# A complete analysis of the laser beam deflection systems used in cantilever-based systems.

**ABSTRACT** A working model has been developed which can be used to significantly increase the accuracy of cantilever deflection measurements using optical beam techniques (used in cantilever-based sensors and atomic force microscopes), while simultaneously simplifying their use. By using elementary geometric optics and standard vector analysis it is possible, without any fitted or adjustable parameters, to completely and accurately describe the relationship between the cantilever deflection and the signal measured by a position sensitive photo-detector. By arranging the geometry of the cantilever/optical beam, it is possible to tailor the detection system to make it more sensitive at different stages of the cantilever deflection or to simply linearize the relationship between the cantilever deflection and the measured detector signal. Supporting material and software has been made available for download at http://www.physics.mun.ca/beauliu_lab/papers/cantilever_analysis.htm so that the reader may take full advantage of the model presented herein with minimal effort.

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**ABSTRACT:**The impressive developments in micro / nano-electro-mechanical-systems (MEMS; NEMS) have led to a new class of chemical and biological sensors based on micro and nano cantilevers. This work focuses on fabrication challenges of flat cantilevers exhibiting well-controlled, uniform and reproducible mechanical performance. Our experimental study is based on cantilevers made of crystalline silicon (c-Si), using SOI wafers as the starting material and using bulk micromachining. Experimental results on fabrication and characterization of composite porous silicon-crystalline silicon microcantilevers made of SOI wafers are also presented, where the porous silicon surface provides an excellent interface for immobilization of the biosensing layer. The optimal geometric design of microcantilevers depending on the application as well on the selected sensing mode (static or dynamic) is considered. The innovative aspects and open issues of NEMS/MEMS cantilever-based biosensors are addressed.Proc SPIE 01/2008; - SourceAvailable from: P.J. FrenchChung-Kai Yang, Andre Bossche, Paddy J. French, H. Sadeghian, J. F. L. Goosen, F. van Keulen, K. B. Gavan, E. W. J. M. van der Drift[Show abstract] [Hide abstract]

**ABSTRACT:**Laser beam deflection is a well known method commonly used in detecting resonance frequencies in atomic force microscopes and in mass/force sensing. The method focuses a laser spot on the surface of cantilevers to be measured, which might change the mechanical properties of the cantilevers and affect the measurement accuracy. In this work we showed that the joule heating of the laser, besides other extrinsic effects such as surface contamination, can cause a significant amount of shift in the resonator. The longer and softer the cantilever is, the more significant the effect. We suggest that the laser effects on the resonant response of sensors have to be taken into account.01/2009; - Futures. 01/2011; 43(10):1041-1043.

Page 1

Ultramicroscopy 107 (2007) 422–430

A complete analysis of the laser beam deflection systems

used in cantilever-based systems

L.Y. Beaulieua,?, Michel Godinb, Olivier Larochec, Vincent Tabard-Cossac, Peter Gru ¨ tterc

aDepartment of Physics and Physical Oceanography, Memorial University. St. John’s, NL., Canada A1B 3X7

bDivision of Biological Engineering, Media Laboratory, Massachusetts Institute of Technology, 20 Ames Street, Cambridge, MA 02139, USA

cPhysics Department, McGill University, Montreal, QC., Canada H3A 2T8

Received 20 February 2006; received in revised form 2 November 2006; accepted 2 November 2006

Abstract

A working model has been developed which can be used to significantly increase the accuracy of cantilever deflection measurements

using optical beam techniques (used in cantilever-based sensors and atomic force microscopes), while simultaneously simplifying their

use. By using elementary geometric optics and standard vector analysis it is possible, without any fitted or adjustable parameters, to

completely and accurately describe the relationship between the cantilever deflection and the signal measured by a position sensitive

photo-detector. By arranging the geometry of the cantilever/optical beam, it is possible to tailor the detection system to make it more

sensitive at different stages of the cantilever deflection or to simply linearize the relationship between the cantilever deflection and the

measured detector signal. Supporting material and software has been made available for download at http://www.physics.mun.ca/

beaulieu_lab/papers/cantilever_analysis.htm so that the reader may take full advantage of the model presented herein with minimal

effort.

r 2006 Elsevier B.V. All rights reserved.

Keywords: Cantilever; Optical beam deflection; Cantilever sensor; AFM

1. Introduction

Micro-cantilevers are small V-shaped or rectangular

cantilevers (typically made of silicon nitride (SiNx) or

silicon (Si)) which are of the order of 400mm long, 50mm

wide and 1mm thick. Although micro-cantilevers are

typically used in atomic force microscopes (AFM) for

surface imaging, they have also been employed as ultra-

sensitive sensors to detect various phenomena such as

changes in temperature [1], changes in mass [2] and the

detection of chemical reactions through changes in surface

stress [3].

Properly measuring the cantilever deflection is at the

heart of acquiring precision measurements when perform-

ing cantilever-based sensing or atomic force spectroscopy

experiments. The most common method to measure the

cantilever deflection is by using an optical beam deflection

system [4]. Although there have been some discussions in

the literature regarding the laser detection scheme [5–15],

we have not seen addressed anywhere the subject of how to

obtained a well-defined relationship between the actual

cantilever deflection and the PSD signal. Nor have we seen

anywhere discussed the influence of different optical beam

deflection geometries on the relationship between the

cantilever deflection and the PSD signal. This is the subject

of this paper.

In a recent paper, we have reported how, by properly

designing a cantilever-based instrument, it is possible to

linearize the relationship between the cantilever deflection

and the measurement made with a photo-sensitive detector

(PSD) [16]. In this paper, we describe in full detail

the mathematical model used to properly characterize the

cantilever/laser beam deflection system so that potential

users may adapt and use the model with their instruments.

We also provide a web link where the reader can down-

load software and other supporting material related to

this paper.

ARTICLE IN PRESS

www.elsevier.com/locate/ultramic

0304-3991/$-see front matter r 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.ultramic.2006.11.001

?Corresponding author. Tel.: +17097376203; fax: +17097378739.

E-mail address: beaulieu@physics.mun.ca (L.Y. Beaulieu).

Page 2

2. Theory

Fig. 1 shows a schematic representation that describes

the geometry of a cantilever/laser beam detection system.

In this diagram, the cantilever surface is in the x?y plane

and is oriented in the positive x direction. The cantilever

chip is fixed in space and does not move. An incident laser

hits the cantilever at a distance D from the base of the

cantilever chip. The incident laser is fixed at an angle of

inclination y with respect to the x?y plane and at an

azimuthal angle f measured from the positive x-axis. The

laser reflects off the free end of the cantilever into a

position sensitive detector (PSD) held at an initial distance

L from the cantilever. The PSD is itself inclined at an angle

x also with respect to the x?y plane. In Fig. 1, the line

labeled Nc, is the vector normal to the surface of the

cantilever and is used to calculate the reflected laser beam

direction in accordance with the law of reflection.

The vector equation for a straight line in three

dimensions is given by

I ¼ Ipþ tIa,

where Ipis any point on the line, Iais the unit direction

vector and t is any scalar. In this case Ipis defined as the

initial point where the incident laser hits the cantilever at

Ip¼ ðD; 0; 0Þ. Given the geometry shown in Fig. 1, the

direction vector Iais defined as

(1)

Ia¼ ðcosðyÞ cosðp ? fÞ; cosðyÞ sinðp ? fÞ; ?sinðyÞÞ.

During cantilever deflections, the incident laser beam

reflects away from the surface normal in accordance to the

law of reflection. Therefore it is imperative that the proper

cantilever curvature be taken into consideration. In

cantilever sensor experiments, it has been reported that

the cantilever undergoes a deflection as if it were subjected

to an end-moment [17,18]. For this type of deflection the

curvature of the cantilever is described by the following

equation:

(2)

zðxÞ ¼ gx2. (3)

In this equation, g is a constant defined by the physical

parameters of the cantilever multiplied by the applied end

moment force. In the following calculations, the end

deflection of the cantilever is controlled which in turn

governs the value of g. Therefore, at each calculated

deflection, g is defined as

g ¼zðxmaxÞ

x2

max

.(4)

The value of xmaxis determined by solving the integral

equation, which defines the total length CL of the cantilever.

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

@x

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x2

max

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4zðxmaxÞ

x2

maxlnð2zðxmaxÞ þ

4zðxmaxÞ

Solving for xmaxfrom Eq. (5) is done numerically.

During AFM force spectroscopy experiments, a point

load is applied to the end of the lever. Such a force causes a

cantilever to deflect with a curvature as described by

Eq. (6) [19]

CL ¼

Zxmax

Zxmax

0

1 þ

@z

?

?

?2

q

!

v

v

u

u

u

u

t

t

dx

¼

0

1 þ

2zðxmaxÞ

x

?2

!

dx

¼

2zðxmaxÞx2

maxþ 4zðxmaxÞ2

? x2

maxlnðxmaxÞ

þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x2

maxþ 4zðxmaxÞ2

q

Þ

ð5Þ

zðxÞ ¼Fx2

where F is the applied force, E is the Young’s Modulus of

elasticity, I is the area moment of inertia, and a is the

location of the applied force. As before, in order to

perform the simulations, the cantilever curvature needs

only to be described using Eq. (7).

6EIðx ? 3aÞ,(6)

zðxÞ ¼ gx2ðx ? 3aÞ,

where g is determined by the maximum deflection.

(7)

g ¼

zðxmaxÞ

max? 3ax2

The value of xmaxis determined by solving the length

equation

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

qx

Zxmax

the solution of which is too lengthy to show here. Solving

this equation numerically gives the coordinate of the end of

the lever (xmax, zend).

When the cantilever deflects, the intersection point W

between the incident laser and the cantilever surface

must be calculated. This is accomplished by solving the

x3

max

.(8)

CL ¼

Zxmax

0

1 þ

qz

??2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

!

v

r

u

u

t

dx

¼

0

1 þ gð3x2? 6axÞðÞ2

??

dxð9Þ

ARTICLE IN PRESS

Fig. 1. Schematic representation of a laser reflecting from a cantilever into

a PSD detector.

L.Y. Beaulieu et al. / Ultramicroscopy 107 (2007) 422–430

423

Page 3

vector equation

I ¼ Ipþ tIa¼ ðWx;Wy;gW2

ðEnd moment deflectionÞ,

I ¼ Ipþ tIa¼ ðWx;Wy;gW2

ðPoint load deflectionÞ.

At the contact point W between the incident laser and

the cantilever, it is necessary to obtain the vector normal Nc

to the cantilever surface. The slope mNCof the vector Ncis

given by the negative of the reciprocal of the slope of the

cantilever at the point W. This can be written as

xÞÞ ¼ W

xðWx? 3aÞÞ ¼ W

ð10Þ

mNC¼ ?

1

2gWx

ðEnd moment deflectionÞ, (11)

mNC¼ ?

1

gð3W2

x? 6aWxÞ

ðPoint load deflectionÞ.(12)

Defining the angle ? ¼ arctanðmNCÞ, the vector Ncmay be

written as

Nc¼ ðcosð?Þ;0; sinð?ÞÞ.

The direction vector of the reflected beam Ra is by

definition, the reflection of Iaacross Ncin the Ia?Ncplane.

This is a well-known formula [20] given by Eq. (14).

(13)

Ra¼ Ia? ð2Ia? NcÞNc.

As in the case of the incident beam, the vector line of the

reflected beam R is given by the vector equation

(14)

R ¼ W þ tRa.

As one of the adjustable parameters, the PSD/cantilever

separation is given by the value L. The value of L is used to

define the initial contact point PSDpof the reflected laser

on the PSD

(15)

PSDp¼ Ipþ LRa.(16)

The vector PSDpis determined at the beginning of the

algorithm and remains fixed for the duration of the

calculations. The active area of the PSD is described as a

plane in space. Describing such a plane requires a point

PSDpand a normal vector PSDn(see Fig. 2). The normal

vector PSDn is determined by two orthogonal vectors

defined by the geometry in Fig. 1. Fig. 2 shows a schematic

representation of the geometry used to describe the PSD.

In this diagram, the vector PSDnis normal to the PSD

plane while the vectors PSD1and PSD2are in the PSD

plane. Also, since the PSD plane is inclined with respect to

the x?y plane, the vector PSDnmay not necessarily be

parallel to the reflected laser beam vector R. In all

calculations, the vectors I, R, and PSDnall lie in the same

plane. The vectors describing the PSD plane are shown in

Table 1 for three specific cases.

Finally, the vector normal to the PSD plane is given by

the following vector product:

PSDn¼ PSD2? PSD1.

Once the PSD plane has been defined, it is then necessary

to determine the intersection point rPSD between the PSD

plane and the reflected laser R. This is done by using the

following equation [20]:

(20)

rPSD ¼ W þðPSDp? WÞ ? PSDn

Ra? PSDn

Ra. (21)

The algorithm described above is performed for a range

of cantilever deflections (i.e. from 0 to some end deflection).

Since the PSD plane is inclined in 3D space, it is necessary

to rotate it such as to transform it into 2D space. This is

shown schematically in Fig. 3.

The general rotation matrix M used to rotate any vector by

an angle b about a vector u is given by the following [20]:

2

M ¼

u2

xþ Cð1 ? u2

uxuyð1 ? CÞ þ uzS

uxuzð1 ? CÞ ? uyS

xÞ

uxuyð1 ? CÞ ? uzS

u2

uxuzð1 ? CÞ þ uyS

uyuzð1 ? CÞ ? uxS

u2

yþ Cð1 ? u2

uyuzð1 ? CÞ þ uxS

yÞ

zþ Cð1 ? u2

zÞ

664

3

775,

(22)

where C?cos(b) and S?sin(b).

Although b ¼ p/2 ? x, the value of u changes depending

on which system is being studied. Table 2 shows the

different values of u.

Throughout this paper the linearity of the PSD meas-

urement versus cantilever deflection curves is discussed.

ARTICLE IN PRESS

Fig. 2. Schematic representation of the vector geometry involved in

characterizing the plane of the PSD. The vector PSDnis normal to the

PSD plane while the vectors PSD1and PSD2are in the PSD plane. In this

diagram, the vector R represents the reflected laser beam, which is incident

on the PSD at the point PSDp. The vector R subtends an angle b with

respect to the PSD surface normal.

Table 1

For f ¼ 01

PSD1¼ ðcosðxÞ;0; sinðxÞÞ,

PSD2¼ ð0; 1; 0Þð17Þ

For f ¼ 901

PSD1¼ ð0;?cosðxÞ; sinðxÞÞ,

PSD2¼ ð1; 0; 0Þð18Þ

For f ¼ 1801

PSD1¼ ð?cosðxÞ; 0; sinðxÞÞ,

PSD2¼ ð0; ?1; 0Þð19Þ

L.Y. Beaulieu et al. / Ultramicroscopy 107 (2007) 422–430

424

Page 4

In order to compare the linearity of different data we will

quote the value of w2given by the following:

X

where rPSD znðxmaxÞ

for a specific cantilever deflection znðxmaxÞ at the end point

of the cantilever xmaxand m is the slope of the best fit line.

w2¼

n

ðrPSD znðxmaxÞðÞ ? mznðxmaxÞÞ2, (23)

ðÞ is the calculated PSD measurement

3. Equipment

Matching experiments performed to validate the above

model were conducted on a macro-sized cantilever made

in-house. The rectangular-shaped cantilever was con-

structed from a 3.17mm thick aluminum piece, 25.4mm

wide with an extended unsupported length of 410mm. The

cantilever was held horizontally approximately 2mm from

a solid aluminum plate in which holes capable of accepting

dowel pins were machined at precise locations such as to

force the cantilever to bend with the desired curvature as

required by different bending mechanisms. A standard

undergraduate laboratory HeNe laser was used to monitor

the deflection of the cantilever. In order to promote the

reflection of the laser off the cantilever surface, the free end

of the lever was machined polished to a mirror finish. The

large size of this system made the measurement of each

parameter D, L, CL, f, y, and x possible with high

precision.

4. Results

Fig. 4 shows experimental (symbols) and calculated data

(solid lines) of the PSD displacement versus cantilever

deflection for system 3 described in Table 2. The PSD

signal is given in units of length which describes the

displacement of the incident laser spot on the PSD. Most

cantilever-based instruments use a PSD where a measured

voltage is linearly proportional to the displacement of the

laser on the PSD. Hence the PSD displacement versus

cantilever deflection data as shown here is consistent with

current technology. The two plots 4 (a) and (b) show data

taken where the values of D, L, f and y are fixed and x

varies from 01 to 901. As the data shows, excellent

agreement is obtained between the experimental and

calculated data. It is important to stress that the above

model completely and accurately describes the measured

data without the use of any adjustable or fitted parameters.

The values of D, L, CL, f, y, and x uniquely defines the

cantilever/laser beam detection system. Fig. 4 shows how

the concavity of the data changes as the value of x is

changed (similar results occur if D, L, f and x are fixed and

y is varied). The PSD versus cantilever deflection curve

goes from concave to convex as the PSD angle is changed

from 0.01 to 90.01, respectively. The change in curvature

can easily be rationalized when the geometry of the

reflected laser and the angle of the PSD are taken into

consideration. As the PSD/Cantilever deflection curve goes

ARTICLE IN PRESS

Fig. 3. Schematic representation showing how the PSD plane in 3D is

rotated by an angle d ¼ p/2 ? x into a 2D plane.

0 10203040

Cantilever Deflection (mm)

0

100

200

300

400

PSD Signal (mm)

0

100

200

300

ξ= 0.0°

ξ = 16.1°

ξ = 22.0°

ξ = 39.17°

ξ = 50.16°

ξ = 67.0°

ξ = 90.0°

a

b

Fig. 4. Comparison between experimental (symbols) and calculated data

(solid lines) for the PSD displacement versus cantilever deflection of

system 3 described in Table 2 for different values of PSD angle (a) x ¼

0:01; 16:11; 22:01; 39:171 and (b) x ¼ 50:161; 67:01; 90:01. The macro-

cantilever was forced to deflect by an end-moment. The fixed geometrical

parameters of the system were: D ¼ 382:5 ? 0:5mm, L ¼ 603:7 ? 0:5mm,

CL ¼ 410:0 ? 0:5mm, f ¼ 180.070.51, y ¼ 69.970.51.

Table 2

SystemValue of f

Vector u

1

2

3

f ¼ 01

f ¼ 901

f ¼ 1801

u ¼ ð0; ?1; 0Þ

u ¼ ð?1; 0; 0Þ

u ¼ ð0; 1; 0Þ

L.Y. Beaulieu et al. / Ultramicroscopy 107 (2007) 422–430

425

Page 5

from concave to convex, it passes a region where the

relationship is nearly linear. We have found that for any

geometry D, L, CL, f, y, and bending curvature, it is

possible to find a PSD angle x (or x fixed and y is varied)

such that the linearity of the relationship between the PSD

and the cantilever deflection is maximized.

As an example, the geometry: CL ¼ 375mm, D ¼

350mm, L ¼ 3:0cm, f ¼ 1801 (system 3), and y ¼ 701 is

characteristic of a typical cantilever sensor setup used in

our laboratory. For this geometry, assuming a maximum

cantilever deflection of 40mm, it can be found by least-

squares-fit that the optimized PSD angle is x ¼ 37:251.

Fig. 5a shows a comparison between calculated data (gray

crosses) and a linear fit (solid line) showing the linearity of

the PSD signal. Fig. 5b shows, on a much-reduced scale,

the difference between the optimized linear PSD/cantilever

curve and the straight line fit. At most, a difference of

40nm corresponds, as shown in Fig. 5c to an error of

0.08% of the maximum PSD measurement. In fact,

calculations have shown that regardless of the values of

D, L, f, and y, it is always possible to find a value of x that

maximizes the linearity of the relationship between the

PSD signal and the cantilever deflection.

When attempting to linearize the PSD versus cantilever

deflection relationship, it is important to note that not only

does the value of x depend on the parameters D, L, CL, f,

and y, but it also depends on the maximum cantilever

deflection (range). Fig. 6a) shows how the optimized PSD

angle x varies with the maximum cantilever deflection. The

reason for this dependence stems from the fact that

the relationship between the PSD displacement and the

cantilever deflection is not perfectly linear as shown in Fig.

5b. The PSD displacement/cantilever deflection curve

oscillates, although minutely, about the best-fit line used

to obtain the PSD angle x. Therefore, depending on the

deflection range used in the analysis, the value of x will

differ. However the data shows that the value of x becomes

stationary for small cantilever deflections (o3mm) which is

the range where most cantilever-based experiments occur.

Fig. 6b reveals that w2also decreases as the cantilever

deflection range decreases. This indicates that the accuracy

of the fit increases when smaller cantilever deflections are

considered. However, even for large deflections, the

linearity of the optimized geometry is still adequate as

shown by the small value of w2.

As the cantilever deflects, the position where the laser

beam hits the cantilever also changes. As expected for

system 3 (Table 2), the x-position (see Fig. 1) of the

intersection point of the incident laser beam on the

cantilever (the x-coordinate of the vector W) increases

with cantilever deflection. Our model can be used to

completely describe the position of the laser spot on the

cantilever for any deflection. Fig. 7 shows a plot of the

displacement of the intersection point between the incident

laser beam and the cantilever as measured directly along

the length of the lever versus cantilever deflection. The

graph shows a comparison between measured data (black

ARTICLE IN PRESS

0 1020 3040

Cantilever Defelction (μm)

0

4

8

12

16

PSD

Displacement (mm)

-0.04

0.00

0.04

Difference

(μm)

0

0.08

0.16

Percent

Error

a

b

c

Fig. 5. (a) Calculated data (gray crosses) and linear fit (solid line) for an

optimized setup with CL ¼ 375mm, D ¼ 350mm, L ¼ 3:0cm, f ¼ 1801,

y ¼ 701 and x ¼ 37:251. (b) Difference between the calculated data and

best line of fit. (c) Percent error of the linearity fit.

0510 1520 253035 40

Maximum Cantilever

Deflection (μm)

24

28

32

36

40

PSD Angle

ξ(deg.)

0

0.02

0.04

0.06

0.08

χ2

a

b

Fig. 6. (a) Calculated results showing how for the following geometry,

CL ¼ 375mm, D ¼ 350mm, L ¼ 3:0cm, f ¼ 1801, y ¼ 701, the optimized

angle of the PSD changes with the maximum cantilever deflection. (b)

Calculated results showing how w2increases when the maximum cantilever

deflection increases. This result indicates that the relationship between the

PSD signal and the cantilever deflection for an optimized geometry

becomes increasing linear when small maximum cantilever deflections are

considered.

L.Y. Beaulieu et al. / Ultramicroscopy 107 (2007) 422–430

426

Page 6

circles) as taken with the macro-cantilever and the results

of the model (solid gray line). Clearly the model completely

and accurately describes the position of the laser spot on

the cantilever. Again it is important to emphasize that no

adjustable parameters are used anywhere in this analysis.

For the case of a laser reflecting off a cantilever in a typical

AFM, an end deflection of 10mm would cause the incident

laser to undergo a total displacement of approximately

3mm on the cantilever surface.

Although system 3 (Table 2) is the most commonly used

geometry for cantilever-based instruments, there are other

geometries that can be used to monitor the cantilever

deflection. Fig. 8 shows experimental data (symbols) taken

with the macro-cantilever and calculated curves from the

model of the PSD signal versus cantilever deflection for the

following geometries: f ¼ 01, y ¼ 69:91, L ¼ 603:7mm,

D ¼ 383:5mm, CL ¼ 410mm and x ¼ 9:551; 39:751; and

59:641. As in the previous case the model perfectly

describes the f ¼ 01 system. The main difference between

systems 1 and 3 is in the relationship between the incident

laser beam I and the normal vector Nc. In the case of

system 3, the angle between I and Nc continuously

increases while for system 1 the angle continuously

decreases during cantilever deflections. As a consequence,

system 1 imposes more physical restrictions for it requires

that the PSD and the laser focusers be in close proximity to

each other since the angle of incidence (and reflection)

becomes smaller as the cantilever bends. A second

consequence imposed by physical restriction is that it is

more difficult to obtain a strongly convex relationship

between the PSD signal and the cantilever deflection then

for system 3. However the opposite is true for producing a

concave relationship.

As in the case for f ¼ 1801, it is possible to linearize the

relationship between the PSD displacement and cantilever

deflection for system 1 (f ¼ 01) by optimizing the PSD

angle x (or incident laser angle y). Fig. 9 shows calculated

data for f ¼ 01, y ¼ 701, L ¼ 3cm, D ¼ 325mm, CL ¼

350mm and an optimized PSD angle of x ¼ 13:891. These

values are typical dimensions for actual cantilever sensors.

Fig. 9a shows the calculated PSD displacement (gray

symbols) versus cantilever deflection for the optimized

geometry along with a linear best-fit line (solid line). The

difference between these two curves is shown on a reduced

scale in Fig. 9b and the relative percent error in Fig. 9c.

With a w2¼ 0:04, these values are comparable to those

found for the f ¼ 1801 system and represent an uncer-

tainty well within experimental noise.

Also similar to the f ¼ 1801 system, it is possible to

model the motion of the laser spot on the cantilever as the

cantilever is forced to deflect. For this case however, as the

cantilever deflects the laser spot moves up the length of the

cantilever (closer to the chip). As in the previous case,

excellent agreement was obtained between experimental

and calculated values.

Both systems 1 and 3 (Table 2) are very similar and differ

only by their geometry. Specifically both show similar

behavior and accuracy regardless of the types of cantilever

deflections used. However, system 2 characterized by f ¼

901 (see Fig. 1) is very different than systems 1 and 3

(f ¼ 01 and 1801, respectively) where the length of the

cantilever, the incident and reflected laser are all restricted

to the x–z plane. For the case of f ¼ 901, the incident laser

is parallel to the y–z plane. Because of this, the PSD signal

is no longer a straight line but a curve as shown

schematically in Fig. 10. Notice that for the f ¼ 901

system, the PSD must be rotated so that the active region

of the detector (long axis) is oriented in the PSD2direction.

In contrast, for the f ¼ 01 and 1801 systems, the PSD is

oriented so that the long axis of the PSD is in the PSD1

direction. The above model can be used to simulate the

trace made on the PSD by the reflected laser as the

cantilever is made to deflect. Fig. 11a shows calculated data

of the long axis (PSD2 direction) versus the short axis

ARTICLE IN PRESS

010 20 3040

Cantilever Deflection (mm)

380

384

388

392

396

400

Position on

Cantilever (mm)

Fig. 7. Position of the laser spot on the cantilever versus cantilever

deflection. Dots show experimental data taken with a macro-cantilever

while the solid curve is calculated data from the model discussed within the

text. In this data, the geometry of the macro cantilever was set to

f ¼ 1801, y ¼ 69:91, D ¼ 383:5mm and CL ¼ 410mm.

010 2030 40

Cantilever Deflection (mm)

0

100

200

300

400

PSD Displacement (mm)

ξ = 9.55°

ξ = 39.75°

ξ = 59.64°

Fig. 8. Experimental (symbols) and calculated PSD displacement (solid

curves) versus cantilever deflection for the following geometries: f ¼ 01,

y ¼ 69:91,

9:551; 39:751 and 59.641.

L ¼ 603:7mm,

D ¼ 383:5mm,CL ¼ 410mmand

x ¼

L.Y. Beaulieu et al. / Ultramicroscopy 107 (2007) 422–430

427

Page 7

(-PSD1direction) of the PSD displacement (see Fig. 10) for

the following geometries: f ¼ 901, y ¼ 701, L ¼ 3cm,

D ¼ 325mm, CL ¼ 350mm and x ¼ 10:01; x ¼ 50:01; x ¼

60:01; x ¼ 70:01andx ¼ 80:5041. For these calculations the

cantilever was made to deflect as if subjected to an end

moment to a maximum deflection of 40mm. As shown, the

curvature of the PSD trace increases with increasing values

of x. The same results are obtained for calculations

performed with a point load type deflection. However for

the case of the point load, the curvature at the free end of

the cantilever where the laser is made to reflect is much

smaller then for end moment-type deflections. As a

consequence the curving effects are slightly smaller.

The geometry described by system 2 has to date only

been used in cantilever-based sensor instruments [21]. The

PSD used in most cantilever sensors are only sensitive to

the displacement of the reflected laser along the length

(long axis) of the PSD. Therefore, provided the reflected

laser spot remains within the active area of the PSD

(approx. 10 (long)?2 (wide) mm) the measurement is

unaffected by the curvature of the trace made by the

reflected laser. Taking this under consideration it is

possible to optimize the f ¼ 901 system by linearizing the

long-axis component of the trace made by the reflected

laser versus the cantilever deflection. Fig. 11b shows the

long-axis component of the PSD trace versus cantilever

deflection of the same data shown in Fig. 11a. Although

each curve is close to being linear, the optimized PSD angle

is found to be 80.5041 with a w2¼ 7:23 ? 10?7. Fig. 11c

shows the small differences between the five different

curves shown in Fig. 11b. It is only at large cantilever

deflections such as 40mm that the difference between the

five curves is appreciable. In most cantilever-based experi-

ments the cantilever is generally only made to deflect a few

micrometers. For the case of a system with f ¼ 901,

y ¼ 701, L ¼ 3cm, D ¼ 325mm, CL ¼ 350mm with a

cantilever subjected to a point load at 300mm, the

optimized PSD angle is x ¼ 81:041 with w2¼ 5:55 ? 10?7.

In order to further validate the above model, we show in

Fig. 12 a comparison of the PSD displacement of both the

short- and long-axis components versus cantilever deflec-

tion between experimental measurements (symbols) col-

lected with a macro-cantilever and calculated data (solid

curves) obtained from the model. The geometry of the

ARTICLE IN PRESS

05 10152025 3035 40

Cantilever Deflection (μm)

0

2

4

6

8

10

12

14

PSD

Displacement (mm)

-0.04

-0.02

0.00

0.02

0.04

Difference

(μm)

0

0.04

0.08

Percent

Error

a

b

c

Fig. 9. (a) Calculated data (gray crosses) and linear fit (solid line) of PSD

displacement versus cantilever deflection for the optimized geometry:

f ¼ 01, y ¼ 701, L ¼ 3cm, D ¼ 325mm, CL ¼ 350mm and an optimized

PSD angle of x ¼ 13:881. (b) Difference and (c) relative percent error

between the calculated data and the linear fit shown in (a).

Fig. 10. Schematic representation showing the trace made on the PSD by

the reflected laser for the f ¼ 901 system. In this geometry the PSD must

be rotated so that the active region of the PSD is along the PSD2direction.

0 0.51 1.52

PSD Displacement (mm)

(- PSD1 direction)

0

2

4

6

8

10

12

PSD Displacement (mm)

(PSD2 direction)

010203040

Cantilever Deflection (μm)

ξ = 10.0°

ξ = 50.0°

ξ = 60.0°

ξ = 70.0°

ξ = 80.504°

3638

Cantilever

Deflection (μm)

40 42

11.2

11.6

12

12.4

12.8

PSD

Displacement (mm)

c

ba

Fig. 11. (a) Calculated results showing the trace of the incident laser beam

on the PSD during cantilever deflections for various PSD angles x for the

following geometries: f ¼ 901, y ¼ 701, L ¼ 3cm, D ¼ 325mm, CL ¼

350mm and x ¼ 10:01; 50:01; 60:01; 70:01 and 80.5041. (b) The long axis

component (PSD2direction) of the laser beam trace on the PSD as a

function of cantilever deflection for the same PSD angles shown in panel

(a). (c) Expanded view of the PSD versus cantilever deflection curves

shown in panel (b) with the associated PSD angles x.

L.Y. Beaulieu et al. / Ultramicroscopy 107 (2007) 422–430

428

Page 8

system was as follows: f ¼ 901, y ¼ 69:91, L ¼ 603:7mm,

D ¼ 383:5mm, CL ¼ 410mm and x ¼ 9:551; 49:111 and

69:91. As before, the agreement between the experimental

data and our model is very good. In this case however,

there is some noise in the experimental data, which

originates from the inability to reduce torque effects on

the macro-cantilever when forcing deflections.

In this paper, we have stressed the fact that for any kind

of cantilever deflection and cantilever/laser beam detection

system geometry it is possible to adjust the incident laser

angle or the PSD angle such that the relationship between

the PSD displacement and the cantilever deflection is

nearly linear. It is sometimes desirable that this relationship

not be linear but curved. For example, Figs. 4 a and b show

how a PSD held at an angle x ¼ 0:01 produces a concave

curvature while a PSD held at an angle x ¼ 90:01 produces

a convex curvature. Clearly the concave curvature will

provide more sensitive data at larger deflections while the

convex curvature will provide more sensitive measurements

at smaller cantilever deflections. This can be very useful

when attempting to detect small deflection signal within a

larger measurement range. Also due to system restrictions,

it is not always possible to orient the incident laser angle or

PSD angle in order to linearize the system. However as

long as it is possible to obtain the values of D, L, CL, f, y,

and x (for example, from manufacture specifications), the

above model can be used to obtained the exact relationship

between the cantilever deflection and PSD signal provided

the proper cantilever curvature is taken into consideration.

This last condition is not very stringent because end-

moment, point-load, and/or uniform load type deflections

are induced by very different physical phenomena.

The above model has been incorporated into an excel

spreadsheet and a visual basic program for end-moment

and point load type deflections, respectively. We have also

included a visual basic program to solve for xmaxin both

Eqs. (5) and (9). This supporting material has been made

available for download off our website at: http://www.

physics.mun.ca/beaulieu_lab/papers/cantilever_analysis.htm.

We welcome contact from users that would like to

implement the above model into their instrument or as

part of their data analysis.

5. Conclusion

We have shown by using elementary geometric optics

and vector analysis that it is possible to completely

characterize the cantilever/laser beam deflection system so

as to obtain the exact relationship between the cantilever

deflection and the PSD measurement. We have also shown

that by adjusting either the incident laser angle or the PSD

angle it is possible to tailor the relationship between the

cantilever deflection and PSD displacement so as to make it

more sensitive at different cantilever deflections or nearly

linear to increase its ease of use. Elsewhere we have shown

how the above model can be incorporated with new

equipment design so as to obtain a self-calibrating system

[16]. We believe that the analysis presented here and in

Ref. [16] should be strongly considered when designing the

next generation of cantilever-based sensor instruments or

atomic force microscopes.

Acknowledgments

The authors would like to thank NSERC, CFI, IRIF,

Memorial University, Genome

University for funding through various programs, grants

and fellowships.

Quebec, and McGill

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ARTICLE IN PRESS

05 10 15 2025303540 45

Cantilever Deflection (mm)

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50

100

150

200

250

PSD Displacement (mm)

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obtained from the model described within the text. The geometry of the

system was as follows: f ¼ 901, y ¼ 69:91, L ¼ 603:7mm, D ¼ 383:5mm,

CL ¼ 410mm and x ¼ 9:551; 49:111 and 69.91.

L.Y. Beaulieu et al. / Ultramicroscopy 107 (2007) 422–430

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