Design of large scale plasmonic nanoslit arrays for ...dabm/441.pdf · Design of large scale plasmonic nanoslit arrays for arbitrary mode conversion and demultiplexing Pierre Wahl,1,3,*

Design of large scale plasmonic nanoslitarrays for arbitrary mode conversion

and demultiplexing

Pierre Wahl,1,3,* Takuo Tanemura,2 Nathalie Vermeulen,1Jürgen Van Erps,1 David A. B. Miller,3 and Hugo Thienpont1

1Brussels Photonics Team B-PHOT,Department of Applied Physics and Photonics,Vrije Universiteit Brussel, Brussels, Belgium

2Research Center for Advanced Science and Technology,University of Tokyo, Tokyo, 153-8904, Japan

3Department of Electrical Engineering,Stanford University, Stanford, CA 940305, USA

*[email protected]

Abstract: We present an iterative design method for the coupling andthe mode conversion of arbitrary modes to focused surface plasmonsusing a large array of aperiodically randomly located slits in a thin metalfilm. As the distance between the slits is small and the number of slits islarge, significant mutual coupling occurs between the slits which makesan accurate computation of the field scattered by the slits difficult. Weuse an accurate modal source radiator model to efficiently compute thefields in a significantly shorter time compared with three-dimensional (3D)full-field rigorous simulations, so that iterative optimization is efficientlyachieved. Since our model accounts for mutual coupling between the slits,the scattering by the slits of both the source wave and the focused surfaceplasmon can be incorporated in the optimization scheme. We apply thismethod to the design of various types of couplers for arbitrary fiber modesand a mode demultiplexer that focuses three orthogonal fiber modes to threedifferent foci. Finally, we validate our design results using fully vectorial3D finite-difference time-domain (FDTD) simulations.

© 2014 Optical Society of America

OCIS codes: (060.4230) Multiplexing; (130.3120) Integrated optics devices; (240.6680) Sur-face plasmons; (350.4238) Nanophotonics and photonic crystals.

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#197391 - $15.00 USD Received 10 Sep 2013; revised 26 Oct 2013; accepted 28 Oct 2013; published 6 Jan 2014(C) 2014 OSA 13 January 2014 | Vol. 22, No. 1 | DOI:10.1364/OE.22.000646 | OPTICS EXPRESS 646

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1. Introduction

Over the last decade, surface plasmon polaritons have generated a substantial amount of re-search interest because of their unique property of being confined to a metallic surface and theassociated ability of metallic structures to confine light to sub-wavelength volumes. Surfaceplasmons have been used in a broad area of applications such as subwavelength focusing [1, 2],light-harvesting [3], and photodetection [4]. Among those applications, mode coupling anddemultiplexing are very interesting as any linear optical component can be described as a con-verter between two sets of orthogonal optical modes [5]. This problem of mode splitting orconversion has recently become very topical because of the possibility of using multiple modesin optical fibers. Various techniques have been explored in the literature to separate modes op-tically, including fiber or waveguide coupler-based devices [6, 7, 8, 9, 10, 11], phase plates orspatial light modulators with free-space optics [12, 13, 14, 15, 16, 17, 18], spatial samplinginto planar light-wave circuits or silicon photonics with subsequent waveguide interferome-ters [19, 20], and holographic approaches [21]. Previous work with dielectric structures hasshown the possibility of arbitrary mode coupling in photonic-crystal-like structures with cus-tom regions [22, 23]. With the additional incorporation of detectors and local feedback loopstogether with phase shifters and interferometers, schemes have been proposed that can auto-matically convert between arbitrary modes [24, 25, 26, 27].

Structures that couple and demultiplex surface plasmons have mostly been based on metal-lic films perforated with subwavelength apertures or grooves [see Fig. 1]. Those have beenused to show a variety of physical phenomena such as extraordinary optical transmission(EOT) [28], beam steering [29], beam steering with polarization control [30, 31] and negativerefraction [32]. While most work on these arrays has made use of periodic arrays, aperiodicarrays of metallic slits have also been used to achieve binary surface plasmon polariton fo-


x

y

z

Mode 1Mode 2Mode 3

Multimode FiberMode 1 Mode 2

Mode 3

Mode 1

Mod

e 2

Mode 3

(x f 1,y f 1)

(x f 2,y f 2)

(x f 3,y f 3)

Fig. 1. Illustration of the plasmonic mode coupler/demultiplexer that consists of a thin metallayer grown on top of an oxide in which slits are etched. Well optimized slit locations canfocus a particular mode to a single location or can demultiplex several orthogonal modesto different focal points.

cusing [33] and spectrally discrete wavelength demultiplexing [34, 35], which is not readilypossible with periodic approaches.

In an array of slits, mutual coupling between the slits occurs as surface plasmons gener-ated by one slit can be scattered by another slit and couple back into the first one. This phe-nomenon is numerically tractable for periodic arrays through the use of periodic boundaryconditions [36] and for 1-dimsional (1D) and quasi-1D slit arrays [37, 38, 39]. However, anaccurate modeling of the mutual coupling in aperiodic arrays of slits has often required rigor-ous exact 3-D fully vectorial methods such as Finite Difference Time Domain (FDTD) [40],Finite Elements Method (FEM) [39], rigorous coupled wave analysis [41] or Green’s Tensorapproaches [42, 43]. Consequently, previous work on aperiodic arrays was operated in a regimewhere the slits are located relatively sparsely, so that the mutual coupling between slits couldbe safely neglected [34].

Recently, we have developed a novel semi-analytic model, namely the modal source radiatormodel, which allows us to compute the mutual coupling among arbitrarily located apertureswith a significantly reduced computational cost [44]. We present a design method based onthe modal source radiator model that, in contrast to the schemes used in [34], can account forthe scattering by the slits of both the direct excitation and the surface plasmons all the slitsgenerate. This knowledge allows us to calculate the optimal position of slits that scatter thesurface plasmon waves more than they scatter the excitation beam. In addition, we are able todetect when a slit is “blocking” a focused surface plasmon and is basically scattering the fieldone is trying to focus; a better device is achieved by removing this particular slit.

This paper is structured as follows: first we provide a brief description of the modal sourceradiator model which allows us to calculate the scattered field in a quantitatively accurate way.Next we describe our design method which is built on the modal source radiator model. Weconclude with some design examples: the first of these allows the focusing of a higher-order


l s

ws

x

y

z x

z

y

Bα Aα

Oxide

Air

AgEx

TE01

ExTE02

II z = z2

I z = z1

Fig. 2. The field inside the slits is decomposed into TE01 and TE02 eigenmodes of the slits.

L31 fiber mode to an arbitrary location and the second is a structure that demultiplexes three or-thogonal fiber modes to three different foci as depicted in Fig. 1. The latter design is confirmedwith full-field FDTD simulations to prove the validity of our calculations.

2. Radiator source model

In the modal source radiator model, the field inside the slits is decomposed into a superpositionof a finite number of eigenmodes, such as TE01-like and TE02-like modes [see Fig. 2], whichare the guided modes that would propagate in the z-direction inside the slit if the metal wasinfinitely thick [44]. Those modes can also be seen as the plasmonic equivalents of TE modes inrectangular waveguides made of a perfect conductor, where Ez = 0. In addition, in the exampleswe consider below, we limited this decomposition to the TE01-like and TE02-like modes shownin Fig. 2. We can therefore write the |E〉 and |H〉 fields inside the slit using the Dirac notationas

|E〉= ∑α(Aα eiqα z|e+α 〉+Bα e−iqα z|e−α 〉)

|H〉=∑α(Aα eiqα z|h+

α 〉+Bα e−iqα z|h−α 〉)(1)

In Eq. (1), α runs over all the eigenmodes in all the slits, qα is the propagation constant of theslit mode, Aα and Bα are the complex amplitudes of the downward and the upward propagatingmodes respectively and |e±α 〉(|h±α 〉) represents the mode profiles of the slits. The modal sourceradiator model is constructed based on the fact that the transverse magnetic field at the topinterface |HI

t〉 can be expressed in two different ways using the following modal basis:

|HIt〉 = ∑

α(Aα eiqα z1 −Bα e−iqα z1)|htα〉 (2)

|HIt〉= |HI

0〉+∑α(Aα eiqα z1GI

H|e+α 〉+Bα e−iqα z1GIH|e−α 〉) (3)

where Eq. (2) is only valid for the area covered by the apertures of the slits. In Eqs. (2) and (3)|htα〉 is the transverse part of the magnetic field of the slit modes, |HI

0〉 is the field that wouldbe present at the top interface in the absence of any slits and GI

H|e±α 〉 is the transverse magneticfield of the radiation pattern at the top metal interface, which is generated by the downward andupward propagating slit modes. Similar equations can be written for the bottom metal interface.Using Eqs. (2) and (3) at the top interface and analog expressions at the bottom interface, Aαand Bα can be calculated for all slits using a low-rank linear set of equations whose coefficients


can be numerically calculated for arbitrary slit locations using a set of numerical simulations ona single slit [44]. The transverse magnetic and the normal electric field can be very efficientlycalculated at any point on the interface using Eq. (3) and

|EIz〉= |EI

z0〉+∑α(Aα eiqα z1GI

E|e+α 〉+Bα e−iqα z1GIE|e−α 〉) (4)

respectively, where GIE|e±α 〉 can be calculated for every field component using one simulation

on a single slit [44].

3. Design method

Figures 3 and 4 show the optimization algorithm we employ to design the mode demultiplexer,which couples multiple orthogonal optical modes to SPPs and focuses each mode towards adifferent position on either the top or bottom interface of the metal layer. Our iterative designmethod, which is illustrated in Fig. 3, uses the modal source radiator model at every iterationstep. We define ns as the total number of slits and n f as the total number of modes and aim tofocus the optical mode q onto the desired location rF

q = (xFq ,y

Fq ).

First, in Step 1 of Fig. 3, we compute the full field of the slit array under illumination ofeach optical mode q, using the modal source radiator model, which we have described in theprevious section. Consequently, for a certain slit configuration, Aα and Bα are known for everymode in every slit and the electric field in the normal direction |EI/II

z 〉 can be calculated at bothinterfaces in a very computationally efficient way using Eq. (4).

Then, in Step 2, the complex contribution Ψq,k of each slit to the field at each focal point isisolated by performing only a summation of all the modes α in slit k in Eq. (4) and evaluatingat rF

q as shown in Eq. (5).

Ψq,k = |EIz0〉(rF

q )+ ∑α(in slit k)

(Aα eiqα z1GIE|e+α 〉(rF

q )+Bα e−iqα z1GIE|e−α 〉(rF

q )) (5)

The goal of the design method is to place the slits in such a way that for each focus q, all thens contributions Ψq,k would be in phase. Naturally, it is not possible to do this perfectly for ahigher number of foci, or when strong coupling occurs between the slits, and we have to find acompromise using an optimization scheme that aims at minimizing the phase difference:

∆nΦq,k = arg(Ψq,k)−φ re fq +2nπ (6)

where arg(Ψq,k) is the phase of Ψq,k and

φ re fq = arg(exp(i2π(q−1)/n f )) (7)

is the target phase for each focus and is chosen to be fixed throughout the entire optimizationprocess. The choice of the target phase is arbitrary in the example cases we study below, wherewe choose to optimize the intensity. Nevertheless, an arbitrary target phase can be set using ouralgorithm, should that be of importance for a particular application. Also, we found that for theexamples we use in this paper, setting fixing the phase does affect the final position of the slits,but the overall performance remains the same. Using Eqs. (5)-(7), ∆nΦq,k is calculated for allslits at step 2.

In step 3, we calculate for each slit k the position rmink where ∆Φq,k is as small as possible,

compromising between all foci q. This is done by finding the minimum of a cost functionCFk(r(x,y)) that is set up to be minimal when ∆nΦq,k is small for each focus in an area close tothe slit and not in the immediate vicinity of other slits. In addition, CFk needs to take the field


Initial slit position

Compute the full field using themodal source radiator model

(Step 1)

For k = 1 to ns

Calculate the slit contributionΨq,k on each focus q and

the phase difference ∆Φq,k

(Step 2)

Derive the new position rmaxk

for slit k that would minimizethe cost function CFk

(Step 3)

Update the weight functionwq for each mode q (Step 4)

Completed nsciterations?

Calculate Pk foreach slit k and remove

the slit if Pk < 0(Step 5)

Move all slits to the new positionrnew

k = (1−w)rk +wrmink

Remove colliding slits(Step 6)

Reachedconvergence?

Done

Yes

No

No

Fig. 3. Illustration of the different steps in one iteration of the design method.


Sk

S1

Sns

F1,Ψ1,1..ns ,∆Φ1,1..ns

Rk1

Fq,Ψq,1..ns ,∆Φq,1..ns

Rkq

Fn f ,Ψn f ,1..ns ,∆Φn f ,1..ns

Rknf

δ1 δ2

x

y

z

Fig. 4. Illustration of the design method. We assume that the surface plasmon propagateswith phase velocity kspp. For slit Sk, the dashed blue circles represent the locations where∆Φq,k = arg(Ψq,k) and the full black circles represent the locations where ∆Φq,k = 2nπ .Also, δ1 = ∆Φq,k/kspp and δ2 = (2π −∆Φq,k)/kspp . The design method has the goal toplace the slit in a location where ∆Φq,k ≈ 2nπ , like the area inside the red circle.

intensity in each focus into account so as to optimize towards equal intensities in each focus.We choose CFk as :

CFk(r) =

n f

∑q=1

(min

n

{∣∣∣Rkq−|rFq − r|− ∆nΦq,k

kspp

∣∣∣}wq

)Gn

k(r)

Gk(r)(8)

where kspp denotes the surface plasmon phase velocity at the metallic interface and

minn

{∣∣∣Rkq−|rFq − r|− ∆nΦq,k

kspp

∣∣∣} denotes the shortest distance between r and the circles nearslit k where ∆Φq,k = 2nπ , as depicted by full black circles in Fig. 4. Naturally, we cannot checkthis for all n in practice and only perform the minimization for |n|<2 throughout this work. InEq. (8) we also introduce

Gk(r) = exp(−|r− rk|2/σ2g ) (9)

which represents a Gaussian distribution of width σg around the slit k so that the minimum ofCFk would be in the neighborhood of slit k located at rk . In addition, we introduce

Gnk(r) = 1+

ns

∑l 6=k

exp(−|r− rl |2/σ2nn) (10)

which is a sum over Gaussian distributions with width σnn centered around all the slits l (exceptslit k); we do this to avoid having several slits evolve towards the same optimum, where theywould collide.

To obtain a uniform intensity distribution along the n f foci we introduce a weight factor wqin Eq. (8) to give more weight to a focus with a relatively lower intensity. wq is initially definedas 1 for all q and is updated in step 4 according to the relation

wnewq = wold

q

(max(|Ez(rF

q )|2)|Ez(rF

q )|

)v

(11)


where max denotes the maximum over all foci and v is chosen as 0.1 throughout this work toavoid instability in the optimization[34]. In the examples we discuss later, we found that whenν << 0.1 we converged to a solution where one mode would be significantly favoured overthe other modes, while in the case that ν >> 0.1, the algorithm would not converge at all as itwould oscillate between solutions where one mode would be favoured over the others.

Throughout the optimization, some slits will lie in the path of the focused surface plasmongenerated by other slits and will consequently scatter the surface plasmon more than they scatterexcitation source. This surface plasmon already has the correct phase and the scattering causedby this slit will not improve the merit function, as those slits are just “blocking” the propagationof the surface plasmon. Such slits can be detected as ∆nΦq,k does not tend to converge to amultiple of 2π . For this reason every nsc iterations we remove the slits for which

Pk =

n f

∑q=1

[ℜ(Ψq,k)ℜ(φ re f

q )+ℑ(Ψq,k)ℑ(φ re fq )]< 0 (12)

where ℜ and ℑ denote the real and the imaginary part of a complex number respectively. Pk isthe sum over all foci of the projections in the complex plane of Ψq,k, i.e., the contribution ofeach slit to focus q onto φ re f

q , which is the target phase for each focus. The positions of the slitsthat remain will continue to be updated until the iteration number reaches the next multiple ofnsc. Throughout this work we performed this operation (step 5) every thirty iterations (nsc = 30).

Finally, in step 6 , we move each slit k to a new position rnewk given by

rnewk = (1−w)rk +wrmin

k (13)

where rmink = min(CFk(r)) is the minimum of the merit function for each slit k and w = 0.1 was

used throughout this work. In the examples we discuss later, we found that when w << 0.1 weconverged to a solution, albeit very slowly while in the case that w >> 0.1, the algorithm wouldnot converge at all. During this operation it may happen that despite the introduction of Gn

k(r),two or more slits still converge to the same location when all potential local minima have beentaken by other slits. To avoid that, after moving the slits to rnew

k , we check for slit pairs that aretoo close to each other and remove one of the two.

4. Examples

In this section, we apply our optimization method to two different cases with arrays of slitswith a width ws = 60 nm and a length ls = 300 nm in a silver layer of 200 nm thick on topof a silicon oxide layer with refractive index 1.45. We use a wavelength of 800 nm. First, weillustrate that our design method can be successfully used to design slit arrays that can focusan arbitrary mode onto an arbitrary spot on bottom side of the metallic sheet at the dielectricmetal interface. To this end, we design two different slit arrays that focus an L31 mode of aCorning® SMF-28e® optical fiber which has a diameter of 8.2 µm, a core index of 1.4585 anda cladding index of 1.4533 [Fig. 5(a)] onto foci at the oxide-metal interface at rF

1 = (0,0)µmand rF

1 = (12,0)µm respectively. In both cases we start with a 40 x 34 array of slits that isinitially centered around (0,0) and with an initial slit separation of 400 nm and 500 nm in thex and y direction respectively.

Also, in Eq. (8) we use σg = 0.5 µm, σnn = 0.05 µm ; kspp was computed using FDTD sim-ulations and the mode profile itself was numerically calculated using COMSOL Multiphysics3.5a. Our algorithm converged to a solution within 40 iterations and took about 50 seconds periteration on a regular desktop computer with a Intel(R) Xeon(R) CPU X5650-2.67GHz proces-sor and using 8 GB of RAM. In Figs. 5(b) and 6(a), the intensity of the normal electric field atthe oxide metal interface is depicted for both cases. This intensity is normalized to the average


(b)

−10 −5 0 5 10−10

−5

0

5

10

x (µm)

y(µ

m)

0

40

80

120

160(a)

−10 0 10−10

0

10

L31 Mode

0 50 100 150

(c)

|Ez|2

Fig. 5. Design of a slit array that focuses an L31 fiber mode (a) at rF1 = (0,0)µm . In (b)

the |Ez|2 field just below the metal layer is depicted and the white lines show the optimizedpositions of the slits. The cross-section of |Ez|2 through the focus and along the yellowdashed line is depicted in (c). All the fields are normalized to the average field intensity inthe fiber core.

−10 0 10−10

−5

0

5

10(a)

x (µm)

y(µ

m)

0

10

20

0 10 20

(b)

|Ez|2

Fig. 6. Design of a slit array that focuses an L31 fiber mode at rF1 = (12,0)µm . In (a) the

|Ez|2 field just below the metal layer is depicted and the white lines show the optimizedpositions of the slits. The cross-section of |Ez|2 through the focus and along the yellowdashed line is depicted in (b). All the fields are normalized to the average field intensity inthe fiber core.


0 10 20 30 400

50

100

150 (a)

Iteration #|E

z(rF 1

)|20 10 20 30 40

0

10

20(b)

Iteration #

|Ez(

rF 1)|2

Fig. 7. |Ez(rF1 )|2 at each iteration step for rF

1 = (0,0)µm (a) and for rF1 = (12,0)µm (b).

One clearly observes the improvement in field intensity obtained at iteration 30 when weremove the slits for which Pk < 0. All the fields are normalized to the average field intensityin the fiber core.

intensity of the electric field inside the fiber core. The final positions of the slits are drawn aswhite lines in Figs. 5(b) and 6(a). The normalized field intensity at the focal point at each iter-ation step is depicted in Fig. 7. For both cases, it can be seen that the field intensity increasessignificantly throughout the optimization before saturating. The discontinuous improvementthat occurs at the 30th iteration is caused by the removal of the slits for which Pk < 0, whichcan be thought of as “blocking” the focused surface plasmon as explained earlier. This canalso be seen in Fig. 5(b), where slits have been removed around the focus and Fig. 6(a) whereslits have been removed in the area where the focused surface plasmon has already reachedsubstantial intensity as it is being focused onto the focal point.

In a second example, we apply our optimization method to design a slit array aiming atdemultiplexing three orthogonal fiber modes. In addition, to validate our method, we confirmthe field patterns with full-field FDTD simulations. In order to keep the FDTD simulation sizereasonable, we choose a smaller fiber core and a smaller array of slits. More specifically, wechoose to demultiplex the L01, L21 and L22 [see Figs. 8(a)-8(c)] modes of a multimode fiberwith a diameter of 2.2 µm, a core index of 2 and a cladding index of 1.6 onto three foci locatedat rF

1 = (7,0)µm, rF2 = (7,−3)µm and rF

3 = (7,3)µm respectively. We start with a 13 x 11array of slits that is initially centered around (0,0) and with a initial slit separation of 400nm and 500 nm in the x and y direction respectively. All other parameters are equal to thoseused in the previous example. As can be seen from Fig. 9, our design method converged to asolution containing 48 slits after 110 iterations and three clearly distinct intensity peaks canbe observed at x = 7 µm corresponding to the yellow dotted line in Figs. 8(d)-8(f). To confirmthe correct implementation of the modal source radiator model, we perform a full-field 3D-FDTD simulation of the area illuminated by the source. In order to limit the simulation sizethe focus plane was not included in the simulation area and only the area inside the yellowdotted box in Figs. 8(d)-8(f) was simulated with a cell size of 5 nm. This resulted in an FDTDgrid of 1200x1200x400 cells. The FDTD simulation was performed using Belgium CAliforniaLight Machine[45] on 16 NVIDIA Tesla M2070 GPUs simultaneously. In this configuration,a simulation of 40000 time-steps completed in less than 140 minutes. As can be seen fromFig. 10, an almost perfect match is obtained between the modal source radiator model and thefull-field FDTD, which confirms the validity of our model.

5. Estimating the overall efficiency

An exact calculation of the overall efficiency is difficult, since the modes to which we coupleare not well defined nor did we include a material in which the coupled light would be ab-


(a)

−2 0 2

−2

0

2

y(µ

m)

L01 Mode

(b)

−2 0 2

−2

0

2

y(µ

m)

L21 Mode

(c)

−2 0 2

−2

0

2

x (µm)

y(µ

m)

L22 Mode

(d)FDTD Zone

−4 −2 0 2 4 6 8−4

−2

0

2

4

0

0.2

0.4

>0.6

(e)FDTD Zone

−4 −2 0 2 4 6 8−4

−2

0

2

4

0

0.2

0.4

>0.6

(f)FDTD Zone

−4 −2 0 2 4 6 8−4

−2

0

2

4

x (µm)

0

0.2

0.4

>0.6

Fig. 8. Ez field intensity just below the metal layer. The white lines show the optimizedpositions of the slits for demultiplexing the L01, L21 and L22 fiber modes. The fields arenormalized to the average field intensity in the fiber core.

0 20 40 60 80 100 1200

0.2

0.4 (a)

Iteration #

|Ez|2

atrF

−4 −2 0 2 40

0.15

0.3 (b)

y (µm)

|Ez|2

L21 ModeL22 ModeL01 Mode

Fig. 9. (a) Field intensity |Ez(rFq )|2 at each iteration step when demultiplexing the L01, L21

and L22 fiber modes and (b) Cross-section at x = 7 µm of the |Ez|2 field intensity for theoptimized structure shown in Figs. 8(d)–8(f) along the dashed yellow lines. The fields arenormalized to the average field intensity in the fiber core.


−4 −2 0 2 40

2

4

6

y (µm)

L21 Mode

FDTDModel

−4 −2 0 2 40

2

4

6

y (µm)

L22 Mode

−4 −2 0 2 40

2

4

6

y (µm)

norm

aliz

ed|E

z|2

L01 Mode

Fig. 10. Cross section of the |Ez|2 field intensity for the optimized structure shown in Fig. 8along the full line at x = 0 µm for comparison with a full-field FDTD simulation. TheFDTD simulation is run on a domain containing all the slits enclosed by the dotted box inFig. 8. The fields are normalized to the average field intensity in the fiber core.

sorbed. Nevertheless, it is possible to estimate the efficiency from the field intensity |Ez(r)|2as presumably most of the energy at the metal oxide interface is carried by surface plasmons.Therefore, we assume that the in-plane Poynting vector

P‖(x,y) =ˆ −∞

z=z2

(P(x,y,z)×z)dz (14)

which denotes the integrated components of the Poynting vector that are parallel to the metalsheet over the z direction, is proportional to |Ez(r)|2 or

|P‖(x,y)|= τ|Ez(x,y)|2 (15)

In the Eq. (14), P(x,y,z) is the Poynting vector and z denotes a unit vector along the z direction.We calculate the proportionality factor τ in Eq. (15) numerically using a simulated surfaceplasmon propagating in the x direction and then estimate the efficiency η from

η =

´ y2y1

τ|Ez(xF ,y)|2dy˜Psource

z dxdy(16)

where˜

Psourcez dxdy denotes the integrated Poynting vector of the excitation source and y1

and y2 are chosen at 1 µm around at either side of yF . Using this method we estimated totalefficiencies of between 0.8 percent for the designs depicted in Figs. 6 and 8 and 4 percent for thedesign depicted in Fig. 5. As is shown in Figs. 5 and 6, most of the surface plasmon energy isconcentrated in the focus. However, for the demultiplexing design we observe multiple “peaks”of surface plasmon energy. While the main focus still contains most surface plasmon energy, wecalculated that ηp ≈ 40% , where ηp is the ratio of the surface-plasmon energy that is actuallyconcentrated in the focus, over the integrated surface plasmon energy around the slits. For thisparticular integration, we use a square with side length 2x f centered at (0,0). Also, η and ηpremain stable when using a larger number of slits as the slits are already covering most of theilluminated area. In addition, η and ηp remain stable when we permute the location of the fociin our design as shown in Fig. 11, where the intensity is plotted for a design where we optimize


(a)

0 5−4

−2

0

2

4

x (µm)

y(µ

m)

L01 Mode

(b)

0 5x (µm)

L21 Mode

(c)

0 5x (µm)

L22 Mode

0

0.3

>0.6

Fig. 11. Ez field intensity just below the metal layer. The white lines show the optimizedpositions of the slits for demultiplexing the L01, L21 and L22 fiber modes focusing to rF

1 =(7,3)µm, rF

2 = (7,0)µm, rF3 = (7,−3)µm The fields are normalized to the average field

intensity in the fiber core.

the slit positions for demultiplexing the L01, L21 and L22 fiber modes to rF1 = (7,3)µm, rF

2 =(7,0)µm, rF

3 = (7,−3)µm, respectively. The behaviour of this design is completely analogousto the one shown in Fig. 8 and we obtained efficiencies η ≈ 0.8% and ηp ≈ 40% . From Figs. 8and 11, we observe that most of the non-focussed light is back scattered. As better unidirectionalperiodic grating couplers can be designed under illumination at an off-axis angle[46, 47], it maybe possible to improve ηp and η by optimizing the design for illumination at an angle.

We emphasize that the goal of this paper is to explore an optimization scheme for modecouplers and demultiplexers that is based on the source radiator model for a certain slit. Con-sequently, the slits’ size, metal thickness and wavelength are fixed and those parameters weresomewhat arbitrarily chosen to illustrate the optimization of the positions of a number of givenslits. In our structures, most of the light is still coupled into the oxide or reflected back andhigher efficiencies may be obtained by optimizing the slit size and metal thickness for a spe-cific wavelength [48]. Also we could consider using grooves instead of slits to prevent transmis-sion when coupling at the top interface [49], or coating of the metal with a dielectric grating,trapping the light at the metal surface. The algorithms presented here could be used for theabove-mentioned topologies, as those topologies could be modeled accurately with the sourceradiator model.

6. Discussion

We can compare the complexity of our designs here with previous analysis of the complexityrequired to make a particular optical component. As an example, we take our second design ofa mode splitter above. Ref. [50] gives the complexity number ND of real numbers that need tobe specified for a given component to be

ND = 2Mc

(MI +MO−MC−

12

)(17)

if we do not care about the absolute phase of the outputs [Eq. (11) of [50]]; in Eq. (17), MI isthe number of input modes (the dimensionality of the input space), here 3, MC is the numberof channels to be coupled through the device (here 3 also), and MO is the dimensionality of theoutput space. We can see from Fig. 9(b) that we are able to place our outputs along one line,with each output spot spaced approximately by one spot width from the neighboring spots. Wemight reasonably interpret this as saying we could place any one output at any of 5 distinctpositions at the output along this line. In our device, with slits oriented in a “top-bottom” axisin Fig. (8), we can design to route light to the left and right faces, essentially. Light scatteringinto the “top-bottom” direction is forbidden by the polarization of the light and the symmetry


of the slits. Since we could design this device to give its outputs along either the left or theright “faces” of the device, we could argue that our device has at least MO ∼ 10 different outputmodes it can choose from. Substituting these values in Eq. (17) gives a required minimumnumber of degrees of freedom of ND= 57 . In our design, we choose the x and y positions of 48slits, corresponding to 96 degrees of freedom, just under a factor of 2 larger than ND. We foundthat approximately this number of slits was the smallest that we could use and still effectivelyseparate the modes into the desired output spots. We should expect to use more degrees offreedom than the minimum number ND in a real design because that minimum number doesnot account for any control over the precise form of the output beams. To control that form,we should expect to be using a larger dimensional output space in practice. Our work here issuggesting a factor of ∼ 2 increase in output dimensionality is sufficient to allow the formationof convenient forms of output beams.

7. Conclusion

In this paper we successfully used a quantitatively accurate semi-analytic modal source radiatormodel for arbitrary two-dimensional non-periodic arrays of slits in order to design arbitrarymode couplers and demultiplexers. The use of the modal source radiator model allows us tocalculate the contribution of each slit taking both the direct scattering of the source and thescattering of the surface plasmon into account in a quantitatively accurate and computationallyefficient way. This ability enables the design of large arrays of closely-spaced slits. Our designmethod aims at focusing an arbitrary set of orthogonal modes onto different foci, by iterativelymoving slits to minimize the phase difference between the contributions of each slit in eachfocus. Thanks to our model, we can detect the slits that are scattering the focusing surfaceplasmon more than the source and remove them. We successfully applied our method to designtwo large aperiodic slit arrays that focus a L31 mode in the middle of the array itself or to a focalpoint outside they array. Also we used our design method to calculate aperiodic slit array thatfocuses the L01, L21 and L22 fiber modes onto 3 separated foci and confirmed the correctnessof our calculations using full field FDTD simulations.

Acknowledgments

The authors acknowledge the support of the Belgian American Education Foundation, theMethusalem and Hercules Foundations, IAP, FWO-Vlaanderen. This project was supportedby funds from Duke University under an award from the DARPA InPho program, and by theAFOSR Robust and Complex On-Chip Nanophotonics MURI. The authors expres their grati-tude to Andrew V. Adinetz, Jiri Kraus and Dirk Pleiter for their help in running the B-CALMsimulations on JUDGE, a GPU-cluster in Forschungszentrum Jülich.


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Design of large scale plasmonic nanoslit arrays for ...dabm/441.pdf · Design of large scale plasmonic nanoslit arrays for arbitrary mode conversion and demultiplexing Pierre Wahl,1,3,*