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Fundamental Limitations to the Ultimate Kerr Nonlinear Performance of Plasmonic Waveguides Guangyuan Li, Martijn de Sterke, and Stefano Palomba ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.7b01331 • Publication Date (Web): 12 Jan 2018 Downloaded from http://pubs.acs.org on January 13, 2018

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Fundamental Limitations to the Ultimate Kerr Nonlinear Performance of Plasmonic Waveguides Guangyuan Li,∗,†,‡ C. Martijn de Sterke,†,‡ and Stefano Palomba∗,†,¶ †Institute of Photonics and Optical Science (IPOS), School of Physics, The University of Sydney, NSW 2006, Australia ‡Centre for Ultrahigh bandwidth Devices for Optical Systems (CUDOS), School of Physics, The University of Sydney, NSW 2006, Australia ¶Sydney Nano Institute, The University of Sydney, NSW 2006, Australia E-mail: [email protected]; [email protected]

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Abstract Plasmonic waveguides can greatly enhance nonlinear light-matter interactions through strong field confinement. However, achieving high performance nonlinear plasmonic devices remains challenging because of optical losses and material damage. Here we investigate the ultimate Kerr nonlinear performance of plasmonic waveguides. We account for optical damage by requiring that the local electric field intensity does not exceed the damage threshold of the nonlinear material. This allows us to factorize the fundamental limitations into those stemming from the constituent materials’ linear and nonlinear properties, and from the mode characteristics. We define quality coefficients for the metal and for the nonlinear dielectric so that these materials can be selected appropriately, and illustrate their utility by application to surface plasmon polaritons (SPPs). We further propose the concept of nonlinear effectiveness in order to quantify a mode’s ability to exploit the material’s nonlinearity. We find that the full exploitation of the material’s maximum nonlinearity requires a uniform field in addition to slow light effects. This is exemplified by the discovery that the maximum nonlinearity of Metal-Dielectric-Metal structures can be stronger than that of the bulk material. These counterintuitive insights provide deep understanding into the ultimate performance of nonlinear waveguides, and point to novel approaches to achieve practical, high performance nonlinear plasmonic devices.

Keywords Kerr effect, maximum nonlinearity, plasmonic, waveguide, factorization, slow light

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Nonlinear photonic devices are attractive for signal generation and processing, both at the classical and the quantum levels, as well as in sensing and imaging applications, because they combine an ultrafast response time and high bandwidth, with low energy consumption. 1–3 Plasmonics has emerged as an exciting new platform for Kerr nonlinear devices since it allows subwavelength light confinement and hence greatly enhanced nonlinear effects. 4 Calculations show that the nonlinear coefficient γ of plasmonic waveguides can be orders of magnitude larger than in all-dielectric waveguides. 5 However considering only γ has the misleading implication that the nonlinear effects can increase indefinitely with power. In practice, for any real material there is an upper limit to the driving power and thus to the nonlinear refractive index change that can be induced optically, which may be due to material damage. 6 As illustrated in Figure 1, for a plane wave in bulk nonlinear material (blue curve), the strength of the nonlinear effect (measured by the nonlinear phase shift ∆ΦNL per unit length) initially rises linearly with driving power P0 , i.e., ∆ΦNL /L = k0 n2 P0 /A0 , where k0 = 2π/λ, n2 is the nonlinear refractive index of the material, and A0 = P0 /I is the beam size with I the light intensity. When the electric field intensity reaches the maximum allowed value of |Ebulk,max |2 before the material is damaged, the material reaches its maximum nonlinear index change ∆nmax , which in turn determines the maximum achievable nonlinear effect ∆ΦNL /L = k0 ∆nmax (blue dashed line). A nonlinear optical waveguide (red curve) has similar nonlinear behavior as bulk material. The low-power slope of the waveguide, i.e., the nonlinear coefficient γ = k0 n2 /Aeff , can be much larger than in bulk thanks to field confinement (mode effective area Aeff A0 ) 7 and slow light effects. 8–11 This nonlinear enhancement, defined as the ratio of low-power slopes for a waveguide and for bulk, i.e.,

ENHNL ≡

A0 γ = , k0 n2 /A0 Aeff

(1)

is important for achieving low driving power operation and/or short device length. 7 This is

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the major drive for pursuing ever smaller Aeff in nonlinear guided-wave optics. On the other hand, strong confinement also lowers the maximum admissible input power P0,max , for which the electric field intensity in the nonlinear material reaches |Ebulk,max |2 , and thus the nonlinear index change reaches its maximum, ∆nmax , before damage occurs. This is particularly so for plasmonic waveguides since, even at modest driving power, the strongly enhanced local field may cause damage. 12 Therefore, in contrast to most work in Kerr nonlinear guided-wave optics, which concentrates on the low-power slope γ, we focus on the maximum achievable nonlinear effects in waveguides, γP0,max . NL

/

L

Bulk

W aveguide

Maximum achievable nonlinear effects

k

0

n A

2

P

0,max

n

max

k

0

P

0

0

Figure 1: Nonlinear phase shift per unit length ∆ΦNL /L versus driving power P0 for bulk material (blue), and for a waveguide (red). Both shows an initial linear increase with slope k0 n2 /A0 , and γ, respectively. The maximum achievable nonlinear effect is limited by optical damage (dashed). By further including the limited interaction length because of metal absorption, 12,13 we earlier showed that the ultimate Kerr nonlinear performance of plasmonic waveguides can be characterized by the Figure of Merit (FOM), 14

F ≡ γP0,max Latt ,

(2)

where Latt is the attenuation length. This FOM provides the upper limit to the nonlinear performance of lossy waveguides: 14 at the optimal length that balances the Kerr effect and linear loss, Lopt = ln (3)Latt ≈ 1.1Latt , the maximum conversion efficiency of degenerate fourwave mixing (DFWM) is ηmax = 4F 2 /27 and the nonlinear phase shift is ∆ΦNL,opt = 2F/3. 4


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Constrained by the largest nonlinear refractive index change and the limited interaction length, Kerr nonlinear plasmonic waveguides, proposed or demonstrated, therefore have had modest performance to date. 15–18 The best-explored way to address these constraints is by using novel materials. These include replacing noble metals by alternative plasmonic materials, 19,20 and utilizing materials 21–25 or metamaterials 26,27 with a very strong nonlinearity. However, efforts to improve the nonlinear performance of plasmonic waveguides are not only confused by the rich choices of metallic and nonlinear materials, but also by the large variety of geometries. 28 This is because γ, P0,max and Latt in eq 2 all depend on both the constituent materials and the waveguide geometries, as illustrated in a limited way by comparing two types of plasmonic waveguides, 14 making it challenging to understand why one structure performs better than another. In this work, we provide understanding on how the ultimate Kerr nonlinear performance, i.e., F of plasmonic waveguides is fundamentally limited by the properties of constituent materials and by the mode characteristics. To understand the limitations due to the metal and the nonlinear dielectric, we illustrate with surface plasmon polaritons (SPPs) along the interface between these two materials. To illustrate the modal limitations, we consider additional five elementary one-dimensional (1D) plasmonic modes. In order to quantify a mode’s capability to exploit the material’s nonlinearity without damaging it, we introduce the nonlinear effectiveness. Since it is a modal property it applies to both lossy and lossless nonlinear waveguides. We report the unexpected finding that the maximum nonlinearity of plasmonic Metal-Dielectric-Metal geometry can be larger than that of the bulk material thanks to slow-light effects, which do not rely on a structural resonance and are therefore extremely broadband. These striking findings provide the necessary foundation for identifying novel ways to enable high performance nonlinear plasmonic devices for optical signal processing, sensing, imaging and quantum applications.

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RESULTS AND DISSCUSSION Limitations from materials We consider wavelengths away from the plasma wavelength of the metal to avoid prohibitive absorption, and take all dielectrics to be lossless so that optical loss is entirely due to the presence of metal. We first study the simplest plasmonic mode, the SPP along a metaldielectric interface, as illustrated in Figure 2a. Although all constituent materials contribute to some degree to the overall nonlinearity, the metal’s nonlinear response dominates only when the dielectric is weakly nonlinear because the electric field is mainly confined to the dielectric and rapidly decays in the metal. 29 Since we are interested in the ultimate performance of plasmonic structures, we consider highly nonlinear dielectrics (denoted as “D”) so that the metal’s nonlinear contribution can be neglected (see Supporting Information). We also drive the SPP mode at its maximum admissible power P0,max to avoid damage in the nonlinear medium. This implies that we are not limited by input power. For the SPP mode, eq 2 reduces to (see Supporting Information)

FSPP =

1 |εr,M | |εr,M |2 ∆nmax · 00 q · , 3 2 εr,M n 0 2 2 00 2 2 0 (|εr,M | − n0 ) + (εr,M ) − n0

(3)

where εr,M = ε0r,M + iε00r,M is the metal’s relative permittivity, and n0 is the linear refractive index of the nonlinear dielectric "D". Moreover, if |ε0r,M | n20 , we obtain

FSPP ≈

1 |εr,M |2 ∆nmax · · . 2 ε00r,M n30

(4)

Equation 4 allows us to factorize the contributions of the modal profile (1/2), the metal (|εr,M |2 /ε00r,M ) and the nonlinear dielectric (∆nmax /n30 ) to the SPP Figure of Merit. Therefore

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Re{

M

(c)

y

ex} x

(d)

D

t

M

t

L

H

c

2

D

0 5

1.0

MDM 0 8

m

0 0 2

=

1

0 8

0.5

SPP 60 30

0 3

0 c

10

0 80

1

t

8 1

MDH

MDM

c

600 1.5

M

t

400

MLD

MLD

(f)

m)

LR-SPP

c

M

MDH

(

SR-SPP

c

D

opt

3

D

M

D

t

200

D

SR-SPP

(e)

0

M

z

L

(g)

LR-SPP D

0

(rad)

D

ez}

NL,opt

(b)

SPP Im{

SPP

(a)

F / F

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0.0 1

M

L

att

/

L

2

3

att,SPP

Figure 2: (a)–(f) Schematics of elementary 1D plasmonic modes considered: SPP, LR- and SR-SPPs, hybrid plasmonic modes in MLD and MDH, and MDM gap plasmon mode. Red and blue curves: modal electric fields Im{ez } and Re{ex } (scaled individually in metal and dielectric regions), respectively. tc : central layer thickness. (g) Bottom-left axes: FOM versus attenuation length, both normalized to those of the SPP; top-right axes: nonlinear phase shift versus optimal length. The tangent slope m (the dotted line) is specified with respect to the bottom-left axes. Numbers indicate tc in nanometers, and arrows indicate increasing tc . All results are obtained for λ = 1, 550 nm, nH = 2.0 (Si3 N4 ), and nL = 1.37 (MgF2 ), silver, 30 and highly nonlinear DDMEBT with n0 = 1.8, 21 and assumed ∆nmax /n0 = 10−3 . we can introduce quality coefficients for the metal and for the nonlinear dielectric as

QM ≡

|εr,M |2 , ε00r,M

(5)

QD ≡

∆nmax , n30

(6)

and

respectively, neither of which were reported previously. We can thus write FSPP = 1/2 · QM · QD . The modal contribution in terms of the prefactor 1/2 will be studied later. The rigorously derived expression for QM (defined in eq 5) differs slightly from that used for linear SPP waveguides, 19,31 QSPP = |ε0r,M |2 /ε00r,M , corresponding to the ratio of the real and imaginary parts of the propagation wavevector. If |ε0r,M | ε00r,M , then QM ≈ QSPP and thus future improvements in the properties of metals for linear waveguiding applications 32 will be directly applicable to Kerr nonlinear applications. Based on QM , comparison of noble metals 7


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(gold, silver and copper), aluminum, and five alternative plasmonic materials 33 shows that silver is the best metal in the visible and near-infrared, and that aluminum is the best in the ultraviolet (see Figure S1 in Supporting Information). Fortuitously, because QM ≈ QSPP , our results for Kerr nonlinear SPPs are similar to those based on QSPP for linear SPPs. 19 According to eq 4 and eq 6, QD and FSPP are proportional to ∆nmax /n30 , and thus they can be improved by utilizing larger ∆nmax or, more significantly, smaller n0 . The magnitude of the improvement of the DFWM conversion efficiency is more striking since 2 ηmax,SPP = 4FSPP /27 ∝ (∆nmax )2 /n60 . For example, given the same ∆nmax , if we were able

to substitute a nonlinear dielectric of n0 = 1.8 with another nonlinear dielectric of n0 = 1.4, ηmax,SPP would increase by a factor of (1.8/1.4)6 ≈ 4.5. These results suggest that the ultimate nonlinear performance of the SPP shows a cubic dependence on the linear property of the nonlinear dielectric, and a linear dependence on the maximum nonlinear index change of the material. This counterintuitive finding is encouraging since it points to a promising alternative approach to achieve high performance nonlinear SPP waveguides by exploring nonlinear materials or metamaterials of low linear refractive index.

Limitations from mode characteristics We now study the limitations to the ultimate performance arising from the mode characteristics. Besides the SPP, we also consider another five 1D plasmonic modes, as shown in Figure 2b-f: the long-range and short-range SPPs (LR- and SR-SPPs) supported by a Dielectric-Metal-Dielectric (DMD) structure, hybrid plasmonic modes supported by a MetalLow index spacer-Dielectric (MLD) and a Metal-Dielectric-High index superstrate (MDH), and the Metal-Dielectric-Metal (MDM). All these modes are based on the same metal (“M”) and highly nonlinear dielectric (“D”) as the SPP, and they are composed of two semi-infinite layers sandwiching a central layer of thickness tc , so that the fields can be calculated analytically. Based on these modes sophisticated plasmonic structures with two or three dimensional confinement can be understood, as will be exemplified later. We take tc to start at 20 nm for 8


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the DMD LR-SPPs, to start at 0.5 nm for the DMD SR-SPPs and the MDM gap plasmons, and to start at 2 nm for the other modes. As in the analysis of SPPs, we consider highly nonlinear dielectric in order to study the ultimate nonlinear performance, so that the metal’s nonlinear contribution is negligible (see Figure S2 and Figure S3 in Supporting Information). Figure 2g shows the FOM versus the attenuation length (bottom and left axes), both normalized to those of the SPP so as to focus on the limitations from the mode properties. The corresponding physical quantities, the nonlinear phase shift versus the optimal device length (top and right axes), are also shown. It shows that the ultimate nonlinear performance of most of these plasmonic modes simply scales with their attenuation lengths. Exceptions are the MDH structure, which performs worse, and the MDM, which can generate the same nonlinear effect as the SPP or the SR-SPP in a shorter length. This behavior can be quantified by the tangent slope m in Figure 2g with respect to the bottom-left axes, which corresponds to FSPP m F =m ≈ ∆nmax k0 , Latt Latt,SPP 2

(7)

where for the SPP m = 1, corresponding to the prefactor 1/2 in eq 4, and the approximation assumes that |0r,M | 00r,M and |0r,M | n20 (see Supporting Information). The proportionality to ∆nmax k0 can be understood in analogy to plane wave in bulk nonlinear dielectric (blue dashed line in Figure 1). Since ∆ΦNL,opt = 2F/3 and Lopt = ln (3)Latt , the slope in Figure 2g with respect to the top-right axes is ∆ΦNL,opt 2/3 F m = ≈ ∆nmax k0 . Lopt ln (3) Latt 3 ln (3)

(8)

Evidently, therefore, the prefactors m/2 in eq 7 and m/(3 ln (3)) in eq 8 express the degree to which a mode exploits the maximum nonlinear index change afforded by the nonlinear material. To formalize this idea, we introduce the concept of nonlinear effectiveness EFFNL , which allows us to understand the ultimate nonlinear performance of not only different plasmonic

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waveguide modes, but, as we shall see, also all-dielectric waveguide modes. We define it through EFFNL ≡

∆ΦNL,WG , ∆ΦNL,bulk

(9)

where the numerator and the denominator are the nonlinear phase shifts for the waveguide mode and for a plane wave in bulk nonlinear material, respectively, both driven by their own maximum admissible input powers P0,max , so that for the plane wave in bulk EFFNL = 1. For plasmonic waveguide modes, ∆ΦNL,WG = ∆ΦNL,opt and ∆ΦNL,bulk = ∆nmax k0 Lopt where we take the device length to be the optimal length Lopt , and thus EFFNL = m/(3 ln (3)) from eq 8. For all-dielectric waveguide modes, which we take to be lossless, ∆ΦNL,WG = γP0,max L and ∆ΦNL,bulk = ∆nmax k0 L with L the device length. Therefore, eq 9 can be written as (see Supporting Information)

EFFNL = f` ·

cε0

R NL

n20 |e|2 · (|e|2 /max{|e|2 }) dA R , n0 ∞ (e × h∗ ) · zˆdA

(10)

where for lossy modes f` ≡ 2/(3 ln (3)) to include the effect of absorption loss at the optimal R R device length (see eq 8), whereas f` ≡ 1 for lossless modes. NL (·)dA and ∞ (·)dA are integrals over the nonlinear material and the entire cross section, respectively, and {e, h} are the mode profiles. The weight factor |e|2 /max{|e|2 } corresponds to the field uniformity in the nonlinear material. In order to understand the physics of EFFNL , we rewrite eq 10 in an intuitive way as the product of factors that express the effects of losses, of slow light, and of the electric energy overlap with the nonlinear medium,

EFFNL = f` · S · ρNL ,

(11)

where S ≡ vpw /ve is the slow-down factor with vpw ≡ c/n0 the velocity of a plane wave in bulk, ve = P/W the energy velocity of the mode, and the mode energy per unit length

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W =

1 4

R ∞

[µ0 |h|2 + ε0 [∂(ωε0r )/∂ω]ω0 |e|2 ] dA.

In eq 11, ρNL (0 ≤ ρNL ≤ 1) quantifies the electric energy ratio in the nonlinear material weighted by the field uniformity (|e|2 /max{|e|2 }) since, R

ε n2 |e|2 NL 0 0

· (|e|2 /max{|e|2 }) dA 2W R 2 2 2 n |e| · (|e| /max{|e|2 }) dA NL R 0 , ≈ [∂(ωε0r )/∂ω]ω0 |e|2 dA ∞

ρNL =

(12)

For dispersive metal [∂(ωε0r,M )/∂ω]ω0 = ε0r,M +ω0 (∂ε0r,M /∂ω)ω0 whereas [∂(ωε0r )/∂ω]ω0 = n20 for the dispersionless nonlinear material. Equation 12 implies that a large ρNL requires strong and uniform electric energy confinement in the nonlinear material. 10

800

0

(b)

(c) 10

Si-Slot

25

5

200

Im{

Air

ez}

Re{

ex }

Si

10

SPP

-1

SR-SPP

10

t t t

Si

D

1

c

Si

Si

Air

S

Bulk

MDH

10

10 10

t

c

100

(nm)

0

-1

Si-Slot

-2

1

10

10

MDM 85

1

EFFNL

(a)

NL

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1,000

0

1

10

t

c

100

1,000

10

-2

1

(nm)

10

t

c

100

1,000

(nm)

Figure 3: (a) Weighted electric energy ratio, (b) slow-down factor, and (c) nonlinear effectiveness versus central layer thickness. The horizontal dotted lines indicate unity, the value for a plane wave in bulk, and the dashed lines indicate the values for the SPP. We take tc to start at 10 nm for the Si-Slot TM0 mode. The MDH mode has a cutoff at tc = 85 nm, so its curve terminates at this point (square). Results for the Si-Slot are obtained with tSi = 100 nm and nSi = 3.5. To validate these results, we illustrate with four plasmonic modes, the SPP, the SR-SPP, the MDH and the MDM, and with one all-dielectric waveguide, a Si-Slot waveguide (inset of Figure 3b). The LR-SPP and MLD modes are not shown for clarity since they have almost the same values as the SPP. The Si-Slot waveguide is considered since it can also confine light to tens of nanometers. Since we are interested in nonlinear contributions of the highly nonlinear dielectric “D”, the nonlinear contributions of Si are neglected. According to Figure 3a the MDM and the SR-SPP behave differently from all the other modes, plasmonic or all-dielectric. It shows that for all the other modes ρNL is of order 11


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unity and drops to zero as tc → 0 since the electric energy is no longer confined to the nonlinear material. Similarly, Figure 3b shows that S ≈ 1 for all these modes, indicating that slow-light effects are weak or absent. In contrast, for the MDM ρNL remains large even when tc is as small as 0.5 nm, a thickness for which classical theories still approximately apply. 34 This is because the MDM gap plasmon mode can have indefinitely strong and uniform energy confinement even as tc → 0. 35 In addition, the MDM has substantial slowlight effects for small values of tc , leading to a longer effective path length through the nonlinear dielectric, thus increasing the nonlinear effects. According to eq 11 the nonlinear effectiveness in Figure 3c can be understood as the product of ρNL in Figure 3a and S in Figure 3b, multiplied by the constant f` . For large tc , the characteristics of the SR-SPP and the SPP modes are very similar. As tc → 0, the slow-down factor S for the SR-SPPs and the MDM gap plasmons are the same, because in this limit these modes have the same dispersion relation. 36,37 However, the weighted electric energy ratio ρNL for the MDM is approximately twice as large as that for the SR-SPP because the electric energy in the nonlinear “D” region is approximately uniform in the MDM, but decays exponentially for the SR-SPP, resulting in an additional factor 1/2 for the SR-SPP. Thus the nonlinear effectiveness EFFNL of the MDM is also twice as large as that of the SR-SPP. Therefore, the MDM with small tc is thus particularly advantageous for nonlinear optical applications, since it is unique in that it combines strong slow-light effects with strong and uniform energy confinement in the nonlinear dielectric. Although the Si-Slot geometry can strongly and uniformly confine the electric field in a slot that is tens of nanometers wide, 38 the electric energy is not confined well to the dielectric "D" thus ρNL → 0 for such small tc . This striking result confirms our insight from eq 12 that a large ρNL requires strong and uniform electric energy confinement (rather than electric field confinement) in the nonlinear material. The slow-light effects in the MDM arise because the energy propagates forwards in the nonlinear dielectric, but backwards in the metal due to metal’s negative permittivity, as

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1.0

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0.6

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(b)

0.2

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z(

m)

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EFF

120

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30

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60 30 0.5

0.0

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NL

20

EFF

x (nm)

(a)

S

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ACS Photonics

10 1.0

1.5

( m)

Figure 4: (a) Normalized electric energy density (color) and Poynting vector (arrows, scaled individually in metal and dielectric regions) for tc = 25 nm. (b) Broadband characteristics of slow-light effects and nonlinear effectiveness for tc = 0.5 nm. illustrated in Figure 4a. This mechanism differs from that for slow-light effects in conventional photonic crystals or gratings, which rely on a structural resonance and are thus narrow band. 8–10 Since the permittivities of the dielectric and the metal are opposite in sign for a large wavelength range, the slow-light effects and thus the nonlinear effectiveness for the MDM are intrinsically broadband. Figure 4b shows that, as the wavelength decreases and becomes closer to the plasma wavelength of the metal, both the slow-down factor and the nonlinear effectiveness increase strongly. This larger-than-unity nonlinear effectiveness over extremely broad band, which arises from strong slow-light effects and high, uniform electric energy confinement in the nonlinear dielectric, makes the plasmonic MDM very attractive in nonlinear optics.

Discussion We emphasize that the nonlinear effectiveness, introduced for the first time in this work, is distinct from the nonlinear enhancement defined by eq 1. The nonlinear effectiveness, defined as the ratio of the maximum achievable nonlinear effects in Figure 1, quantifies the mode 13


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ability to exploit a material’s maximum nonlinearity. In contrast, the nonlinear enhancement, defined as the ratio of the low-power slopes in Figure 1, quantifies the enhancement of the low-power strength of nonlinear effects due to field confinement in waveguides. According to eq 11 and 0 ≤ ρNL ≤ 1, the nonlinear effectiveness is usually smaller than unity. A large nonlinear effectiveness requires strong and uniform electric energy confinement (rather than field confinement) in the nonlinear material, and it can only exceed unity through slow light effects. In contrast, the nonlinear enhancement can be significantly larger than unity through field confinement of waveguide 7 or slow light effects in periodic structures, 9,10 and a large nonlinear enhancement requires strongly confined (usually non-uniform) fields 11 according to eq 1. Equation 11 shows that the nonlinear effectiveness is proportional to S, whereas it is known that γ and thus the nonlinear enhancement are proportional to S 2 (see Supporting Information). 9–11 This different dependence on slow-light effects arises because the nonlinear effectiveness is proportional to γP0,max and P0,max ∝ 1/S (see Supporting Information). Though we only illustrated with a few layered plasmonic waveguides that are invariant in the propagation direction, the concept and the physics of the nonlinear effectiveness are general and apply to any type of Kerr nonlinear structure, plasmonic or all-dielectric, 1D, 2D or 3D. As a first example, a nonlinear dielectric waveguide doped with silver nanoparticles was shown to have modest nonlinear performance, even for very small absorption loss or very long Latt . 12 Our insight allows us to understand this result: the extremely localized (non-uniform) electric energy distribution in the nonlinear material, as shown by Figure 2 in ref ( 12), leads to high γ, but small ρNL (eq 12) and thus small EFFNL (eq 11) and small F (eq 8). As a second example, when the 1D MDM configuration is extended into a variety of 2D structures, including V-groove channeled plasmon polaritons, 39 the high confinement may remain, but the field in the gap is likely to lose uniformity. 35 These changes in the linear properties result in significant differences in the Kerr nonlinear characteristics: even though γ may remain high, EFFNL may be reduced. Comparisons and understanding of different 2D Kerr nonlinear waveguides will be further discussed elsewhere.

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For the LR- and SR-SPP modes, the metal requires a certain minimum thickness to be fabricated as a continuous film, 40 and quantum size effects become important for metal thickness approaching a few nanometer. 41,42 However, the inclusion of quantum size effects is outside the scope of this work. For the other modes considered here, quantum size effects are negligible at these dimensions and thus classical theories apply to good approximation.

CONCLUSIONS In conclusion, our analytic investigation provides deep understanding of the fundamental limitations to the ultimate Kerr nonlinear performance of plasmonic waveguides and points to novel approaches for improving the ultimate performance. We find that the ultimate performance of the SPP mode is inversely proportional to the third power of the linear refractive index of the nonlinear dielectric. The novel concept of nonlinear effectiveness quantifies a mode’s capability to exploit the material’s maximum nonlinearity and requires a combination of strong, uniform electric energy confinement (rather than electric field confinement) and slow light effects. This concept is general and is applicable to any Kerr nonlinear waveguide. We also find that plasmonic MDM waveguides are particularly attractive for providing much more maximum nonlinearity than the bulk material can offer over a broad spectral range. With our understanding of two- and three-layer plasmonic waveguide modes established, we are in an ideal position to consider more complicated waveguides in order to find optimal performing designs. Our findings thus advance the development of high performance Kerr nonlinear nanophotonic devices and open the way to practical applications.

AUTHOR INFORMATION Corresponding Author *E-mail: [email protected]; [email protected] ORCID Guangyuan Li: 0000-0002-5969-9157 15


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Notes The authors declare no competing financial interest.

Acknowledgement This work was funded by the Australian Research Council (Discovery Projects scheme, DP150100779). The authors thank Benjamin J. Eggleton and Andrea Blanco-Redondo for fruitful discussions.

Supporting Information Available Section S1: Derivation of the Figure of Merit for the SPP mode; Section S2: Discussion on the nonlinear contributions from metals; Section S3: Derivation of the nonlinear effectiveness and discussion of slow-light effects. This material is available free of charge via the Internet at http://pubs.acs.org.

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For Table of Contents Use Only Fundamental Limitations to the Ultimate Kerr Nonlinear Performance of Plasmonic Waveguides Guangyuan Li, C. Martijn de Sterke, and Stefano Palomba This graph shows the Metal-Nonlinear Dielectric-Metal structure harvests more nonlinearity than bulk material over extremely broad band thanks to slow-light effect and high, uniform electric energy confinement. (b) 180

Metal

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Fundamental Limitations to the Ultimate Kerr ... - ACS Publications

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