Vibration sensitivity of the scanning near-field optical microscope with a tapered optical fiber probe

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Ultramicroscopy 102 (2005) 85–92

Vibration sensitivity of the scanning near-ﬁeldoptical

microscope with a taperedoptical ﬁber probe

Win-Jin Chang

a,

, Te-Hua Fang

b

, Haw-Long Lee

a

, Yu-Ching Yang

a

Department of Mechanical Engineering, Kun Shan University of Technology, Tainan 710, Taiwan

b

Department of Mechanical Engineering, Southern Taiwan University of Technology, Tainan 710, Taiwan Received3 April 2004; receivedin revisedform 11 August 2004; accepted24 August 2004

Abstract

In this paper the Rayleigh–Ritz methodwas usedto study the scanning near-fieldoptical microscope (SNOM) with a taperedoptical fiber probe’s flexural andaxial sensitivity to vibration. Not only the contact stiffness but also the geometric parameters of the probe can influence the flexural andaxial sensitivity to vibration. According to the analysis, the lateral andaxial contact stiffness hada significant effect on the sensitivity of vibration of the SNOM’s probe, each mode had a different level of sensitivity and in the first mode the tapered optical fiber probe was the most acceptive to higher levels of flexural andaxial vibration. Generally, when the contact stiffness was lower, the taperedprobe was more sensitive to higher levels of both axial andflexural vibration than the uniform probe. However, the situation was reversedwhen the contact stiffness was larger. Furthermore, the effect that the probe’s length andits taperedangle had on the SNOM’s probe axial andflexural vibration were significant andthese two conditions shouldbe incorporatedinto the design of new SNOM probes.

PACS: 68.35.Ja; 07.79.Fc; 61.16.Ch

Keywords: Scanning near-ﬁeld optical microscope; Optical ﬁber probe; Sensitivity; Flexural vibration mode; Axial vibration mode

1. Introduction

The atomic force microscope (AFM) was developed for studying the surface topography of both conductive and insulating samples on a 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. Therefore, SNOM has become a powerful tool for investigating the photophysical andphotoche-mical properties of materials on a nanometer scale

[5–8].

An optical ﬁber probe cantileveredat one endis the key element of the SNOM. In a typical SNOM,

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the optical fiber probe is perpendicular to the sample’s surface. The probe allows simultaneous measurement of the topography andoptical transmission of the sample surface with a high level of lateral resolution[9–12]. In general, when the probe scans the sample’s surface, nonlinear axial andlateral interactive forces occur between the tip andthe sample surface. The interaction forces can influence the sensitivity of the SNOM probe. However, few works have studied the effect the interaction force has on the probe perfor-mances. Recently, Fang andChang[13] assumed that the optical fiber probe hada uniform circular cross section and studied the dynamic responses of the probe, including the analyses of the cylindrical probe sensitivity to axial andflexural vibration. They found that each mode had a different mode shape anda different sensitivity, affectedby the rigidity of the probe and the surface properties. In order to obtain the highest contrast for imaging, the most sensitive modes in the system with the

highest level of vibration shouldbe foundand used.

In this paper, the dynamic responses of an SNOM’s taperedoptical fiber probe axial and flexural vibrations were considered. The simple analysis was that the amplitude of the surface motion was not very large, so a linearizedresponse couldbe assumed. The modal sensitivities for axial and flexural vibrations were derived, and an approximate solution was obtainedusing the Rayleigh–Ritz method.

2. Analysis

A schematic diagram of an SNOM with an optical ﬁber probe that is cantileveredat one endis depicted inFig. 1. The cantilever probe length is L and includes a section of the uniform cylinder radius R andthe taperedsection with a tapered angle a anda sharpenedtip with radius r. To

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Fig. 1. Schematic diagram of SNOM apparatus with an optical ﬁber probe cantilevered at one end. The interaction with the sample is modeled by an axial spring stiffness, ka; anda lateral spring stiffness, kl.

W.-J. Chang et al. / Ultramicroscopy 102 (2005) 85–92 86

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increase the optical reﬂectivity the probe was coatedwith a layer of aluminum or aurum. For simplifying the problem in this paper, the coating of the probe was not taken into account. The interaction with the sample was modeled by an axial spring stiffness, ka, anda lateral spring

stiffness, kl. The probe experiencedﬂexural

andaxial vibrations while scanning the sample surface.

2.1. Flexural vibration

MACROBUTTON MTEditEquationSection

Equation Section 1The optical ﬁber probe includes a taperedsection andits cross section was a function of x. When the probe scans the sample surface, the lateral interaction force between the probe andthe sample surface was inducedcausing the probe to vibrate ﬂexurally. The linear differ-ential equation of motion for free vibration of the probe is[14] @2 @x2 EI ðxÞ @2yðx; tÞ @x2 þmðxÞ@ 2_{yðx; tÞ} @t2 ¼0; (1)

where E is the modulus of elasticity, I(x) is the area moment of inertia, and m(x) is the mass density, which is the product of the volume density r and the cross-sectional area of the taperedoptical ﬁber probe A(x).

The corresponding boundary conditions are

yð0; tÞ ¼ 0; (2) @yð0; tÞ @x ¼0; (3) EI ðxÞ@ 2_{yðx; tÞ} @x2 x¼L ¼0; (4) @ @x EI ðxÞ @2yðx; tÞ @x2 x¼L ¼klyðx; tÞjx¼L; (5)

where the probe is assumeda ﬁxedendat x=0. When the probe is considered to be a cantilever beam, then the boundary conditions given by Eqs. (2) and(3) correspondto conditions of zero displacement and zero slope. The boundary conditions given by Eqs. (4) and (5) corres-pondto zero momentum at x=L andthe

force is balancedwith the interaction force between the tip andthe sample. Because a linear model is used to describe the inter-action force, the probe was restrictedto small displacements.

Eqs. (1)–(5) give an eigenvalue problem in the form of the fourth-order ordinary differential equation and have four boundary conditions. The harmonic solution of the form can be expressedas yðx; tÞ ¼ Y ðxÞeiolt_{: For the tapered}

probe, the bending stiffness EI(x) andmass density m(x) are functions of positions along the probe. The Rayleigh–Ritz method [15,16]can be usedto deal with the problem.

Applying the Rayleigh–Ritz method, the com-ponents of the stiffness andthe mass matrices are expressedas follows: Kij¼ Z L 0 EI ðxÞ d 2_u jðxÞ dx2 ! d2ujðxÞ dx2 ! dx þkluiðLÞuiðLÞ ð6Þ and Mij¼ Z L 0 mðxÞuiðxÞujðxÞdx; (7)

where u(x) is a set of admissible functions which are chosen to satisfy the geometric boundary conditions. The matrices may be normalized as follows: e Kij¼ Kij EI0=L3 ¼ Z 1 0 EI ð_exÞ EI0 d2ujðxÞe dx_e2 ! d2ujðxÞe d_ex2 ! d_ex þb_luið1Þuið1Þ ð8Þ and e Mij¼ Mij m0 ¼ Z 1 0 mðxÞ_e m0 uiðxÞue jðxÞde ex; (9)

where EI0 is the bending stiffness and m0 is the

mass density at x ¼ 0; b_l¼kl=ðEI0=L3Þ and x ¼e

x=L:

In equations (8) and(9), let the terms EI ð_exÞ=EI0 ¼1 and mðxÞ=me 0¼1; these can be

appliedto calculate the uniform cross-section probe. In terms of the normalized

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measurement frequency resolution by 0.14% when b_a¼0:001; while it increasedthe resolution by 46.8% when b_a¼1:

4. Conclusion

The sensitivity of flexural andaxial vibration modes of SNOM with a tapered optical fiber probe has been analyzedusing the Rayleigh–Ritz meth-od. The results showed that the low-order flexural andaxial vibration modes were more sensitive than the high-order modes and the first mode was the most sensitive when the contact stiffness was low. The flexural andaxial sensitivities of the taperedprobe were more sensitive than the uniform probe except for the axial sensitivity of mode 1 when the contact stiffness was low. However, the situation was reversedwhen the contact stiffness became higher. The flexural and axial sensitivity of the taperedprobe decreasedas the taperedangle increasedwhen the lateral contact stiffness was low. However, when the contact stiffness was higher, the flexural andaxial sensitivities of the probe apparently increasedas the angle increased. According to the analysis, increasing the taperedangle from 91 to 151 decreased the flexural sensitivity by 8.41% when b_l¼0:1 andthe axial sensitivity by 0.41% when b_a¼0:001; while it increasedthe flexural sensitiv-ity by 60.09% when b_l¼10 andthe axial sensitivity by 74.1% when b_a¼1: Furthermore, the flexural andaxial sensitivities of the probe apparently increasedas the probe length was increasedwhen the contact stiffness was higher. Increasing the probe length from 1400 to 1800 mm decreased the flexural sensitivity by 3.83% when bl¼0:1 andthe axial sensitivity by 0.14% when

ba¼0:001; while it increasedthe ﬂexural

sensitiv-ity by 34.71% when bl¼10 andthe axial

sensitivity by 46.8% when b_a¼1:

Acknowledgement

This work was partially supportedby the National Science Council of Taiwan, under Grant nos. NSC 93-2212-E-168-020, NSC92-2218-E218-002 andNSC92-2212-E-218-006.

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