Journal of Sensor Technology, 2012, 2, 196-205
http://dx.doi.org/10.4236/jst.2012.24027 Published Online December 2012 (http://www.SciRP.org/journal/jst)
Nonlinear Response of Multi-Segmented Photodetectors
Used for Measurements of Microcantilever
Motion over Large Dynamic Ranges
Asit Kar, Mi c hae l George
Laboratory for Materials and Surface Science, Department of Chemistry,
University of Alabama in Huntsville, Huntsville, USA
Received September 8, 2012; revised October 10, 2012; accepted November 12, 2012
ABSTRACT
The use of multi-segmented Position Sensitive Photodiodes (PSD) to measure microcantilever deflections have been
found to produce nonlinear signal output, especially when the dynamic range is large. The reflected beam of the micro-
cantilever may undergo intensity and shape modifications prior to reaching the PSD. In a multi-microcantilever sensor
system the variation in the size of the individual spots plays an additional role contributing to the nonlinearities of de-
tector output. Irrespective of the range of operation the merits of intensity normalization have been discussed. We show
that the output is proportional to the width of the spot along the split line of the detector. This enables the determination
of the shape of a spot. We show that the microcantilever vibrational spectrum can be obtained just using a single seg-
ment photodetector instead of using multiple segmented PSDs. These concepts will greatly facilitate interpretation of
sensor data acquired from either single or multi-microcantilever experimental platforms.
Keywords: Photodetectors; Microcantilever Sensors; Signal Artifacts; Laser Spot Shape; AFM
1. Introduction
The use of a reflected laser spot from the microcantilever
to a PSD [1,2] to measure microcantilever motion has
been one of the established techniques for the study of
chemical [3-9], biological [10-13] and physical [14-18]
processes for scanning force microscopy [19,20] as well
as microcantilever sensors. In recent years, the necessity
of the use of a multi-microcantilever readout system as a
sensing device has been realized. [21] A number of re-
search teams has come up with possible solutions to de-
velop a sensor device containing multiple microcanti-
levers working on the principle of optical beam deflec-
tion [3,22,23]. In usual scanning force microscopes the
deflection of the cantilever beam is small and is limited
within a few tens of nanometers depending on the rough-
ness of the sample surface. While in microcantilever based
sensor experiments, the cantilever can deflect extending
from a few hundred nanometers [24-27] to several mi-
crometers. [28] With the intrinsic sensitivity of micro-
cantilever chemical and biological sensors, it is vital to
maintain the linearity of the differential deflection of
microcantilever systems, especially where identical mi-
crocantilevers are employed for sensor and reference sig-
nals [21].
Calibration or the position sensitivity of a two segment
PSD is changed if the size and shape of the spot are al-
tered depending on optical alignment, shape of the free
end of a microcantilever or any random factor. For any
shape other than a rectangle, the signal output varies with
the geometry of the spot and its location on the detector.
With a few nanometers of deflection, an oval shaped spot
can approximately behave like a rectangle thus producing
a near linear output signal as is the case for SFM. This is
usually not the case for microcantilever based sensor
experiments where the spot can move much more due to
considerable changes in surface stress and free energy
during the sensing event. Therefore, when calibrating the
detector, the full range of spot displacement must be
measured and recorded. Lack of prior knowledge about
the nature of spot shapes and concurrent photodiode out-
put may lead to misinterpretation of the readout signals.
The effect of noise in an optical deflection technique
with respect to the minimum detectable displacement has
been discussed before [29-31]. Gustafsson and Clarke [32]
have demonstrated that appropriate design of cantilevers
results in lower shot-noise improving performance. Schäf-
fer [33] has discussed the aspects of linearity and dyna-
mic range at large cantilever deflections. It has been shown
that a detector made of a linear arrangement of multiple
photodiode segments [34] can provide a large dynamical
range. D’Costa and Hoh [35] have experienced the effect
C
opyright © 2012 SciRes. JST

A. KAR, M. GEORGE 197
of spot-shape on the sensitivity. In all previous studies,
little or no attention has been paid to the effects of the
size, shape and intensity of the laser spot on the linear
behavior of the detector in larger dynamic deflection range.
Here we focus on the issues of nonlinear sensitivities of
static and dynamic deflections in a microcantilever sen-
sor system providing an overall visualization of their ef-
fect on linearity.
2. Anomalous Behavior in Multi-Segment
Photodetector Readout
Either in a single microcantilever or in a multi-micro-
cantilever platform, we have frequently noticed that the
bending signal goes beyond the detection range of the
detector. The signal saturates after sometime when the
stimulus to the cantilever becomes large and prolonged.
Often this saturation is not caused by the maximum
change due to surface stress.
Figure 1 demonstrates the result of an experiment in
an unfavorable condition where the deflections of six
identical gold coated silicon microcantilevers are simul-
taneously read out in six position sensitive photodetec-
tors. In the experiment, six gold-coated silicon cantile-
vers are placed in a small fluid cell with an effective in-
ternal volume of approximately 90 µl. A constant flow
rate of 30 µl/min of distilled water has been maintained
in the chamber throughout the time period of experiment.
The holder has been heated at a constant ramping rate of
1 ˚C/min. With increase in temperature the microcantile-
vers bend due to bimetallic effect which should result in
a linear deflection signal. As shown in Figure 1(a), the
temperature increases with constant slope. The responses
of six detectors are supposed to be same apparently being
in the same experimental condition. But they are quite
different as shown in Figure 1(b). Though the deflec-
tions start from different offset, they are asymmetric, dis-
similar, and not coincident around the position of zero-
deflection. At first, the possible effects of annealing pro-
cesses of the polycrystalline gold film were considered.
However, the films were annealed to a higher tempera-
ture after deposition. As shown in Figure 1(a), the in-
crease in temperature is steady with no fluctuation, and
there is no cantilever specific chemical or physical acti-
vity around the microcantilevers, therefore, the reason of
distortion may be attributed to some properties of the shape
of the microcantilever tip or of the optics used to project
the reflected beam from the microcantilever to the PSD.
The features of the signal output in Figure 1(b) re-
semble the changes in slope that occur for a sensing event
for chemical sensors. If the reasons are overlooked, they
can suggest positive responses for some experiment where
certain specific chemical or physical activities are excepted.
As seen in the Figure, the differential signal of any two
Figure 1. Linear increase in temperature of six identical
sample microcantilevers; (a) produces six different nonlin-
ear deflection characteristics; (b) in an unclean experimen-
tal condition.
curves can produce bumps and troughs completely un-
correlated to any chemical activity. It can be critical for
experiments where multiple microcantilevers are used and
particularly so when only one sensor and one reference mi-
crocantilever are used. In the following discussion, we will
demonstrate how the size and shape of the reflected spot
can influence the PSD output signal of a microcantilever.
3. Multi-Segment Photodetector Signal
A position sensitive photo detector counts total illumi-
nated area; it does not recognize the exact shape of the
spot. It is possible to draw an equivalent illuminated sur-
face area observed by a detector. By restricting our dis-
cussion only to the vertical deflection of a microcantile-
ver, the equivalent area of the illuminated surface should
be confined in a single closed contour with no opening
inside and should be symmetric about a vertical axis. The
behavior of a detector concerning the actual and equiva-
lent shape of a spot can be understood from the illustra-
tions exemplified in Figure 2.
An actual irregular shape of a spot is treated by the
detector in the same way as a virtually regular equivalent
shape is (Figure 2(a)). If the illuminated area looks rag-
ged and torn, all the disconnected spots can horizontally
be merged together and aligned symmetrically along the
vertical axis. Maximum height does not vary but the
width may change. So if a couple of spots are both hori-

A. KAR, M. GEORGE
198
zontally and vertically disconnected, the detector verti-
cally treats them separate as can be noticed in Figure
2(b). Horizontally separate spots can be coalesced toge-
ther to form a symmetric structure. Accordingly even if a
spot does not produce a mirror image across the diode
split line, but its width follows some kind of regularity
along the vertical direction, it is treated like a regular struc-
ture as shown Figure 2(c).
4. Photo Detector Response with Shape of a
Reflected Spot
As discussed in section III, the measurement of the spot
shape can be assumed to be symmetrical about a vertical
axis. It can be shown that the position sensitivity of a
PSD for a specific beam spot is related to the shape of
the spot. This can be proved from Figure 3. The Figure
depicts an irregular boundary of a laser spot on the XY
plane of the detector. The split line of the detector is as-
sumed to be coincident with the abscissa and the spot
moves along the Y-direction. The total height of the spot
is subdivided in N equal segments of interval Δy, where
N is very high and Δy is extremely small. The width of
the spot at any vertical point is represented by xm, where
m is the segment number starting from top.
Figure 2. Transverse equivalence of illuminated area of
coverage of a beam spot to a split detector: any arbitrary
shape of spot (left column) can be treated equivalent to a
structure with the same area of coverage distributed sym-
Figure 3. Contour map of a beam spot of arbitrary shap
To an approximation assuming a rectangular distribu-
tio
y
e
projected on a PSD draw n in XY plane. The split line of the
detector is assumed to be coincide nt w ith absc issa.
n of laser intensity (constant photon density p) on the
detector surface, the difference signal (D) at any position
m is proportional to the product of the photon density and
the difference of the top and bottom areas:
mN

11
mnn
nnm
Dpx x







(1)
Similarly, the signal at (m + 1) is
y

1mN

1
12
mn n
nnm
Dpx x







So the change in detector signals
11
2
mm
DD Dm px y


  (2)
Hence, the position sensitivity
1
1
2m
m
D

px
y




 (3)
Again the total area of the spot can be given by
1
n
n
N
A
xy




The sum signal (S) of a photodetector is the summa-
tion of the signals from the top and bottom parts of the
split detector and is constant so long as the spot is within
the active area of the detector. This is proportional to the
total light intensity on the detector which in turn depends
on the photon density and the total illuminated area of the
detector. So the sum signal can be expressed as:
SpA

(4)
metrically along a vertical axis (right column).

A. KAR, M. GEORGE 199
The difference signal of a P
by
SD is usually normalized
the sum signal. If D is sum-normalized, Equation (1)
takes the form,

Sum Norm11
1mN
mn
nnm
n
xxy
A




 (5)
So if we consider the sensitivity of sum-no
di
D
rmalized
fference signal, Equation (3) becomes,
1
1Sum Norm
2Dm
m
x
yA






 (6)
Equations (3) and (6) imply that, at any position of the
sp
determining the shape of
r Spot
a simplified form of an elliptical or




ot the sensitivity of the detector is proportional to the
width of the spot along the split line. So if the sensitivity
profile of a PSD for an unknown beam is known, one can
approximate the possible equivalent shape of the beam
spot. The sensitivity curve is directly obtained from the
positional derivative of the bending profile.
5. Shape Determination
To have an intuitive idea about
a spot, in the following we consider three simple fre-
quently encountered ideal geometric structures presum-
ably produced by the similar shaped free ends of micro-
cantilevers.
5.1. Circula
A circular structure is
oval spot shape where both width and slope of the boun-
dary-line of the spot change along vertical direction. This
is presented in Figure 4. Figure 4(a) shows a circular spot
of radius r. Vertical displacement of the spot is repre-
sented by the variable y; it signifies the coordinate of the
center C of the circle with respect to an origin on the split
line OX. The difference signal of the circle can be ex-
pressed by:
2
221
π2cos1
yy
Dpr ryr
rr



  

 

 

 



(7)
A graphical presentation of the above equation is shown
in Figure 4(b) in arbitrary units assuming the radius of
the circle to be 1 unit. From top to bottom total dis-
placement of the circle is 2 units. Data have been plotted
with the displacement axis horizontally aligned. A slightly
S-shape of the deflection curve is characteristic of any
oval shaped illumination spot. Figure 4(c) shows the nu-
merical derivative of the deflection curve in Figure 4(b).
This gives the PSD sensitivity characteristic of a circular
spot. It is clear from this Figure that the position sensiti-
vity is never constant for an elliptical spot. It has a mini-
mum at the top and bottom and goes to a maximum along
the diameter. The extremes correspond to the minimum
and maximum width of the circle. Dif- ferentiating Equa-
tion (7) with respect to y and simplifying give,
22
2
1d 1
d
Dy
yr




a

 (8)
where a is a constant. This is the equatio
the curve of Figure 4(c) shows. Substituting y with x (the
n of an ellipse as
width of the circle at any point along vertical direction)
gives,
dD2
dpx y (9)
Equation (9) is exactly the sam
rived for a generalized shape.
s half of the original
sh
iangle as shown in Figure 5(a).
of the spot changes along ver-
e as Equation (3) de-
The semi-oval shaped curve is an indication of the
shape of the spot. But it appears a
ape. The reason has been explained before. The detec-
tor does not recognize the exact shape but an equivalent
shape. To get back the original shape it is necessary to
redistribute the sensitivity values symmetrically along the
displacement axis. As the sensitivity values are not ab-
solute, the same task can be performed by taking a mirror
image of the profile and viewing both together. Finally it
is necessary to rotate the whole image by –90˚ to have
the displacement axis vertical and the original orientation
of the spot. The operations yield an ellipse instead of a
circle due to the unknown proportionality constant.
5.2. Triangular Spot
We consider a bilateral tr
In this example the width
tical direction but the slope of the tangent at any point on
the boundary remains constant.
Figure 4. Spot shape determi nation from PSD out put for; (a)
Circular spot; (b) Deflection characteristic and; (c) Sensiti-
vity profile. Sensitivity curve in conjunction with its mirror
reflection produces an elliptical shape imitating the spot
shape. Upward movement of spot in (a) corresponds to a right-
ward movement in (b).

A. KAR, M. GEORGE
200
In the Figure, b is the base and h is the height of the
triangle.
The difference signal can be calculated as the triangle
moves from the bottom of the split line OX to the top by
observing the displacement y of the center C (height wise)
of the triangle. The signal in the Figure can be given by,
21
22
bh
Dpy hb
h









(10)
The equation describes a parabola. It i
ure 5(b) assuming both height and base of th
be
In a rectangular shape (Figure 6(a)) the width and the
e along the vertical direction
s plotted in Fig-
e triangle to
2 units. The nature of the deflection is quite different
from that of a circular spot. The corresponding sensitivity
curve is shown in Figure 5(c). It is linear as is obvious
from Equation (10). The curve shows that the sensitivity
goes to a minimum at the apex (top corner) of the train-
gle where the width is the smallest and a maximum at the
base of the triangle where the width is the largest. It also
shows that the sensitivity is proportional to the width.
Differentiation of Equation (10) with y and replacing y
with x give the same expression as Equation (9). As de-
scribed for a circular spot, the curve together with its
mirror image forms a bilateral triangular structure. When
this is rotated clockwise by an angle of 90˚ it reproduces
a shape similar to the original spot.
5.3. Rectangular Spot
slope of the boundary lin
remain unaltered. Considering the vertical movement of
the center C of the rectangle across the split line OX, the
difference signal is given by,
2Dpxy (11)
Figure 5. A triangular spot shape. (a) Produces a deflection
curve like; (b) The position sensitivity profile; (c) Togethe
with its mirror image indicates the shape of the origina
The deflection characteristic of a microcantilever mea-
r
According to Equation (1), the difference signal produced
photon density. If for any
r
l
beam spot.
The bending profile of the rectangular spot is linear
(Figure 6(b)). As the width x of a rectangle is constant,
the spot yields a constant PSD output as shown in Figure
6(c). Differentiation of Equation (11) also gives the same
expression as Equation (9). The sensitivity curve coupled
with its mirror image and rotated by –90˚, reproduces a
rectangular structure.
From the above three examples it is clear how the cur-
vature of the edge of a spot or, specifically, the width of
the spot at the split line controls the PSD output signal.
In other words it is possible to determine the shape of a
spot from the output signal characteristics of the PSD.
6. Microcantilever Bending Characteristics
sured with a PSD depends on nature of the reflected lase
beam from the back of microcantilever. Assuming an
ideal situation of a constant bending rate of microcanti-
lever sensors, the deflection profiles can be non-uniform
due to the variation in certain properties of the beam spot
on the detector. The situations will be explained from the
concepts discussed above.
6.1. Intensity Dependence
by a spot is proportional to the
reason the spot intensity fluctuates, the difference signal
also fluctuates. The problem is avoided by the normali-
zation process where the difference is divided by the sum
of top and bottom signals making it independent of the
incident photon density as in Equation (4). In an un-
normalized multi-detector system, if the spots are other-
wise identical, the variation in light intensity between
different spots produces different bending profiles and
each profile may have some waviness based on the
Figure 6. (a) A rectangular spot; (b) Its deflection profile.
The sensitivity profile; (c) Coupled with its mirror imae
produces a shape similar to (a). g

A. KAR, M. GEORGE 201
frequency of intensity fluctuation. Intensity or sum nor-
malization can also eliminate the problem.
6.2. Size Dependence
In a multi-detector multi-microcantilever system, the same
free ends should amount bending of all microcantilever
produce the same linear displacements of the spots on the
detectors. If spot sizes differ, the bending signals of the
spots saturate at different times and at different values of
deflection. The bending profiles show different slopes due
to their difference in illumination area. The effect of size
variation is counter to detector calibration unless each
detector is individually calibrated. Ideally the deflections
and the time of saturation of identical cantilevers should
be the same. In an experiment where various chemically
sensitized microcantilevers are present in the same sensor
system, an assumption of equal spot size and shape may
create the impression that the different output signals are
in response to chemical sensor processes when in actua-
lity they are due to variations in the reflected spots.
In Figure 7, PSD signal variations from three circular
spots of different diameters are illustrated. The difference
in height of the spots (Figure 7(a)) leads to a different
saturation time. This is apparent in both the unnorma-
lized (Figure 7(b)) and normalized (Figure 7(d)) PSD
output features. The shortest spot saturates first. Deflec-
tion of the cantilever becomes constant when the spot to-
tally crosses the split line of the detector. The spots de-
monstrate different degrees of position sensitivity (Figure
7(c)) due to the variation of slopes of the bending curves.
Intensity normalization forces the total deflection to be
confined within ±1 irrespective of the size of the spots as
shown in Figure 7(d). Though, spots of different size
have different deflections, they appear to have the same
deflections after the biggest spot saturates. Again their
slopes remain different because of their difference in the
area of illumination. In this case spots of smaller diame-
ters go to saturation faster than the larger ones. Thus it
exaggerates the deflection and sensitivity of the smaller
spots where the smaller spots appear to be more sensitive
than the larger ones (Figure 7(e)). Intensity normalization
now also turns out to be normalization of area and hence
the total deflection, as the sum signal contains both the
terms of photon density and area as in Equation (4).
To eliminate the discrepancy, the slopes of the bending
profiles can be normalized to the slope of a calibrated
cantilever. Figure 7(f) shows a set of normalized curves
where the curves in Figure 7(d) have been normalized
by their slope at zero position. Slope normalization can
make the traces coincident to some extent depending upon
the shape of a spot. The deviation from coincidence at a
certain displacement is proportional to the slope of the
spot boundary with respect to the vertical axis. Mutual
differences between two slope-normalized curves have
Figure 7. Spot size dependence of PSD signals: dynamical
range and sensitivity both change with the size of a spot. (a)
Circular spots of different diameters; (b) Unnormalized d
lope corrected traces of circular spots of different radii
er is affected by the
een thoroughly discussed in
n the concepts of shape de-
e-
flection [D] signals of three spots; (c) Corresponding posi-
tion sensitivity curves [dD/dy]; (d) Sum normalized deflec-
tion signals [ND]; (e) Sensitivity curves [d(ND)/dy] of sum
normalized deflection signals; (f) Slope normalized deflec-
tions [ND/S0] of the curves in (d); (g) Mutual difference of
slope normalized curves [DSNC] in (f) within the range of
smallest spot height.
been shown in Figure 7(g) within the range of displace-
ment of the height of the smallest circle. It shows that
s
are never coincident at any other point except the point
of correction. But based on the above idea the traces of
rectangular spots of different sizes can be made perfectly
coincident up to the smallest height.
6.3. Shape Dependence
How the bending profile of a cantilev
shape of a beam spot has b
sections VI and V. Based o
termination developed in section V, the equivalent shapes
of the spots in the experiment corresponding to Figure 1

A. KAR, M. GEORGE
202
have been calculated from their respective sum-norma-
lized deflection characteristics. Assuming a constant rate
of bending of a microcantilever, the vertical displace-
ment has alternatively been considered as proportional to
time. The spot shapes, shown in Figure 8, appear dis-
torted and non-uniform. They deviate considerably from
the expectation of identical uniform size and shape. Oc-
currence of the top and bottom tails of the spots is due to
non-rectangular distribution of light intensity on the de-
tector. The gap between the left and right parts of the
curves is arbitrary. So the constrictions in Figure 8(II)
and Figure 8(III) may mean the spots are broken. The
example reveals the extreme distortions of the spots that
can occur for real experimental conditions.
Though the nature of shape dependence mentioned by
D’Costa and Hoh [32] agrees with our calculations, the
linear nature of the variation of optical lever sensitivity
w
odes
n experiment the laser
sp
odes and har-
nal i.e. how much light is
e number of peaks in-
ith photodiode shift voltage in their work appears am-
biguous. It shows that the sensitivity is a minimum
around the split line of the detector where it usually be-
comes a maximum. According to the sensitivity curve,
the spot shape should have exponential boundaries.
7. Resonance Spectrum of a Microcantilever
A microcantilever always resonates at its natural m
of frequencies induced by ambient thermal energy. At
constant ambient temperature and pressure it can be as-
sumed to vibrate at the same amplitude at any bending
position. The resonance of the microcantilever is reflec-
ted in the fluctuation of the difference signal (Equations
(1) and (4)). With higher values of signal fluctuation (Equ-
ation (2)) the quality of the thermal resonance improves.
The quality of a spectrum is understood by the appear-
ance of intensified characteristic peaks of vibration with
low background noise. The appearance or disappearance
of spectral features depend on the reflected beam inten-
sity or sum signal, the width and location of the spot on
the detector, the position of the point of reflection of the
incident beam on the cantilever, the area of cantilever
free end etc. These properties are independent of the elec-
tronic processing condition [33].
The dependence of the quality of a resonance spectrum
on the shape and location of the spot on the detector has
been illustrated in Figure 9. In a
ot on the photodetector has been slowly moved from
the bottom to the top of the PSD using a vertical motion
micrometer. With movement, the signals from the top (A)
and the bottom (B) part of the split detector, and their
difference (D) and sum (S) signals have been recorded
(Figure 9(a)). The intensity of the fundamental mode
and the number of characteristic peaks in the thermal
vibrational spectrum of the microcantilever have also
been recorded as shown in Figure 9(b), where nonlinear
characteristic can easily be realized. In the Figure, the
deflection signal (D) is sum-normalized (Equation (4))
but the spectrum has been simultaneously acquired from
its unnormalized phase (Equation (1)). The arrows and
arrow-heads on the D-signal indicate different points of
interest as the spot moves along the detector. The left
most arrow head corresponds to a location of the spot
where it has approximately entered at the bottom of the
detector by half of its total area. The relative values of A,
B and S also justify the position. It can be noticed that
around this position, the peak intensities and the number
of observed peaks go to maxima. The same is true when
the spot goes out by half of its area (right most arrow-
head) at the top edge of the detector. When the spot
moves to a position just beneath or just above the center
(split line) of the detector, corresponding to the left and
right arrows respectively, the intensity and number of
peaks go to a minimum. The maxima dominate again
when the spot is at the middle of the detector (central
arrow head). Inset in Figure 9(a) shows that the spot is
oval shaped as derived from the D-signal.
7.1. Intensity Dependence
It has been observed that higher order m
monics depend on the sum sig
being reflected to the detector. Th
creases with the increase in sum signal. Higher beam
intensity produces a higher sum signal and also a higher
fluctuation in the difference signal. From Equation (2) it
can be noted that the change in the difference signal is
proportional to the photon density and the elementary
area of the spot. So the increase in sum signal can be
related to either of these characteristics or both. Assum-
ing constant laser intensity, the photon density depends
on the reflectivity of the surface of the microcantilever.
Hence a spectrum obtained from a gold-coated micro-
cantilever gives a better signal than an uncoated one. The
illuminated area depends on the focusing of the laser
beam on the free end of microcantilever. The factors
Figure 8. Possible equivalent spot shapes of Figure 1, as
observed by six detectors, after the laser beam is reflected
off the free ends of six microcantilevers.

A. KAR, M. GEORGE 203
Figure 9. Characteristic features of the two segment PSD of
an optical head. (a) PSD output signals—A and B: signals
from top and bottom parts respectively, S = A + B, D = α(
n on the microcantilever, the available
Figure 9(b) indicates that the detector has been able to
the cantilever even when the
of a spectrum depends on the value of
th
s been shown that the linear
eristics of the PSD output, whether
al, are distorted with the variation in
rted by the NASA
A
– B)/S, α is an amplification factor. Inset shows the derived
shape of the spot on the detector; neglecting the tails it has
an oval shape; (b) Variation of the intensity of the first
resonance mode and the number of characteristic peaks of a
microcantilever with change in position of the spot on the
detector.
result in optimum illumination are the location of the
point of reflectio
area in the proximity of the point of reflection, at least to
cover the cross sectional area of the beam, and the posi-
tion of the detector to capture the whole reflected spot.
Normalization makes the spectrum independent of the
light intensity (Equation (5)) which results in reduced am-
plitude resolution and poor spectrum quality.
7.2. Shape Dependence
characterize the vibration of
spot is positioned only at the top or bottom part of the
detector, where either A or B signal is totally absent. There
the fluctuation of difference signal means the fluctuation
of either top or bottom signal with half of the amplitude
of Equation (2). The same fluctuating signal now appears
as the sum signal. So if the difference signal is norma-
lized, one cannot observe any vibrational peaks in its spec-
trum but can see the same in the spectrum from S-signal.
However, with an unnormalized D-signal the same vibra-
tional information can be found in S as in D. At the edges
the difference signal itself does not make any sense, but
the change in difference signal has significant importance.
If one only intends to determine the frequency of vibra-
tion of a microcantilever, it is not necessary to use a split
detector. A monolithic photodetector can function as a
vibration detector. For best performance, the center of the
spot should coincide with either the top or the bottom
edge of the detector. This helps reduce some circuit com-
plexity and cost.
Equation 2 implies that the change in the difference
signal is proportional to the width of the spot. The am-
plitude resolution
is fluctuation of the D-segment signal. Hence a wide
strip of laser spot produces a better spectrum than a nar-
row strip. When an oval shaped or a circular spot moves
transversely with the bending of the microcantilever, the
width of the spot changes continuously at the split line of
the detector. Accordingly the nature of the spectrum also
changes giving rise to nonlinearity. This is clearly de-
monstrated in Figure 9(b). The spectrum is rich (at maxi-
ma) along the central width or diameter. But with con-
tinuous unidirectional displacement, as the width de-
creases, the fluctuation amplitude also diminishes. This is
true whether the spot is on the split line or on the top or
bottom edge of the detector. When the spot is just above
or below the split line of the detector, the width of the
spot becomes zero and no spectral features are observed
as implied by the minima in the Figure. If the vertical
length of a spot is less than half of the vertical length of
the detector, spectral features do not appear until the spot
touches an edge of the detector. A rectangular spot should
not show any minimum because the width does not vary;
the spectral features are expected to remain almost un-
changed and hence linear.
8. Conclusion
In the above discussion it ha
deflection charact
static or vibration
geometry of the laser spot. While some features of the
PSD output signal are independent of the range of opera-
tion, the effect of the spot size and geometry are pro-
nounced for a large dynamical range of microcantilever
deflection. The influence though may not be appreciable
for small operational ranges. The effect of intensity vari-
ation can be reduced, however, the geometry dependence
is hard to eliminate for a large range of operation. The
use of array detectors instead of two-segment detectors
may be a possible solution [30,31].
9. Acknowledgements
This research was financially suppo

A. KAR, M. GEORGE
204
EPSCOR project contract number NCC5-580. We thank
[1] B. Culshaw, “Photodetectors and Photodetection,”
sors and Actua986, pp. 263-285.
doi:10.1016/02
Ashwini Kadam for coating the microcantilevers with
gold. We also sincerely acknowledge the helpful discus-
sion with Greg Nordin and Thomas Thundat during con-
struction of a microcantilever array sensor system.
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