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Direct and alignment-insensitive measurement of cantilever curvature
Rodolfo I. Hermans∗ and Gabriel Aeppli†
London Centre for Nanotechnology, University College London andDepartment of Physics and Astronomy, University College London
Joe M. BaileyCentre for Mathematics and Physics in the Life Sciences and Experimental Biology, University College London and
London Centre for Nanotechnology, University College London(Dated: July 18, 2013)
We analytically derive and experimentally demonstrate a method for the simultaneous measure-ment of deflection for large arrays of cantilevers. The Fresnel diffraction patterns of a cantileverindependently reveals tilt, curvature, cubic and higher order bending of the cantilever. It providesa calibrated absolute measurement of the polynomial coefficients describing the cantilever shape,without careful alignment and could be applied to several cantilevers simultaneously with no addedcomplexity. We show that the method is easily implemented, works in both liquid mediums and inair, for a broad range of displacements and is especially suited to the requirements for multi-markerbiosensors.
Silicon-based microfabrication has enabled not onlythe electronics revolution but also made micromechan-ics nearly as ubiquitous, with applications from motionsensing to biochemical analysis[1, 2]. The atomic forcemicroscope [3, 4], where the motion of a small tip at theend of a cantilever traces nanoscale features on surfaces,provides the fundamental paradigms for nanomechanicalmetrology, including optical readouts of cantilever dis-placement and bending. Such readout typically requirescareful and costly alignment of mechanical and opticalelements. Here we demonstrate an alternative approachbased on the recognition that in many applications weare interested not so much in the motion of a tip as inthe overall curvature of a cantilever. In particular, nearfield imaging of entire cantilevers yields diffraction pat-terns providing precise measures of the tilt, curvature andhigher order bending components of cantilevers. Evenwhile we have used very inexpensive components and nocareful alignment is required, we obtain sensitivity to nm-scale motion of the cantilever end.
There are many methods to measure the displace-ment of microstructures such as AFM cantilevers[4].They are based on various physical principles, includingoptics[5], piezoresistance[6, 7], field-effect transistors[8]and capacitance[9]. The method most commonly used,and still the most sensitive and reliable, is the opticallever or optical beam deflection technique (OBDT)[10,11] –implemented in the AFM market– and optical inter-ferometry [12–14]. No detection method is optimal for alltypes of measurements and the growing use of cantileversas multiplexed biosensors imposes its own challenges[15],particularly in the case of arrays of several cantilevers.Cantilever arrays are used to obtain simultaneous detec-tion of different targets, increase statistical significance,in-situ control and for differential measurements[16, 17].
∗ [email protected]† [email protected]
They require detection systems with a complexity thattypically scales with the number of cantilevers in the ar-ray. Examples of such systems include arrays of illumi-nating lasers with a single multiplexed detector[18] or2D scanners of a single laser beam employing voice-coilactuators[19], which are simply extensions of OBDT forcantilever arrays.
The main drawback of the common optical techniquesis the difficulty of accurately aligning each illuminat-ing source with its corresponding cantilever and detec-tor, making it difficult to measure large numbers of can-tilevers without an elaborate pre-measurement protocol.A further limitation, particularly of OBDT (Fig 1a),comes from the fact that the observed quantity is thelocal change in angle and displacement of the lever atthe point of illumination, and therefore cantilever tiltand bending cannot be distinguished in a single mea-surement. Also, a bending model must be assumed toestimate the true beam curvature. Efforts to measurethe whole cantilever profile have partially addressed thelatter issue using scanning or an array of Light-EmittingDiodes (LED)[20]. Research has revealed that com-monly assumed bending profiles may not be as realis-tic as expected[21], providing further justification for thedevelopment of a simple technique for imaging cantileverbending.
In this Letter we describe a method for the direct andcalibrated measurement of cantilever bending profilesthat requires no scanning or alignment of the illumina-tion and does not assume particular bending models. Weshow that an out-of-focus image of the whole cantileverarray is enough to estimate at least the first four coeffi-cients of a polynomial describing the cantilever shape foreach of the cantilevers in an array independently, makingit suitable for multi-marker biological essays. We namethis technique NANOBE for No-Alignment Nearfield Op-tical Bending Estimation.
To detect bending profiles we illuminate a cantileverarray with a homogeneous monochromatic plane wave
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a) b)
FIG. 1. Concepts for standard and proposed optical detectionmethods. a) The optical beam deflection technique (OBDT)requires a focused beam carefully aligned to a cantilever andthe center segmented photo-diode. b) In the No-AlignmentNearfield Optical Bending Estimation (NANOBE) technique,a broad beam illuminates the whole cantilever array gener-ating a diffraction pattern projected onto a CMOS detector.The intensity profile of the diffraction pattern is insensitiveto misalignments but sensitive to the details of the cantileverbending.
and measure the intensity pattern of the reflected light(Fig 1b). Based on the Huygens-Fresnel principle, wemodel the reflected component from the finite-size can-tilever as a rectangular source in the plane (ξ, η). Weexpress the wave amplitude in the observing plane (x, y)as the convolution integral [22]
U(x, y) =eikz
iλz
∞x
−∞U(ξ, η) ei
πλz [(x−ξ)2+(y−η)2]dξdη. (1)
where U(ξ, η) is the function defining the amplitudeand phase at the source. We model the shape of the can-tilever as a rectangle of dimensions (w, l) and the bend-ing profile by a polynomial function P (ξ) =
∑ciξ
i withi ≥ 1 ; although here we assume bending only along theξ direction which is the long axis of the cantilever, ourresults can be generalized to P (ξ, η) taking account of ar-bitrary curvature. The bending of the surface introducesa difference in the optical path, which introduces a phaseφ = 4πλ−1P (ξ). The field amplitude at the source is welldescribed by
U(ξ, η) = rect
(ξ
l
)e
4πiλ P (ξ) rect
( ηw
)(2)
where rect(x) = 1 for |x| < 1/2 and zero otherwise.We concentrate attention on the x coordinate along thecantilever and rewrite equation 1 to separate variablesU(x, y) = −i exp(ikz) I(x)I(y) and considering that the
observed intensity I(x, y) = |U(x, y)|2 = |I(x)|2 |I(y)|2we analyze the value of
-1
1
0
-1
1
0
-1
1
0
-1
1
0
103 104102101100 0.25 0.50-0.25-0.5
0.05 0.100-0.05-0.10 0.15 0.300-0.15-0.30
Intensity [I/I0]Fresnel Number Tilt [L-1]
Curvature [L-2] Cubic bending [L-3]
Intensity [I/I0]
Pos
ition
[x/L
]P
ositi
on [x
/L]
Pos
ition
[x/L
]P
ositi
on [x
/L]
Intensity [I/I0]
Intensity [I/I0]
a) b)
c) d)
FIG. 2. Intensity map I(x), where x is pixel position alongcantilever direction on the detector, for a cantilever as a func-tion of different relevant variables. Insets are vertical cross-sections of the intensity map showing the intensity profiles forspecific abscissa values. The horizontal axes are: a) Fresnelnumber F = L2(λz)−1, where L is the cantilever length, and zthe distance from the cantilever at which the intensity is mea-sured, b) c1 the cantilever tilt, c) c2 the cantilever curvatureand d) c3 the cubic bending.
I(x) =1√λz
L/2∫−L/2
exp
[iπ
λz(x− ξ)2 + i
4π
λP (ξ)
]dξ (3)
We now use the first two terms of P (ξ) to complete asquared binomial for ξ and for convenience rewrite equa-tion 3 as
I(x) = A
L/2∫−L/2
ei πλz
(m( x−sm −ξ)
2+4z
∑i≥3
ciξi
)dξ. (4)
with m = 1 + 4c2z , s = 2c1z and
A = (λz)−12 exp
[i
4π
λm
(c2x
2 + c1x− c21z)]. (5)
If we neglect cubic and higher order terms in the poly-nomial description of the cantilever curvature i.e. assum-ing ci ≈ 0 ∀ i > 2, we can introduce a change of variablesand a corresponding change in integration limits to finda familiar result. Using
3
α(ξ) =
√2m
λz
(x− sm− ξ)
(6)
we obtain the solution for a rectangular slit
I(x) =
((C(α2)− C(α1))2 + (S(α2)− S(α1))2
)2m
, (7)
where C(α) and S(α) are Fresnel Integrals de-fined as C(αi) =
∫ αi0
cos(πα2/2
)dα, and S(αi) =∫ αi
0sin(πα2/2
)dα and the integration limits α1 =
α(−L/2) and α2 = α(L/2)
α1 =√
2
√4c2z + 1
λz
(−L
2− x− 2c1z
4c2z + 1
)(8)
α2 =√
2
√4c2z + 1
λz
(L
2− x− 2c1z
4c2z + 1
)(9)
The inset of figure 1b shows calculated 2D patterns givenby I(x, y) = |I(x)|2 |I(y)|2 resembling experimentallyobserved patterns. Figure 2a shows I(x) for a rectangu-lar slit or a flat cantilever i.e. ci = 0 ∀i as a functionof the Fresnel number F = L2(λz)−1, where L is thecantilever length, and z the distance from the cantileverat which the intensity is measured (insets are intensityprofiles for specific abscissa value). In what follows weanalyze the influence of each coefficient ci 6= 0 on theobserved diffraction pattern.
From the equation 6 itself and the figure 2b we see thata non vanishing tilt c1 results in a shift s = 2c1z of thediffraction pattern, as expected from the tilt of any re-flecting surface. Similarly from figure 2c, provided z 6= 0,a non-zero c2 causes a magnification m = 1 + 4c2z in thepattern size and m−1 in the intensity. The magnificationdoes not come from the negligible displacement of thecantilever ends, but is intrinsic to the curvature of thereflecting surface. Both effects do not otherwise distortthe shape of the diffraction pattern.
Higher order contributions to the curvature are moreeasily studied by numerically integrating equation 4. Fig-ure 2d shows the Fresnel diffraction pattern for cantileverwith a deflection profile P (x) ∝ x3. Cubic bendingshown in figure 2d features intensity gradients and pat-tern shifts, both proportional to c3. Quartic bending (notshown) creates a curvature in the intensity profile as wellas a change in the pattern length.
Figure 3a illustrates the experiment where a broad(16.8mm) collimated laser beam (λ = 660nm) illumi-nates a cantilever array. A 4X microscope objective anda CMOS sensor focused in a plane at a distance z fromthe cantilever maps the diffraction pattern from the can-tilevers. The array (purchased from IBM) consisted ofup to eight rectangular silicon (100) cantilevers, eachmeasuring 500µm in length, 100µm in width, and 0.9µmin thickness and a nominal spring constant of 0.02N/m.
330
340
350
360
370
380
390
400
410
420
430
pH 4.8pH 4.8 pH 9.0
Pat
tern
Siz
e [p
ixel
s]
Cur
vatu
re [m
-1]
Dis
plac
emen
t [µm
]
100 150 200 250 300 350 400
Time [s]
789
11121314
1617181920
10
15
2.0
3.0
3.5
4.0
4.5
5.0
2.5
3210
Cantilever Displacement [µm]
Pat
tern
Siz
e R
atio
0.5
0.6
0.7
0.8
0.9
1.0(1+4 b z) L
L
CCD
Microscopeobjective
Beam splitter
Cantilever array
Piezoelectricactuator
Glass tip
Broad beam laser
e)
d)c)
b)a)
FIG. 3. Experimental measurements using NANOBEmethod. a) Setup schematics: A cantilever array is illumi-nated with a broad laser beam through a cubic beam-splitterand the reflected light captured by a lens and CCD. b) A sin-gle cantilever is pushed with a glass tip mounted over a cali-brated piezoelectric actuator. c) Diffraction images (640×528px) of the pushed cantilever (top pattern) are shorter than thepatterns from the relaxed cantilever by a factor of (1 + 4bz).This corresponds to a lensing effect from the curved mirrorformed by the cantilever. d) The experimentally measuredpattern size changes linearly with the cantilever tip displace-ment closely overlapping the model. e) The curvatures ofsix cantilevers as a function of time. Cantilevers functional-ized with MHA (top) and HDT (bottom) have different staticbending and different sensitivity to pH changes.
The cantilevers were coated on one side with a 2nm tita-nium adhesion layer and then 20nm of gold. The diffrac-tion pattern size is calculated in units of pixels by soft-ware calculating the distance between intensity thresholdcrossings or, more preferable, using the second centralmoment of the intensity profile I(j) [23].
So = 2
√3
∑I(j)(j − µ)2∑
I(j)(10)
4
0
10
-10
-20
-30
-40
-50
0.00
0.05
-0.05
-0.10
-0.15
-0.20
+0.05°C +0.10°C +0.15°C +0.20°C
5000 15001000 25002000 35003000Time [s]
Displacem
ent [nm]
# CountsC
urva
ture
[m-1
]
FIG. 4. The cantilever bending caused by small temperaturechanges are easily resolvable. The temperature protocol wasset to 25◦C,25.05◦C,25.10◦C,25.15◦C and 25.20◦C. The lighttrace corresponds to the raw data and the darker trace is a10 point moving average. The right hand side graph shows akernel density estimation of raw data demonstrating that themeasured curvatures feature perfectly distinguishable distri-butions.”
We test the method experimentally for deflections inexcess of one micron by measuring the diffraction patternas a function of cantilever bending where we poke thecantilever end in air with a glass tip (Fig 3b) attached toa well-calibrated piezoelectric actuator (P-363 PicoCubeXYZ Piezo Scanner by Physik Instrumente GmbH). Theconcentrated load at the free end is expected to causea bending profile proportional to x2(3L − x), where Lis the cantilever length[24]. Along with the controlleddisplacement we observe both a change of the size ofthe diffraction pattern as well as a gradient in the inten-sity, characteristic of both parabolic and cubic bending(Fig 3c top pattern). The change in pattern size displaysthe approximately linear predicted relationship with thedisplacement caused by the actuator. Figure 3d showsthe overlap of experimental data (dots) and numericallyintegrated model (line).
We further test the method in fluid by functionalizingthe surface of individual cantilevers with either mercap-tohexadecanoic acid HS(CH2)15COOH, here abbreviatedMHA, or hexadecanethiol HS(CH2)15CH3, here abbrevi-ated HDT, via incubation in an array of glass micro-capillaries. The protonation/deprotonation of MHA atdifferent pH causes a differential surface stress inducingcantilever bending. HDT has identical chain length andsimilar packing density to MHA but differs in the ter-minal methyl group which is non-ionizable and thereforeis used as reference [25]. The cantilever array is thenmounted in a liquid cell with a sapphire window to allowillumination and imaging. An automated system of sy-ringes and valves was developed to control the delivery ofsodium phosphate mono and dibasic solution pH 4.8 andpH 9.0 at a constant rate of 43µL/min and controlledtemperature of 25.00± 0.01◦C. The supplementary ma-terials [23] show a schematic as well as photograph of
the instrumental prototype which we have constructedfor these measurements. Figure 3e shows the distinctivechanges in pattern size as we alternately flow solutionswith pH 4.8 and pH 9.0. The main changes in curvatureare attributed to the differential stress caused by proto-nation/deprotonation. The mean relative deflection forMHA cantilevers is calculated to be 249.8± 8.6nm, con-sistent with previous results [26]. The mean relative de-flection for HDT is somehow smaller than previous results(137.2±4.8nm) giving a differential stress between MHAand HDT-coated cantilevers of around 22.0±1.9mN/m,52% bigger than expected [27]. Observing the initial cur-vatures in figure 3e, we see that NANOBE reveals thatHDT and MHA functionalization causes different staticbending of the cantilevers as well as different sensitivi-ties to pH. The variation in static and dynamic bend-ing among cantilevers with the same functionalizations iscommonly observed and reveals the difficulties in prepar-ing consistent surface modification in small areas. Theinconsistency upon functionalization of nominally iden-tical cantilevers remains an open question of utmost im-portance but is beyond the scope of this work. Our im-plementation of NANOBE allows direct observation ofunexpected objects on the surface, lateral torsion andsome but not all other sources of error typically unde-tectable on OBDT.
To demonstrate the resolution of small deflections inliquid we step the temperature of the cantilever in in-crements of 0.05 ± 0.01◦C around 25◦C. The differentthermal expansion of silicon and gold causes the micro-structure to bend. Figure 4 shows that the measureddeflection is well resolved even for these small changes intemperature. The inset on the right hand side shows thedensity of measured deflections, featuring well-resolvedmaxima and negligible overlap of the individual distribu-tions. The cantilever is found to bend around 150nm/◦C,in good agreement with previous measurements and mod-els [28].
The main output of the NANOBE method is the sizeof a diffraction pattern which is independent of the regionof the detector that is used to capture it, and thereforeinsensitive to small misalignments of the light source ordetector and independent of the orientation of the can-tilever. The pattern size measured in units of pixels isdirectly translated to an absolute value for the curvatureusing
c2 = (So − Sc)/(4zSc) (11)
where So is the observed pattern size in pixels, Sc =ρLc, where Lc is the cantilever length and ρ resolutionof the sensor in pixels per length, and z is the distanceto focus. In contrast to the standard laser beam de-flection method where the optical lever arm needs to becarefully fixed and measured, our method is entirely self-calibrated, i.e. the known geometry of the cantilever ar-ray gives the conversion ratio between length and pixels,and the cross section of the diffraction pattern reveals
5
the effective value of distance z. Nonetheless, if thereare any doubts, these values are also available to the ex-perimentalist by physically measuring the geometry ofthe experimental setup. The changes of the tilt c1 arealso available from the centroid of the diffraction pat-tern, and the higher orders of curvature c3 and above,can be estimated from the overall intensity gradients foreach pattern.
Provided the Fresnel number is large enough (Fig 2a),the diffraction patterns will be localized even for narrowcantilevers (∼ 10µm) with negligible cross talking be-tween adjacent cantilevers, therefore allowing indepen-dent analysis of simultaneously imaged patterns.
The resolving power of the system is proportional tothe magnitude of z, to the number of pixels covered bythe pattern and inversely to the noise level of the de-tector. Our experiments were performed with a sim-ple USB 1280 × 1024 px Monochrome CMOS (ThorlabsDCC1545M) at 1 frame per second where each patterncovered only around 400 pixels; therefore, a much im-proved resolution can be expected from a high-end de-tection system and higher sampling rates.
We have invented a near-field method for measuringcantilever bending which is much more robust and sim-pler to implement than the standard optical beam de-
flection methods. The method is model-free and givesindependent values for tilt, curvature and higher orderbending. It relies on curved mirror diffraction and is in-sensitive to misalignment, opening the way to a varietyof devices for nanometrology, inexpensive from both themanufacturing and operational points of view. We en-vision future devices ranging from force microscopes tobiochemical assays where cantilevers of various geome-tries that could even be weakly tethered (or untethered)to substrates using inexpensive hardware comparable toa portable CD player and a computer webcam.
ACKNOWLEDGMENTS
G.A. and R.H. thank support from grant EPSRCEP/G062064/1, “Multi-marker Nanosensors for HIV”,May 2009. R.H and G.A on 28 March 2012 fileda patent application (“Measurement of micro-structurecurvature” 12054921.2) for the methods here described.All authors thank Dr. Joseph Ndieyira and Dr. Samad-han Patil for metal-coating some of the cantilever arraysused in this work.
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