supplementary information - nature...entire system. using a simple parallel -plate capacitor model,...

SUPPLEMENTARY INFORMATIONDOI: 10.1038/NPHOTON.2015.127

NATURE PHOTONICS | www.nature.com/naturephotonics 1

Supplementary Information All-Plasmonic Mach-Zehnder Modulator Enabling Optical High-Speed

Communication at the Microscale Authors: C. Haffner1*, W. Heni1, Y. Fedoryshyn1, J. Niegemann1, A. Melikyan2, D. L. Elder3, B. Baeuerle1, Y. Salamin1, A. Josten1, U. Koch1, C. Hoessbacher1, F. Ducry1, L. Juchli1, A. Emboras1, D. Hillerkuss1, M. Kohl2, L. R. Dalton3, C. Hafner1, and J. Leuthold1**.

Affiliations:

1Institute of Electromagnetic Fields (IEF), ETH Zurich, 8092 Zurich, Switzerland.

2Institute IMT, Karlsruhe Institute of Technology (KIT), 76131 Karlsruhe, Germany

3Department of Chemistry, University of Washington, Seattle, WA 98195-1700, United States

Correspondence to * [email protected], **[email protected]

I. OPERATION PRINCIPLE ............................................................................................. 2

II. LIGHT-MATTER INTERACTION BY COUPLED WAVE EQUATION ................ 4

III. OPTIMAL GEOMETRY OF THE PHASE SHIFTER ............................................ 9

IV. GEOMETRY OF FABRICATED DEVICE AND SIMULATION PARAMETERS ...................................................................................................................... 12

V. RF CHARACTERIZATION ......................................................................................... 14

VI. OPTICAL CHARACTERIZATION ......................................................................... 16

VII. COMPARISON WITH PRIOR ART ....................................................................... 18

VIII. REFERENCES ............................................................................................................ 20

© 2015 Macmillan Publishers Limited. All rights reserved

http://dx.doi.org/10.1038/nphoton.2015.127

2 NATURE PHOTONICS | www.nature.com/naturephotonics

SUPPLEMENTARY INFORMATION DOI: 10.1038/NPHOTON.2015.127

I. Operation Principle a - Off-state -5.6 V

b - On-state 4.4 V Sym. mode

Asym. mode

1

0

-1

Ex [a.u.]

① PPI ② Plasmonic phase shifters ③ PPI -8 -4 0 4 8

-35

-25

-15

-5

DC Voltage [V]

Tran

smitt

ance

[dB

]

ER ≈ 26 dB

c

Figure S 1 - 2-D FEM simulations illustrating the operation principle. Depending on the applied voltage the modulator can be switched off or on. a, In the off-state the SPPs in the arms of the MZM recombine such that the signal is coupled to an evanescent mode.b, In the on-state the two SPPs recombine and are mapped to a guided mode. c,The transmittance as a function of the applied voltage is shown in subplot.

The operation scheme of the plasmonic MZM is depicted in Figure S 1. Here, the field

distribution is obtained from full wave 2-D finite element method (FEM) simulations for the on

and the off states. Two in-phase plasmonic modes are exited at the beginning of the lower and

upper arm by means of the left PPI. When they propagate along the arms a relative phase shift

is induced by a built-in asymmetry of the arms width. The Upper arm is ~100 nm broad while

the lower arm is 90 nm. Thus, the phase is offset by 𝜋𝜋/2, even though no voltage is applied,

tuning the MZM to an operation around the quadrature point. This is the optimal operation point

for data transmission relying on on-off keying as the power transfer function in this point is

linear. Figure S 1-a shows the off-state. Here a negative voltage is applied to the island inducing

a relative phase shift in both arms which adds up to 𝜋𝜋. As a consequence an asymmetric mode

is excited in the right PPI. This mode couples to evanescent modes and is lost as the subsequent

silicon waveguide supports only the fundamental symmetric mode and does not guide the odd-

mode as it is in cut-off. In contrast, Figure S 1-b depicts the on-state. In this case, a positive

voltage is applied which counteracts the natural induced phase shift by the linear-electro optic

effect. Thus, SPPs are in-phase at the end of the phase-modulators and a symmetric mode is

excited which is mapped completely to the guided fundamental photonic mode. An extinction

ratio of 6 dB wax found for a peak voltage of 3 V. Higher extinction ratios would be





advantageous though. Since we have introduce a new coupler scheme it needs to be verified if

high extinction ratios can be achieved at all. Figure S 1-c shows the simulated extinction ratios

over the applied voltage for the asymmetric structure as used in this experiment. Extinction

ratios above 20 dB should be achievable.





II. Light-Matter Interaction by Coupled Wave Equation Here, we derive equations that shed light on the origin of the large linear-electro optic

effect in our modulator. We will thereby rely on a coupled wave equation formalism for (2)

nonlinear processes in lossy systems 1. In the plasmonic slot waveguide we describe the

propagating field as a linear combination of a finite set of waveguide modes with slowly varying

envelopes, i.e.

𝑬𝑬(𝑥𝑥, 𝑦𝑦, 𝑧𝑧) = ∑ 𝐴𝐴𝑘𝑘(𝑧𝑧)𝑬𝑬𝑘𝑘(𝑥𝑥, 𝑦𝑦)𝑒𝑒𝑖𝑖(𝛽𝛽𝑘𝑘+𝑖𝑖𝛼𝛼𝑘𝑘2 )𝑧𝑧

𝑘𝑘. (1)

Here, 𝑬𝑬𝑘𝑘(𝑥𝑥, 𝑦𝑦) denotes the mode profile, while 𝛼𝛼𝑘𝑘 and 𝛽𝛽𝑘𝑘 correspond to the mode’s power

attenuation coefficient and the mode’s propagation constant, respectively. The coefficient

𝐴𝐴𝑘𝑘(𝑧𝑧) signifies the envelope as a function of the propagation distance 𝑧𝑧. For our particular

setup, we initially consider four distinct modes in the slot. Specifically, we have an incoming

optical carrier (OC) signal with frequency 𝜔𝜔𝑂𝑂𝑂𝑂 and an applied microwave (RF) signal with 𝜔𝜔RF.

Since the material within the slot has a second order nonlinearity, we generate signals with the

difference frequency (DF) 𝜔𝜔DF = 𝜔𝜔OC − 𝜔𝜔RF and also signals with the sum frequency (SF) at

𝜔𝜔SF = 𝜔𝜔OC + 𝜔𝜔RF, see Figure S 2.

Figure S 2 - New frequency generation. Optical spectrum obtained for a 55 GHz sinusoidal RF-signal modulating the optical carrier (OC). The side bands are created by difference frequency (DF) and sum frequency (SF)





generation. The efficiency of the new frequency generation is defined as the average ratio between side bands and OC.

Following the formalism introduced in Ref. 1 we can derive a set of coupled ordinary

differential equations describing the evolution of the envelopes

𝑑𝑑𝐴𝐴OC𝑑𝑑𝑑𝑑 = −𝛼𝛼OC𝐴𝐴OC + 𝑖𝑖𝜔𝜔OC𝜅𝜅OC,1𝐴𝐴DF𝐴𝐴RF 𝑒𝑒−iΔ𝛽𝛽1𝑧𝑧 + 𝑖𝑖𝜔𝜔OC𝜅𝜅OC,2𝐴𝐴SF𝐴𝐴RF

∗ 𝑒𝑒−iΔ𝛽𝛽2𝑧𝑧,𝑑𝑑𝐴𝐴RF

𝑑𝑑𝑑𝑑 = −𝛼𝛼RF𝐴𝐴RF + 𝑖𝑖𝜔𝜔RF𝜅𝜅RF,1𝐴𝐴OC𝐴𝐴DF∗ 𝑒𝑒iΔ𝛽𝛽1𝑧𝑧 + 𝑖𝑖𝜔𝜔RF𝜅𝜅RF,2𝐴𝐴OC

∗ 𝐴𝐴SF𝑒𝑒−iΔ𝛽𝛽2𝑧𝑧,𝑑𝑑𝐴𝐴DF

𝑑𝑑𝑑𝑑 = −𝛼𝛼DF𝐴𝐴DF + 𝑖𝑖𝜔𝜔DF𝜅𝜅DF𝐴𝐴OC𝐴𝐴RF∗ 𝑒𝑒iΔ𝛽𝛽1𝑧𝑧,

𝑑𝑑𝐴𝐴SF𝑑𝑑𝑑𝑑 = −𝛼𝛼SF𝐴𝐴DF + 𝑖𝑖𝜔𝜔SF𝜅𝜅SF 𝐴𝐴OC 𝐴𝐴RF 𝑒𝑒𝑖𝑖Δ𝛽𝛽2𝑧𝑧.

(2)

Here, the phase mismatch between different modes is included by Δ𝛽𝛽1 = −𝛽𝛽DF − 𝛽𝛽RF + 𝛽𝛽OC

and Δβ2 = −𝛽𝛽SF + 𝛽𝛽RF + 𝛽𝛽OC and the coupling constants 𝜅𝜅 are given by the following overlap

integrals:

𝜅𝜅OC,1 =∬ 𝜖𝜖0𝜒𝜒𝑥𝑥𝑥𝑥𝑥𝑥

(2) ∙ 𝐸𝐸RF,𝑥𝑥 ∙ 𝐸𝐸OC,𝑥𝑥 ⋅ 𝐸𝐸DF,𝑥𝑥𝑆𝑆slot 𝑑𝑑𝜎𝜎

2 ∬ (𝑬𝑬OC,𝑡𝑡 × 𝑯𝑯OC,𝑡𝑡) ∙ �̂�𝒛 𝑑𝑑𝜎𝜎 𝑆𝑆

,

𝜅𝜅OC,2 =∬ 𝜖𝜖0𝜒𝜒𝑥𝑥𝑥𝑥𝑥𝑥

(2) ∙ 𝐸𝐸RF,𝑥𝑥∗ ∙ EOC,𝑥𝑥 ∙ ESF,𝑥𝑥𝑆𝑆slot

𝑑𝑑𝜎𝜎


,

𝜅𝜅RF,1 =∬ 𝜖𝜖0𝜒𝜒𝑥𝑥𝑥𝑥𝑥𝑥

(2) ⋅ 𝐸𝐸RF,𝑥𝑥 ∙ 𝐸𝐸OC,𝑥𝑥 ∙ 𝐸𝐸DF,𝑥𝑥∗

𝑆𝑆slot 𝑑𝑑𝜎𝜎


,

𝜅𝜅RF,2 =∬ 𝜖𝜖0𝜒𝜒𝑥𝑥𝑥𝑥𝑥𝑥

(2) ∙ 𝐸𝐸RF,𝑥𝑥 ∙ 𝐸𝐸SF,𝑥𝑥 ∙ 𝐸𝐸OC,𝑥𝑥∗

𝑆𝑆slot 𝑑𝑑𝜎𝜎


,

𝜅𝜅DF =∬ 𝜖𝜖0𝜒𝜒𝑥𝑥𝑥𝑥𝑥𝑥

(2) ∙ 𝐸𝐸𝑅𝑅𝑅𝑅,𝑥𝑥∗ ∙ 𝐸𝐸OC,𝑥𝑥 ∙ 𝐸𝐸DF,𝑥𝑥𝑆𝑆slot

𝑑𝑑𝜎𝜎


,

𝜅𝜅SF =∬ 𝜖𝜖0𝜒𝜒𝑥𝑥𝑥𝑥𝑥𝑥

(2) ∙ 𝐸𝐸RF,𝑥𝑥 ∙ 𝐸𝐸OC,𝑥𝑥 ∙ 𝐸𝐸SF,𝑥𝑥𝑆𝑆slot 𝑑𝑑𝜎𝜎


.

(3)

In our experiment, only the 𝜒𝜒𝑥𝑥𝑥𝑥𝑥𝑥(2) element of the poled chromophore’s second order

susceptibility tensor contributes to the nonlinear conversion processes 2. Thus, the overlap

integrals in the numerator only contain 𝑥𝑥-field components. In contrast to Ref. 1, we do not

assume the modes to be normalized. Instead, we assume the coefficient 𝐴𝐴𝑂𝑂𝑂𝑂(0) always to be





unity. Thus, the coupling coefficient is depending on the z-directional Poynting vector in the

denominator. This is used below to highlight the physical origin of the nonlinear interaction

and it will help later on to get a more intuitive understanding.

Although the equation system (2) can be solved numerically, it does not yield an

intuitive understanding of the physical effects. Therefore, we apply a series of well-justified

approximations to simplify the equations. First, since our modulator is very short compared to

the wavelength of the microwave signal, we assume 𝐴𝐴𝑅𝑅𝑅𝑅 to be constant and purely real over the

entire system. Using a simple parallel-plate capacitor model, we can write 𝐴𝐴RF = 𝑈𝑈peak/𝑑𝑑/√2,

where d donates the slot width and 𝑈𝑈peak the amplitude of the sinusoidal RF-signal. We further

assume that the mode profiles, the attenuation constants and the propagation constants are

identical for modes oscillating with frequencies 𝜔𝜔𝑂𝑂𝑂𝑂, 𝜔𝜔𝐷𝐷𝑅𝑅 and 𝜔𝜔𝑆𝑆𝑅𝑅. Simulations show

differences are indeed negligible for 𝜔𝜔𝑅𝑅𝑅𝑅 < 100 GHz. The coupling coefficients can then be

written as

𝜅𝜅 ≔ 𝜅𝜅𝑂𝑂𝑂𝑂 ≈ 𝜅𝜅𝑂𝑂𝑂𝑂,1 ≈ 𝜅𝜅𝑂𝑂𝑂𝑂,2 ≈ 𝜅𝜅𝑆𝑆𝑅𝑅 ≈ 𝜅𝜅𝐷𝐷𝑅𝑅 ≈∬ 𝜖𝜖0𝜒𝜒𝑥𝑥𝑥𝑥𝑥𝑥

(2) ∙ 𝐸𝐸RF,𝑥𝑥 ∙ 𝐸𝐸OC,𝑥𝑥2

𝑆𝑆𝑔𝑔𝑔𝑔𝑔𝑔𝑑𝑑𝑑𝑑

2 ∬ (𝑬𝑬OC,𝑡𝑡 × 𝑯𝑯OC,𝑡𝑡) ∙ �̂�𝒛 𝑑𝑑𝑑𝑑 𝑆𝑆

. (4)

Thus, the simplified set of coupled equations then reads

𝑑𝑑𝐴𝐴OC𝑑𝑑𝑑𝑑 = −𝛼𝛼𝐴𝐴OC + 𝑖𝑖𝜔𝜔OC𝜅𝜅𝐴𝐴DF

𝑈𝑈peak

√2𝑑𝑑 + 𝑖𝑖𝜔𝜔OC𝜅𝜅𝐴𝐴SF

𝑈𝑈peak

√2𝑑𝑑,

𝑑𝑑𝐴𝐴DF𝑑𝑑𝑑𝑑 = −𝛼𝛼𝐴𝐴DF + 𝑖𝑖𝜔𝜔DF𝜅𝜅𝐴𝐴OC

𝑈𝑈peak

√2𝑑𝑑,

𝑑𝑑𝐴𝐴SF𝑑𝑑𝑑𝑑 = −𝛼𝛼𝐴𝐴𝑆𝑆𝑅𝑅 + 𝑖𝑖𝜔𝜔SF𝜅𝜅 𝐴𝐴OC

𝑈𝑈peak

√2𝑑𝑑.

(5)

These equations can be solved analytically. We find

𝐴𝐴OC(𝑑𝑑) = 𝐴𝐴0 ∙ 𝑒𝑒−𝛼𝛼𝛼𝛼 cos (𝑈𝑈peak

√2𝑑𝑑∙ 𝑘𝑘 ∙ 𝑑𝑑) ,

𝐴𝐴DF(𝑑𝑑) = 𝑖𝑖 𝐴𝐴0 ∙ ωDF𝜅𝜅𝑘𝑘 ∙ 𝑒𝑒−𝛼𝛼𝛼𝛼 sin (

𝑈𝑈peak

√2𝑑𝑑∙ 𝑘𝑘 ∙ 𝑑𝑑) ,

𝐴𝐴SF(z) = 𝑖𝑖 𝐴𝐴0 ∙ ωSF𝜅𝜅𝑘𝑘 ∙ 𝑒𝑒−𝛼𝛼𝛼𝛼 sin (

𝑈𝑈peak

√2𝑑𝑑∙ 𝑘𝑘 ∙ 𝑑𝑑).

(6)

The conversion vector 𝑘𝑘 is given by





𝑘𝑘 = 𝜅𝜅√𝜔𝜔OC(𝜔𝜔DF + 𝜔𝜔SF) = 𝜅𝜅 𝜔𝜔OC√2 (7)

In summary, the propagating field in the plasmonic slot waveguide is given by a linear

combination of carrier and the generated fields. These oscillate at 𝜔𝜔OC, 𝜔𝜔SF and 𝜔𝜔DF, whereby

only the generated fields are phase shifted by ±𝜋𝜋/2.The analytic solution shows how energy is

transferred back and forth from the optical carrier to the DF and SF signals. A complete

conversion and phase shift of 𝜋𝜋/2 is performed over the length

𝐿𝐿𝜋𝜋/2 = 𝜋𝜋2

𝑑𝑑√2 ∙ 𝑈𝑈𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 ∙ 𝑘𝑘 (8)

With the nomenclature of the Pockels effect, a phase shift of 𝜋𝜋/2 is achieved after a length

𝐿𝐿𝜋𝜋/2 = 𝜋𝜋2𝛥𝛥𝛽𝛽𝑝𝑝𝑒𝑒𝑒𝑒

= 𝜋𝜋2

𝜆𝜆OC𝛥𝛥𝑛𝑛𝑝𝑝𝑒𝑒𝑒𝑒 ∙ 2𝜋𝜋. (9)

The change of the effective refractive group index Δ𝑛𝑛𝑝𝑝𝑒𝑒𝑒𝑒 is useful in quantifying the nonlinear

interactions. Thus, to relate it with the product of the coupling coefficient (𝜅𝜅) and the carrier

frequency (𝜔𝜔OC) we assume small RF-frequencies. Furthermore, using the definition of the

coupling coefficient in (7) one obtains for the change in the effective refractive group index

Δ𝑛𝑛eff = 2 ∙ 𝜆𝜆𝑂𝑂𝑂𝑂2𝜋𝜋 ∙

𝑈𝑈𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑑𝑑 ∙ 𝜔𝜔OC ∙ 𝜅𝜅 = 𝑐𝑐0 ∙

𝑈𝑈𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑑𝑑 ∙

∬ 𝜖𝜖0𝜒𝜒𝑥𝑥𝑥𝑥𝑥𝑥(2) ∙ 𝐸𝐸RF,𝑥𝑥 ∙ 𝐸𝐸OC,𝑥𝑥

2𝑆𝑆𝑔𝑔𝑔𝑔𝑔𝑔

𝑑𝑑𝑑𝑑

∬ (𝑬𝑬OC,t × 𝑯𝑯OC,𝑡𝑡) ∙ �̂�𝒛 𝑑𝑑𝑑𝑑 𝑆𝑆

. (10)

The larger Δ𝑛𝑛eff is the shorter a modulator can be. Equation (10) shows that the effective

refractive group index change can be increased by optimizing the overlap of the electric RF-

field and the optical carrier field in the gap. It would be beneficial if (10) could be further

simplified to the point where more insight is revealed.

More insight can be gained by making additional simplifying assumptions and rewriting the

expression. In a first step we assume that the susceptibility is homogenous and the RF field is

constant in the slot. Then, the 𝐸𝐸RF and (2) – term can be extracted from the coupling coefficient

in (10). Next we extend (10) by an expression similar to a modal energy density 3





∬ (𝜖𝜖0𝑑𝑑(𝜖𝜖(𝜔𝜔)𝜔𝜔)

𝑑𝑑𝜔𝜔 𝑬𝑬OC ∙ 𝑬𝑬OC+ − 𝜇𝜇0𝑯𝑯OC ∙ 𝑯𝑯OC

+ ) 𝑑𝑑𝑑𝑑,𝑆𝑆

(11)

where 𝑬𝑬+ = 𝐸𝐸x𝒙𝒙 + 𝐸𝐸y�̂�𝒚 − 𝐸𝐸z�̂�𝒛 and 𝑯𝑯+ = −𝐻𝐻x�̂�𝒙 − 𝐻𝐻y�̂�𝒚 + 𝐻𝐻z�̂�𝒛 are the adjoint fields related to

a backward propagating waves. This allows us to rewrite (10) as a product of three factors

Δ𝑛𝑛𝑒𝑒𝑒𝑒𝑒𝑒 = 𝛤𝛤 ∙Δ𝑛𝑛mat𝑛𝑛mat

∙ 𝑛𝑛slow, (12)

where 𝛤𝛤 is now the field energy interaction factor given by

𝛤𝛤 ≅𝜖𝜖𝑟𝑟 ∬ (𝜖𝜖0 E𝑂𝑂𝑂𝑂,𝑥𝑥

2𝑆𝑆slot

)𝑑𝑑𝑑𝑑

∬ (𝜖𝜖0𝑑𝑑(𝜖𝜖(𝜔𝜔)𝜔𝜔)

𝑑𝑑𝜔𝜔 𝐄𝐄OC𝐄𝐄OC+ − 𝜇𝜇0𝐇𝐇OC𝐇𝐇OC

+ ) 𝑑𝑑𝑑𝑑 𝑆𝑆

. (13)

Further, the relative change of the nonlinear material’s refractive index can be expressed

with the help of the standard expressions of the Pockels effect by 4:

Δ𝑛𝑛mat𝑛𝑛mat

= 1√𝜖𝜖r

𝜒𝜒𝑥𝑥𝑥𝑥𝑥𝑥(2)

√𝜖𝜖r

𝑈𝑈peak𝑑𝑑 = 1

√𝜖𝜖rΔ𝑛𝑛Pockels.

(14)

Finally, we define a slowdown factor.

𝑛𝑛slow = 𝑐𝑐0𝑣𝑣energy

= 𝑐𝑐0∬ (𝜖𝜖0

𝑑𝑑(𝜖𝜖(𝜔𝜔)𝜔𝜔)𝑑𝑑𝜔𝜔 𝑬𝑬OC𝑬𝑬OC

+ − 𝜇𝜇0𝑯𝑯OC𝑯𝑯OC+ ) 𝑑𝑑𝑑𝑑 𝑆𝑆

∬ (𝑬𝑬OC,t × 𝑯𝑯OC,t) ∙ �̂�𝒛 𝑑𝑑𝑑𝑑 𝑆𝑆 , (15)

which is related to the energy slow down due to the surface-plasmon polariton dispersion.





III. Optimal Geometry of the Phase Shifter It is the goal to optimize the plasmonic modulator geometry for the largest extinction

ratios and the lowest optical and electrical losses. For this we can vary the width, the length and

the height of the modulator section. Subsequently, we keep the modulator height and show the

influence of the modulator width and length.

The extinction ratio of the MZM becomes infinite when a full phase shift is induced.

The length that is required to obtain a phase shift is 𝑙𝑙𝜋𝜋 and it depends inversely on the applied

electrical field. It also depends on the slot width 𝑤𝑤slot. In the following discussion we assume

the phase shifter to have the length 𝑙𝑙𝜋𝜋(𝐸𝐸, 𝑤𝑤𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠).

The optical plasmonic (Lo,dB) losses for a modulator of length l then are

𝐿𝐿𝑠𝑠,𝑑𝑑𝑑𝑑( 𝑤𝑤𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠) = 10 log10(exp−2∙𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼(𝑛𝑛𝑒𝑒𝑒𝑒𝑒𝑒( 𝑤𝑤𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠))∙2𝜋𝜋𝜆𝜆 ∙𝑠𝑠𝜋𝜋( 𝐸𝐸,𝑤𝑤𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠)),

(16)

where 𝐿𝐿𝑠𝑠,𝑑𝑑𝑑𝑑 is normalized with respect to the optical input power. From (16) one can see that

the optical losses depend on the slot length, vacuum wavelength λ, the electrical field E and the

imaginary part of the effective refractive index 𝑛𝑛𝑒𝑒𝑒𝑒𝑒𝑒( 𝑤𝑤𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠). To also include losses from the

island taper we mimic these potential losses by extending the slot length by 1 m.

The electrical losses are resulting from charging and discharging of the capacity5. To

obtain a relative quantity of the electrical losses 𝐿𝐿𝑒𝑒,𝑑𝑑𝑑𝑑 we normalize them by the electrical losses

of a phase shifter having a width of 𝑤𝑤𝑟𝑟𝑒𝑒𝑒𝑒 = 150 nm

𝐿𝐿𝑒𝑒,𝑑𝑑𝑑𝑑 = 10 ∙ log10

14 𝐶𝐶(𝑙𝑙𝜋𝜋(𝐸𝐸, 𝑤𝑤𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠), 𝑤𝑤𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠)𝑈𝑈(𝐸𝐸, 𝑤𝑤𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠)2

14 𝐶𝐶(𝑙𝑙𝜋𝜋(𝐸𝐸, wref)wref)𝑈𝑈(wref)2

(17)

Here, the capacitance C is assumed to have the form of a plate capacitor that depends on the

phase shifter length and width. The applied voltage which is used to charge and discharge the

capacitor is a function of the slot width and the applied electrical field.





(a) First, we consider the case where a constant voltage (e.g. 10V) is applied. The corresponding

optical and electrical losses are depicted in Figure S 3-a and Figure S 4-a as a function of the

slot width. Optical losses decrease when reducing the slot widths - indicating that the increasing

electrical field and with it the increasing nonlinear effect allows to fabricate an even shorter

modulator which overcompensates the increasing plasmonic propagation losses. The electrical

power consumption can almost be neglected. Only a 3 dB penalty is observed when narrowing

the slot from 150 nm to 30 nm.

0 50 100 150 2002

3

4

5

6

L O,d

B [d

B]

Slot width [nm]0 50 100 150 200

1

2

3

4

5

L O,d

B [d

B]

Slot width [nm]

ba

constant voltage (10V) constant electrical field (1 MV/cm)

Figure S 3 The normalized optical losses of a plasmonic phase-shifter are given as a function of the slot width for a constant voltage (a) and a constant electrical field (b). The optical losses have been normalized with respect to the optical input power.

(b) In the second case the electrical field is fixed to 1 MV/cm for all slot widths. This case is

off relevance when the device is operated close to the breakdown. The optical and electrical

losses are shown in Figure S 3-b and Figure S 4-b as a function of the slot width. Optical losses

are minimal for a slot width of 150 nm. They increase for smallest slot widths as the propagation

losses can no longer be overcompensated by increasing the electrical field. However, a constant

electrical field leads to reduced voltages, and thus, it leads to reduced electrical power





consumption when the slot width is reduced. A reduction from 150nm to 30nm brings a 9 dB

benefit for the electrical power consumption.

-12

-8

-4

0

4

Slot width [nm]0 50 100 150 200

L E,d

B [d

B]

-12

-8

-4

0

4

Slot width [nm]0 50 100 150 200

L E,d

B [d

B]

ba

constant voltage (10V) constant electrical field (1 MV/cm)

Figure S 4 The normalized electrical losses of a plasmonic phase-shifter are given as a function of the slot width for a constant voltage (a) and a constant electrical field (b).The electrical losses have been normalized by the electrical losses for a phase shifter of 150 nm width.

In conclusions, we find that the smallest slot width enables best performance with

regards to electrical and optical power consumption – no matter if the voltage or the electrical

field is kept constant. A limit is only set by the capability of producing a smallest width phase

shifter with a high yield and the breakthrough field. Based on our experience in fabrication we

decided for slot widths of 90 nm.





IV. Geometry of Fabricated Device and Simulation Parameters The finite element method solver COMSOL Multiphysics was used to find the optimum

geometry. The parameters of the plasmonic phase shifters are based on the above given

discussion. The taper angles were determined by FEM simulations and are in agreement with

expected values from theory 6,7. The dimensions for the silicon waveguides are adapted from

standard literature values to achieve single mode guiding8. For reasons of computational powers

we constrained ourselves mostly to 2-D simulation. These are adapted by factors to include 3-

D effects. The factors are obtained from comparison of 2-D and 3-D simulations. Using these

simulations we determined the optimum geometry of the structure and found the parameters

listed in Table S 1.

DLD-164 Au Si SiO2

b

x

y

wslotwisland wrail

wSi

a tapera taper

Lslot

a

z

x

hBridge hAu

3 μm

tBridge

Figure S 5 – Schematic. Top view, a and cross section, b, scheme of the plasmonic Mach-Zehnder modulator. The different colors corresponds to the different materials.

Table S 1 - Geometrical Parameters of fabricated device

wisland wslot wSi lslot ltip-to-tip αtaper hDLD-164 hAu hSi

410 nm 100/ 90 nm 450 nm 5 μm 500 nm 12° 800 nm 220 nm 200 nm

Furthermore, all materials are assumed to be nonmagnetic (𝜇𝜇𝑟𝑟 = 1). The dielectrics’

optical properties are assumed to be constant over the considered wavelengths: 𝑛𝑛Si = 3.48;

𝑛𝑛SiO2 = 1.44 and 𝑛𝑛DLD−164 = 1.83.

Gold is strongly dispersive and we need the complex permittivity (𝜖𝜖 = 𝜖𝜖′ − 𝑗𝑗𝜖𝜖′′) over a

broad spectrum. We experimentally determined the complex permittivity from a 200 nm layer





of gold by ellipsometry measurements, see Figure S 6-a. Our results are given by the solid lines.

The reference measurements from Johnson & Christy (1972) are given by the cross and

symbols. The Figure shows that our measurements deviate from the values given in standard

literature 9. Especially, the imaginary part 𝜖𝜖′′at the telecommunication wavelength of 1550 nm

shows a reduction by almost a factor of two. The influence of this deviation was cross checked

by comparing calculated and measured propagation losses. For this we first plotted the

theoretical waveguide losses for a plasmonic slot waveguide using the data by Johnson and

Christy (cross) and the ones measured by us (solid lines), see Figure S 6-b. We then directly

measured the losses in our phase-shifter for a waveguide width of 80 ±10 nm. We found losses

of ~0.5 dB m-1. The results from the fabricated device are in agreement with the ellipsometry

data values from Figure S 6-a and other recent literature values9,10.

Figure S 6 – Optical properties of gold. Measured permittivity values of a 200 nm gold layer. a, shows the complex permittivity 𝜖𝜖 = 𝜖𝜖′ − 𝑗𝑗𝜖𝜖′′ as a function of the wavelength for standard literature values (cross) and our measured values (solid line). b, displays the calculated propagation losses of the plasmonic phase shifter as a function of the slot width. The calculations are based on our values obtained from ellipsometry measurements (red line) and the standard literature values from Johnson and Christy (1972). Comparison with an experimentally determined in-device propagation loss of a plasmonic phase shifter (black cross) is in agreement with the measured data.





V. RF Characterization The electrical bandwidth of the device is mainly restricted by RC limitations that are

due to the measurements set-up, i.e., the pads that are needed to contact the device under test

(DUT) with prober needles. The full structure is depicted in Figure S 7-a, including the DUT

structure and its contact pads as highlighted by the gold color. Furthermore, the figure shows

the simulated electrical field indicating that the GND-Signal-GND contact pads dominate the

overall DUT capacitance due to their large area. However, the large area contact pads forming

the electrodes of the modulator would not be necessary in a real device. We needed them to

apply the RF signal through commercially available RF probes (Picoprobe 67A). Hence, full

wave 3-D simulations were carried out to retrieve the individual resistive and capacitive terms

for the device with and without electrical contacts using a commercially available software

package (CST Microwave Studio). Figure S 7 shows the simulated electric field distribution for

a 72 GHz RF source. As seen in the inset, the field is strongest inside the plasmonic slot

waveguide. With 1 V applied between the signal and ground electrodes, the field strength in the

slots are 107 V/m and 1.11 x 107 V/m as expected for slot widths’ of 100 nm and 90 nm,

respectively. The complex circuit impedance, which for this circuit is given by 𝑍𝑍 = 𝑅𝑅 +

𝑗𝑗(𝜔𝜔𝜔𝜔 + 1/𝜔𝜔𝜔𝜔) was obtained from the simulated reflection parameter. Since the device is purely

capacitive, the real and imaginary terms of the complex impedance correspond to the circuit

resistive and capacitive term, respectively. For the modulator itself, the nm cross-section of the

bridge is causing a resistance Rbridge of 0.36 Ω while the nm-scaled slots of the modulator lead

to a capacity Cisland of 2.81 fF. Thus, a 3 dB bandwidth of 1.1 THz is estimated for a 50 Ω source

impedance. If the contact pads are considered, the total resistance (Rbridge and Rpad) and

capacitance (Cisland and Cpad) will become 0.87 Ω and 32.6 fF, respectively. Introducing these

values into the electrical model depicted in the inset of Fig. S 7-b, the 3-dB bandwidth limit for

the device with the electrical contacts decreases to 96 GHz. Figure S 7-b shows the





corresponding frequency responses. It is clear that the non-optimized contact pads limit the

overall electrical bandwidth.

Simulations also indicate that the asymmetric arrangement of the contact electrode also

leads to a distinct frequency response in the reflection coefficient. The asymmetry leads to an

unbalanced transmission line and introduces an LC resonance. The measured S11 parameter

from the DUT are plotted in Figure S 7-c.

a

b

Am

plit

ude

[dB

]

MZM w/o padMZM w/ pad

10 100 1000Frequency [GHz]

-6-3036

Ampl

itude

[dB

]

20 40 60-3

-2

-1

0

Frequency [GHz]

S11

[dB

]

c

Cpad

RpadVAC

50 Ω

Cisland

Rbridge

GNDGND Signal

DUT

Figure S 7 - RF characterization of the device. Subplot a depicts the numerical calculated electrical field for the modulator and the contact pads (GND-Signal-GND). The signal pad is integrated into the left contact pad (GND) as for reasons of the used GSG-probes. This results in a large pad capacity as can be seen by the electrical field distribution. The influence of the contact pad capacity on the electrical bandwidth is shown in, b. Without pads (black) a bandwidth in the THz regime is predicted, while with pads the 3 dB cut off is 96 GHz. c, the reflection parameter shows a dip at 50 GHz which is related to the asymmetric contact pads and is not caused by the modulator itself.





VI. Optical Characterization An external cavity laser at a wavelength of 1534 nm was used as coherent source. The

CW laser was amplified and filtered before it was coupled to the chip by grating couplers. Light

was fed to the device by 1.5 mm long silicon access waveguides.

To characterize the electro-optic coefficient we performed a sweep of the DC bias

applied to the island and measured the voltage dependence of the transmitted power. The bias

was applied by a ground signal ground (GSG) electrical probe contacting signal and ground

pads. The DC-bias was limited to ± 2.5 V as preliminary experiments showed a depolarization

for larger voltages, whereas, no depolarization was observed for high-speed RF-signals.

Passive behavior of the modulator was characterized by tuning the laser’s wavelength

from 1520 nm to 1620 nm and recording the spectrum with an optical spectrum analyzer.

For high speed data experiments we generated a 2-amplitude-shift-keying data stream.

The bit sequence was generated by a standard pseudo random binary sequence generator (215-

1) that was analog converted by a 6-bit 72 GSa s-1 digital to analog converter. The electrical

signal was then amplified to 1.5 Vp measured with a 50 Ω RF power head at 54 GBd and then

fed to the modulator by GSG-probes. At 72 GBd the electrical amplifier caused a strong bit

pattern dependency of the voltage. The modulated optical signal after the device was amplified

and received coherently (Agilent - optical modulation analyzer). In a first experiment, a

54 Gbit s-1 signal with a root raised cosine pulse shape (roll-off: 0.25) was fed to the modulator.

In a second experiment a 72 Gbit s-1 non-return to zero signal was applied similar to the first

experiment. Finally, to estimate the device performance we calculated the bit-error-rate from

7 million symbol counts. To mitigate the frequency dependence of our driver amplifier at such

speed, we applied corrections by performing pre-distortion and post-equalization of the

electronic signal. Due to the limited speed of the DAC of 72GSA s-1 at 72Gbit s-1 no pre-

distortion was applied.





The insertion loss of the modulator have been determined by a comparison with a

reference waveguides of the same length on the same chip. Insertion losses of 8 dB±1 dB have

been found. The 8 dB losses are attributed to 2.5 dB of losses in the plasmonic phase-shifter

section and to 2.75 dB losses in each PPI section. The phase-shifter losses have been

experimentally measured with plasmonic slot waveguides of similar geometry and found to be

0.5 dB m-1. These losses are low when compared to expected losses of ~0.8 dB m-1 obtained

for simulations performed with standard literature values9. We have therefore cross-checked

both the material parameters and waveguides losses. The losses in the PPI section have been

derived from simulations and agree with measured losses of passive test structures. To couple

the light to the chip and to feed the signals into silicon waveguides we used grating couplers

with losses of 14.5 dB each. Additional losses come from the silicon access waveguides. The

access waveguides were as long as 1.5 mm to enable convenient coupling – but added another

2.3 dB to the total losses. The total losses with the grating couplers and the feeding waveguides

resulted in fiber-to-fiber losses of ≈39.3 dB. It should be noted though that in practice grating

coupler losses below 2 dB are achievable when an effort is made towards their optimization 12.





VII. Comparison with Prior Art In the following the all-plasmonic modulator is compared with prior art. The central

focus lies on the comparison with Silicon based MZMs13,14, Silicon-organic-hybrid (SOH)

MZMs15 and hybrid plasmonic phase modulators16. In this discussion, we restricted ourselves

to MZMs as MZMs enable a chirp-free and linear coding of an optical carrier by an electrical

signal over a large operation range.

Table S 2 – Comparison with state-of-the-art: Within the table important properties of a MZM are

given for different type of modulators such as the Si-type, Silicon-organic-hybrid type and the all-plasmonic type.

A hybrid plasmonic phase modulator is included to emphasize the progress made within plasmonics over the last

year.

Type Si13 Si14 SOH15 Hybrid plasmonic16

All- plasmonic

Modulation effect Free carrier (depletion

mode)

Free carrier (injection

mode)

Pockels effect

Pockels effect

Pockels effect

VL [Vmm] 2.4 0.36 0.52 2.3 0.06

Phase mod. length [m]

750 200 1000 24 5

Elect. bandwidth [GHz] 55 not stated 18 > 65 > 70

Symbol rate [GBd] 70 10 40 40 72

Driving voltage Vpeak

5.32 3.5 at 10 GBd

2.1 at 40 GBd

2.35 at 40 GBd

1.5 at 54 GBd

Elect. energy consumption

[fJ bit-1] not stated 5000

at 10 GBd 420

at 40GBd 60

at 40 GBd 25

at 54 GBd

Insertion loss [dB] 4 12 6.6 12 8

Table S 2 shows a comparison of important parameters for the various modulator concepts. The

all-plasmonic modulator outperforms the other devices in most of the important characteristics,

such as the VL, symbol rate, electrical energy consumption and device size.

The small voltage-length product of the all-plasmonic modulator can be traced back to

multiple sources. To do so we compare the plasmonic modulator individually with the other





modulator concepts. First, compared to the silicon based modulators a higher light-matter

interaction is achieved by relying on highly nonlinear organics with electro-optics coefficients

of 180 pm/V rather than the plasma dispersion effect. Furthermore, the nonlinear organic

materials feature a linear change of phase with the applied field while for Si modulators the

phase changes nonlinearly with the applied field (Soref, R.A. and Bennett, B.R. Kramers-

Kronig Analysis of Electro-Optical Switching In Silicon, SPIE Proceedings, 0704, (1987)).

Second, compared to the SOH approach plasmonics is not diffraction limited and allows for a

sub-wavelength confinement resulting in a dramatic increase of the nonlinear light-matter

interaction, see Figure 2 in the manuscript. Third, when comparing the new result with our

previous result in Ref. 15, one finds a voltage-length product (VL) which is improved by as

much as 43 times and now is as little as 60V m. The improvement can be traced back to a

modified geometry of the slot waveguide leading to higher confinement and higher electrical

fields (factor 2.5). An efficient poling of an improved nonlinear material reduced the VL by a

factor 8.6. A further reduction by a factor 2 was achieved by using a MZM configuration rather

than a single phase-shifter.

Another dramatic reduction is obtained in the total length of the phase shifter. This is

due to the increased nonlinear interaction and leads to a reduction from hundreds of m for the

Si and SOH MZMS to a few m for the all-plasmonic MZM. This massive shrinking can be

observed for the hybrid plasmonic and the all-plasmonic approach. But, there is a huge

difference between the achievable footprint for an MZM based on the conventional hybrid

plasmonic approach and the footprint of the new all-plasmonic MZM. The configuration

presented in this paper enabled us to completely embed a whole MZM – including the splitters,

the photonic-to-plasmonic converters and the phase-modulator section – on a smaller space than

the single plasmonic phase shifter of Ref. 15. Such a reduction in size was made possible by

3- D patterning of metals and the above mentioned technical improvements.





The achieved electrical bandwidth of >70 GHz for the all-plasmonic approach is one of

the largest. It should be noted that the expected bandwidth is one order of magnitude higher.

The Silicon and SOH modulators are limited by the finite resistivity of the silicon.

The large electrical bandwidth enables to operate the all-plasmonic modulator at highest

symbol rates up to 72 GBd and potentially beyond. It is remarkable that a similar rate has been

achieved for Si modulators (column 1) – with a driving voltage of 5.32 Vpeak though. This was

made possible by creating an electrical resonance enhancement in the RF strip lines. The

plasmonic modulators do not need any RF engineering. They are so short that spatial walk-offs,

and RF-losses do not cause distortions. The plasmonic MZM can be operated open circuit

enabling a small driving voltages of 1.5 Vpeak at 54 GBd.

Due to the small driving voltages and short phase shifter lengths the energy consumption

is estimated to be 25 fJ Bit-1 at highest symbol rate. The reported power consumptions of Si and

SOH MZMs are at least one order of magnitude larger for such high symbol rates.

Being a relative young technology plasmonics suffer still from non-optimized structures

and thus from high coupling losses of individual components. That is why insertion loss of 8 dB

are reported for MZM and further optimization is required to achieve values similar to 4 dB as

reported for the best competing technologies. As another challenge that needs to be solved one

should mention that the organic material properties may degrade if the devices are heated up

beyond 65 oC.

VIII. References 1 Ruan, Z., Veronis, G., Vodopyanov, K.L., Fejer, M.M. & Fan, S. Enhancement of

optics-to-THz conversion efficiency by metallic slot waveguides. Opt. Express 17, 13502-13515, (2009).

2 Dalton, L. Nonlinear Optical Polymeric Materials: From Chromophore Design to Commercial Applications. Vol. 158 1 (Springer Berlin, Heidelberg, 2002).

3 Chen, P.Y. et al. Group velocity in lossy periodic structured media. Phys. Rev. A 82, 053825, (2010).

4 Boyd, R.W. Nonlinear Optics. (Academica Press,London, 2008). 5 Miller, D.A.B. Energy consumption in optical modulators for interconnects. Opt.

Express 20, A293-A308, (2012).





6 Gramotnev, D.K. & Bozhevolnyi, S.I. Nanofocusing of electromagnetic radiation. Nat. Photonics 8, 13-22, (2014).

7 Tian, J., Yu, S., Yan, W. & Qiu, M. Broadband high-efficiency surface-plasmon-polariton coupler with silicon-metal interface. Appl. Phys. Lett. 95, 013504, (2009).

8 Dan-Xia, X. et al. Silicon Photonic Integration Platform - Have We Found the Sweet Spot? IEEE J. Sel. Top. Quantum Electron. 20, 189-205, (2014).

9 Johnson, P.B. & Christy, R.W. Optical Constants of the Noble Metals. Phys. Rev. B 6, 4370-4379, (1972).

10 Babar, S. & Weaver, J.H. Optical constants of Cu, Ag, and Au revisited. Appl. Opt. 54, 477-481, (2015).

11 McPeak, K.M. et al. Plasmonic Films Can Easily Be Better: Rules and Recipes. ACS Photonics 3, 326-333, (2015).

12 Tang, Y., Wang, Z., Wosinski, L., Westergren, U. & He, S. Highly efficient nonuniform grating coupler for silicon-on-insulator nanophotonic circuits. Opt. Lett. 35, 1290-1292, (2010).

13 Xu, H. et al. High-speed silicon modulator with band equalization. Opt. Lett. 39, 4839-4842, (2014).

14 Green, W.M., Rooks, M.J., Sekaric, L. & Vlasov, Y.A. Ultra-compact, low RF power, 10 Gb/s silicon Mach-Zehnder modulator. Opt. Express 15, 17106-17113, (2007).

15 Palmer, R. et al. High-Speed, Low Drive-Voltage Silicon-Organic Hybrid Modulator Based on a Binary-Chromophore Electro-Optic Material. J. Lightwave Technol. 32, 2726-2734, (2014).

16 Melikyan, A. et al. High-speed plasmonic phase modulators. Nat. Photonics 8, 229-233, (2014).



supplementary information - nature...entire system. using a simple parallel -plate capacitor model,...

Documents