Nonlinear Response of Multi-Segmented Photodetectors Used for Measurements of Microcantilever Motion over Large Dynamic Ranges

doi:10.4236/jst.2012.24027

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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

Email: [email protected]

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

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

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:



mnn

nnm

Dpx x















(1)

Similarly, the signal at (m + 1) is



1mN



nnm

Dpx x















So the change in detector signals

DD Dm

px y





  (2)

Hence, the position sensitivity



px

y









 (3)

Again the total area of the spot can be given by







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

SD is usually normalized

the sum signal. If D is sum-normalized, Equation (1)

takes the form,



Sum Norm11

1mN

nnm

xxy

A









 (5)

So if we consider the sensitivity of sum-no

rmalized

fference signal, Equation (3) becomes,

1Sum Norm

2Dm











 (6)

Equations (3) and (6) imply that, at any position of the

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:

221

π2cos1

Dpr ryr









 



 



 



 







(7)

A graphical presentation of the above equation is shown

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,

1d 1







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

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,

Dpy hb



















(10)

The equation describes a parabola. It i

ure 5(b) assuming both height and base of th

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-

According to Equation (1), the difference signal produced

photon density. If for any

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-

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

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

odes

n experiment the laser

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

s been shown that the linear

eristics of the PSD output, whether

al, are distorted with the variation in

rted by the NASA

– 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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