silicon nanocrystals inside thecomposite allows us to write the effective third-ordersusceptibility tensors entering equation (6) in the form of theproduct [58, 60, 36]

χ(3)K (ωµ;ωµ,−ωµ′ , ωµ′) = χµµ′Kklmn, (7)

where

χµµ′ = cε0εeff(ωµ)

(n2 + i

βµµ′

TPA

2kµ

)ξ(ωµ) (8a)

2

J. Opt. 16 (2014) 015207 I D Rukhlenko

and the tensor

Kklmn =8+ 7ρ

45(δklδmn + δkmδln + δknδlm)

+1− ρ

9δklδlmδmn (8b)

is written in the reference frame associated with thecomposite. Here c is the speed of light in vacuum, ε0

is the permittivity of vacuum, εeff = n2eff is the effective

permittivity of the silicon-nanocrystal composite defined inequation (25), n2 is the nonlinear Kerr parameter, βµµTPA is the

TPA coefficient, βµµ′

TPA(µ 6= µ′) is the cross-TPA coefficient,

kµ = ωµ/c, ξ is the third-order susceptibility attenuationfactor defined in equation (26b), and ρ is the nonlinearanisotropy factor (see section 4). Note that the parameters n2,

βµµ′

TPA, and ρ refer to a single silicon nanocrystal.Substitution of equations (8b) and (7) into (6) leads to the

following vector form for the Kerr polarization:

PKωµ= ε0χµµ

[8+ 7ρ

45

(2|Eωµ |

2Eωµ + E2ωµ

E∗ωµ

)+

1− ρ9

∑η

E2ηE∗ηη

]

+ 2ε0

∑µ′ 6=µ

χµµ′

{8+ 7ρ

45

[∣∣Eωµ′ ∣∣2Eωµ

+(EωµEωµ′

)E∗ωµ′ +

(EωµE∗ωµ′

)Eωµ′

]+

1− ρ9

∑η

EηE′∗η E′ηη

}, (9)

where Eη is the ηth component of vector Eωµ , E′η is the ηthcomponent of vector Eωµ′ , and η is the unit vector in thedirection of the ηth Cartesian axis.

Let us for simplicity assume that there are only twodifferent modes polarized along either x or y axes (i.e., ν = xor y). Then using equations (3a), (4a), and (9), we can evaluatethe Kerr contribution to the double integral on the right-handside of equation (1) as

e−iβµνz√Nµν

∫∫ (e∗µν · PK

ωµ

)dr⊥

= ε0χµµ

[8+ 7ρ

45

∑ν′

(20µµ

νν′|aµν′ |

2aµν

+ 3µµ

νν′a2µν′a

∗µνe2i(βµν′−βµν )z

)+

1− ρ9

0µµνν |aµν |2aµν

]+ 2ε0

∑µ′ 6=µ

χµµ′

[8+ 7ρ

45

∑ν′

(0µµ′

νν′|aµ′ν′ |

2aµν

+ 3µµ′

νν′aµν′aµ′ν′a

∗

µ′νei(βµν′+βµ′ν′−βµ′ν−βµν )z

+ 9µµ′

νν′aµν′a

∗

µ′ν′aµ′νei(βµν′+βµ′ν−βµ′ν′−βµν )z)

+1− ρ

93µµ

′

νν |aµ′ν |2aµν

], (10)

where

0µµ′

νν′=

∫∫|eµν |2|eµ′ν′ |2

NµνNµ′ν′dr⊥, (11a)

3µµ′

νν′=

∫∫(e∗µν · e∗

µ′ν)(eµν′ · eµ′ν′)√

NµνNµ′νNµν′Nµ′ν′dr⊥, (11b)

and

9µµ′

νν′=

∫∫(e∗µν · eµ′ν)(eµν′ · e∗

µ′ν′)√

NµνNµ′νNµν′Nµ′ν′dr⊥. (11c)

2.2. Raman polarization

The strength of Raman interaction between two waves ofcarrier frequencies ωµ and ωµ′ critically depends on theirfrequency detuning ωµµ′ = ωµ −ωµ′ . Like in [1], the Ramancontribution to the material polarization can be represented as

PRωµ(r, t) = ε0ξ

∫ t

−∞

dt1 H(t − t1)∑µ′ 6=µ

RklmnEωµ(r, t1)

× E∗ωµ′ (r, t1)Eωµ′ (r, t) eiωµµ′ (t−t1), (12)

where

Rklmn =2945(δkmδln + δknδlm)−

1645δklδmn

−29δklδlmδmn (13a)

and

H(t) ≈ 4cε0εeff(ωµ)(gR/kµ) 0R e−0R t sin(�R t), (13b)

where gR is the Raman gain coefficient of silicon nanocrystals,�R is the Raman shift, and 0R is half of the amplificationbandwidth.

Using equations (4a), (12), and (13a), we obtain

e−iβµνz

ε0ξ√

Nµν

∫∫ (e∗µν · PR

ωµ

)dr⊥

=

∑µ′ 6=µ

∑ν′

[2945 3

µµ′

νν′ei(βµν′+βµ′ν′−βµ′ν−βµν )zaµ′ν′(t)

×

∫ t

−∞

a∗µ′ν(t1) aµν′(t1)H(t − t1) eiωµµ′ (t−t1) dt1

+2945 9

µµ′

νν′ei(βµν′+βµ′ν−βµ′ν′−βµν )zaµ′ν(t)

×

∫ t

−∞

a∗µ′ν′(t1) aµν′(t1)H(t − t1) eiωµµ′ (t−t1) dt1

−1645 0

µµ′

νν′aµ′ν′(t)

×

∫ t

−∞

a∗µ′ν′(t1) aµν(t1)H(t − t1) eiωµµ′ (t−t1) dt1

−29 δνν′3

µµ′

νν aµ′ν(t)

×

∫ t

−∞

a∗µ′ν(t1) aµν(t1)H(t − t1) eiωµµ′ (t−t1) dt1

]. (14)

3

J. Opt. 16 (2014) 015207 I D Rukhlenko

2.3. Free-carrier polarization

Before evaluating the polarization term describing thefree-carrier effects of absorption and dispersion, we make onemore simplification. Since the silicon-nanocrystal compositeitself does not provide strong confinement of the propagatingmodes [49], we assume the modal profiles, eµν and hµν , tobe purely transverse, in which case it follows from Maxwell’sequations that

e∗µν × hµν =βµν

µ0ωµe∗µν × z× eµν =

βµν

µ0ωµ|eµν |2z (15)

and thus

Nµν =nµν

2µ0c

∫∫|eµν |2 dr⊥, (16)

where nµν = βµν/kµ is the modal refractive index.By denoting the number density of free electron–hole

pairs generated as a result of TPA and cross-TPA as N(z, t)and using equation (16), we can write the double integralresponsible in equation (1) for the free-carrier nonlinearpolarization in the form [1, 60, 36]

e−iβµνz√Nµν

∫∫ (e∗µν · PFC

ωµ

)dr⊥

= −4ζc

neff

nµν

(ω0

ωµ

)2 (σn − i

σα

2kµ

)N(z, t) aµν, (17)

where ζ is the linear susceptibility attenuation factor definedin equation (26a), neff =

√εeff is the effective refractive index

of the composite (defined in equation (25)), ω0 is the referencefrequency corresponding to the wavelength 1550 nm, σn is thefree-carrier dispersion coefficient, and σα is the free-carrierabsorption cross-section (see section 4).

2.4. The rate equation

The electrons and holes generated inside the nanocrystalseventually recombine at a certain rate, which is commonlycharacterized by the effective free-carrier lifetime τeff. Thetwo counteracting processes of generation and recombinationare described by the rate equation

∂N

∂t= −

N

τeff−

∑µ

12hωµ

∑ν

1

Aµνeff

∂|aµν |2

∂z, (18)

in which the effective area of mode ν is defined as [61]

Aµνeff = ANL

∫∫∞

|eµν |2 dr⊥

(∫∫NL|eµν |2 dr⊥

)−1

. (19)

In the case of nonlinear propagation through a silicon-nanocrystal waveguide, ANL is the cross-section area of thewaveguide’s core filled with silicon nanocrystals, and symbolsNL and ∞ denote integrations over this core and the entiretransverse plane, respectively. However, in the situation wherea beam travels through an unbounded silicon-nanocrystalcomposite, ANL is the characteristic area of the beam’scross-section and NL denotes integration over this area. In

both cases, N(z, t) and Iµν = |aµν |2/Aµνeff are the average

free-carrier density and average field intensity inside areaANL. It should be noted that the above definition of theeffective mode area differs from the common one [36, 4, 62].

In order to evaluate the rate of TPA-induced powerdissipation of mode ν at frequency ωµ, we neglect thetime derivatives in equation (1) and retain only the Kerrcontribution to the polarization integral. By discarding thespatially varying terms in equation (10), which arise dueto the waveguide birefringence (nµν 6= nµν′) and dispersion(nµν 6= nµ′ν) and do not contribute to the power dissipation,and assuming ρ to be real, we obtain

−∂|aµν |2

∂z=

14

c2ε20εeffξβ

µµTPA

[8+ 7ρ

45

∑ν′

20µµνν′|aµν′ |

2

+

(8+7ρ

453µµνν +

1−ρ9

0µµνν

)|aµν |

2

]|aµν |

2

+12

c2ε20εeffξ

∑µ′ 6=µ

βµµ′

TPA

[8+ 7ρ

45

×

∑ν′

0µµ′

νν′|aµ′ν′ |

2+

(8+ 7ρ

459µµ

′

νν

+13+ 2ρ

453µµ

′

νν

)|aµ′ν |

2

]|aµν |

2. (20)

The rate of power dissipation is higher for opticalfields interacting inside an unbounded silicon-nanocrystalcomposite, where the propagation constant depends solelyon carrier frequency, βµν = neff(ωµ) kµ, and all fields areperfectly phase matched, 1β = 1β± = 0. Taking this intoaccount in equations (1) and (10) shows that the right-handside of equation (20) in this case should include additionalterms

14

c2ε20εeffξ

8+ 7ρ45

Re[(βµµTPA − 2ikµn2

)×

∑ν′ 6=ν

3µµ

νν′

(aµν′a

∗µν

)2+ 2

∑µ′ 6=µ

(βµµ′

TPA − 2ikµn2

)∑ν′ 6=ν

(3µµ′

νν′aµ′ν′a

∗

µ′ν

+ 9µµ′

νν′a∗µ′ν′aµ′ν

)aµν′a

∗µν

]. (21)

3. Generalized coupled-amplitude equations

The generalized coupled-amplitude equations can be conve-niently written using the following three kinds of dimension-less mode-overlap factors defined similarly to the coefficientsin equation (11):

γµµ′

νν′= Aµ′ν′

eff

n2eff

nµνnµ′ν′

∫∫|eµν |2|eµ′ν′ |2 dr⊥∫∫

|eµν |2 dr⊥∫∫|eµ′ν′ |2 dr⊥

, (22a)

λµµ′

νν′= Aµ′ν′

eff

n2eff

nµνnµ′ν′

∫∫(e∗µν · e∗

µ′ν)(eµν′ · eµ′ν′) dr⊥∫∫

|eµν |2 dr⊥∫∫|eµ′ν′ |2 dr⊥

, (22b)

4

J. Opt. 16 (2014) 015207 I D Rukhlenko

and

ψµµ′

νν′= Aµ′ν′

eff

n2eff

nµνnµ′ν′

∫∫(e∗µν · eµ′ν)(eµν′ · e∗

µ′ν′) dr⊥∫∫

|eµν |2 dr⊥∫∫|eµ′ν′ |2 dr⊥

. (22c)

Using equations (16) and (22) and substituting equations(10), (14), and (17) into (1), on whose right-hand side wediscard the time derivative for simplicity, yields(∂

∂z−

∞∑n=1

in+1β(n)µν

n!

∂n

∂tn+αµν

2

)aµνξ

=

(in2kµ −

βµµTPA

2

)[8+ 7ρ

45

∑ν′

(2γ µµνν′

Iµν′aµν

+ λµµ

νν′a2µν′a

∗µνe2i1βz)

+1− ρ

9γ µµνν Iµνaµν

]+

∑µ′ 6=µ

(2in2kµ − β

µµ′

TPA

) [8+ 7ρ45

∑ν′

(γµµ′

νν′Iµ′ν′aµν

+ λµµ′

νν′aµν′aµ′ν′a

∗

µ′νei1β+z

+ ψµµ′

νν′aµν′a

∗

µ′ν′aµ′νei1β−z)

+1− ρ

9λµµ

′

νν Iµ′νaµν

]+

ikµcε0εeff

×

∑µ′ 6=µ

∑ν′

[2945 λ

µµ′

νν′ei1β+zaµ′ν′(t)

×

∫ t

−∞

a∗µ′ν(t1) aµν′(t1)H(t − t1) eiωµµ′ (t−t1) dt1

+2945 ψ

µµ′

νν′ei1β−zaµ′ν(t)

×

∫ t

−∞

a∗µ′ν′(t1) aµν′(t1)H(t − t1) eiωµµ′ (t−t1) dt1

−1645 γ

µµ′

νν′aµ′ν′(t)

×

∫ t

−∞

a∗µ′ν′(t1) aµν(t1)H(t − t1) eiωµµ′ (t−t1) dt1

−29 δνν′λ

µµ′

νν aµ′ν(t)

×

∫ t

−∞

a∗µ′ν(t1) aµν(t1)H(t − t1) eiωµµ′ (t−t1) dt1

]

− ζneff

nµν

(ω0

ωµ

)2 (σα

2+ iσnkµ

)N

ξaµν (23)

where 1β = βµν′ − βµν and 1β± = βµν′ ± βµ′ν′ ∓ βµ′ν −βµν . The last term in the parentheses on the left-handside of this equation was introduced to take into accountlinear waveguide losses through the polarization-dependentabsorption coefficients αµν .

The equations for the coupled amplitudes are comple-mented by the rate equation, which in view of equations (18),(20), and (21) acquires the form(∂

∂t+

1τeff

)N

ξ

=

∑µ

βµµTPA

2hωµ

∑ν

[8+ 7ρ

45

∑ν′

2γ µµνν′

Iµν′

+

(8+ 7ρ

45λµµνν +

1− ρ9

γ µµνν

)Iµν

]Iµν

+

∑µ

∑µ′ 6=µ

βµµ′

TPA

hωµ

∑ν

[8+ 7ρ

45

∑ν′

γµµ′

νν′Iµ′ν′

+

(8+ 7ρ

45ψµµ

′

νν +13+ 2ρ

45λµµ

′

νν

)Iµ′ν

]Iµν

× ϑ8+ 7ρ

45

∑µ

∑ν

Re

[βµµTPA − 2ikµn2

2hωµ

×

∑ν′ 6=ν

λµµ

νν′

(aµν′a∗µν

)2Aµν′

eff Aµνeff

+

∑µ′ 6=µ

βµµ′

TPA − 2ikµn2

hωµ

×

∑ν′ 6=ν

(λµµ′

νν′aµ′ν′a

∗

µ′ν + ψµµ′

νν′a∗µ′ν′aµ′ν

) aµν′a∗µν

Aµ′ν′

eff Aµνeff

],

(24)

where ϑ is equal to unity for phase-matched fieldspropagating in silicon-nanocrystal composites and zerootherwise.

Equations (23) and (24) constitute a basis for modelingnonlinear optical phenomena in silicon-nanocrystal compos-ites and silicon-nanocrystal waveguides. The meaning of theirdifferent terms is as follows.

The term αµν/2 on the left-hand side of equation (23)accounts for linear propagation losses due to absorption andscattering in the silicon-nanocrystal waveguide/composite.The terms in the first square brackets on the right-hand sideof equation (23) describe the effects of self-phase modulation(SPM) and two-photon absorption (TPA) of mode ν atfrequency ωµ, as well as the effects of cross-phase modulation(XPM) and cross-TPA at this frequency due to the interactionbetween the νth mode and other modes (ν′ 6= ν) at ωµ.The terms in the second square brackets take into accountXPM and cross-TPA between fields of different frequencies(µ′ 6= µ) propagating either in the same mode ν or in differentmodes (ν′ 6= ν). The third square brackets are responsiblefor the Raman-mediated energy exchange between fields ofdifferent frequencies, either of the same mode or differentmodes. Finally, the term proportional to the free-carrierdensity allows for the effects of free-carrier dispersion (FCD)and free-carrier absorption (FCA) exhibited by field µ inmode ν due to the electrons and holes generated throughTPA and cross-TPA at all field frequencies and in all possiblemodes.

The presence of factors like (8 + 7ρ)/45 and29/45, together with coefficients ζ and ξ , which appeardue to the modification of the light–matter interactioninside silicon-nanocrystal composite as compared to thatin silicon, causes the derived equations to differ fromtheir well-known analogs in silicon-on-insulator waveguides[63, 60, 55, 35, 56, 64].

4. Material parameters

The material parameters entering equations (23) and (24)deserve special consideration.

5

J. Opt. 16 (2014) 015207 I D Rukhlenko

The first-order dispersion parameter is the inverse ofthe group velocity, β(1)µν = (v

µνg )−1, which in the case of

a single pulse describes its propagation in the z directionas a whole. Both the group velocity dispersion (GVD),β(2)µν , and the third-order dispersion (TOD) coefficient,

β(3)µν , strongly depend on the geometry and composition of

the silicon-nanocrystal waveguide [65, 62]. For example,the values of these coefficients for a quasi-TM modepredominantly confined within a gap 200 nm wide and 50nm thick of a Si–SiNCs/SiO2–Si waveguide that is 450 nmthick are −1.052 ps2 m−1 and 0.025 ps3 m−1 at the 1550 nmwavelength [50]. If the total waveguide thickness is reducedto 290 nm, these parameter values become 7.5 ps2 m−1 and0.055 ps3 m−1.

It should be noted that dispersion flattening in silicon-nanocrystal waveguides is quite a difficult engineeringproblem due to the tight optical confinement provided bythem and the strong waveguide dispersion. One solution tothis problem was recently suggested by Zhang et al [66],who used a transition from the strip mode to the slot modeof a silicon–SiNCs/SiO2–silica slot waveguide to realize anextremely flat waveguide dispersion of 0± 16 ps nm−1 km−1

over a 553 nm wavelength range (from 1562 to 2115 nm).Linear propagation losses in the silicon-nanocrystal

waveguide/composite may vary in a wide range whiledepending critically on the operating wavelength, waveg-uide geometry, mode polarization, and silicon-nanocrystaldeposition techniques [6, 67–69]. For example, for ahorizontal slot waveguide fabricated by Martinez et al [6]using the low-pressure chemical vapor deposition (LPCVD)technique, these losses were about 20 dB cm−1. If a similarwaveguide is fabricated using the plasma-enhanced chemicalvapor deposition (PECVD) technique [67], the linear lossesare reduced to 4 dB cm−1 and may be even lower attelecommunication wavelengths [70].

In estimating the values of the TPA coefficients

βµµTPA, cross-TPA coefficients βµµ

′

TPA, and Kerr coefficient n2,one needs to remember that they characterize individualnanocrystals rather than the entire composite. These

coefficients are related to respective quantities 〈βµµTPA〉, 〈βµµ′

TPA〉,and 〈n2〉 characterizing the composite as a whole through the

attenuation factor ξ via βµµTPA = 〈βµµTPA〉/ξ , βµµ

′

TPA = 〈βµµ′

TPA〉/ξ ,and n2 = 〈n2〉/ξ . Using the values 〈βµµTPA〉 ≈ 7 cm GW−1

reported by Spano et al [71] for a volume fraction of siliconnanocrystals of 8% [ξ−1(0.08) ≈ 320], we obtain β

µµTPA ≈

2240 cm GW−1. While similar values of 〈βµµTPA〉were reportedby other experimental groups [42, 50, 43, 72], it should benoted that the TPA coefficient of the composite depends onthe optical intensity and doping of silicon nanocrystals, andmay reach values as high as 1.7 m GW−1 [73]. As for theKerr coefficient, which was measured to be 〈n2〉 ≈ (2 ±1) × 10−12 cm2 W−1 [42] for the composite with a siliconconcentration of about 54 atomic per cent (f ≈ 27%), we findthat n2 ≈ (5.5± 2.7)× 10−11 cm2 W−1.

Like for the case of linear losses, the values ofthe TPA and Kerr coefficients depend on the method ofdeposition of the silicon nanocrystals [44]. For instance,

Martinez et al [6, 74] obtained 〈n2〉 = 4×10−13 cm2 W−1 and〈βµµTPA〉 = 5 cm GW−1 for their silicon-nanocrystal composite

with 0.08 silicon excess, and 〈n2〉 = 2 × 10−12 cm2 W−1

and 〈βµµTPA〉 = 50 cm GW−1 for the sample with 0.17 siliconexcess. These values correspond to coefficients n2 ≈ 12.8 ×10−11 cm2 W−1 and βµµTPA ≈ 1600 cm GW−1, and n2 ≈ 16×10−11 cm2 W−1 and βµµTPA ≈ 4080 cm GW−1 for individualcrystallites. It is seen that the TPA and Kerr coefficients ofsilicon nanocrystals may exceed those of bulk silicon by morethan two orders of magnitude.

The values of n2 and βTPA can be calculated fora wide range of experimental conditions and materialparameters—including annealing temperature, silicon excess,excitation power and wavelength—using equation (26b) andthe values of 〈n2〉 and 〈βTPA〉measured by Minissale et al [85]for the silicon-rich oxide and silicon-rich nitride composites.

The enhancement of stimulated Raman scattering(SRS) in low-dimensional silicon is even larger than theenhancement of the TPA and Kerr effects, and was reportedto be given by a factor of 105 [51, 41]. In particular, 〈gR〉 =

438 000 cm GW−1 for the silicon-nanocrystal composite ofeffective refractive index neff ≈ 2 (f ≈ 32%), which impliesthat gR = 〈gR〉/ξ ≈ 7.6 × 106 cm GW−1. This value is 106

and 109 times larger than the Raman gain coefficients in bulksilicon and fused silica, respectively [35, 36]. As for the shapeand position of the Raman gain spectrum, it gets broadenedfrom 105 GHz in bulk silicon to 2 THz in 2 nm siliconnanocrystals and, at the same time, red-shifted from 15.6 THzdown to about 15 THz [75, 76, 86].

An estimate of the absorption cross-section of siliconnanocrystals at 1535 nm, σα = 4 × 10−19 cm2, was given byYuan et al [78]. A much higher value of 2.6 × 10−17 cm2

was later reported at 1550 nm wavelength by Creazzoet al [77]. The measurement of the free-carrier dispersion(FCD) coefficient performed by Spano et al [71] showedthat, much like the TPA coefficient, it depends on the opticalintensity, with a low-intensity value of 〈σn〉 = (1.2 ± 0.3) ×10−22 cm3 for f = 8%, which corresponds to σn = 〈σn〉/ζ ≈

(3.4± 0.85)× 10−21 cm3.The large surface-to-volume ratio of silicon nanocrystals

(due to their small size) enhances the scattering of free carriersand hastens their recombination as compared to the casesfor bulk silicon and silicon-on-insulator waveguides. Theresulting effective free-carrier lifetime in silicon nanocrystalsmay be 10–100 times smaller than that in silicon, thusranging from 10 to 100 ps [79–81]. Together with thereduced absorption cross-section of free carriers inside thenanocrystals, faster recombination of TPA-generated andcross-TPA-generated electrons and holes may significantlyreduce the detrimental impact of FCA on optical propagationin silicon-nanocrystal waveguides/composites.

The number and values of relevant effective modeindices and mode-overlap factors critically depend on thewave propagation conditions [83]. If a pair of opticalfields, say, pump (µ = p) and signal (µ = s), interact inthe form of linearly polarized beams inside an unboundedsilicon-nanocrystal composite, then there are 48 generally

different mode-overlap factors (γ µµ′

νν′, λµµ

′

νν′, and ψµµ

′

νν′with

6

J. Opt. 16 (2014) 015207 I D Rukhlenko

µ,µ′ = p, s and ν, ν′ = x, y) and four generally differenteffective mode indices (npx, npy, nsx, and nsy). However, if thecomposite is placed in the slot between two silicon slabs toenhance the electrical field perpendicular to the slot (aligned,say, in the y direction), then the resulting waveguide supportsonly quasi-TM modes and is characterized by only fourmode-overlap factors of each type (e.g. γ pp

yy , γpsyy , γ sp

yy , and γ ssyy )

and two effective mode indices (npy and nsy) [50]. Numericalsimulations show that in this instance they range from about0.1 to 1 and from 1 to 2.4, respectively, and are functions ofthe slot width and thickness, and the volume fraction of thesilicon nanocrystals [83].

The increase in the refractive index of fused silica dueto its doping with silicon nanocrystals can be accuratelymodeled using the approximation of the effective medium[52, 87, 82, 88]. This approximation leads to the well-knownexpression for the effective refractive index of the silicon-nanocrystal composite [40, 82],

neff =12

{(3f − 1) ε1 + (2− 3f ) ε2

+

√[(3f − 1) ε1 + (2− 3f ) ε2]2

+ 8ε1ε2

}1/2

, (25)

in which ε1 ≈ (3.48)2 is the permittivity of silicon and ε2 ≈

(1.45)2 is the permittivity of silica. Since the fraction ofsilicon nanocrystals in the composites is typically below 35%,neff may lie anywhere between 1.45 and 2.1 [61].

The factors describing the attenuation of linear andnonlinear effects inside the silicon-nanocrystal composite aregiven by [1, 40]

ζ =

√ε1

εeff

(3f − 1) εeff + ε2{[(3f − 1) ε1 + (2− 3f ) ε2]2

+ 8ε1ε2}1/2

(26a)

and

ξ =[(3f − 1) εeff + ε2]2

f{[(3f − 1) ε1 + (2− 3f ) ε2]2

+ 8ε1ε2} . (26b)

Given the above values of permittivities ε1 and ε2, thesefactors can lie anywhere from zero to 0.27 and 0.07,respectively. When the filling factor is of the order ofor less than 1%, equation (26) approximately yields ζ ≈9ε2√ε1ε2/(ε1 + 2ε2)

2f and ξ ≈ (3ε2)4/(ε1 + 2ε2)

4f .The physical meaning of the factor ξ follows from its

definition, according to which it is the ratio of the third-ordersusceptibility tensor components of the silicon-nanocrystalcomposite to the respective components of the third-ordersusceptibility tensor of silicon [40]. Since in derivingequation (26b) we neglected the third-order susceptibility ofsilica and assumed that the nonlinear optical response ofthe composite stems solely from the third-order nonlinearityof the silicon nanocrystals, the factor ξ characterizes theweakening of the nonlinear effects of the silicon nanocrystalsdue to their embedding in fused silica.

The nonlinear anisotropy factor ρ is related tothe components of the electronic susceptibility tensor

of silicon calculated at the degenerate frequency ω asρ = 3χe

xxyy(−ω;ω,−ω,ω)/χexxxx(−ω;ω,−ω,ω) [40, 36].

Experiments show that ρ ≈ 1.27 near the referencewavelength of 1.55 µm [84].

The typical values of the material parameters discussedare summarized in table 1. Some parameter values are seento span over broad ranges, which indicates that they dependheavily on the experimental conditions, such as the annealingtemperature, excitation wavelength, and excess of silicon infused silica. The experimental conditions in each instance canbe found from the references provided in the last column oftable 1.

5. The interaction of continuous waves

Equations (23) and (24) are significantly simplified forcontinuous waves, whose amplitudes are independent of time.By introducing the Raman gain profile [36]

gR(ω) =2igR0R�R

�2R − 2i0Rω − ω2

, (27)

we may evaluate the integrals in equation (23) as∫ t

−∞

H(t − t1) eiωµµ′ (t−t1) dt1 ≈2cε0εeff

ikµgR(ωµµ′). (28)

If the optical fields interact inside a silicon-nanocrystalwaveguide, then, by assuming the absence of phase matchingbetween them and neglecting the spatially varying exponentialterms in equation (23), we arrive at the following equations:

1aµν

(∂

∂z+αµν

2

)aµνξ

=

(in2kµ −

βµµTPA

2

)[8+ 7ρ

45

∑ν′

2γ µµνν′

Iµν′

+

(8+ 7ρ

45λµµνν +

1− ρ9

γ µµνν

)Iµν

]

+

∑µ′ 6=µ

(2in2kµ − β

µµ′

TPA

)[8+ 7ρ45

∑ν′

γµµ′

νν′Iµ′ν′

+

(8+ 7ρ

45ψµµ

′

νν +13+ 2ρ

45λµµ

′

νν

)Iµ′ν

]

+ 2∑µ′ 6=µ

gR(ωµµ′

) [−

1645

∑ν′

γµµ′

νν′Iµ′ν′

+

(1945 λ

µµ′

νν +2945 ψ

µµ′

νν

)Iµ′ν

]

− ζ τeffneff

nµν

(ω0

ωµ

)2 (σα

2+ iσnkµ

){∑µ

βµµTPA

2hωµ

×

∑ν

[8+ 7ρ

45

∑ν′

2γ µµνν′

Iµν′ +(

8+ 7ρ45

λµµνν

+1− ρ

9γ µµνν

)Iµν

]Iµν +

∑µ

∑µ′ 6=µ

βµµ′

TPA

hωµ

7

J. Opt. 16 (2014) 015207 I D Rukhlenko

Table 1. Material parameters of the coupled-amplitude and rate equations, given either at or near the 1.55 µm wavelength. See thereferences in the last column for the experimental conditions.

Parameter Description Values References

β(2)µνGroup velocity dispersion coefficient of silicon-nanocrystalwaveguide

7± 10 ps2 m−1 [50, 65]

0± 16 ps nm−1 km−1 [66]β(3)µν Third-order dispersion coefficient of silicon-nanocrystal

waveguide−0.05± 0.12 ps3 m−1 [50]

αµν Linear loss coefficient of silicon-nanocrystal waveguide 1–20 dB cm−1 [74, 6, 68, 69]

βµµTPA Two-photon absorption coefficient of silicon nanocrystals 16–40 m GW−1 [42, 6, 71, 72]

n2 Nonlinear Kerr coefficient of silicon nanocrystals 3–16 mm2 GW−1 [42, 43, 45]

gR Raman gain coefficient of silicon nanocrystals 106–107 cm GW−1 [51, 41]�R Raman shift in silicon nanocrystals 15–15.6 THz [75, 76, 36]20R Bandwidth of Raman gain spectrum in silicon nanocrystals 0.1–2 THz [75, 76]σα Absorption cross-section of free carriers in silicon nanocrystals 10−19–10−17 cm2 [77, 78]σn Free-carrier dispersion coefficient in silicon nanocrystals 3.4± 0.85 nm3 [71]τeff Effective free-carrier lifetime in silicon nanocrystals 10–100 ps [50, 79–81]neff Effective refractive index of silicon-nanocrystal composite 1.45–2.1 [40, 49, 82]nµν Effective mode index of mode ν at frequency ωµ 1–2.4 [83, 61]

γµµ′

νν′Mode-overlap factor in silicon-nanocrystal waveguide 0.1–1 [83]

ζ Linear susceptibility attenuation factor, ζ ≈ 0.36f for f � 1 0–0.27 [1]ξ Third-order susceptibility attenuation factor, ξ ≈ 0.022f for

f � 10–0.07 [40]

ρ Nonlinear anisotropy factor in silicon nanocrystals 1.27 [63, 36, 84]ω0 Reference frequency ω0 = 2πc/λ0 λ0 = 1.55 µm [36]

×

∑ν

[8+ 7ρ

45

∑ν′

γµµ′

νν′Iµ′ν′ +

(8+ 7ρ

45ψµµ

′

νν

+13+ 2ρ

45λµµ

′

νν

)Iµ′ν

]Iµν

}. (29)

By assuming further that γ µµ′

νν = ψµµ′

νν = λµµ′

νν , wemay rewrite these equations in a compact form using onlyreal-valued field intensities as

1Iµν

(∂

∂z+ αµν

)Iµνξ

= − 2∑µ′

(Aµµ′

∑ν′ 6=ν

γµµ′

νν′Iµ′ν′ + Bµµ′γ

µµ′

νν Iµ′ν)

− ζ τeffσαneff

nµν

(ω0

ωµ

)2∑µ

∑µ′

∑ν

(Cµµ′

∑ν′ 6=ν

γµµ′

νν′Iµ′ν′

+ Dµµ′γµµ′

νν Iµ′ν)

Iµν, (30)

where

Aµµ′ =8+ 7ρ

45βµµ′

TPA +3245

Re[gR(ωµµ′

)] (1− δµµ′

)(31a)

Bµµ′ =29+ 16ρ

45βµµ′

TPA

(2− δµµ′

)−

6445 Re

[gR(ωµµ′

)] (1− δµµ′

), (31b)

Cµµ′ =8+ 7ρ

45

βµµ′

TPA

hωµ, (31c)

and

Dµµ′ =29+ 16ρ

45

βµµ′

TPA

2hωµ(2− δµµ′). (31d)

A different set of equations describes the interaction ofoptical beams inside the silicon-nanocrystal composite. It iseasy to show that the mode-overlap factors in this case areequal to unity. If we assume that all beams have equal lateralprofiles and are, therefore, characterized by the same effectivemode area, Aeff, then the coupled-mode equations become(∂

∂z+αµν

2

)bµνξ

=

(in2kµ −

βµµTPA

2

)[8+ 7ρ

45

∑ν′

(2Iµν′bµν + b2µν′b

∗µν)

+1− ρ

9Iµνbµν

]+

∑µ′ 6=µ

(2in2kµ − β

µµ′

TPA

) [8+ 7ρ45

×

∑ν′

(Iµ′ν′bµν + bµν′bµ′ν′b

∗

µ′ν + bµν′b∗

µ′ν′bµ′ν)

+1− ρ

9Iµ′νbµν

]+ 2

∑µ′ 6=µ

gR(ωµµ′)

×

[∑ν′

(2945 bµν′bµ′ν′b

∗

µ′ν

+2945 bµν′b

∗

µ′ν′bµ′ν −1645 Iµ′ν′bµν

)−

29

Iµ′νbµν

]− ζ τeff

(ω0

ωµ

)2 (σα

2+ iσnkµ

)bµν

8

J. Opt. 16 (2014) 015207 I D Rukhlenko

×

∑µ

∑ν

{βµµTPA

2hωµ

(8+ 7ρ

45

∑ν′

2Iµν′

+13+ 2ρ

45Iµν

)Iµν

+

∑µ′ 6=µ

βµµ′

TPA

hωµ

(8+ 7ρ

45

∑ν′

Iµ′ν′ +7+ 3ρ

15Iµ′ν

)Iµν

+8+ 7ρ

45Re

[βµµTPA − 2ikµn2

2hωµ

∑ν′ 6=ν

(bµν′b

∗µν

)2+

∑µ′ 6=µ

βµµ′

TPA − 2ikµn2

hωµ

×

∑ν′ 6=ν

(bµ′ν′b

∗

µ′ν + b∗µ′ν′bµ′ν)

bµν′b∗µν

]}, (32)

where bµν = aµν/√

Aeff and Iµν = |bµν |2.

6. Conclusions

We have generalized the recently developed theory ofnonlinear propagation of two quasi-monochromatic opticalfields through silicon-nanocrystal waveguides to the caseof an arbitrary number of quasi-monochromatic opticalfields of arbitrary polarizations. The new theory takesinto account the nonlinear effects of self-phase modulation(SPM) and two-photon absorption (TPA) of different fieldspropagating in different modes, as well as the effectsof cross-phase modulation (XPM), cross-TPA, stimulatedRaman scattering (SRS), free-carrier absorption (FCA),and free-carrier dispersion (FCD) due to the fields ofall frequencies propagating in all possible modes. Ourtheory can be used for computationally efficient modelingof a wide variety of nonlinear optical phenomena inboth silicon-nanocrystal composites and silicon-nanocrystalwaveguides with a uniform distribution of the crystallites’axes in space.

Acknowledgment

This work was sponsored by the Australian ResearchCouncil, through its Discovery Early Career ResearcherAward DE120100055.

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