Machine learning-based design of meta-plasmonic biosensors

with negative index metamaterials

SUPPORTING INFORMATION

Gwiyeong Moon1*, Jong-ryul Choi2*, Changhun Lee1, Youngjin Oh3, Kyung Hwan Kim3, and

Donghyun Kim1†

1 School of Electrical and Electronic Engineering, Yonsei University, Seoul, Korea, 03722

2 Medical Device Development Center, Daegu-Gyeongbuk Medical Innovation Foundation

(DGMIF), 80 Cheombok-ro, Dong-gu, Daegu, Korea, 41061

3 OLED Division, Samsung Display, Asan, Chungcheongnam-do 31454, Korea

4 Department of Biomedical Engineering, Yonsei University, Wonju, Korea, 26493

S1. Model estimation of the meta-plasmonic biosensor

S2. Effect of ambience on the detection sensitivity of DNA hybridization

S3. Structure of multilayer perceptron and autoencoder

S4. Definition of α

S5. Design of double-negative index metamaterials

S6. Definition of mean-square error (MSE) and error of resonance angle (ERA)

S7. Experimental confirmation of reduced MSE and ERA using machine learning

S8. Reflection curves of 20 clusters C1, C2, …, and C20

S9. Optimization of k in k-means clustering of C5

* These authors contributed equally to this work.† Corresponding author: kimd@yonsei.ac.kr

1

S1. Model estimation of the meta-plasmonic biosensor

Consider a metamaterial shown in Figure 1(c) with effective permittivity ε eff=εeff' + jε eff

' ' = (neff + jκeff)2

and permeability μeff , where neff and κeff refer to refractive index and absorption coefficient. The

metamaterials may consist of underlying 2D periodic structures to give rise to the desired effective

index using, for example, the second-order EMT (Haggans et al., 1993; Kim et al., 2007; Lalanne et

al., 1998; Moon et al., 2006; Rytov, 1956), i.e.,

ε eff=ε0TM+ π2

3f 2 (1−f )2( 1

εm− 1

εamb )2

(ε 0TM )3 ε0

TE ( Λλ )

2

. (S1.1)

Here, f is the filling factor of metal and Λ the period. m and amb are the permittivity of metal and

ambience. Zeroth-order effective permittivities for TM and TE polarization are given by

ε 0TM=εm εamb / [ f εamb+ (1−f ) εm ] and ε 0

TE= f εm+ (1−f ) εamb. In the simple model based on the second-

order EMT given by Eq. (S1.1), f, Λ and the layer thickness tmeta are the structural parameters, i.e.,

machine learning can be performed in the three-dimensional parameter space in the range of 0 < f < 1

and 0 < tmeta 70 nm in the long wavelength limit kΛ < 1 (k: wave number). EMT has been developed

for magnetic permeability with magnetic/dielectric materials and also various non-magnetic meta-

structures (Alù, 2011; Jin et al., 2009; Lerat et al., 2006; Smith et al., 2006; Smith et al., 2000; Wu et

al., 2006; Zhang et al., 2015), wherein effective permeability μeff is often found numerically, rather

than in a closed analytical form. Note that this work is not limited to a specific EMT so that an

effective medium optimized for SPR detection may be implemented using any EMT suited to existing

fabrication technology and materials.

To calculate far-field optical characteristics produced by metamaterial structures, multi-layer ITMA

was custom-coded on MathematicaTM (Wolfram Research, Champaign, IL, United States). The results

from the transfer matrix algorithm were verified by finite element method. With random assignment

of a parameter set in the parameter space, optical characteristics and the degree of sensitivity

enhancement are calculated based on the transfer matrix approach. The parameter space is spanned by

reduced three dimensional data, i.e., ε eff' , μeff , and the metal thickness tgold were spanned in the range

2

of -5 < ε eff' < -0.5, -3.5 < μeff < -0.5, and the metal thickness 10 nm < tgold < 70 nm, while ε eff

' ' is fixed

at ε eff' ' =−0.001, which is taken from Ishimaru et al, 2005, and the meta-layer thickness tmeta was

enforced to be t = λ/4 tgold.

References

Alù, A., 2011. Phys. Rev. B 84 (7), 075153.

Haggans, C.W., Li, L., Kostuk, R.K., 1993. J. Opt. Soc. Am. A 10 (10), 2217–2225.

Ishimaru, A., Jaruwatanadilok, S., Kuga, Y., 2005. Prog. Electromagn. Res. 51, 139–152.

Jin, J., Liu, S., Lin, Z., Chui, S.T., 2009. Phys. Rev. B 80 (11), 115101.

Kim, D., Yoon, S.J., 2007. Appl. Opt., 46 (6), 872–880.

Lalanne, P., Hugonin, J.P., 1998. J. Opt. Soc. Am. A 15 (7), 1843–1851.

Lerat, J.-M., Malléjac, N., Acher, O., 2006. J. Appl. Phys. 100 (8), 084908.

Moon, S., Kim, D., 2006. J. Opt. Soc. Am. A 23 (1), 199–207.

Rytov, S.M., 1956. Sov. Phys. J. Exp. Theor. Phys. 2, 466–475.

Smith, D.R., Pendry, J.B., 2006. J. Opt. Soc. Am. B 23 (3), 391–403.

Smith, D.R., Vier, D.C., Kroll, N., Schultz, S., 2000. Appl. Phys. Lett. 77 (14), 2246.

Wu, Y., Li, J., Zhang, Z.-Q., Chan, C.T., 2006. Phys. Rev. B 74 (8), 085111.

Zhang, X., Wu, Y., 2015. Sci. Rep. 5, 7892.

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S2. Effect of ambience on the detection sensitivity of DNA hybridization

To the first degree, it is expected that the higher the refractive index of buffer ambience is, the higher

the sensitivity would be. On the other hand, if the buffer index is too high, optical signature that can

be obtained from the presence of DNA binding would be low when we measure the signature based

upon the overlap of light fields and DNA molecules.

Because DNA hybridization of probe ssDNA would be extremely difficult to measure experimentally

while changing ambient solvents (and therefore ambient refractive indices), we have calculated

detection sensitivity Shy=( δθ/δn )hybridization in an implicit ambience model assuming 28-mer DNA

oligomers hybridizing on 50-nm thick gold and SF10 glass substrate.

The results observed in these data confirm the expectation above. In response to the reviewer

question, we expect that the sensitivity would improve if we use an optical dense solvent as ambient

buffer, instead of water, with a refractive index higher than 1.33 and lower than 1.5, assuming that

complementary DNA strands can hybridize in a stable manner in the solvent.

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S3. Structure of multilayer perceptron and autoencoder

Figure S3 (a) The structure of multilayer perceptron (MLP) to predict optical characteristic from

meta-plasmonic structure parameters. The number of nodes in hidden layers was set to 500. Structural

parameters were used as an input to the MLP. Optical characteristics sampled in the angular range of

θ = 40 ~ 89° at an interval of 0.1° were used as an output. (b) The structure of Autoencoder (AE) to

reduce the dimension of reflectance curves from 197 to 8 dimensions. AE was trained by setting

output to be the same as input, which is a reflection curve.

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S4. Definition of α

As a quantitative measure of resonance characteristics, we have defined a quality factor as follows:

first, we define an average around the resonance minimum such that

Average = ∑θspr−0.5°

θspr+0.5°

R (θ¿)/ N θ (S4.1)

In Eq. (S4.1), spr represents the resonance angle. R(in) is the reflectance at an angle of incidence that

is scanned from spr – 0.5 to spr + 0.5 in a step that is not specified. N denotes the number of total

samples that is averaged. then becomes

= Average – R(spr) . (S4.2)

A reflectance curve with a large may be considered to have efficient SPR feature characteristics

with good resonance contrast.

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S5. Design of double-negative index metamaterials

In this work, we focus on effective parameters for enhanced SPR detection, rather than the design of

metamaterial per se to achieve specific sets of effective parameters. Nonetheless, metamaterial design

can be important to tailor characteristics of negative-index metamaterial depending on applications.

There are metamaterial designs that can be used to achieve double-negative index, for example, gold

square hole array (SHA) with 400-nm period and 360-nm side length and gold circular hole array

(CHA) with 400-nm period and 320-nm diameter. Among these, we present two schematic designs as

presented in Figure S5.

Figure S5 Designs of double-negative index metamaterial based on for gold square hole array (SHA)

and gold circular hole array (CHA).

The effective optical parameters of these designs (ε eff=εeff' + jε eff

' ' and μeff = μeff' + j μeff

' ' ) can be

obtained from the reflection and transmission characteristics (Kildishev et al., 2006). The reflection

and transmission characteristics of the two designs on an BK7 substrate were obtained at normal

incidence using 3D finite element method (FEM) on COMSOLTM. In Table S5, the effective

permittivity and permeability at λ = 632.8 nm are listed.

Table S5 Effective optical parameters obtained for square hole array (SHA) and circular hole array

(CHA)

ε eff' μeff

'

SHA -1.7127 -0.8237CHA -1.7905 -0.8372

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The real part of effective permittivity and permeability of both designs were negative resulting in

double-negative index metamaterial. In addition, other references provide basis upon which double

negative index may be implemented, e.g., using double stripes, closed square ring structure which

consists of a single silver layer (García-Meca et al., 2007), and fishnet structure with three layers of

alumina separated by silver layers (Chettiar et al., 2008).

References

Chettiar, U. K., Xiao, S., Kildishev, A. V., Cai, W., Yuan, H. K., Drachev, V. P., Shalaev, V. M., 2008. MRS Bulletin, 33(10), 921-926.

García-Meca, C., Ortuño, R., Salvador, R., Martínez, A.,Martí, J., 2007. Opt. Express 15(15), 9320-9325.

Kildishev, A. V., Cai, W., Chettiar, U. K., Yuan, H. K., Sarychev, A. K., Drachev, V. P., Shalaev, V. M., 2006. J. Opt. Soc. Am. B 23(3), 423-433.

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S6. Definition of mean-square error (MSE) and error of resonance angle (ERA)

As a quantitative measure of accuracy of machine learning, we have introduced MSE and ERA. MSE

and ERA can be described as follows:

MSEML=1N ∑

n=1

N

∑θ

¿ t n (θ )−pn (θ )∨¿2¿ (S6.1)

MSENe=1N ∑

n=1

N

∑θ

¿ tn (θ )−nrn (θ )∨¿2 ¿ (S6.2)

MSE¿=1N ∑

n=1

N

∑θ

¿tn (θ )−tin (θ )∨¿2 ¿ (S6.3)

ERA ML=1N ∑

n=1

N

∑θ

¿argmin(tn (θ ))−argmin ( pn (θ ))∨¿2¿ (S6.4)

ERA Ne=1N ∑

n=1

N

∑θ

¿argmin (t n (θ ))−argmin (nrn (θ ))∨¿2 ¿ (S6.5)

ERA¿=1N ∑

n=1

N

∑θ

¿argmin (t n (θ ))−argmin (tin (θ ))∨¿2 ¿ (S6.6)

In Eq. (S6.1) – (S6.6), tn(θ), pn(θ), nrn(θ), and tin(θ) denote angular reflectance, respectively, in the test

set, predicted by the machine learning algorithm, estimated by the nearest interpolation, and one

estimated by the linear tetrahedral interpolation with the training dataset. θ represents an incident

angle and N is a total number of test datasets (N = 1,500). argmin() is a function that exports the

incident angle with the minimum relative intensity of reflection. The average MSEs and ERAs are

presented as follows with standard deviation (s.d. n = 1500 for both tables below):

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Mean (standard deviation

)

bare immobilization hybridization

MSEML 0.2146 ± 0.0392 0.1685 ± 0.0225 0.1739 ± 0.0214

MSENe 2.2124 ± 0.2167 2.4315 ± 0.2559 2.4893 ± 0.2614

MSEIn 1.4530 ± 0.1488 1.5146 ± 0.1437 1.4959 ± 0.1351

ERAML 0.7322 ± 0.0837 0.6314 ± 0.0720 0.8239 ± 0.0807

ERANe 1.4514 ± 0.1334 1.6271 ± 0.1477 1.6560 ± 0.1516

ERAIn 1.5078 ± 0.1403 1.6583 ± 0.1537 1.5722 ± 0.1500

MSEs and ERAs obtained for DNA biosensing are presented as a ratio for comparison as listed in the

following table.

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bare immobilization hybridization

MSENe/MSEML 10.3094 ± 2.1368 14.4303 ± 2.4534 14.3145 ± 2.3157

MSEIn/MSEML 6.7707 ± 1.4179 8.9887 ± 1.4724 8.6021 ± 1.3131

ERANe/ERAML 1.9822 ± 0.2908 2.5770 ± 0.3756 2.0100 ± 0.2965

ERAIn/ERAML 2.0593 ± 0.3035 2.6264 ± 0.3859 1.9082 ± 0.2609

S7. Experimental confirmation of reduced MSE and ERA using machine learning

For experimental confirmation of machine learning to produce much smaller error with higher

accuracy, we have employed experimental SPR curve data that we had measured of metallic (gold)

grating structure. Although this in itself does not form double negative metamaterials, the structure

may be interpreted as a meta-plasmonic layer. In this regard, we think that it may be used to show the

utility of machine learning in meta-plasmonic structures. For these data, samples were fabricated first

by evaporating 2-nm thick chromium adhesion layer and 40-nm gold film on an SF10 substrate. Then,

negative resist ARN-7520 was spin-coated for e-beam lithography. After developing resist, 20-nm

gold film was evaporated and resist was removed for lift-off. The period and width of grating are 400

nm and 200 nm. For more details, readers are suggested to check (Oh et al., 2014). Figure S7(a)

shows SEM images of the sample structures. For measurement of SPR curves, He-Ne laser (𝜆 =

632.8 nm, Melles-Griot, Carlsbad, CA, USA) was used as a light source using two dual motorized

stages (URS75PP, Newport, Irvine CA, U.S.A) for angle-scanning. The reflection intensity of a

sample was measured by a p-i-n photodiode (818-UV, Newport, Irvine, CA). SPR curves with

immobilization of 1-μM probe DNA oligomers were measured and presented in blue in Figure S7(b)

The incident angle is controlled from 55° to 67.99° at an interval 0.01°. In this particular

measurement, the resonance angle was obtained at 61.05°. The metallic structure in combination with

the DNA interaction layer can be approximated by a film with effective permittivity (ε eff=εeff' + jε eff

' ' )

and permeability (μeff ) according to the effective medium theory (EMT). The effective parameters

were obtained in many ways, e.g., phenomenologically by fitting an SPR curve (Moon et al., 2006).

For this particular sample, ε eff' ' and the thickness (teff) of the effective medium were fixed at 1 and

28.16 nm. Then ε eff' and μeff were obtained as -5.6798 and -0.3532. The corresponding reflection

intensity can be obtained using transfer matrices as presented in orange in Figure S7(b). The

resonance angle was 60.82°, which is almost identical to experimental results.

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Figure S7 (a) SEM images of fabricated plasmonic nanograting samples, the reflectance of which

were used to test the algorithm. (b) Experimental SPR curve measured of metallic (gold) grating to

form a meta-plasmonic layer after immobilization of 1-μM probe DNA oligomers (black line). Also

shown is the reflectance obtained with an effective medium. (c) Reflectance curves produced with

EMT (black), prediction with MLP (red), nearest interpolation (blue), and linear tetrahedral

interpolation (green).

An SPR curve was obtained for the three-dimensional parameter space in the range of -8 < ε eff' < -2, -

3.5< μeff < -0.1, and 10 nm < teff < 50 nm to train the multilayer perceptron (MLP) network for

prediction of reflectance from the three parameters. The number of SPR curves for training was

10,000. After training the MLP network, we have compared the performance of MLP network for

prediction of reflectance with nearest interpolation and linear tetrahedral interpolation. The

reflectance curves were obtained for the parameter set ε eff' = -5.6798 + 1j and μeff = -0.3532 at teff =

29.32 nm using the three methods and presented in Figure S7(c). The reflection curve predicted by the

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MLP is in good agreement with that of EMT, while interpolated curves show noticeable differences.

For quantitative comparison, the mean squared error (MSE) and error of resonance angle (ERA) were

calculated as follows:

Table S7. MSE and ERA for three methods: MLP, nearest interpolation and linear tetrahedral

interpolation.

MLP Nearest interpolation

Tetrahedral interpolation

MSE 0.1310 4.5281 4.0373ERA 0.0600 1.2300 1.0200

Table S7 lists the MSE and ERA obtained by the three methods, i.e., MLP, nearest interpolation, and

linear tetrahedral interpolation. MSE and ERA obtained with the MLP were the smallest, resulting in

MSENE/MSEMLP = 34.56, MSETETRA/MSEMLP = 30.81, ERANE/ERAMLP = 20.5, and MSENE/MSEMLP = 17.

These data coincide with the results that we observed in Section 3.2 and support the utility of machine

learning from an experimental perspective.

References

Moon, S. and Kim, D. 2006. J. Opt. Soc. Am. A 23(1), 199-207.

Oh, Y., Lee, W., Kim, Y. and Kim, D., 2014. Biosens. Bioelectron. 51, 401-407.

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S8. Reflection curves of 20 clusters C1, C2, …, and C20

Figure S8 Reflectance curves that belong to each of the 20 clusters (C1 ~ C20) are presented with in

the inset. It is shown that reflectance curves can be clustered by the shape based on machine learning

with as an indicator.

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S9. Optimization of k in k-means clustering of C5

Figure S9 k-means clustering and reflectance curves with respect to the cluster number: (a) k = 2 and

(b) k = 3. (c) Average silhouette width over different k.

The k-means clustering with k = 2 shows clustered reflectance curves with mixed SPR features in C5-1

vs. curves with little SPR feature characteristics in C5-2, as shown in Figure S9(a). In comparison with

the results of k = 4 presented in Figure 3(g), C5-2 and C5-4 may be interpreted as combined into C5-1 in

the case of k = 2. The results clearly suggest that k should be improved. With k = 3, C5-2 and C5-3 still

have curves with mixed features. An average silhouette width over a different k shown in Figure S9(c)

presents a maximum value at k = 4, indicating that an optimum number of clusters may be 4

(Kodinariva et al., 2013).

Reference

Kodinariva, T.M., Makwana, P.R., 2013. Int. J. Adv. Res. Compute. Sci. Manag. Stud. 1 (6), 90–95.

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