1

Nonlinear reconstruction of multilayer media inscanning microwave microscopy

Zhun Wei, Rui Chen, and Xudong Chen

Abstract—Scanning microwave microscopy (SMM) is ananoscale characterization technique widely used to image elec-trical or magnetic properties of materials at microwave frequen-cies. Nevertheless, it is challenging to reconstruct the properties ofmaterials from measured SMM signals and it is even difficult toretrieve multilayer media directly from measured signals. Thispaper proposes a nonlinear inversion scheme to quantitativelyreconstruct multilayer media. In the forward solver, hybridboundary integral-finite element (HBI-FE) method is used toreduce the computational cost. In the inverse problems, capaci-tance or conductance derivative signals at different probe-sampledistances are used to retrieve multilayer media. Reconstructedresults show that the proposed approach is able to reconstructboth permittivity and conductivity distributions under noisyenvironment in three dimensional samples. Most importantly, itis found that the resolution has been significantly improved inthe retrieved images compared with capacitance or conductancesignals. To the best of authors’ knowledge, it is the first time thatthe multilayer media in three dimensional media is reconstructedin SMM.

Index Terms—Nonlinear inversion, quantitative reconstruction,scanning microwave microscopy, resolution improvement, multi-layer media.

I. INTRODUCTION

SCANNING microwave microscopy (SMM) has attractedintense interests in the past decade due to its considerable

abilities to determine the physical properties of nanoscalematerials and devices such as magnetic materials, dielectricsamples, nanoparticles and even two-dimensional electron gasat microwave frequencies [1]–[3]. Numerous studies have beenaddressed on system design [2], [4], probe-sample interaction(interaction between electromagnetic waves and materials)[2], [5], [6], characterization of dielectric parameter [7]–[9],and high resolution imaging [10] for SMM. In addition tothe topography image, the SMM provides also images ofthe complex impedance of materials from which complexpermittivity, permeability, or conductivity of semi-conductorscan be extracted [5], [11], [12]. In SMM, as is illustrated inFig. 1, a metallic probe is used to scan across various pointson the surface of a sample mounted on an electrical groundsurface under either a contact mode or maintaining a constantprobe-sample spacing (z).

The capacitance or conductance between the probe and theground surface changes when the probe is near or upon aperturbation (permittivity or conductivity variations) in thesample [13], [14]. In general, the vector network analyzer in

Z. Wei, Rui Chen, and X. Chen are with Department of Electrical andComputer Engineering, National University of Singapore, 4 Engineering Drive3, Singapore 117583, Singapore (e-mail: elechenx@nus.edu.sg).

Fig. 1. A simplified schematic of the interaction between a probe andmaterials, where the dashed box represents effective region I . In the outputchannels, Re and Im correspond to the real part and imaginary part of theprobe-sample admittance variation, respectively [19], [20].

SMM measures the response of sample perturbation which isproportional to the capacitance or conductance related signal(Details of analysis are included in Appendix). Thus, theproperties of materials, such as permittivities and conductivi-ties, are contained in the measured signals. Although SMMcan receive signals that are related to physical propertiesof objects under test, most of the studies are limited toqualitative detection. Extracting physical information fromreceived signals is still a very challenging task, especially forthree dimensional inhomogeneous samples. Till now, existingstudies based on the line-fitting method have been used toextract a single dielectric constant parameter [7], [9], [15],[16] and also dopant concentrations [17]. In [18], skin effectsat different frequencies are also used to detect the buriedperturbations or defects [18]. However, when the perturbationsor defects themselves are multilayered with unknown shapesand properties, the mentioned methods can hardly distinguishor reconstruct the shapes or properties of these perturbations.

In [5], we have proposed a forward solver to deal withprobe-sample interaction problem in microwave impedancemicroscopy (MIM), and this solver is also found to be fastand accurate in calculating capacitance or conductance relatedsignals. Based on the forward model, a nonlinear recon-struction algorithm based on multi-probe-sample distancesin SMM is proposed in this paper. The proposed inversionapproach is utilized to reconstruct pixel-based permittivity andconductivity distributions in inhomogeneous samples insteadof extracting parameters by fitting methods. Moreover, pertur-bations distributed in different layers can be reconstructed. Itis also found that, the proposed approach is able to provideimproved image resolution compared with the capacitance orconductance signals.

2

II. FORMULATION OF THE PROBLEM

The geometry and parameters used in this paper to presentour method is illustrated in Fig. 1. we consider a widely usedcone-sphere probe which is depicted by the height of the wholeprobe H , radius of the probe r, and half cone angle θ. Itis noted that the probe can be of arbitrary geometry for theproposed approach and we choose the cone-sphere geometryas an example to present the results. The perturbation P isdistributed at some position inside the substrate with the sizeof lp × lp × hp1. In this paper, frequency is set to be 1 GHzunless otherwise stated. In the near field region, the sizes ofprobe and sample, as well as their gaps, are much smallerthan the wavelength, and thus the probe-sample interaction isactually in the electro-quasistatic regime.

It is noted that, since the resolution in SMM is related withmany factors such as probe radius, probe-sample distance,and materials under test, we are not exploring the limit ofresolution for SMM in this paper. Instead, our purpose is toreconstruct the distribution of permittivity or conductivity ofmaterials and improve the image resolution given the receivedsignals from a specific setup. Consequently, we use somerepresentative setups, as introduced above, to present ourmethod.

A. forward model

The forward model aims at calculating capacitance orconductance signal between probe and sample under a givenprobe-sample bias with known sample properties such aspermittivity and conductivity. According to our previous work[5], [21], it is found that only a limited effective region I(Dashed box in Fig. 1) beneath the probe contributes to theprobe-sample capacitance variation in SMM, where the sizeof the region is affected by setups of probe-sample systems,such as dielectric constant and probe geometry. Thus, we areable to decompose the whole 3-D simulation domain intotwo regions, as depicted in Fig. 1. For the region outside ofdomain I, we use the boundary integral method (BIM) to dealwith the potential, whereas the potential φ(r) in domain I iscalculated using finite element method (FEM) and it satisfiesthe following equations:

∇ · (ε(r)∇φ(r)) = 0 (1)

For dielectric samples, ε(r) is a real value representing per-mittivity of sample, whereas for conductive materials, ε(r) isreplaced by ε(r)− jσ(r)/ω with σ(r) and ω to be electricalconductivity and angular frequency, respectively. Followingthe finite element method [22], domain I is discretized intorectangular brick elements, and (1) can be discretized as:

K · φ−B · qb = 0 (2)

where K and B are evaluated as integral over domain I ele-ment and its boundary element respectively. qb is correspond-ing to potential derivative at the boundary with qb = ∂φb

∂n′ ,where φb and n′ are the potential on the boundary and inwardnormal direction of the boundary, respectively. According to

Green’s Theorem, the electrical potential in the exterior regionof domain I satisfies the following equation [23]:

φ(r) = φi(r) +

∮s

[G(r, r′)ε(r′)∂φ(r′)

∂n′

−φ(r′)ε(r′)∂G(r, r′)

∂n′]dr′ (3)

where G(r, r′) is the potential due to a unit charge (Green’sfunction) in the background medium, i.e., when there is noperturbation presented in the sample. s and n′ are the boundaryof domain I and its inward normal direction, respectively.Following the discretization method in FEM, the potential φbon boundary of domain I satisfies the following equation byapplying collocation method to (3):

H · φb +G · qb = b (4)

where H and G are calculated as integrals ofε(r′)∂G(r, r′)/∂n′ and −ε(r′)G(r, r′) over boundaryelement of domain I, respectively, and b is corresponding toφi(r) on the boundary of domain I. By combining (2) and(4), the potential on the boundary of domain can be easilysolved.

A homogeneous sample which excludes the perturbation ischosen as a reference model to obtain reference capacitanceCref between probe and ground. Contrast capacitance (de-noted as ∆C), which is defined as the difference betweencapacitance in the presence of perturbation and Cref , isevaluated in this paper. According to (3), contrast capacitancebetween probe and ground due to the presence of perturbationin sample can be expressed as:

∆C =

∮s

[Gc(r, r′)ε(r′)

∂φ(r′)

∂n′

−φ(r′)ε(r′)∂Gc(r, r

′)

∂n′]dr′ (5)

where Gc(r, r′) and ∂Gc(r, r′)/∂n′ can be calculated as the

total charge on the probe due to a unit charge (Green’sfunction) and dipole in background, respectively. For (5), usingthe same process in discretizing (3), contrast capacitance onprobe can be evaluated as:

∆C = −LT · φb −MT · qb (6)

The matrices M and L are calculated as integrals of−ε(r′)Gc(r, r′) and ε(r′)∂Gc(r, r

′)/∂n′ over boundary ele-ment of domain I, and the superscript T denotes the transposeoperator. For the perturbation with conductive materials pre-sented, capacitance variation will be frequency dependent. Therelationship between charge on the probe and capacitance is:

Q(ω) =I(ω)

jω= V (ω)(−jGts(ω)

ω+ C(ω)) (7)

where Gts is the conductance between probe and sample.Under a probe-sample bias amplitude of 1 V , capacitance isequal to the real part of Q(ω), and combined with (6), wehave:

3

Fig. 2. The procedure for the multi-probe-sample distances scanning basednonlinear reconstruction algorithm. n is the iteration number; L is the totalnumber of recordings for different probe-sample distances scanning.

∆C(ω) = Re(−LT · φb −MT · qb) (8)

Similarly, considering the relationship between Q(ω) andGts, contrast conductance is obtained from (6):

∆Gts(ω) = Im[(LT · φb +M

T · qb)ω] (9)

Normally, a complex signal ∆S(ω) = −LT ·φb−MT ·qb can

be obtained by measuring capacitance and conductance signalat two separate channels in SMM. It is noted that, under mostcases, Green’s function in the forward solver has no analyticalsolution in SMM, but it can be evaluated numerically easilyusing commercial software. The most convenient approach isto save it in the library. In this paper, COMSOL Multiphysicssoftware is used to conduct the simulations. Note that thenumerical evaluation of the Green’s function is needed onlyonce for a given experimental setup, and will not changeduring the scanning process.

B. Inverse formulation

To solve the potential at the boundary of domain I , (2)and (4) is firstly transformed to the following forms to avoidseverely ill condition in inverse process.

Kt · φ−Bt · qtb = 0 (10)

H · φb +Gt · qtb = b (11)

where Kt = t2K, Bt = t1t2B, Gt = t1G, and qtb = qb/t1with t1 and t2 be constant values (t1 = 107 and t2 = 106 inthis paper).

Defining the matrix P , which picks up the boundary nodesout of all nodes (details are included in Appendix), the

potential at the boundary φb is obtained by combining (10)and (11):

φb = P · φ = P ·Kε

−1·Bt ·Gt

−1· b (12)

where Kε = Kt + Bt ·Gt−1·H · P . Therefore, the contrast

signal ∆Si at the ith scanning point is obtained by

∆Si = −LTφb−MTqb = (M1+M3)·K

i

ε

−1·M2+M4 (13)

where M1 = −LT · P , M2 = Bt · Gt−1· b, M3 = t1M

T ·Gt−1·H , and M4 = −t1M

T ·Gt−1· b. Here, K

i

ε is the valueof Kε at the ith scanning point, and it is also the only termswhich relates to the unknown complex permittivity ε in (13).Then, the residue in (13) can be calculated as

∆ie = (M1 +M3) ·K

i

ε

−1·M2 +M4 −∆Sri (14)

where ∆Sri represents the received contrast signal at the ithscanning point with notation “r” representing “received”. Aleast square cost function is defined:

fz =

N∑i=1

||∆ie||

2/||∆Sri ||2 (15)

where N and notation z represent the total scanning points andprobe-sample distance, respectively. The gradient of objectivefunction fz with respect to the complex permittivity εe at aspecific element e is calculated as:

∂fz∂εe

=

N∑i=1

−2[(M1+M3)·Ki

ε

−1·∂K

i

ε

∂εe·K

i

ε

−1·M2]∗∆i

e/||∆Sri ||2

(16)where the superscript “*” denotes the transpose conjugateoperator.

In SMM, except contrast signals ∆Si, the derivative ofsignals (dSri,z = −∂∆Sri,z/∂z) are widely measured throughlock-in to a probe-sample distance (z) modulation to reducethe effects of stray capacitance or conductance [24], whichcome from the contribution of cantilever and upper conepart of the probe. In this paper, the derivative of signals areconsidered, and based on the residue definition in (14), thecost function corresponding to dSri,z is expressed as:

f1z =

N∑i=1

||(∆Siz−dz −∆Siz+dz)/2dz − dSri,z||2/||dSri,z||2

(17)where ∆Sz−dzi and ∆Sz+dzi are ∆Si in (13) calculated atprobe-sample distances z− dz and z+ dz, respectively. Here,a finite difference is considered with dSi

z = (∆Siz−dz −

∆Siz+dz)/2dz. Since in practice the measured signal is noise

contaminated, to balance in data fitting and stability of so-lution, Tikhonov regularization is used, and the final costfunction is defined as:

F 1z = f1z + α||ε||2 (18)

where α is a constant regularization parameter. With (16),(17), and (18), the gradient of F 1

z can be easily calculated.The Tikhonov regularization in (18) is used to balance in

4

data fitting and stability of the solution. As α tends to zero,the solution tends to the solution fitting objective functionf1z , but the solution becomes unstable. In this sense, thesolution provided by the Tikhonov regularization scheme, asa minimizer of the Tikhonov functional, keeps the residuef1z small and is stabilized through the penalty term α||ε||2.According to our experience, α is chosen around the ratio ofthe gradient of the first part to the second part on right handside of (18).

C. Multi-probe-sample distances scanning based implementa-tion procedures

In this paper, conjugate gradient (CG) method is usedto minimize the objective function in (18). To reconstructmultilayer media, values of dSri,z corresponding to a totalnumber of L different probe-sample distances are measured.The implementation procedures of the multi-probe-sampledistances scanning based inverse problem, as is shown in Fig.2, are detailed as follows:• Step 1) Initial step, n = 1; Give an initial guess of εn =εb, and εb is the background permittivity (Initial value ofε0 can be of arbitrary one).

• Step 2) Let l = 1, and the probe-sample distance z = zl.• Step 3) Calculate M1, M2, M3, and M4 in (13) at zl.• Step 4) Determine the search direction: Calculate the

matrix term Ki

ε, objective function F 1zl

(εn), and gradientof objective function g(εn) = ∂F 1

zl(εn)/∂εn. Then de-

termine the Polak-Ribiere-Polyak (PRP) directions [25]:If n=1, the search direction ρ1 is ρ1 = −g1. Otherwise,ρn = −gn + (Re[(gn − gn−1)∗ · gn]/||gn−1||2)ρn−1.

• Step 5) Determine the search length ln according to Wolfeconditions [19] (Initialize m=0):

– 5.1) Calculate f1zl(ε+ γmρn) and g(ε+ γmρn),where γ is a positive constant parameter smaller that1 (γ = 0.4 in this paper).

– 5.2) If f1zl(ε)−f1zl

(ε+ γmρn) ≥ −δγmgTnρn, whereδ is a parameter adjusted in optimization and 0 <δ < 1, let ln = γm. Otherwise, m = m+ 1, and goto Step 5.1).

• Step 6) Update εn+1: εn+1 = εn + lnρn.• Step 7) If l = L, go to step 8). Otherwise, let l = l + 1

and z = zl, and go to step 3).• Step 8) If n = Nm, stop iteration (Nm = 150 in this

paper). Otherwise, let n = n+ 1, and go to step 2).It is noted that, at each iteration, εn+1 is updated accordingto dSri,z with smallest probe-sample distance since dSri,zwith smaller probe-sample distance has better resolution andcontains more local information. Therefore, we have assumedz1 > z2 > ... > zL in the implementation procedures.

III. NUMERICAL EXAMPLES

This section presents some numerical results to evaluate theperformance of the proposed nonlinear reconstruction methodin this paper. In all the numerical results, we use a probewith H = 5 µm, r = 50 nm, and θ = 20◦. The substratethickness hs = 0.8 µm and background of the sample is filled

with SiO2 with permittivity of εb = 3.9, as is illustratedin Fig. 1. Three different sample structures including singlelayered dielectric media, two layered dielectric media and twolayered conductive media are considered in this section. Apriori information is known that perturbations are within acube with 0.5 µm × 0.5 µm × 0.15 µm, which means thathp1 = 0.15 µm and lp = 0.5 µm in Fig . 1. To avoid inversecrime, we use a finer mesh in the forward process and a coarsemesh in the inverse process. In these examples, additive whiteGaussian noise (AWGN) is added to the measured voltages,and is quantified by (||n||/||dSz||)×100%, where ||·|| denotesFrobenius norm. For the examples in this section, 1% Gaussiannoises are added in the received signals.

A. Mechanism of choosing effective region

As is shown in Fig. 3(a), to collect signals at different points,the probe scans around the perturbation over the sample.In practical, one can hardly know the exact boundary ofperturbation, thus we assume that perturbations are distributedin a cuboid region P . For a given setup, an effective region(I) is firstly approximately established, and one only needsto calculate the perturbation from the effective region insteadof the whole sample at each scanning point [5]. The moti-vation is that, for different scanning points in Fig. 3(a), theregion that perturbs the probe-ground capacitance is limitedto a box (dashed box in Fig. 3(a)) beneath the probe and,consequently, for the materials outside this region we can treatthem as known homogeneous material due to their negligiblecontribution to contrast capacitance.

Fig. 3. (a) Schematic of choosing effective region mechanism, wherethe I that is inside the dashed square denotes effective region, P denotesperturbation presented region, and the cross sign represents the center of theprobe. (b) Normalized received signal (NdS) varying with perturbation widthlp.

5

Fig. 4. Single layered dielectric material example:(a) A three-dimensionalsample with an “H” shape perturbation presented with Ws = 6 µm,hs = 1 µm, hp = 0.2 µm, hp1 = 0.15 µm, Wh = 100 nm, Ls = 400 nm,εb = 3.9 and ε1 = 16; (b) Top view of exact distribution of relativepermittivity in (a); (c) The simulated received signal; (d) Reconstruction ofrelative permittivity distribution from the signal in (c). It is noted that thedashed lines denote the exact boundary of stripes in (a).

The width of effective region lI is chosen as the smaller onefrom two parameters. The first parameter is saturation widthlpm. To approximately determine lpm, scanning point is firstlymove to the center of domain P in Fig. 3(a). Perturbations withconstant permittivity εt are filled in region P , then receivedsignals are calculated with the size of lp gradually increasing.Fig. 3(b) presents the normalized received signal NdS varyingwith the value of lp, where NdS is defined as the signal dS(z)normalized to its maximum value dS(z = 16 µm). It suggeststhat NdS increases when the size of perturbation is enlarged,but it saturates when lp reaches some certain value lpm. FromFig. 3(b), one can tell that lpm is approximately equal to1.5 µm. It is noted that the value of εt can slightly affectthe value lpm, and we usually choose a larger value such asεt = 16 in simulation to ensure that lpm is not underestimated.

The second parameter is the full cover width lIm. If regionI with a width lIm can cover all the region of P when probescans at four corners of P , then the calculated dS will beaccurate without any approximation. It can be easily seen fromFig. 3(a) that lIm = 2lp. In this paper, since full cover widthlIm = 1 µm is smaller than saturation width lpm = 1.5 µm,we directly choose the width of effective region lI as 1 µm.

B. Single layered dielectric material

Fig. 4(a) presents a three dimensional “H” shape perturba-tion presented sample. The total sample size is Ws×Ws×hswith Ws = 6 µm and hs = 0.8 µm. The “H” shape perturba-tion is distributed in a top layer layer of the sample with thethickness hp1 = 0.15 µm, width Wh = 100 nm , and lengthLs = 400 nm. As is illustrated in Fig. 4(a), except the “H”perturbation shape, all the other regions of the sample havea permittivity of εb = 3.9. In this example, the perturbationsare presented in a single layer so that the unknowns are only

Fig. 5. (a) Top view (b) side view of a two layered media with hs = 0.8 µm,hp = 0.2 µm, hp1 = 150 nm, hp2 = 100 nm, ε1 = 16, and ε1 = 6.Exact distribution of relative permittivity of (c) bottom (d) top layer withLs1 = 400 nm, Ws1 = 100 nm, and ds1 = 200 nm. Exact distributionof relative permittivity of (e) bottom (f) top layer with Ls1 = 300 nm,Ws1 = 50 nm, and ds1 = 100 nm.

two dimensional distribution of permittivity. The top view ofexact distribution of permittivity is depicted in Fig. 4(b). Toreconstruct the relative permittivity distribution, the derivativesignal at z = 7.5 nm is used, which is shown in Fig. 4(c).It is seen that the“H” feature is hardly recognized althoughthe position and size of the “H” shape are roughly displayed.Whereas, the permittivity of the sample can be retrievedfrom the received signals. Fig. 4(d) presents the reconstructedrelative permittivity from the received signal in Fig. 4(c). Itsuggests that, by reconstructing the relative permittivity pixelby pixel, the “H” pattern is retrieved in Fig. 4(d), and imagingresolution is highly improved. It is noted that, with the increaseof noise, the qualities of both simulated received image andreconstructed image would be highly degraded.

C. Multi-layered dielectric material

In the second example, a two-layer dielectric media isconsidered. The total size of the sample is the same as thatof the first example, whereas the shape of perturbation aremore complicated. As is illustrated in Fig. 5, four stripes areformed as a square shape from the top view of the sample, andthey are presented in two different layers of the sample. Tworeconstruct the multilayer media, dSzi at three different probe-sample distances (z1 = 17.5 nm, z2 = 12.5 nm, z3 = 7.5 nm)is calculated. It is noted that the values of the distances can bechosen arbitrarily, but they are better below 20 nm since thelocal information will be reduced in the received signals with

6

Fig. 6. Multi-layered dielectric material example: (a) The simulated received derivative signal at z3 = 7.5 nm with Ws1 = 100 nm, ds1 = 200 nm, andLs1 = 400 nm; Reconstructed distribution of (b) bottom and (c) top layer relative permittivity from received signals. (d) The simulated received derivativesignal at z3 = 7.5 nm with Ws1 = 50 nm, ds1 = 100 nm, and Ls1 = 300 nm; Reconstructed distribution of (e) bottom and (f) top layer relative permittivityfrom received signals. It is noted that the dashed lines in (b), (c), (e), and (f) denote the exact boundary of stripes in Fig. 5.

a larger probe-sample distance. Fig. 6(a) presents the receivedsignal at z3 = 7.5 nm with the stripes length Ls1 = 400 nm,width Ws1 = 100 nm, and distance ds1 = 200 nm. It isseen that one can barely observe a very obscure square fromFig. 6(a), and it is impossible to distinguish different layersmaterials from the received signal. In Figs. 6(b) and (c), thereconstructed relative permittivity distribution for bottom andtop layers are displayed, where the dash lines denote the exactboundary of stripes. It is found that the four stripes are clearlyretrieved in reconstructed images. In Figs. 6(d), (e), and (f),a more challenging case is considered, in which the stripeshave length Ls1 = 300 nm, width Ws1 = 50 nm, and distanceds1 = 100 nm. It can be seen that the received signals displayno useful information of materials at all when two stripes areput closer to each other. On contrary, in Figs. 6(e) and (f), thedistribution of relative permittivity for both layers are clearlydisplayed for reconstructed image, although the results are notas good as the previous case.

D. Multi-layered conductive material

In the third example, we consider a three dimensionalsample with conductive perturbation presented, which is muchmore complicated compared with dielectric cases in SMM. Asis depicted in Fig. 7, the structure is similar with that of thefirst example, and three stripes are presented forming a “H”shape from the top view. Different from the first case, threestripes are presented in two different layers and the bottomstripe is conductive material instead of dielectric. Similar to thesecond example, dSzi at three different probe-sample distances(z1 = 17.5 nm, z2 = 12.5 nm, z3 = 7.5 nm) is calculated.Figs. 8(a) and (d) presented the simulated capacitance andconductance derivative signal at z3 = 7.5 nm, respectively.In SMM, the capacitance and conductance derivative signal

Fig. 7. (a) Top view of a two layered media with ls2 = 400 nm, Ws2 =100 nm, ds2 = 200 nm, ε1 = 16 − jσ1/(ω1ε0), and ε2 = 6, whereσ1 = 0.2 and ε0 is the permittivity in vacuum. The the stripe at bottom layeris conductive, whereas two stripes at top layer are dielectric. Exact distributionof relative permittivity of (b) bottom (c) top layer. (d) Exact distribution ofconductivity of bottom layer.

correspond to two separate channels, which are perpendicularto each other [19], [20]. It is seen from Figs. 8(a) and (d)that it is impossible to distinguish the distribution of multi-layered materials from the measured signal directly. Figs.8(b) and (e) present the reconstructed distribution of relativepermittivity and conductivity at bottom layer from receivedcomplex signals, respectively. It is shown that the shapes ofboth permittivity and conductivity of the stripe presented atbottom layer are clearly restored in the retrieved image. As isshown in Figs. 8(c) , the two dielectric stripes in the top layerare also restored from the reconstructed relative permittivity

7

Fig. 8. Multi-layered conductive material example: The simulated (a) capacitance (d) conductance derivative signal at z3 = 7.5 nm; Reconstructed distributionof (b) relative permittivity (e) conductivity at bottom layer from received complex signals; Reconstructed distribution of (c) relative permittivity (f) conductivitytop layer from received complex signals. It is noted that the dashed lines in (b), (c), and (e) denote the exact boundary of stripes.

distribution.

E. Discussion of noise contaminations

In inverse problems, since practical measured signals inexperiment are usually contaminated with noise, Gaussiannoise is usually added to the simulated signal to test theperformance of the inversion method. When higher Gaussiannoise such as 5% or 10% is added, the qualities of bothsimulated received imaging and reconstructed imaging willbe degraded, which can seen from the results in Fig. 9.Specifically, different with the case in Fig. 4(c) with 1%Gaussian noise, 5% and 10% Gaussian noise are added into thereceived signals in Figs. 9(a) and (b), respectively. In Figs. 9(c)and (d), we present the reconstructed results from the receivedsignals with 5% and 10% noise contaminated. It is found that,with the increase of noise, the qualities of both simulatedreceived and reconstructed images are degraded, which is alsoa common rule in inverse problems.

IV. CONCLUSION

In this paper, by using the received signals in scanning mi-crowave microscopy at different probe-sample distances witha specific order, a nonlinear reconstruction scheme is proposedbased on conjugate gradient algorithm. The proposed approachis verified for single layered dielectric, multi-layered dielectric,and multi-layered conductive materials. It is concluded fromthe results that, compared with received signals, the resolutionsin retrieved images are significantly improved given a specificsetup. Most importantly, the proposed nonlinear reconstructionscheme is able to reconstruct multi-layered media from the re-ceived signals, which is significant for microwave microscopyimaging. In the next step, the experimental part of the inversionmethod will be focused to verify that the proposed methodis able to improve resolution in experiment. In addition todifficulties in numerical inversion, there will be also some

Fig. 9. The simulated received signals of “H” shape model in Fig. 4 with(a) 5% and (b) 10% noise; Reconstruction of relative permittivity distributionfrom the signals with (c) 5% and (d) 10% noise.

experimental tasks, such as the compensation of drift errorsand noise and the calibration of various samples.

ACKNOWLEDGMENT

This research was supported by the National ResearchFoundation, Prime Minister’s Office, Singapore under its Com-petitive Research Program (CRP Award No. NRF-CRP15-2015-03).

APPENDIX

The first part of the appendix introduces the principles toestablish the relationship between measured signals in a typicalscanning microwave microscopy, i.e., MIM [5], [19], [20], and

8

impedance variations between tip and sample. For the MIMsystem in Fig. 1, a reference point is defined as the positionbetween Z-match and microwave electronics, and the outputsignals from Re and Im channels are represented as SR andSI , respectively. For the linear MIM microwave electronicsystem, the relationship between SR and SI signals and thevariations of reflection coefficient ∆S11 at reference point canbe expressed as [5], [19], [20]:

SR +j SI ∝ ∆S11 (19)

where ∆S11 is the variations due to the perturbation in sampleand can be calculated as S11(Y )−S11(Y0) with Y0 being thereference impedance, i.e., the impedance between referencepoint and ground without perturbation presented. Specifically,Y0 consists of impedances from Z-match circuit, and probe-sample interaction.

Take Taylor expansion on S11(Y ):

S11(Y ) = S11(Y0) + S′

11(Y0)(Y − Y0) + . . . (20)

in which the difference between Y and Y0 is a tiny perturbationcompared to the whole impedance. Then, from (20), one canget:

∆S11 ∝ ∆Y (21)

Here, ∆Y = ∆1/R + jω∆C with ∆1/R and ∆C beconductance and capacitance variation between probe andground. Therefore, by combing (19) and (21), it is seen that thereceived MIM-Re and MIM-Im have an approximately linearrelationship with the variations of impedance between tip andsample:

SR +j SI ∝ ∆1/R+ jω∆C (22)

Thus, in data interpretation process of MIM, one onlyneeds to firstly do a calibration to find the linear coefficientbetween measured signals and impedance variations for furthersample information analysis [5]. Specifically, in [5], we taketip-sample approach curves at the same scanning point byboth electrostatic force microscopy (EFM) and microwaveimpedance microscopy (MIM), then the approach curves arematched between them by a scaling factor. Consequently, oneis able to directly convert measured signals in MIM intocapacitance images when tip scans across the sample usingthe scaling factor obtained above.

In the second part of this appendix, some details of deriva-tion in Section II(B) are offered. To derive (12) from (10) and(11), qtb is expressed as:

qtb = Gt−1· b−Gt

−1·H · φb (23)

based on (11). Substitute (23) into (10) and consider that φb =

P · φ, (12) can be obtained.Here, the matrix P is defined as picking up the boundary

nodes out of all nodes with the size of Nb × Nd, where Ndand Nb denotes the number of total discretization nodes ofthe domain and nodes on the boundary, respectively. Morespecifically, P contains only 0 and 1 elements and at each row,only the index corresponding to the boundary node elementis 1. A very simple example would be: if the potential of thewhole domain is φ = [φ1, φ2, φ3, φ4]T with T denoting matrix

transpose, and we know that φ1 and φ3 are on the boundary,then one can easily obtain that

[φ1, φ3]T = P · φ =

[1 0 0 00 0 1 0

]· φ (24)

REFERENCES

[1] E. Y. Ma, Y.-T. Cui, K. Ueda, S. Tang, K. Chen, N. Tamura, P. M. Wu,J. Fujioka, Y. Tokura, and Z.-X. Shen, “Mobile metallic domain wallsin an all-in-all-out magnetic insulator,” Science, vol. 350, no. 6260, pp.538–541, 2015.

[2] H. Bakli, K. Haddadi, and T. Lasri, “Interferometric technique for scan-ning near-field microwave microscopy applications,” IEEE Transactionson Instrumentation and Measurement, vol. 63, no. 5, pp. 1281–1286,2014.

[3] M. F. Crdoba-Erazo and T. M. Weller, “Noncontact electrical char-acterization of printed resistors using microwave microscopy,” IEEETransactions on Instrumentation and Measurement, vol. 64, no. 2, pp.509–515, 2015.

[4] K. Haddadi and T. Lasri, “Scanning microwave near-field microscopebased on the multiport technology,” IEEE Transactions on Instrumenta-tion and Measurement, vol. 62, no. 12, pp. 3189–3193, 2013.

[5] Z. Wei, Y. T. Cui, E. Y. Ma, S. Johnston, Y. Yang, R. Chen, M. Kelly,Z. X. Shen, and X. Chen, “Quantitative theory for probe-sample interac-tion with inhomogeneous perturbation in near-field scanning microwavemicroscopy,” IEEE Transactions on Microwave Theory and Techniques,vol. 64, no. 5, pp. 1402–1408, 2016.

[6] S. J. Gu, K. Haddadi, A. El Fellahi, and T. Lasri, “Setting parametersinfluence on accuracy and stability of near-field scanning microwavemicroscopy platform,” IEEE Transactions on Instrumentation and Mea-surement, vol. 65, no. 4, pp. 890–897, 2016.

[7] R. Joffe, R. Shavit, and E. Kamenetskii, “Multiresonance measurementmethod for microwave microscopy,” IEEE Transactions on Instrumen-tation and Measurement, vol. 66, no. 8, pp. 2174–2180, 2017.

[8] T. Monti, O. B. Udoudo, K. A. Sperin, C. Dodds, S. W. Kingman,and T. J. Jackson, “Statistical description of inhomogeneous samplesby scanning microwave microscopy,” IEEE Transactions on MicrowaveTheory and Techniques, vol. 65, no. 6, pp. 2162–2170, 2017.

[9] L. Fumagalli, M. A. Edwards, and G. Gomila, “Quantitative electrostaticforce microscopy with sharp silicon tips,” Nanotechnology, vol. 25,no. 49, p. 9, 2014.

[10] G. Gramse, A. Klker, T. Lim, T. J. Z. Stock, H. Solanki, S. R.Schofield, E. Brinciotti, G. Aeppli, F. Kienberger, and N. J. Curson,“Nondestructive imaging of atomically thin nanostructures buried insilicon,” Science Advances, vol. 3, no. 6, 2017.

[11] Z. Wu, W.-q. Sun, T. Feng, S. W. Tang, G. Li, K.-l. Jiang, S.-y. Xu, andC. K. Ong, “Imaging of soft material with carbon nanotube tip usingnear-field scanning microwave microscopy,” Ultramicroscopy, vol. 148,pp. 75–80, 2015.

[12] K. Haddadi, S. Gu, and T. Lasri, “Sensing of liquid droplets with ascanning near-field microwave microscope,” Sensors and Actuators A:Physical, vol. 230, pp. 170–174, 2015.

[13] F. Wang, N. Clement, D. Ducatteau, D. Troadec, H. Tanbakuchi,B. Legrand, G. Dambrine, and D. Theron, “Quantitative impedancecharacterization of sub-10nm scale capacitors and tunnel junctions withan interferometric scanning microwave microscope,” Nanotechnology,vol. 25, no. 40, p. 7, 2014.

[14] A. Guadarrama-Santana and A. Garcia-Valenzuela, “Obtaining the di-electric constant of solids from capacitance measurements with a pointerelectrode,” Review of Scientific Instruments, vol. 80, no. 10, p. 3, 2009.

[15] G. Gramse, G. Gomila, and L. Fumagalli, “Quantifying the dielectricconstant of thick insulators by electrostatic force microscopy: effects ofthe microscopic parts of the probe,” Nanotechnology, vol. 23, no. 20,p. 7, 2012.

[16] L. Fumagalli, G. Gramse, D. Esteban-Ferrer, M. A. Edwards, andG. Gomila, “Quantifying the dielectric constant of thick insulators usingelectrostatic force microscopy,” Applied Physics Letters, vol. 96, no. 18,p. 3, 2010.

[17] H. Huber, I. Humer, M. Hochleitner, M. Fenner, M. Moertelmaier,C. Rankl, A. Imtiaz, T. Wallis, H. Tanbakuchi, and P. Hinterdorfer,“Calibrated nanoscale dopant profiling using a scanning microwavemicroscope,” Journal of Applied Physics, vol. 111, no. 1, p. 014301,2012.

9

[18] C. Plassard, E. Bourillot, J. Rossignol, Y. Lacroute, E. Lepleux,L. Pacheco, and E. Lesniewska, “Detection of defects buried in metal-lic samples by scanning microwave microscopy,” Physical Review B,vol. 83, no. 12, p. 121409, 2011.

[19] K. Lai, W. Kundhikanjana, M. Kelly, and Z. X. Shen, “Modeling andcharacterization of a cantilever-based near-field scanning microwaveimpedance microscope,” Review of Scientific Instruments, vol. 79, no. 6,p. 6, 2008.

[20] W. Kundhikanjana, “Imaging nanoscale electronic inhomogeneity withmicrowave impedance microscopy,” Ph.D. dissertation, Stanford Univer-sity, 2013.

[21] W. Zhun and C. Xudong, “Numerical study of resolution in near fieldmicroscopy for dielectric samples,” pp. 910–911, 19-24 July 2015 2015.

[22] J. Jin, The finite element method in electromagnetics. New York;GreatBritain;: John Wiley and Sons, 2002, vol. 2nd.

[23] J. Jackson, Classical Electrodynamics. Wiley, 1998.[24] Z. Wei, E. Y. Ma, Y. T. Cui, S. Johnston, Y. L. Yang, K. Agarwal,

M. A. Kelly, Z. X. Shen, and X. D. Chen, “Quantitative analysis ofeffective height of probes in microwave impedance microscopy,” Reviewof Scientific Instruments, vol. 87, no. 9, p. 6, 2016.

[25] Y. H. Dai and Y. Yuan, “A nonlinear conjugate gradient method witha strong global convergence property,” Siam Journal on Optimization,vol. 10, no. 1, pp. 177–182, 1999.