Inverse Estimation of the Tapered Probe –Sample Shear Force
of Scanning Near-Field Optical Microscope
Haw-Long Lee
Department of Mechanical Engineering
Kun Shan University
Tainan 710-03, Taiwan
Republic of China
Abstract
In this paper, the conjugate gradient method of minimization with an adjoint equation is
successfully applieded to solve the inverse problem in estimating the shear force between the
tapered probe and sample during the scanning process of scanning near-field optical
microscope (SNOM). While knowing the available deflection at the tapered probe tip, the
determination of the interaction shear force is regarded as an inverse vibration problem. In the
estimating processes, no prior information on the functional form of the unknown quantity is
required. The accuracy of the inverse analysis is examined by using the simulated exact and
inexact measurements of deflection at the tapered probe tip. Numerical results show that good
estimations on the interaction shear force can be obtained for all the test cases considered in
this study.
Keywords: Scanning near-field optical microscope; Inverse vibration problem; Conjugate
gradient method
1
1. Introduction
The atomic force microscope (AFM) can yield the surface topography of both
conductive and insulating samples on nanometer scale [1-4]. Not only the topography
image of the sample surface but also its optical image can be yielded by using a
scanning near-field microscope (SNOM) with an optical fiber probe. When the probe
scans the sample surface, it allows simultaneous measurement of the topography image and
optical transmission of the surface at high resolution [5-7]. According to the direction of
vibration of the oscillator, the SNOM has three vibration modes, which include one axial and
two flexural directions. Each mode results in the optical fiber probe to move in different
direction and to induce the nonlinear interaction forces between the optical fiber probe and
the sample. Generally, the interaction shear forces between the optical fiber probe and the
sample surface are difficult to measure directly, but they can greatly influence the sensitivity
of the probe. Therefore, the high-resolution requirement of the SNOM optical topography
image benefits the investigation of the interaction shear forces.
In this paper, the conjugated gradient method [8-12] was used to estimate the interaction
shear force. The advantage of the conjugated gradient method is that an iterative
regularization is implicitly built in the computational procedures. The method can quickly
obtain the target’s function. It is very powerful and has been used to solve the function
estimation problems by many researchers [13-16].
2. Analysis
The schematic diagram of an SNOM with an optical fiber probe cantilevered at one end
is shown in Fig. 1. When the SNOM scans the sample’s surface, the interaction shear force,
acting on a laterally oscillating tapered probe tip and the sample surface, is induced. The
time-varying shear force depends on the rigidity of the probe and the surface. It can be
modeled as the shear force on the probe. When the shear force is unknown, the system of the
2
probe can be considered as an inverse vibration problem with the shear force as boundary
condition, which is time-dependent. In this paper, the conjugate gradient method, a function
estimation approach, was adopted to solve the inverse problem. The calculation process of the
method includes the following problems: the direct problem, the sensitivity problem, and the
adjoint problem, which are discussed in the following sections.
2.1 Direct problem
The optical fiber probe includes a section of uniform cylinder with radius R, the tapered
angleα, and a sharpened tip with radius r. In order to increase the optical reflectivity, the
probe is coated with a thin layer of aluminum or aurum. To simplify the problem, the coated
layer of the optical fiber is not taken into account in this paper.
The probe experiences flexural vibration during contact with the sample. The flexural
vibration of a cantilever can be depicted by a partial differential equation with its flexural
deflection y ( x* , t * ) depending on the spatial coordinate x* and the time t * . For the
tapered probe, the bending stiffness E I ( x* ) and mass density  A( x* ) are functions of
position x* along the probe. In addition, the unknown shear force F (t * ) acting on the
laterally oscillating tapered probe tip is a function of time t * .The linear differential
equation, and the corresponding boundary and initial conditions in dimensionless forms for
the flexural vibration of the probe are [17].
2
 2 y ( x, t )
 2 y ( x, t )
[
I
(
x
)
]

A
(
x
)
0
x 2
x 2
t 2
(1a)
y (0, t )  0
(1b)
 y (0, t )
0
x
(1c)
3
 2 y (1, t )
0
 x2
(1d)

 2 y( x, t )
I ( x)
]  F (t )
x
x 2
x 1
(1e)
[
y ( x, 0) 
 y ( x, 0)
0
t
(1f)
where y ( x, t ) is the dimensionless flexural deflection, F (t ) is the dimensionless shear
force acting on the tapered probe tip, x is the dimensionless spatial coordinate, and t is
the dimensionless time. I ( x ) and A( x ) are the dimensionless area momentum of
inertia and cross-sectional area of the tapered optical fiber probe, respectively. The
dimensionless variables are defined as
°( x* ) / A , I ( x)  %
x  x* / L, y ( x, t )  %
y ( x* , t * ) / L, A( x)  A
I ( x* ) / I 0 ,
0
°(t * )
F
t  t * /  A0 L4 / EI 0 , F (t ) 
EI 0 / L2
where L is the tapered optical fiber length. I 0 and A0 are the area momentum of
inertia and the cross-sectional area of the uniform section of the tapered optical fiber
probe, respectively. E and  are the elasticity modulus and the density of the
optical fiber probe, respectively. The direct problem considered here is concerned with
the determination of the flexural deflection y ( x, t ) when the shear force F (t ) , probe
material properties, and initial and boundary conditions are known. The Rayleigh-Ritz
Method [18-20] can be applied to solve the direct problem of Eq. (1).
For the inverse problem, the shear force F (t ) is regarded as being unknown,
while everything else in Eq. (1) is known. In addition, deflection readings taken at x =
1 are considered available. The objective of the inverse analysis is to predict the
unknown time-dependent shear force F (t ) from knowledge of these deflection
readings.
Let the measured deflection at position x = 1 and time t be denoted by Y(1, t).
4
Then this inverse problem can be stated as follows: by utilizing the above mentioned
measured deflection data Y(1, t), estimate the unknown F (t ) over the specified time.
The solution of the present inverse problem is to be obtained in such a way that the
following functional is minimizes:
tf
J [ F (t )]   [ y (1, t )  Y (1, t )] dt
2
0
(2)
where t f is the final time of the measurement and y(1, t) is the estimated (or
computed) deflection at the measurement location x = 1. The estimated deflection y(1,
t) is determined from the solution of the direct problem given previously by using an
estimated F K (t ) for the exact F(t). Here F K (t ) denotes the estimated quantities at
the Kth iteration.
2.2 Sensitivity problem
In order to derive the sensitivity problem, it is assumed that when F (t )
undergoes a variation  F (t ) , the deflection y ( x, t ) changes by a corresponding
amount  y . Then replacing in the direct problem F with F   F and y with
y   y , subtracting from the resulting expressions the direct problem, and neglecting
the second-order terms, the sensitivity problem is defined as follows:
[
2
 2 y ( x, t )
 2 y ( x, t )
[
I
(
x
)
]

A
(
x
)
0
x 2
x 2
t 2
(3a)
 y (0, t )  0
(3b)
  y (0, t )
0
x
(3c)
 2  y (1, t )
0
 x2
(3d)

 2 y( x, t )
I ( x)
]  F (t )
x
x 2
x 1
(3e)
5
 y ( x, 0) 
  y ( x, 0)
0
t
(3f)
2.3 Adjoint problem and gradient equation
In order to obtain the adjoint problem, Eq. (1) is multiplied by the Lagrange
multiplier (or adjoint function)  ( x, t ) , and the resulting expression is integrated over
the time and correspondent space domains. Then the result is added to the right-hand
side of Eq. (2) and the following form is obtained:
tf
J [ F (t )]   [ y (1, t )  Y (1, t )]2 dt
0

tf
0

1
0
 ( x, t ){
2
 2 y ( x, t )
 2 y ( x, t )
[
I
(
x
)
]

A
(
x
)
}dxdt
x 2
x 2
 2t
(4)
The variation of  J is obtained by perturbing y ( x, t ) with  y ( x, t ) in Eq. (4)
and then by subtracting it from Eq. (4), the following is obtained:
tf
J [ F (t )]  2 [ y (1, t )  Y (1, t )]y (1, t )dt
0

tf
0
2
 2 y ( x, t )
 2 y ( x, t )

(
x
,
t
){
[
I
(
x
)
]

A
(
x
)
}dxdt
0
x 2
x 2
 2t
1
(5)
Utilizing the boundary and initial conditions of the sensitivity problem in Eq. (3), we
can integrate the double integral term in the above equation by parts. Then J is set
to zero. After some manipulation, the following adjointproblem is obtained:
2
 2 ( x, t )
 2 ( x, t )
[
I
(
x
)
]

A
(
x
)
0
x 2
x 2
t 2
(6a)
 (0, t )  0
(6b)
  (0, t )
0
x
(6c)
 2  (1, t )
0
 x2
(6d)

 2  (1, t )
[ I (1)
]  2[ y (1, t )  Y (1, t )]
x
x 2
  ( x, t f )
 ( x, t f ) 
0
t
6
(6e)
(6f)
The adjoint problem of Eqs. (6a)-(6f) is different from the standard initial-value
problem in that the final time condition at time t = tf is specified instead of the
customary initial condition. However, this problem can be transformed to an
initial-value problem by the transformation of the time variable as   t f  t . Finally
the following integral term is left:
 J 
tf
 ( 1 ,t ) F (t )d t
0
(7)
From the definition used in reference [21], we have:
 J   J  F  t  dt
tf
(8)
0
where J  is the gradient of the functional J, and a comparison of Eq. (7) with Eq. (8)
leads to the following gradient equation:
J    ( 1t, )
(9)
2.4 Conjugate gradient method for minimization
The following iteration process based on the conjugate gradient method is now
used for the estimation of F (t ) by minimizing the above functional J[F(t)]. The
shear force F (t ) at the (K+1)th step can be evaluated by:
F K 1 (t )  F K (t )   K P K (t ) ,
K = 0, 1, 2,…
(10)
where  K is the search step size in going from iteration K to iteration K+1, and P K
is the direction of descent (i.e., search direction) given by
P K ( t ) J K ( t) 
K
PK1 (, t )
K = 1, 2, 3,…
(11)
where  K is the conjugate coefficient and determined from

K



tf
0
tf
0
[ J K (t )]2 dt
[J 
K 1
with  0  0
(12)
2
(t )] dt
The functional J [ F (t )] for iteration K+1 is obtained by rewriting Eq. (2) as:
7
tf
J [ F K 1 (t )]   [ y ( F K   K P K )  Y (1, t )]2 dt
0
(13)
Expanding Eq. (13) into a Taylor series, yields:
tf
J [ F K 1 (t )]    y ( F K )   K  y ( P K )  Y (1, t )  2 dt
0
(14)
The sensitivity function  y ( P K ) is taken as the solution of Eq. (3) at the measured
position by letting F  P K . The search step size  K can be determined by
minimizing the function given by Eq. (14) with respect to  . After rearrangement,
the following expression is obtained:

K


tf
0
 y ( P K ) y[ F( K )Y dt ]

tf
0
(15)
[y (P K )2 dt
]
2.5 Stopping criterion
If the problem contains no measurement errors, the convergence condition for
the minimization of the criterion is:
J [ F K 1 ]  
(16)
where  is a small specified number.
2.6 Computational Procedures
The computational procedure for the solution of this inverse problem may be
summarizes as follows:
Suppose F K (t ) is available at iteration K.
Step 1 Solve the direct problem given by Eqs. (1a)–(1f) for y ( x, t ) .
Step 2 Examine the stopping criterion given by Eq. (16). Continue if not satisfied.
Step 3 Solve the adjoint problem given by Eqs. (6a)–(6f) for  ( x, t ) .
Step 4 Compute the gradient of the functional J  from Eq. (9).
Step 5 Compute the conjugate coefficient  K and direction of decent P K from Eqs.
8
(12) and (11), respectively.
Step 6 Set F (t ) = P K (t ) and solve the sensitivity problem given by Eq. (3) for
y (1, t ) .
Step 7 Compute the search step size  K from Eq. (15).
Step 8 Compute the new estimation for F K 1 (t ) from Eq. (10) and return to Step 1.
3. Results and discussions
The objective of this article is to show the validity of the present approach in
estimating F(t) accurately with no prior information on the functional form of the
unknown quantities, which is the so-called function estimation. In order to illustrate
the accuracy of the present inverse analysis, we consider the simulated exact values
for the shear force F(t) acting on the tapered optical fiber probe as:
F ( t ) 1 . 0 ( s i tn
0 . 2 s tin 2
0 .t 5 s i n 3
)
(17)
The geometry and material parameters of the probe used for the numerical
calculations are listed in Table 1. The simulated exact deflection measurements Y(1,t)
at position x = 1 in Eq. (2) can be obtained by substituting the above F(t) into the
direct problem of Eq.(1).
In the analysis, the Rayleigh-Ritz method was adopted to study the deflection on
the probe for an SNOM scanning a sample surface, and a ten-term polynomial
expansion technique was used in the calculation processes. The deflection
measurement is located at the position x = 1. The total measurement time is chosen as
t f  1. 0 sec and measurement time step is taken 0.005 sec. In order to compare the
results with situations involving random measurement errors, a random noise is added
to the simulated exact deflection values to generate the measured deflection Y, that is
Y  Ye x a ct
(18)
9
where Yexact is the deflection at position x = 1 of the direct problem with the
simulated exact shear force F(t),  is a random variable within -2.576~2.576 for a
99% confidence bounds, and  is the standard deviation of the measurement.
The estimated shear force of F(t), obtained at the 10th iteration with initial guess
F 0  0.3 and measurement errors   0.0, 0.005 , and 0.01 are shown in figures 2-4,
respectively. For a deflection of unity and 99% confidence, these standard deviations
correspond to a measurement error of 0.0, 1.29, and 2.58%, respectively. Figures 2-4
show that, for the case considered in this article, increase in the measurement error
does not cause obvious decrease in the accuracy of the inverse solution. In addition, to
find the effect of the initial guess values F 0 on the accuracy of the estimation, we
then take F 0  0.001 with the measurement error   0.01 . Then the results are
plotted in figure 5, which is estimated at the 10th iteration. It indicates that the
variation of the initial guess values has a small effect on the estimation.
4. Conclusions
The conjugate gradient method, which utilized the function estimation approach,
was successfully used to determine the shear force between the scanning near-field
microscope (SNOM) probe and the sample. The results show that the conjugate
gradient method is not sensitive to the measurement errors. The advantage of the
conjugate gradient method is that the prior information on the functional form of the
unknown quantity is not required. Furthermore, the initial guess for the interaction
shear force can be arbitrarily chosen for the iteration process.
10
References
[1] D. Rugar , P. Hansma, Atomic force microscopy, Physics Today 43 (1990) 23.
[2] M. Radmacher, R.W. Tillmann, M. Fritz , H.E. Gaub, Science 257 (1992) 1900.
[3] K. Holmberg, A. Matthews, Coatings Tribology: Properties, Techniques, and
Applications in Surface Engineering, Elsevier, New York,1994.
[4] F. Aviles, O. Ceh , A.I. Oliva, Rev. Sci. Instrum. 74 (2003) 3356.
[5] M. De Serio, R. Zenobi, V. Deckert, Trends in Analytical Chemistry 22 (2003) 70.
[6] P.J. James, M. Antognozzi, J. Tamayo, T. J. McMaster, J.M. Newton, M.J. Miles,
Langmuir 17 (2001) 349.
[7] G. Kaupp, A. Herrmann, J. Schmeyers, J. Boy, J. Photochemistry, Photobiology A:
Chemistry 139 (2001) 93.
[8] J.V. Beck, B. Blackwell, C.R.St. Clair, Jr., Inverse Heat Conduction: Ill-Posed Problems,
Wiley-Interscience, New York, 1985.
[9] O.M. Alifanov, Inverse Heat Transfer Problem, Springer-Verlag, New York, 1994
[10] A.J. Silva Neto and M.N. O zisik, J. Appl. Phys. 71 (1992) 5357.
[11] Y.C. Yang, U.C. Chen, W.J. Chang, J. of Thermal Stresses 25 (2002) 51.
[12] W.J. Chang, T.H. Fang, Applied Physics B: Lasers and Optics 80 (2005) 373.
[13] W.J. Chang, C.I. Weng, I. J. of Heat and Mass Transfer 42 (1999) 2661.
[14] C.H. Huang, J. Sound. Vib. 248 (2001) 789.
[15] W.J. Chang, T.H. Fang, C.I Weng, Nanotechnology 15 (2004) 427.
[16] Y.C. Yang, S.S. Chu, W.J. Chang, J. of Applied Physics 95 (2004) 5159.
[17] W.J. Chang, T.H. Fang, H.L. Lee, Y.C. Yang, Ultramicroscopy 102 (2005) 85.
11
[18] L. Meirovitch, Analytical Methods in Vibrations, Macmillan, New York, 1967.
[19] J.H. Ginsberg, Mechanical and Structural Vibrations, John Wiley & Sons, New
York 2001.
[20] F. B. Hildebrand, Method of Applied Mathematics, Prentice-Hall, INC.
Englewood Cliffs, N. J., 1983.
[21] O.M. Alifanov, Inverse Heat Transfer Problem, Springer-Verlag, New York,
1994.
12
Table 1. Parameters for an optical fiber SNOM probe.
cylinder length
1600μm
cylinder radius
50μm
tapered angle
12°
tip radius
30 nm
elastic modulus
72.5 GPa
density
2.2 g/cm3
13
Fig.1. Schematic diagram of the SNOM apparatus. The shear force between the optical fiber
probe and the sample surface is modeled as the shear force, F(t), on the probe.
14
Shear Force, F
1.5
+++++
++ ++
+
+
1.2
+
+
+
+
+
+
+
+
+
+
+
+
+
+
0.9
+
+
++++
+
+++ +++
+
++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
0.6
+
+
+
++
+
++
+
+
++
+
++++++++
+
+
+
+
+
0.3 +
+
Exact Solution
+
+
Inverse
Solution
+
+
+
+
+
0+
0
0.2
0.4
0.6
0.8
1
Time, t
Fig. 2. Estimated shear force at 10th iteration with initial guess F 0  0.3 and
  0.0 .
15
1.5
++++++++
++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
0.9
+
+
+
++++++++
+
++
++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
0.6
+
+
+
+
++
+
++
+
+
++
++++++++
+
+
+
+
+
+
0.3 +
Exact Solution
+
+
Inverse
Solution
+
+
+
+
+
+
0+
Shear Force, F
1.2
0
0.2
0.4
0.6
0.8
1
Time, t
Fig. 3 Estimated shear force at 10th iteration with initial guess F 0  0.3 and
  0.005 .
16
Shear Force, F
1.5
+++++
++ ++
+
+
+
+
1.2
+
+
+
+
+
+
+
+
+
+
+
+
+
+
0.9
+
++++++++
+
++
++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
0.6
+
+
++
+
+
+
+
+
++
++++++++
+
+
+
+
+
+
0.3 +
Exact Solution
+
+
Inverse Solution
+
+
+
+
+
+
0+
0
0.2
0.4
0.6
0.8
1
Time, t
Fig. 4 Estimated shear force at 10th iteration with initial guess F 0  0.3 and
  0.01 .
17
Shear Force, F
1.5
+++++
++ ++
+
+
+
+
1.2
+
+
+
+
+
+
+
+
+
+
+
+
+
+
0.9
++++++++
+
++
++
+
+
+
+
+
++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
0.6
+
+
++
+
++
+
+
++
+
++++++++
+
+
+
+
+
0.3 +
Exact Solution
+
+
Inverse Solution
+
+
+
+
+
+
0+
0
0.2
0.4
0.6
0.8
1
Time, t
Fig. 5 Estimated shear force at 10th iteration with initial guess F 0  0.001 and
  0.01 .
18