reflection interference contrast microscopy of arbitrary convex surfaces

Reflection interference contrast microscopy ofarbitrary convex surfaces

Jose C. Contreras-Naranjo, James A. Silas, and Victor M. Ugaz*Artie McFerrin Department of Chemical Engineering, Jack E. Brown Engineering Building,

Texas A&M University, College Station, Texas 77843-3122, USA

*Corresponding author: [email protected]

Received 4 February 2010; revised 7 May 2010; accepted 14 May 2010;posted 18 May 2010 (Doc. ID 123806); published 23 June 2010

Current accurate applications of reflection interference contrast microscopy (RICM) are limited to knowngeometries; when the geometry of the object is unknown, an approximated fringe spacing analysis isusually performed. To complete an accurate RICM analysis in more general situations, we reviewand improve the formulation for intensity calculation based on nonplanar interface image formationtheory and develop a method for its practical implementation in wedges and convex surfaces. In addition,a suitable RICM model for an arbitrary convex surface, with or without a uniform layer such as amembrane or ultrathin coating, is presented. Experimental work with polymer vesicles shows thatthe coupling of the improved RICM image formation theory, the calculation method, and the surfacemodel allow an accurate reconstruction of the convex bottom shape of an object close to the substrateby fitting its experimental intensity pattern. © 2010 Optical Society of AmericaOCIS codes: 070.5010, 080.2720, 100.2650, 100.3010, 100.6890, 180.3170.

1. Introduction

An image formed by reflection interference contrastmicroscopy (RICM) contains precise informationabout the topography of the object under observation.Current applications of RICM are focused on an-alysis of experimental interference patterns fromknown geometries and sizes using sophisticatedmodels of image formation [1–3], or greatly simpli-fied fringe spacing analysis when the geometry ofthe object is unknown [4–9]. For instance, simulta-neous determination of the three-dimensional (3D)positions of multiple spherical particles can be per-formed by direct comparison with simulated interfer-ograms from spheres [2]. On the other hand, RICMadhesion studies of cells and lipid vesicles, where thegeometry is unknown, usually involve determiningtheir approximated contour near the contact regionby inverse cosine transform [4–7]. However, if a so-phisticated RICM image formation model could beapplied to simulate intensity patterns from arbitrary

geometries so that they could be directly comparedwith experimental interferograms, it would be possi-ble to extract much more accurate information fromRICM images (e.g., object shape, separation distance,and membrane thickness).

In the present research, we focus on geometrieswith an arbitrary convex surface with or without auniform second layer such as an ultrathin coatingor a membrane, because these systems are importantin a number of applications. In addition to cells andlipid vesicles, the polymer vesicles employed herecome from the novel use of versatile materials suchas amphiphilic block copolymers that self-assemblein aqueous solutions and are important in the designof drug delivery systems, biosensors, and cell mi-micry [10–13]. Another type of system that can bestudied includes ultrathin polyelectrolyte multilayerfilms adsorbed onto micrometer-size colloidal parti-cles. Following the removal of the particle template,these systems provide tailored hollow capsules suita-ble for diverse applications [14,15]. More generallyspeaking, an accurate RICM analysis fits extraordi-narily well in understanding the deformation be-havior of soft particles near a flat substrate, which

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1 July 2010 / Vol. 49, No. 19 / APPLIED OPTICS 3701

is of paramount importance for the progress incolloid and interface science [16].

Since the first RICM theoretical considerations byGingell and Todd in 1979 [17], important theoreticalimprovements have been accomplished thanks to thework of Sackmann and co-workers during the 1990s[1,18,19]. The nonplanar interface image formationtheory [1], introduced in 1998, represents the mostsophisticated theory yet. This paper presents areview and improvement of this theory, with the for-mulation of a general statement for intensity calcu-lations and a correction in terms of the optical pathlength difference (OPLD) determination in nonpla-nar interfaces. A method for the practical implemen-tation of the improved theory in wedges and convexsurfaces is presented, based on an approximated for-mulation that directly uses the convenient backwardray-tracing technique. Also, a general model for anarbitrary convex surface, with or without a uniformlayer such as a membrane or ultrathin coating, ispresented. In order to speed up the calculationswithout losing accuracy, the appropriate geometricsimplifications in the model are studied using simu-lations. As a result, the experimental data demon-strate the validity of the improved theory, thecalculation method employed in the simulations,and the developed surface model when used togetherto accurately obtain the surface profile of a polymervesicle by fitting its experimental intensity pattern.

2. RICM Image Formation Theory Improvement

RICM image formation is illustrated in Fig. 1(a), fol-lowed by the description of the nonplanar interfaceimage formation theory [1] in Fig. 1(b), which isthe most sophisticated to date. According to this the-ory, the contributions to the intensity at Bðx; yÞ areintegrated incoherently as follows:

Iðx; yÞ ¼�R

E�EdΩRdΩ

�t; ð1Þ

where E is the local electric field, Ω ¼ ðθ;ϕÞ are sphe-rical coordinates denoting angles of incidence at B,and t stands for a time average. The integral inEq. (1) has been rewritten as

Iðx; yÞ ¼ Is þ Ip; ð2Þ

Is;p ¼ ðI0=2ÞR2π0

R αDA0 ½Rs;pðθ;ϕÞ�Rs;pðθ;ϕÞ� sinðθÞdθdϕR

2π0

R αIA0 sinðθÞdθdϕ ;

ð3Þconsidering that s-polarized light cannot interferewith p-polarized light and vice versa; recently, polar-ization effects have been included in a planar modelshowing that they become relevant when a high illu-mination numerical aperture (INA) is used [20]. R isthe effective reflection coefficient, which is a functionof the geometry, refractive indices, and constraints in

the system; and I0 is the intensity of the incidentlight supposed to be homogeneous in θ, in the inter-val ½0; αIA�. According to this formulation, the inte-gration variables ðθ;ϕÞ correspond to ðθ2;ϕ2Þ inFig. 1(b) because they are indexing all rays incidentat B that reenter the microscope after reflection atthe nonplanar interface.

This formulation correctly describes a situationsuch as the one depicted in Fig. 1(b) with two iden-tical beams, I0, originating from the homogeneous il-lumination source, and I2 as the only ray incident atBðx; yÞ after reflection at the nonplanar interface,which gives a single possibility for defining the inte-gration variables ðθ;ϕÞ as ðθ2;ϕ2Þ. However, if wethink about a more complex situation where morethan two complementary rays, I0, interfere, theuse of a consistent indexing ðθ;ϕÞ, based on rays that

Fig. 1. (Color online) (a) RICM image formed due to the interfer-ence of rays reflected from different optical interfaces in thesystem when the object is illuminated from below using monochro-matic light. (b) Nonplanar model in a single-layer system. The il-lumination source is considered monochromatic, pseudocoherent,and angularly limited by the INA of the microscope, which meansall I0 originate from within the cone defined by the maximum il-lumination angle, αIA. The normalized intensity at B in the imageplane, Iðx; yÞ, is calculated integrating over spherical coordinatesðθ2;ϕ2Þ the contributions from rays I1 and I2 incident within thecone of detected light, αDA, determined by the numerical aperture(NA) of the objective. The path length and intensity correspondingto each particular ray are determined by backward ray tracing,taking into account reflection and transmission at every opticalinterface corresponding to a given geometry; however, multiplereflections are not considered.

3702 APPLIED OPTICS / Vol. 49, No. 19 / 1 July 2010

are reflected back from the object, becomes more dif-ficult and even impossible. This is due to the factthat, after reflection at the nonplanar interface, eachcomplementary I0 results in different contributionsI2, I3;… and each one of them is more likely to havea different orientation ðθ2;ϕ2Þ, ðθ3;ϕ3Þ;… when inci-dent at B. Another issue with the ðθ2;ϕ2Þ indexing isthat, in some situations, we may not be able to ac-count for all I1 contributions to the intensity becausethe existence of I1 is determined by the existence ofI2, and this depends on the geometry and constraintsin the system. Because ðθ2;ϕ2Þ are not general vari-ables for integration in Eqs. (1)–(3), we instead con-sider the I1 angles ðθ1;ϕ1Þ, shown in Fig. 1(b), asquantities capable of providing a basis for consistentindexing in any situation. These variables are uniquebecause no other complementary I0 is reflected offthe substrate-layer 1 interface and they exist when-ever there are contributions to the intensity.

Based on these considerations, we express the localintensity, Iðx; yÞ at the image point B, located at thesubstrate-layer 1 interface, in terms of

Iðx; yÞ ¼�R

Ω1E�EdΩRΩ1dΩ

�t

¼ Is þ Ip; ð4Þ

Is;p ¼R2π0

R αIA0 ½Rs;pðθ1;ϕ1Þ�Rs;pðθ1;ϕ1Þ�½I0ðθ1Þ=2� sinðθ1Þdθ1dϕ1R

2π0

R αIA0 sinðθ1Þdθ1dϕ1

; ð5Þ

where the average over the solid angle of the contri-butions to the intensity is done by indexing the inte-gration following I1, with orientation ðθ1;ϕ1Þ, asshown in Fig. 1(b), and its interference with the cor-responding contributions I2, I3;… if any, from com-plementary rays I0. This defines a two-dimensional(2D) set of angles in the (θ1;ϕ1) plane,Ω1, correspond-ing to the solid angle where all contributions to theintensity reside (the differential element of solid an-gle is given by dΩ ¼ sinðθ1Þdθ1dϕ1). It can be seenthat the denominator of Eqs. (4) and (5) is a normal-ization constant directly related to Ω1. Once the INAof the microscope is set to a certain value, the ex-tended illumination source, I0ðθ1Þ, becomes geome-trically constrained within the illumination conedefined by αIA. Consequently, the solid angle overwhich contributions to the intensity are collected,Ω1, is set by θ1max ¼ αIA, the only parameter thatis necessary to obtain the normalization term. Onthe other hand, we see that the numerator can be cal-culated, unambiguously and at least in theory, bytracking all beams I1 within Ω1 and finding all com-plementary rays I0 and their contributions I2, I3;…that satisfy the constraints in the system and com-bine to produce interference. However, the intensity

calculation becomes more challenging in this newscenario because instead of using the convenientbackward ray-tracing method based on ðθ2;ϕ2Þ, wehave to start with ðθ1;ϕ1Þ, and only well-definedgeometries such as wedges and spheres offer the pos-sibility of obtaining an analytical solution.

In addition to the described general formulationfor intensity determination, we find that a correctionis also needed in the OPLD calculation to properlyaccount for nonplanar interfaces. The interferenceof I1 and I2, shown in Fig. 1(b), has been previouslydescribed to have an OPLD, Δ, as follows [1]:

Δ ¼ n1ð �ASþ �SBÞ − n0�CB; �CB ¼ sinðθ1Þ �AB: ð6Þ

However, this equation is only valid in the particularorientation of the 2D situation described in Fig. 1(b).For arbitrary geometries, an expression in terms ofvectors is more appropriate:

Δ ¼ n1ð �ASþ �SBÞ þ n0ðR0 · BAÞ; ð7Þ

whereR0 is the unitary vector (pointing upward) thatdefines the orientation of ray I0 in the glass side andBA is the vector from point B to point A, according toFig. 1(b). Although the vector product term in Eq. (7)is part of the calculation of the OPLD between I1 and

I2, it can be applied to the interference between I1and additional contributions I3, I4… by taking intoaccount that, in general, A represents the pointsource in the image plane of the corresponding com-plementary I0.

In the following section, we describe how to per-form intensity calculations according to Eqs. (4)and (5). For simplicity, we concentrate on geometriessuch as wedges and arbitrary convex surfaces,although these equations are general and can be ap-plied to any arbitrary geometry with any number oflayers. The selected geometries allow us to exploresimplifications to the problem so we can take advan-tage of the backward ray-tracing method to make thecomputations easier and practical.

3. RICM Intensity Calculations in Wedges and ConvexSurfaces

In geometries such as wedges and convex surfaces, aconsistent indexing in ðθi;ϕiÞ, i ≥ 2, is sometimes pos-sible by performing a change of variables. We use thesubscript “i” to generalize a feasible indexing byusing any ray incident at Bðx; yÞ after being reflectedback from any layer of the nonplanar interface. In or-der to be successful, any single or multiple consistent


indexing must be able to account for the contribu-tions of all rays incident at B. Such indexing definesa 2D set of angles in the ðθi;ϕiÞ plane, Ωi, and has tocorrespond to a one-to-one mapping between pointsinside Ω1 and Ωi. Therefore, we formally introducethe map G of Ω1 onto Ωi defined by the change ofvariables

G :

�θi ¼ θiðθ1;ϕ1Þϕi ¼ ϕiðθ1;ϕ1Þ ; ð8Þ

that leads to

Iðx; yÞ ¼�R

ΩiE�EdΩRΩidΩ

�t¼ Is þ Ip; ð9Þ

Is;p ¼�ZZ

Ωi

½Rs;pðθi;ϕiÞ�Rs;pðθi;ϕiÞ�½I0ðθ1ðθi;ϕiÞÞ=2�

× Jðθi;ϕiÞdθidϕi

�=

�ZZΩi

Jðθi;ϕiÞdθidϕi

�: ð10Þ

Notice that the differential element of solid angle isnow given by dΩ ¼ Jðθi;ϕiÞdθidϕi, where Jðθi;ϕiÞ isthe Jacobian of the transformation [21]. Becausethe map G depends on the geometry of the reflectingsurface, this Jacobian is also a function of it and itscalculation requires the first-order partial deriva-tives of the mapping functions to be continuous.

The most important condition for the success ofEqs. (9) and (10) is the complete transformation ofthe integration space Ω1 delimited by αIA. Whenwe use ðθi;ϕiÞ variables, such a solid angle is trans-

formed by the geometry of the reflecting surface andit may be that only a partial transformation into Ωi ispossible. This is illustrated in Fig. 2 using the geome-try of wedges. For the 30° wedge in the top row, themain constraint in both domains is αIA and the inter-action with αDA produces two regions, one with onlyI1 contributions and a second one with interference ofI1 and I2, which can be completely observed in Ω1 andΩ2. The intensity can therefore be calculated usingeither Eqs. (9) and (10) or Eqs. (4) and (5), as shownin the third figure. In the 40° wedge at the bottomrow, Ω2 is cut off by αTIR (total internal reflection lim-it), and from the translation of this limit to Ω1 we cansee that there is amissing region, I1�, outside the bor-ders, with only I1 contributions. In this situation,only Eqs. (4) and (5) give the correct result, whileEqs. (9) and (10) produce a different value, as canbe seen in the intensity profile.

In addition to the possibility of not accounting forall contributions to the intensity when performingthe change of variables, calculation of the Jacobianof the transformation implies that we know the rela-tionship between ðθ1;ϕ1Þ and ðθi;ϕiÞ. This is basicallythe same reason that makes the direct application ofthe general formulation impractical when dealingwith an arbitrary geometry. Therefore, althoughEqs. (9) and (10) are given in terms of ðθi;ϕiÞ vari-ables, we still cannot take full advantage of the back-ward ray-tracing method. However, we can obtain anew formulation for the intensity calculation, wherethe inconvenient Jacobian in Eq. (10) is replaced bysinðθiÞ such that the differential element of solidangle is now given by dΩ0 ¼ sinðθiÞdθidϕi. Conse-quently, we have

Fig. 2. (Color online) Intensity calculation for glass–water–air wedges of 30° (top row) and 40° (bottom row). The integration domains Ω1

andΩ2, shown in the first and second figures for eachwedge, remain the same for all image plane positions where intensities are calculatedand plotted in the third figure, according to ðθ1;ϕ1Þ and ðθ2;ϕ2Þ indexing. To illustrate the transformation of Ω1 into Ω2 and vice versa, itcan be seen how the boundaries of the integration domains, mainly determined by αIA and αTIR; four arbitrary interior points, A, B, C, andD; and some arbitrary interior lines (white dashed lines) translate from one domain to the other in each wedge geometry. The intensityprofiles show that the change of variables is successful for the 30° wedge but not for the 40° wedge because of I1� contributions present inΩ1

but missing in Ω2. The simulations were performed with INA ¼ 0:48, NA ¼ 1:25, and refractive indices nglass ¼ 1:5, nwater ¼ 1:33, andnair ¼ 1. Notice that θ1 ¼ 0 represents a single point E at the Ω2 boundary and ϕ goes only up to π in order to exploit the symmetryof the problem.


Iðx; yÞ ≅�R

ΩiE�EdΩ0RΩidΩ0

�t¼ Is þ Ip; ð11Þ

Is;p ¼�ZZ

Ωi

½Rs;pðθi;ϕiÞ�Rs;pðθi;ϕiÞ�½I0ðθ1ðθi;ϕiÞÞ=2�

× sinðθiÞdθidϕi

�=

�ZZΩi

sinðθiÞdθidϕi

�: ð12Þ

This formulation is not a mathematically correcttransformation of Eqs. (4) and (5) because Jðθi;ϕiÞ ¼sinðθiÞ, only valid in the limiting case of planar par-allel interfaces and we expect to obtain an approxi-mate result that deviates from the correct one asthe inclination of the nonplanar interface increases.However, this approximated formulation yields cor-rect fringe spacings from wedges and convex surfacesbecause it accounts for all the interference betweencomplementary rays I0 with contributions I1, I2, I3…in such geometries. The fact that we might bemissing single I1 contributions only affects the aver-age intensity in the interferogram, similarly to whatis seen in Fig. 2 (bottom row) for the case of thechange of variables.

It is also important to consider how the approxi-mated intensity profile is obtained from Eqs. (11)and (12). The normalization term is no longer simplya constant given by αIA because the absence of thecorrect Jacobian makes it a function that dependsof the geometry of the reflecting surface, and possiblyof position in the image plane. However, a similar is-sue associated with approximating the Jacobian alsoaffects the numerator of Eq. (12), thereby making itpossible to obtain the approximated intensity valuesfrom these equations after normalization. This is il-lustrated in Fig. 3, using simulations from a sphere,where the approximated solution gives acceptable in-tensity values (average relative error of 5.21%) thatmaintain the correct fringe spacing in the interfero-gram; the agreement is even better when smallerINAs are used (data not shown).

Equations (11) and (12) can be directly implemen-ted using simulations by backward ray tracing of therays with ðθi;ϕiÞ orientation incident at Bðx; yÞ, with-out previous knowledge of the relationship betweenðθ1;ϕ1Þ and ðθi;ϕiÞ and applying appropriate normal-ization. However, the most significant aspect of thisformulation is that numerical evaluation of the nu-merator in Eq. (11) allows indirect calculation of theexact intensity from Eqs. (4) and (5). From Fig. 2 wesee that within the boundaries of Ω2 the map G isinvertible; then, whenever Jðθi;ϕiÞ ≠ 0 we can write

G−1 :

�θ1 ¼ θ1ðθi;ϕiÞϕ1 ¼ ϕ1ðθi;ϕiÞ ; ð13Þ

which defines the inverse map G−1 for wedges andconvex surfaces. Therefore, when performing theintegration in Eq. (11), we can proceed as follows:first, we find all contributions possible in Ωi, by back-

ward ray tracing, with their corresponding effectivereflection coefficients and intensities of the incidentlight; next, by using the idea of the inverse mapping,we proceed to relocate that information in Ω1; finally,the existence of missing regions is determined in Ω1so that the missing contributions can be appropri-ately accounted for and the exact calculation accord-ing to Eqs. (4) and (5) can be completed. Moredetailed information about the numerical construc-tion of the inverse mapping and subsequent integra-tion can be found in Appendix A. Figure 3(c) shows acomparison between the exact calculation performedby direct evaluation of Eqs. (4) and (5), the approxi-mated calculation from direct evaluation of Eqs. (11)and (12), and the indirect evaluation of Eqs. (4) and(5) based on the direct evaluation of Eqs. (11) and(12). It can be seen that we have successfully usedthe information collected from the approximatedformulation and, via inverse mapping, performedan indirect evaluation of Eqs. (4) and (5) which isin excellent agreement with the direct computation.This allows calculating exact intensity values in amuch more computationally efficient manner whendealing with an arbitrary convex geometry.

Finally, we note how the previous formulation gi-ven by Eqs. (1)–(3) compares to the different sets offormulations developed here. When we look at thedenominator of Eq. (3), we see that it correspondsto the same normalization term in Eq. (5), the exactgeneral calculation, except that the numerator of Eq.(3) is equivalent to the one in Eq. (12), the approxi-mated solution for wedges and convex surfaces. Con-sequently, because the appropriate normalizationterm is required to obtain either the approximatedor exact intensity, as seen in Fig. 3, it can be con-cluded that the previous formulation does not pro-

Fig. 3. Simulations corresponding to a 6 μm radius latex spherein water and 100nm above the glass surface, with INA ¼ 0:78,NA ¼ 1:25, and nlatex ¼ 1:55, are performed to illustrate the beha-vior and convenience of the approximation given by Eqs. (11) and(12) compared to the exact solution, Eqs. (4) and (5). (a) Intensitywithout normalization is the calculation from the numerators ofthementioned equations, (b) normalization corresponds to the eva-luation of their denominators, and (c) the final intensity value isobtained after normalization. It is important to point out that, inthese calculations, there are no missing contributions in Ω2.


vide correct intensity values. However, we expectEqs. (1)–(3) to accurately reproduce the fringe spa-cing in wedges and convex surfaces and to providethe correct intensity value in the limiting case ofplanar parallel interfaces, in the same way as theapproximated solution.

4. RICM Simulations of Known Geometries

The current mathematical models for wedges andspheres have been reviewed. For these known geome-tries, it is possible to calculate the intensity directlyfrom Eqs. (4) and (5), Eqs. (9) and (10) or Eqs. (11) and(12), with essentially the same computational effort;thus, we can perform simulations to study and com-pare results from these formulations. In addition tothe earlier discussions, we find that the normaliza-tion term in Eq. (12) is a particular constant for dif-ferent inclination angles of the wedge, because theloci of contributions Ω2, illustrated in Fig. 2, remainunchanged for all positions. On the other hand, simu-lations from a sphere in Fig. 3 illustrate how thatnormalization term monotonically decreases as afunction of radial position. This behavior occursdue to the fact that the loci of contributions displacetoward larger θi0s and become smaller as we moveaway from the center of the sphere.

Also, we can use wedges and spheres to directlycompare the results from our improved theory andmodels with previously published results [1].Although, in general, Eqs. (1)–(3) do not producethe correct intensity values, the qualitative behaviorpreviously observed in interferograms from thesegeometries is also obtained with the new general for-mulation. For instance, the mean intensity of fringepatterns from wedge geometries is constant withinthe interferogram [Fig. 2], but it is a function ofthe angle of inclination, while in the case of spheres,the mean intensity varies within the interferogram[Fig. 3]. This confirms that the mean intensitychanges with local inclination of the nonplanar inter-face, an effect that is successfully explained by thedisplacement of the loci of interference of the rays,the region labeled as “I1&I2” inside Ω2 in Fig. 2,and their interaction with constraints such as αDA.

To look at the fringe spacing behavior, we noticethat it is possible to obtain an approximated micro-topography of the sample based on a fringe spacinganalysis by using the simplest theory of RICM imageformation [1,18]; when this is applied to wedges andspheres, the inclination angle, β, and the radius ofcurvature, ρ, can be estimated, respectively. Thedeviation of such estimation from the correct valueshas been studied using simulations, and an empiricalcorrection term, δC, has been introduced as follows[1]:

βC ¼ βUCδC;wedgeðβUCÞ; ð14Þ

ρC ¼ ρUCδC;sphereðρUCÞ; ð15Þwhere the corrected measurement is obtained fromthe uncorrected value, denoted by subscripts UC,

times the correction factor that is expressed as afunction of uncorrected measurements. The state-ment made in Eqs. (4) and (5) does not modify thefringe spacing with respect to Eqs. (1)–(3); however,the correction made to the calculation of Δ in Eq. (7)is expected to have a significant effect. This is shownin Fig. 4, where new correction factors for wedges andspheres are plotted and compared to the ones withthe uncorrected OPLD and previously reportedfittings [see Eqs. (32) and (35) in [1] and the corre-sponding fitting parameters given]. The fact thatthe factors calculated from uncorrected Δ values,Eq. (6), follow previously reported fittings obtainedby performing simulations of Eqs. (1)–(3) shows theiragreement with our simulations. At the same time, itconfirms that the fringe spacing remains invariantwith respect to the formulation used for the intensitycalculation. Finally, we see that the effect of the Δcorrection is basically to reinforce the results fromthe simplest theory as the correction terms arenow closer to unity, especially for the cases of largeinclinations of the nonplanar interface.

Other known geometries such as ellipsoids and cir-cular and elliptic cylinders have been studied, espe-cially in order to determine the effect of using acylindrical geometry to approximate a convex surfacewith nonzero Gaussian curvature. Some relatedresults are discussed in the following section, in ad-dition to simulations of double-layer systems withspherical geometry.

5. RICM of Arbitrary Convex Surfaces

In this section we present an approach to modelgeneral convex reflecting surfaces with or without

Fig. 4. Using the corrected OPLD the correction factors for fringespacing analysis in (a) wedges and (b) spheres have been deter-mined. In addition, the factors calculated from the uncorrectedOPLD are shown; they follow previously reported fittings obtainedby using simulations of Eqs. (1)–(3). The systems studied corre-spond to glass-water-air for wedges and glass-water-latex forspheres contacting the substrate with INA ¼ 0:48 and NA ¼ 1:25.


a thin layer. In addition, appropriate simplificationsare studied that can enable the calculations to be per-formed faster without loss of accuracy. The simula-tions from such systems are performed accordingto the procedure described in Appendix A, using a nu-merically reconstructed inverse map. Then a deform-able polymer vesicle close to the glass surface is usedto provide an experimental system that resembles anarbitrary convex surface.

A. Arbitrary Convex Surface Model

Splines are smooth piecewise polynomial functionsthat can be utilized to model the contour of the objectimmediately above the observation line—the linewhere the intensity profile is measured and anal-yzed. Then, a general way to define a surface fromthis contour is to rotate the spline around a givenaxis. The location of the rotation axis, defined by theradius of rotation (RoR) gives the ability to representthe surface of the object under study that fits differ-ent experimental situations, as shown in Fig. 5(a).This model allows arbitrary surfaces to be describedand is able to exactly represent the geometries ofspheres, with RoR ¼ 0, and wedges and cylinders,with RoR ¼ infinity. The case of RoR ¼ infinity,referred to as the cylindrical approximation, is of par-ticular interest because it has the advantage that a3D geometric problem of finding intersections be-tween the rays and the surface is simplified to a2D problem, making the calculations easier and fas-ter. At the same time, it is expected to be a validapproximation for any convex geometry becausethe RICM image is a strong function of the localheight of the object, which remains unchanged.

In order to evaluate our model, we performed si-mulations of latex spheres in water contacting aglass surface with numerical aperture (NA) = 1.25using the exact geometry, and compared their inter-ferograms with the ones from the cylindrical approx-imation. In general, the same fringe spacings aremaintained across all radial positions and no differ-ence is observed in the initial portion of the intensityprofile, although the magnitude of the intensitycurve obtained from the cylindrical approximationdiverges somewhat from the exact geometry. It isfound that the average of the relative error betweenthe interferograms, taken up to the point where theprofiles decay to the background intensity, follows alinear trend with INA values larger than 0.48; theerror is about 1.66% for INA ¼ 0:48 and 4.76% forINA ¼ 0:98. Several simulations show that this rela-tive error presents small variations with the spheresize and can be as small as 1.03% for INA ¼ 0:28. Thebehavior can be explained given that the loci of con-tributions become larger as the INA value increases,collecting more information about the lateral shapeof the object. Therefore, if the sphere is replaced byan elongated ellipsoid, the error decreases and tendsto zero as the elongation becomes larger. For in-stance, the cylindrical approximation accurately re-presents the interferogram along the x axis of an

elongated ellipsoid with radii 4, 40, and 4 μm alongthe x, y, and z axes, respectively. Because the splinemodel with RoR ¼ infinity does not introduce a sig-nificant error in the calculated intensities, especiallywhen a small INA is used, subsequent simulationsinvolving double-layer systems are performed usingthis model.

B. Model Extension to Double-Layer Systems

To the best of our knowledge, explicit models usingthe nonplanar interface image formation theory indouble-layer systems have not been reported. Thus,we formulated a method to calculate the intensity inthese systems, considering that the second layer re-sembles a membrane or ultrathin coating of arbi-trary convex geometry with a uniform thickness,L, much smaller than the object size. The calculationis based in the indirect evaluation of Eqs. (4) and (5),taking advantage of the backward ray-tracing meth-od, as described in Appendix A for a single-layer sys-tem. Therefore, when calculating the intensity Iðx; yÞat a point B, every individual beam incident at thatpoint and resulting from a reflection at the secondlayer/ambient medium interface, I3, is traced back-ward to a source point D; see Fig. 5(b). In order todo this, the following approximation is used to speedup the calculations: once I3 intersects the spline at apoint Q, the radius of curvature at Q is calculated todefine concentric circular cylinders that model the

Fig. 5. (Color online) (a) Spline model for an arbitrary convex sur-face that fits different experimental situations by adjusting theRoR parameter. (b) Local geometry, approximations, and contribu-tions to the intensity in the spline model extension to double-layersystems.


local curvature and thickness of the layer, as illu-strated in Fig. 5(b). Using this local and well-definedgeometry, the tracing backward of the beam I3 can becompleted to its source I0 at a point D, and the twocomplementary I0 beams with contributions I1 and I2can be determined.

Notice that we trace backward I3 instead of I2. Themain reason for this is to avoid an additional and ne-cessary numerical search for I3 if we select I2; bychoosing I3 and obtaining I0, I2 can be determinedanalytically from the cylindrical geometry after sol-ving a fourth-order polynomial equation. In addition,that choice does not introduce a consistency problembecause the thickness of the second layer is assumedto bemuch smaller than the local radius of curvature,and, consequently, I2 and I3 exist simultaneously.Thisrepresents the situation where amultiple and consis-tent indexing is possible for Eqs. (11) and (12), whichallows the indirect calculation of Eqs. (4) and (5).

To look at the implementation of this double-layermodel, we performed simulations of a 6 μm radiuslatex sphere with an ultrathin coating (<200nmthickness) of refractive index 1.45 in water. For asymmetric particle with a convex geometry at its bot-tom, the intensity at the center of the interferogram,radial position 0 μm, can be calculated from the the-ory for stratified planar structures [1]. Using thisplanar theory, it is possible to account for all reflec-tions from the three interfaces involved, glass/water,water/coating, and coating/latex, in addition to thefinite aperture effect. This is shown in Fig. 6(a) wherethe intensity is plotted as a function of the thicknessof layer one, height of the particle above the glasssurface, and layer two, thickness of the coating. Ifthe height and the thickness are unknown, a singleintensity at the center of the interferogram mightcorrespond to multiple (height, thickness) pairs, ascan be seen from the contours in Fig. 6(a). Therefore,a more detailed analysis involving the complete in-tensity pattern and the geometry of the particle isrequired to obtain a unique solution.

Here we consider that the coated particle is in con-tact with the substrate, so the intensity at the centerof the interferogram, Ið0Þ, can be directly correlatedwith the thickness of the coating Fig. 6(b). These re-sults verify the agreement between the intensity Ið0Þobtained from the planar and nonplanar theories,using the exact model for a sphere and the corre-sponding spline model with RoR ¼ infinity. Also, si-mulations from the nonplanar theory indicate thatthe average intensity of the interferogram, AvgðIÞ,can be correlated with the thickness of the coating.This additional information could be used to discri-minate between thicknesses that give the same in-tensity Ið0Þ. Finally, notice that the AvgðIÞ curveobtained from the cylindrical approximation is invery good agreement with the one from the sphericalgeometry, with an average relative error of 0.75%,confirming that the spline model with RoR ¼infinity is a suitable model for intensity calculationsin double-layer systems.

C. Experimental Section

The experimental RICM setup that facilitated imageacquisition consisted of a Zeiss Axiovert 200M in-verted microscope with a 103W HBO mercury vaporlamp coupled to a Zeiss AxioCam MRm camera; a5nm bandpass filter used to obtain the monochro-matic green light, 546:1nm; and a Zeiss AntiflexEC Plan-Neofluar 63x=1:25 Oil Ph3 objective. Imageprocessing was performed with the software ImageJ1.41o (public domain, National Institutes of Health,USA).

Block copolymer vesicles were the experimentalsystem chosen to resemble an arbitrary convex sur-face with a double layer. These vesicles sponta-neously form in water when the copolymer has onehydrophobic and one hydrophilic block, called amphi-philic diblock copolymers [10,22]. The synthetic blockcopolymer employed was polyethylene oxide-block-polybutadiene (PEO89 − PBD120, MW 10400 g=mol)purchased from Polymer Source, Incorporated(Canada), and the vesicles are obtained by film rehy-dration. A polymer film containing 250 μg of the blockcopolymer was formed by evaporation (8h) at the bot-tom of a 5mL glass scintillation vial. Vesicles wereformed by rehydration of this polymer film during

Fig. 6. RICM in a spherical double-layer system. (a) Intensity atthe center of the interferogram, Ið0Þ, can be easily obtained fromthe planar theory, considering that the particle is close to the sur-face and the thickness of the coating is small. (b) Nonplanar theorygives the same information for Ið0Þ, and it points out that the aver-age intensity of the interferogram, AvgðIÞ, can be correlated withthe thickness of the coating, as can be seen for the case when theparticle is in contact with the glass. The simulations are performedwith INA ¼ 0:48, NA ¼ 1:25, and RoR ¼ infinity in the splinemodel.


24h at 60 °C with 1–2mL of a 300mOsm=kg sucrosesolution (Osmometer model 3320, Advanced Instru-ments, Incorporated, Norwood, Massachusetts).Then, the vesicles were placed in phosphate bufferedsaline (PBS) solution of equal osmotic pressure thantheir interior.

200 μL of the previously prepared vesicle solutionwas placed in a sealed chamber on top of a cover glasscoated with bovine serum albumin (BSA) to avoid ad-hesion and spreading of the vesicles onto the glasssurface. The gravitational force determined by thedensity difference between the sucrose and PBSsolutions pushes the vesicles toward the substrateand allows them to settle, while the BSA coatingand possible electrostatic repulsion due to impuritycharges on the vesicles and the highly negativelycharged glass surface keep them from getting in con-tact with the substrate [23]. As a result, the vesiclesfloat above the glass surface in a slightly deformedconfiguration with respect to their spherical shapewhen they are free in solution. Figure 7(a) showsthe RICM image of one of these vesicles using thesmallest INA available, INA ¼ 0:48.

In this particular experiment, the shape of the ob-ject under observation and its height above the sub-strate are unknown, but they can be determinedbecause we know all the parameters in the system.

The refractive indices of glass, buffer, and sucrose so-lutions are 1.530, 1.335, and 1.351, respectively, fromRICM measurements. The polymer membrane is gi-ven by the hydrophobic core of polybutadiene; accord-ing to the literature, its thickness is about 15nm [24]with a refractive index of 1.51 [25]. Also, two addi-tional effects have been incorporated into the simula-tions: first, the real illumination profile, I0ðθ1Þ, and,second, a cosine fourth-law factor that affects the in-tensity of the rays incident at the position of interest.

The analysis of the experimental data begins withthe determination of the minimum separation dis-tance between the vesicle and the glass surface,hmin. We have previously seen in Fig. 6 that the in-tensity at the center of the interferogram is basicallyobject size- and shape-independent and, given thatthe membrane thickness is known, it can be directlycorrelated with this minimum height using simula-tions from the planar theory that account for theINA and NA effects, as seen in Fig. 7(b). This ishow we find hmin ¼ 59� 1nm, taking into accountthat the intensity continuously increases from thecenter up to the first intensity peak in Fig. 7(c).

Now we determine the bottom shape of the vesiclethat gives the best fit to the experimental intensitypattern. The traditional RICM analysis considersthat the surface profile can be reconstructed by

Fig. 7. (Color online) (a) RICM image of a polymer vesicle in PBS and filled with sucrose solution when it is close to a glass surface coatedwith BSA. (b) Intensity at the center of the interferogram, given by the planar theory, is used to determine the height of the vesicle abovethe glass surface. (c) Intensity profile is obtained from a circular average of the picture shown in (a). Experimental data were fitted by usingthe improved nonplanar interface image formation theory and the splinemodel withRoR ¼ infinity. (d) Reconstructed bottom shape of thevesicle according to the nonplanar and planar models.


inverse cosine transform of the intensity distribution[18]; however, when dealing with a double-layer sys-tem, it has been pointed out that the traditional anal-ysis underestimates the heights by a value h�, in thiscase 37:4nm [26]. Therefore, we use the result illu-strated in Fig. 7(b) to obtain an appropriate fast re-construction of the profile using the planar theory.The incremental height difference between two adja-cent intensity extrema is Δh ¼ 103:8nm, which isclose to the value for normal incidence light, λ=4n1 ¼102:3nm, in the traditional analysis; then, based onthe previously determined hmin, the height for thefirst intensity peak (h� þΔh), and the distance be-tween consecutive extrema obtained from Fig. 7(c),the bottom shape of the vesicle is reconstructedand the result is presented in Fig. 7(d).

By incorporating the improved nonplanar theoryand the arbitrary convex surface model with a thinmembrane, an additional surface profile is recon-structed; see Fig. 7(d). Because the simulated inten-sity profile obtained from the reconstructed contoursuccessfully reproduces the experimental data inFig. 7(c), it can be concluded that it corresponds toan accurate representation of the bottom shape ofthe vesicle under study. Up to a radial position of1:5 μm, there are no significant curvature effectsand the bottom shape given by the planar theoryis within 3% of error when compared to the nonpla-nar result; however, beyond that point the errorquickly increases, reaching 20% at 4 μm. To the bestof our knowledge, this is the first time that an accu-rate RICM analysis is performed in a double-layersystem of unknown geometry.

6. Conclusions

The improvement of the RICM image formation the-ory, with the general formulation for intensity calcu-lations in Eqs. (4) and (5), successfully accounts forcontributions to the intensity that can be missingin previous formulations. But analysis of wedgesand convex surfaces shows that the implementationof this formulation is not trivial, leading us to devel-op a method to obtain the exact result from an ap-proximated, intermediate formulation. In addition,simulations of known geometries, such as wedgesand spheres, allow a direct comparison with previous

models, revealing that the OPLD correction, Eq. (7),becomes important for large inclinations of the non-planar interface and that the fringe spacing analysisis more accurate than previously thought.

RICM can be used to obtain valuable informationfrom the system under study. For instance, by usingsimulations, we explored how to determine the thick-ness of an ultrathin coating (<200nm) on a sphericalparticle. The development of a general model for anarbitrary convex surface, including a uniform secondlayer, which could be a thin coating or a membrane,expands the possible range of systems that can beaccurately studied using RICM. Because RICMexperiments are usually performed using the smal-lest INA available, a particular case of this surfacemodel, the cylindrical approximation, becomes a sui-table model with an expected average relative errorin the calculated intensity profile smaller than 2%. Ifhigh INAs are used, the average relative error wouldbe larger than 5%, but the correct fringe spacing isstill maintained. Our experimental work with poly-mer vesicles shows that the coupling of the improvedRICM image formation theory and the surface modelallows an accurate reconstruction of the convex bot-tom shape of an object close to the substrate by fittingits experimental intensity profile.

A natural direction for expansion of the presentwork would be the study of concave geometries.RICM has already been employed to measure smallcontact angles of droplets using a wedge approxima-tion [1,27] or simple fringe spacing analysis [9,28];however, there is not a clear idea of the error involvedin such measurements. Because it is possible to havemultiple contributions to the intensity coming fromdifferent regions of a concave geometry, a consistentindexing based on rays that are reflected back fromthe object might not be feasible. Therefore, our for-mulation for intensity calculations is expected toplay a critical role in this scenario. Also, the formu-lated method to calculate intensities in double-layersystems could potentially be used to analyze morecomplex systems, such as cells, with different reflec-tion distributions (e.g., due to local refractive indexheterogeneities) on the membrane. These local varia-tions can be incorporated into the intensity calcula-tion via the effective reflection coefficient, R, andtheir effect in interferograms when the object shaperemains unchanged could be evaluated.

Appendix A: Indirect Evaluation of Exact IntensityFormulation from Direct Evaluation of ApproximatedSolution

In this Appendix, we describe the indirect evaluationof Eqs. (4) and (5), exact intensity formulation, fromthe direct evaluation of Eqs. (11) and (12), approxi-mated solution; a comparison between results fromthese sets of equations is presented in Fig. 3(c).The main idea is to take advantage of the backwardray-tracing method to account for all the contribu-tions of rays I2, I3… that are reflected back fromthe nonplanar interface and simplify the calculations

Fig. 8. Typical surface of integrand values approximated by theDelaunay triangulation method.


when dealing with an arbitrary convex surface.Although we focus on single-layer systems, the sameprocedure can be applied to situations involving mul-tiple layers [see the double-layer system depictedin Fig. 6]. The variables used have been defined inFig. 1(b) and throughout the main text of this paper.

Ideally, we would like to know in advance if inter-ference occurs at the position of interest, Bðx; yÞ—inother words, if there is a range of incident angleswhere we expect to have I2 contributions that inter-fere with I1. As seen in Fig. 2, such angles are con-strained, as in most situations, by αIA and αDA.The boundary corresponding to αIA in Ω2 is a functionof the geometry of the reflecting surface and, in gen-eral, also a function of position at image plane.Therefore, because we are interested in arbitraryconvex surfaces, it is not possible to obtain an analy-tical expression for the incident angle range thatwould be valid across all possible geometries and po-sitions. To approximately predict the positions at im-age plane where interference fringes could still existfor an arbitrary convex geometry, we can use an anal-ytical solution for the maximum inclination angle ofa wedge, βmax, to observe interference:

βmax ¼αL1DA þ αL1IA

2; ðA1Þ

where the superscripts L1 indicate that αIA and αDAare given in layer 1, and it is assumed that they arethe only relevant constraints. Then we define generalintegration limits for the ðθ2;ϕ2Þ variables at Bðx; yÞ;θ2 goes from 0 to αDA and ϕ2 ranges from 0 to π toexploit the symmetry of the system. The upper limitof θ2 is defined as αDA, taking into account that we arepursuing the indirect calculation of the exact formu-lation, and beyond that limit there are no contribu-tions from rays reflected back from the nonplanarinterface because they cannot reenter the micro-scope. However, if we want to evaluate an approxi-mated intensity value directly from Eqs. (11) and(12), the upper limit of θ2 would be set to αTIR giventhat I2 rays with θ2 in between αDA and αTIR might beindexing single I1 contributions to the intensity; seeFig. 2. From the following paragraphs it can be seenthat the exact limits for the angles where interfer-ence occurs are implicitly determined during thecourse of our calculations.

The integration can be performed numericallywithin the specified domain, noting that we only needto compute the numerators of the mentioned equa-tions because the denominators of Eqs. (4) and (5)can be easily determined as 2πð1 − cosðαIAÞÞ. In our si-mulations, we make calls to the functions “quad” and“dblquad” in MATLAB R2007b, which use adaptiveSimpson quadrature. When the integrand at a parti-cular point ðθ2;ϕ2Þ has to be calculated, the backwardray-tracing procedure is executed. For those situa-tions where I2 can be traced backward to a sourcepoint A, as shown in Fig. 1(b), we still need to verifythat it will contribute to the intensity. The first andmost important condition is that I0 must come from

the illumination source (θ1 ≤ αIA). If this is not satis-fied, the integrand value can be set to zero and I1 andI2 are not of further interest. Provided that θ1 ≤ αIAwe have, at least, the I1 contribution to the intensityand its possible interference with I2 depends on twoadditional constraints: point A must be inside thefield of view and the OPLD must be smaller thanthe coherence length of waves from the mercuryarc lamp.

The conditions stated above can be incorporatedinto the calculation of the effective reflection coeffi-cients as follows:

Rs;pðθ2;ϕ2Þ ¼ ΘðαIA − θ1Þ½rs;p01 ðθ1ÞþΨðA;ΔÞts;p01 ðθ1Þrs;p12 ðθreflÞts;p10 ðθL12 Þ× expð−ikΔÞ�; ðA2Þ

where θL12 ¼ sin−1ðn0 sinðθ2Þ=n1Þ is the angle θ2 inlayer 1, according to Snell’s law; rij and tij are thereflection and transmission coefficients of the inter-face between the layers i and j, respectively, given bythe Fresnel equations and taken as a function of theangle between the incident ray and the normal to theinterface ði; jÞ; k ¼ 2π=λ is the wavenumber and λ isthe wavelength of the illuminating light; Δ is theOPLD given by Eq. (7); the Heaviside function Θguarantees that the rays come from within the illu-mination cone; and

ΨðA;ΔÞ ¼�1 if jAj < rFieldStop and Δ < 30 μm0 otherwise :

ðA3Þ

At this point we are able to evaluate the numerator ofEqs. (11) and (12), but, because our interest is theexact calculation, it is possible to write

Rs;pðθ1;ϕ1Þ ¼ Rs;pðθ2;ϕ2Þ: ðA4Þ

In other words, the effective reflection coefficient canbe indistinctly considered either as a function ofðθ2;ϕ2Þ or ðθ1;ϕ1Þ. When we do this at every nonzeroRs;pðθ2;ϕ2Þ point calculated while evaluating the ap-proximated integral, we are numerically constructingthe inversemapG−1, Eq. (13), for the convex geometryinvolved. This means that we can calculate integrandvalues for the numerators of Eqs. (4) and (5) that cor-respond to a particular region in the Ω1 domain withpossible interference between I1 and I2 contributions[such a region is labeled as “I1&I2” in Fig. 2].

In order to compute the desired integral, two finalsteps are necessary. First, if the “I1&I2” region doesnot fill the entire Ω1 domain, that implies that thereis a region of single I1 contributions where the inte-gration can be performed directly, using Eqs. (4) and(5), provided the translation of the boundary θ2 ¼αDA from Ω2 into Ω1. Second, because the integrandvalues in “I1&I2” form a nonregular grid in theΩ1 domain, a set of points (θ1;ϕ1, integrand), it is


necessary to find an appropriate method to completethe computation of the desired integral. 2DDelaunaytriangulation [29] performed on a set of ðθ1;ϕ1Þallows us to obtain a 3D approximation of the surfaceintegrand ¼ Fðθ1;ϕ1Þ with a series of flat triangles.Then, the integration can be carried out as a sum-mation of volumes over all the triangles inside Ω1.Figure 8 shows a typical surface of integrand valuesapproximated by the triangulation method.

References1. G. Wiegand, K. R. Neumaier, and E. Sackmann, “Microinter-

ferometry: three-dimensional reconstruction of surface micro-topography for thin-film and wetting studies by reflectioninterference contrast microscopy (RICM),” Appl. Opt. 37,6892–6905 (1998).

2. N. G. Clack and J. T. Groves, “Many-particle tracking withnanometer resolution in three dimensions by reflectioninterference contrast microscopy,” Langmuir 21, 6430–6435(2005).

3. E. W. Gomez, N. G. Clack, H. J. Wu, and J. T. Groves, “Like-charge interactions between colloidal particles are asym-metric with respect to sign,” Soft Matt. 5, 1931–1936 (2009).

4. A. Albersdorfer, T. Feder, and E. Sackmann, “Adhesion-induced domain formation by interplay of long-range repul-sion and short-range attraction force: a model membranestudy,” Biophys. J. 73, 245–257 (1997).

5. A. Kloboucek, A. Behrisch, J. Faix, and E. Sackmann,“Adhesion-induced receptor segregation and adhesion plaqueformation: a model membrane study,” Biophys. J. 77,2311–2328 (1999).

6. E. Sackmann and R. F. Bruinsma, “Cell adhesion as wettingtransition?” Chem. Phys. Chem. 3, 262–269 (2002).

7. E. Sackmann and S. Goennenwein, “Cell adhesion as dynamicinterplay of lock-and-key, generic and elastic forces,” Prog.Theor. Phys. Suppl. 78–99 (2006).

8. S. Moulinet and D. Bartolo, “Life and death of a fakir droplet:impalement transitions on superhydrophobic surfaces,” Eur.Phys. J. E 24, 251–260 (2007).

9. M. Sundberg, A. Mansson, and S. Tagerud, “Contact anglemeasurements by confocal microscopy for non-destructive mi-croscale surface characterization,” J. Colloid Interface Sci.313, 454–460 (2007).

10. B. M. Discher, Y. Y. Won, D. S. Ege, J. C. M. Lee, F. S. Bates, D.E. Discher, and D. A. Hammer, “Polymersomes: tough vesiclesmade from diblock copolymers,” Science 284, 1143–1146(1999).

11. B. M. Discher, H. Bermudez, D. A. Hammer, D. E. Discher, Y. Y.Won, and F. S. Bates, “Cross-linked polymersomemembranes:vesicles with broadly adjustable properties,” J. Phys. Chem. B106, 2848–2854 (2002).

12. K. Kita-Tokarczyk, J. Grumelard, T. Haefele, and W. Meier,“Block copolymer vesicles—using concepts from polymer

chemistry to mimic biomembranes,” Polymer 46, 3540–3563(2005).

13. A. Mecke, C. Dittrich, and W. Meier, “Biomimetic membranesdesigned from amphiphilic block copolymers,” Soft Matt. 2,751–759 (2006).

14. G. B. Sukhorukov, E. Donath, H. Lichtenfeld, E. Knippel, M.Knippel, A. Budde, and H. Mohwald, “Layer-by-layer self as-sembly of polyelectrolytes on colloidal particles,”Colloids Surf.A 137, 253–266 (1998).

15. Y. Wang, A. S. Angelatos, and F. Caruso, “Template synthesisof nanostructured materials via layer-by-layer assembly,”Chem. Mater. 20, 848–858 (2008).

16. K. K. Liu, “Deformation behaviour of soft particles: a review,”J. Phys. D 39, R189–R199 (2006).

17. D. Gingell and I. Todd, “Interference reflection microscopy—aquantitative theory for image interpretation and its applica-tion to cell-substratum separation measurement,” Biophys. J.26, 507–526 (1979).

18. J. Radler and E. Sackmann, “Imaging optical thicknesses andseparation distances of phospholipid vesicles at solid sur-faces,” J. Phys. (Paris) II 3, 727–748 (1993).

19. M. Kuhner and E. Sackmann, “Ultrathin hydrated dextranfilms grafted on glass: Preparation and characterization ofstructural, viscous, and elastic properties by quantitative mi-crointerferometry,” Langmuir 12, 4866–4876 (1996).

20. O. Theodoly, Z.-H. Huang, and M.-P. Valignat, “New modelingof reflection interference contrast microscopy includingpolarization and numerical aperture effects: application tonanometric distance measurements and object profile recon-struction,” Langmuir 26, 1940–1948 (2010).

21. E. W. Weisstein, “Change of variables theorem,” retrievedfrom http://mathworld.wolfram.com/ChangeofVariablesTheorem.html.

22. D. E. Discher and A. Eisenberg, “Polymer vesicles,” Science297, 967–973 (2002).

23. J. Radler and E. Sackmann, “On the measurement ofweak repulsive and frictional colloidal forces by reflection in-terference contrast microscopy,” Langmuir 8, 848–853 (1992).

24. H. Bermudez, A. K. Brannan, D. A. Hammer, F. S. Bates, andD. E. Discher, “Molecular weight dependence of polymersomemembrane structure, elasticity, and stability,” Macromol. 35,8203–8208 (2002).

25. J. E. Mark, Polymer Data Handbook (Oxford U. Press, 1999).26. L. Limozin and K. Sengupta, “Modulation of vesicle adhesion

and spreading kinetics by hyaluronan cushions,” Biophys. J.93, 3300–3313 (2007).

27. G. Wiegand, T. Jaworek, G. Wegner, and E. Sackmann,“Studies of structure and local wetting properties on hetero-geneous, micropatterned solid surfaces by microinterferome-try,” J. Colloid Interface Sci. 196, 299–312 (1997).

28. K. W. Stockelhuber, B. Radoev, and H. J. Schulze, “Some newobservations on line tension of microscopic droplets,” ColloidsSurf. A 156, 323–333 (1999).

29. E. W. Weisstein, “Delaunay triangulation,” retrieved fromhttp://mathworld.wolfram.com/DelaunayTriangulation.html.


http://mathworld.wolfram.com/ChangeofVariablesTheorem.html





http://mathworld.wolfram.com/DelaunayTriangulation.html




reflection interference contrast microscopy of arbitrary convex surfaces

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