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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 17 — Aug. 18, 2008
  • pp: 13476–13485
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Theoretical and experimental study of optothermal expansion and optothermal microactuator

Dongxian Zhang, Haijun Zhang, Chao Liu, and Jianzhong Jiang  »View Author Affiliations


Optics Express, Vol. 16, Issue 17, pp. 13476-13485 (2008)
http://dx.doi.org/10.1364/OE.16.013476


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Abstract

A new type of microactuators based on optothermal (OT) expansion is introduced. The mechanism of the OT expansion is theoretically analyzed, and comprehensive models for OT expansion and bi-direction microactuator are presented in this paper. An expansion arm and a microswitch-like OT microactuator with 1200µm-length are fabricated by an excimer laser micromachining system using single layer material. A laser diode (650nm) is employed as the external power source to activate the arm and the microactuator. Experimental results indicate that the OT expansion increment is approximately linear with the laser power irradiating the expansion arm, coinciding with theoretical predictions quite well. As to the switch-like microactuator, an enlarged bi-direction deflection has been obviously observed. The OT expansion and deflection amplitude that can reach micron scale is generally large enough for most microsystems. The new technique of OT microactuators can be widely applied in those fields where simple structure, easy fabrication, large displacement, wireless and remote controlling are required.

© 2008 Optical Society of America

1. Introduction

Due to the increased requirements of micro electromechanical system (MEMS), it is now crucial to develop and adopt new microactuators, which are often the most critical part for MEMS devices to perform physical function. Up to now, a number of microactuators utilize different driving principles, such as electrostatic force [1

1. J. Kyokane, K. Tsujimoto, Y. Yanagisawa, T. Ueda, and M. Fukuma, “Actuator using electrostriction effect of fullerenol-doped polyurethane elastomer (PUE) films,” IEICE Transactions on Electronics E 87C, 136–141 (2004).

,2

2. J. L. Yeh, C. Y. Hui, and N. C. Tien, “Electrostatic model for an asymmetric comb drive,” Journal of Microelectromechanical Systems 9, 126–130 (2000). [CrossRef]

], piezoelectric force [3

3. D. L. Devoe and A. P. Pisano, “Modeling and optimal design of piezoelectric cantilever microactuators,” Journal of Microelectromechanical Systems 6, 266–270(1997). [CrossRef]

,4

4. X. Huang and H. Shen, “Nonlinear free and forced vibration of simply supported shear deformable laminated plates with piezoelectric actuators,” International Journal of Mechanical Sciences 47, 187–208 (2005). [CrossRef]

], electromagnetic force [5

5. Y. W. Park and D. Y. Kim, “Development of a magnetostrictive microactuator,” Journal of Magnetism and Magnetic Materials E 1765–1766, 272–276 (2004).

], and electrothermal expansion [6

6. C. N. Saikrishna, K. V. Ramaiah, and S. K. Bhaumik, “Effects of thermo-mechanical cycling on the strain response of Ni-Ti-Cu shape memory alloy wire actuator,” Materials Science and Engineering A-Structural Materials Properties Microstructure and Processing 428, 217–224 (2006). [CrossRef]

,7

7. T. Lalinský, M. Držík, J. Chlpík, M. Krnáč, Š Haščík, Ž. Mozolová, and I. Kostič, “Thermo-mechanical characterization of micromachined GaAs-based thermal converter using contactless optical methods,” Sensors and Actuators A 123–124, 99–105 (2005). [CrossRef]

]. Each of them has their own advantages and disadvantages. The choice and the optimization should be made according to the requirements of the applications. Compare to electrostatic or piezoelectric system, thermal actuators allow much higher deflection angles and can exert larger forces on a load. Therefore, the thermal actuators have been widely used in those fields where slow motions but large force/displacements are required.

In this paper, a new method and technique of microactuators based on optothermal (OT) expansion are developed. We theoretically analyzed the mechanism of the OT expansion, and presented the comprehensive models for OT expansion and bi-direction microactuators. Different from electrothermal actuators, some OT expansion arm and microactuators are manufactured with single layer material by using an excimer-laser micromachining system. Experiments are carried out to demonstrate the theoretical model of OT expansion, as well as to check the feasibility of OT microactuators. The relationship between the OT expansion/deflection and the laser power is further investigated.

2. Theoretical model of optothermal expansion and deflection

The basic concept of OT expansion is shown in Fig. 1. When a focused laser beam irradiates a slender OT expansion arm, the laser power will be partly absorbed and stored in the form of internal heat [12

12. Y. L. He, H. J. Zhang, and D. X. Zhang, “Theoretical and experimental study of photo-thermal expansion using an atomic force microscope,” J. Micromech. Microeng. 15, 1637–1640 (2005). [CrossRef]

]. The arm will elongate for ΔL due to its temperature increases and volume expands (Fig. 1(a)). In order to acquire an enlarged lateral deflection, an actuator with two slender arms is proposed as shown in Fig. 1(b). As the free ends of the arms are connected, when the laser irradiates one of the arms, thermal expansion will make the actuator laterally deflect for d. When the other arm being irradiated, the direction of deflection will reverse.

Fig. 1. The basic concept of OT expansion, (a) longitudinal expansion of OT arm, (b) lateral deflection of bi-direction OT microactuator

For a slender OT arm, whose length is much greater than its width and thickness, its OT expansion can be simplified to one-dimensional model for theoretical analysis. Figure 2 shows a simplified one-dimensional coordinate system and an arbitrary infinitesimal transmission element within the slender OT arm.

Fig. 2. The schematic model of the slender arm, (a) one-dimensional coordinate system, (b) an arbitrary infinitesimal OT transmission element within the arm

The slender arm with length L, width W, thickness D and the initial temperature T 0 is connected with the base on the left end. A laser spot with radius R is formed when a laser of power P 0 irradiates the arm. Set the left end of the arm as the coordinate origin (x=0), the center of the spot as x=L 1, the free right end as x=L, and L 2=L-L 1. The power-density distribution along the X direction is named Q (x). The power distribution of the spot is usually Gaussian, however, as the diameter of the spot is much smaller than the length of the arm and the integral compensation, the power-density distribution can be assumed to be constant. Therefore, Q (x) can be expressed as:

Q(x)={ρAq0L1RxL1+R00x<L1R,L1+R<xL
(1)

Where ρ A is the ratio of absorption, q 0 the incident laser power density, i.e., P 0/(πR 2). We choose an arbitrary infinitesimal element with dx length at the position x on the right (Fig. 2 (b), so does on the left) of the spot center to analyze the OT transmission. The element gains heat from the laser irradiation and inner heat conduction, and losses heat through surface heat convection and conduction. The heat balance equation is:

KT(x)DW+Q(x)W·dx=KT(x+dx)DW+[T(x)T0]·(2Whw·dx+2DhD·dx)
(2)

Here K is the thermal conductivity of the material, T(x) the temperature distribution when the arm is irradiated, the temperature rise ΔT(x)=T(x)-T 0, h D is the coefficient of the convective heat transfer on the side surface, h W is that on the up and down surfaces (their h W is nearly the same as the arm thickness is on micrometer scale). Eq. (2) can also be written in the form of second-order differential equation:

d2T(x)dx22(hD+hWW)KDW·ΔT(x)=Q(x)KD
(3)

The general solution of Eq. (3) is:

ΔT(x)=C1·ch(bx)+C2·sh(bx)+1bx0xf(ξ)sh(b(xξ))dξ
(4)

Where b and f (x) are defined by

b2=2(hDD+hWW)KDW,f(x)=Q(x)KD
(5)

x 0 ranges from 0 to L. As the heat environment at the left end (fixed) is different from the right end (free), the coefficients of the convective heat transfer are different, named h 1, h 2 respectively. h 2 is the same with h D. So the thermal boundary condition can be expressed as:

K.dTdxx=0h1ΔT(0)=0K.dTdxx=L+h2ΔT(L)=0
(6)

Solving Eq. (4) with the boundary conditions, the distribution of temperature rise can be given as follows:

ΔT(x)={2ρAq0sh(bR)b2KDC3[Kbch(bL2)+h2sh(bL2)][Kbch(bx)+h1sh(bx)]0x1L1R2ρAq0sh(bR)b2KDC3[Kbch(bL2)+h2sh(bL2)][Kbch(bx)+h1sh(bx)]+ρAq01ch(b(x+RL1))KDb2,L1RxL1+R2ρAq0sh(bR)b2KDC3[Kbch(bL2)+h2sh(bL2)][Kbch(bx)+h1sh(bx)]ρAq02sh(bR)sh(b(xL1))KDb2,L1+RxL
(7)

Here the sh() and ch() are hyperbolic sine and cosine functions, the constant C 3 is

C3=(h1h2+K2b2)sh(bL)+Kb(h1+h2)·ch(bL)
(8)

According to the relation between thermal expansion and temperature rise, the thermal expansion increment ΔL along the X direction can be expressed as:

ΔL=α0LΔT(x)dx
(9)

Where α is the linear thermal expansion coefficient of the material. With Eq. (7) and (8), ΔL can be derived:

ΔL=α2ρAq0RKDb2
2αρAq0h1h2·sh(bR)·(sh(bL1)+sh(bL2))+Kb·sh(bR)·(h2ch(bL1)+h1ch(bL2))b3KD[(h1h2+K2b2)sh(bL)+K(h1+h2)b·ch(bL)]
(10)

Therefore, when the irradiating laser is determined (including the power, the wavelength, the size and the location of the spot) and the geometric parameters and physical properties of the arm are given, the value of OT expansion increment can be uniquely determined and calculated.

Table 1 shows the parameters of the laser and the OT arm studied in this paper.

Table 1. Parameters of the Laser and the OT Arm

table-icon
View This Table

According to Eq. (10) and Table 1, the relationship between the expansion increment and the laser power can be calculated and demonstrated (as shown in Fig. 3). The results indicate that the expansion increment is approximately linear with the laser power.

Fig. 3. Theoretical relationship between the OT expansion increment and the laser power

Like other microactuators, such as the bi-direction electrothermal actuators, the bi-direction OT actuator is able to convert and enlarge longitudinal OT expansion ΔL into lateral deflection d. The deflection can be expressed and calculated by the following Eqs., where the geometrical parameters are indicated in Fig. 4. For clarity, the deflection and bending of the actuator are not to scale.

Fig. 4. Simplified model of the bi-direction deflection of OT microactuator
dAOsinθ
(11)
tgθ=BCCO=[(L+ΔL)L]S
(12)
dAOcosθ×ΔLS
(13)

Where AO’ is approximately equal to the length L of the unirradiated arm, BC is the length difference of the two arms due to OT expansion, CO’, i.e. S, is the gap distance between the two arms. The deflection angle θ is actually quite small, cosθ≈1, so the deflection d could be given:

dΔL×LS
(14)

In Eq.(14), when the actuator’s material, structure, and the environment temperature are given, ΔL is dominantly determined by the laser power. Although the unirradiated arm might slightly reduce the longitudinal expansion ΔL of the irradiated arm, as L is much larger than S, an enlarged lateral deflection could be acquired.

3. Experiments

In this paper, an OT arm and a microswitch-like bi-direction microactuator are fabricated by using a high performance and multifeatured excimer laser micromachining workstation (Optec Promaster, Belgium). The black polypropylene material is chosen to manufacture the microactuators, for it is easy to machine with excimer laser and has good thermal-dynamic properties including low thermal conductivity and high expansion coefficient (as presented in Table 1). The actuators lose less heat while irradiated by laser for its low thermal conductivity, and obtain larger expansion due to its high thermal expansion coefficient. Figure 5(a) is the microscopic image of the OT arm (the simplest OT microactuator) with the parameters presented in Table 1.

Fig. 5. Microscopic images captured from videos showing the different expansions when the power of irradiating laser is (a) 0 mW; (b) 6 mW (Media 1); (c) 10 mW (Media 2)

A laser diode (650nm) is employed as the external power source to activate the microactuators. A light power meter is previously used to calibrate the value of laser power that irradiates the arm of microactuator. By using a self-designed electronic circuit, the value of laser power can be adjusted from 0 to 10 mW. All experiments can be monitored by a charge coupled device (CCD)-combined light microscope system. For briefness, Fig. 5 only shows three typical states of the OT expansion arm. The left images are snapshots from videos using a 4× objective lens, while the zoomed images (right columns) are snapshots from videos using a 40× objective lens, corresponding to the marked areas in the left images. We can precisely measure the data of expansion increment by using a self-designed sub-pixel video analysis software for microactuators.

According to the principle of sub-pixel measurement, the measurement accuracy of our microscopic imaging equipment is on the scale of sub-pixel, which depends on the image magnification and CCD detector resolution. The pixel size of microscopic images captured using a 40× objective lens is about 0.3 µm, therefore, the measurement accuracy (sub-pixel accuracy) is on the scale of less than 0.1 µm.

When the values of laser power are 0, 6, 10 mW, as shown in Fig. 5(a)–5(c), respectively, the measured expansion increments are 0, 2.3, 3.4 µm, correspondingly. More pairs of data are also acquired. The experimental relationship between the OT expansion increment and the laser power is plotted in Fig. 6.

Fig. 6. The experimental results showing the relationship between expansion increment and the laser power

For each given laser power, a group of OT expansions have been measured in the experiments. Each of the experimental point in Fig. 6 represents the average value of the OT expansions. The error bars included in the graph show that the expansions have an uncertainty of ±0.14 µm. This uncertainty is small and tolerable. Furthermore, the experimental data and curve agree well with the theoretical predictions, for the theoretical and experimental curves are nearly of the same shape and trend.

The experiments show that the value of OT expansion is 3.4 µm when the laser power is 10 mW. In this case, the ratio of the expansion to the initial length of the arm is about 0.3% (3.4µm/1200µm). In the experiments, we measure and utilize the absolute expansion, and should also consider the relative ratio. From the experimental videos, obvious OT expansions and deflections can be observed. Although these expansions and deflections might be sufficient for most of the practical applications, we should still reduce the length of the arm while keeping or increasing the expansion level as to obtain a higher ratio. As an alternative way, we have developed various kinds of bi-direction microactuators, which can convert and enlarge longitudinal OT expansion into larger lateral deflection (higher ratio).

As an example of applications, we further designed and fabricated a switch-like OT microactuator of 1200-µm length in order to acquire enlarged bi-direction (lateral) OT deflection. Figure 7(a) gives out its scanning electron microscopy (SEM) image. Two slender arms each with one contact are connected on the free end. Another two contacts are fabricated on the fixed base. The distance between the adjacent contacts is 10 µm. The original state of the OT microactuator is shown in Fig. 7 (b). When the laser (650 nm, 3 mW) irradiates the upper or lower arm, the lower or upper pairs of contacts can totally connect alternatively, with a deflection of at least 10 µm (Fig. 7(c) and 7(d)).

Compared to the OT expansion of the slender arm, the deflection of the switch-like microactuator is significantly magnified, as already proved by both of the theoretical predictions and the experiment results. For the single arm, the expansion value is 3.4 µm when the laser power is 10 mW, while for the bi-direction microactuator, the deflection value is more than 10 µm just when the laser power is 3 mW. Again, the longitudinal OT expansion is converted and enlarged into much larger lateral deflection.

Fig. 7. SEM and microscopic images captured from videos showing the bi-directional deflection, (a) SEM image; (b) original state; (c) laser (3mW) irradiates the upper arm (Media 3); (d) irradiates the lower arm (Media 4)

Regarding the comparisons of the performances/efficiency of bi-direction OT actuator to those of other actuating mechanisms, such as the electrothermal actuators, some actuation values and power levels are concerned. With the same input power (such as 3mW), the deflections of two given electrothermal actuators are about 2 µm (Fig. 4 of Jeffrey T. Butler et al) [13

13. Jeffrey T. Butler, Victor M. Bright, and William D. Cowan, “Average power control and positioning of polysilicon thermal actuators,” Sensors and Actuators 72, 88–97(1999). [CrossRef]

] and 1.5 µm (Fig. 10 of Lijie Li et al.) [14

14. Lijie Li and Deepak Uttamchandani, “Modified asymmetric micro-electrothermal actuator,” J. Micromech. Microeng. 14, 1734–1741(2004). [CrossRef]

], smaller than that of the bi-direction OT actuators (larger than 10 µm). On the other hand, for obtaining a 10-µm deflection, the bi-direction OT actuator requires a 3-mW input power, while the electrothermal actuators require a 12-mW [13

13. Jeffrey T. Butler, Victor M. Bright, and William D. Cowan, “Average power control and positioning of polysilicon thermal actuators,” Sensors and Actuators 72, 88–97(1999). [CrossRef]

] and an 18-mW [14

14. Lijie Li and Deepak Uttamchandani, “Modified asymmetric micro-electrothermal actuator,” J. Micromech. Microeng. 14, 1734–1741(2004). [CrossRef]

] input power. These data show that the OT actuator is of the same or better performances/efficiency with electrothermal actuating mechanisms, and may obtain comparatively larger actuation value with lower power level.

4. Conclusions

This paper proposed a new technique of microactuators based on OT expansion. Through theoretical analysis and mathematical derivation, a comprehensive model has been developed to describe the characteristics of OT expansion, as well as to make the theoretical predictions about the relationship between the expansion increment and the laser power. The feasibility of the bi-direction OT microactuators is proved during experiments. Both of the theoretical and experimental results show that the OT expansion increment is nearly linear with the laser power for the slender expansion arm, matching each other quite well. As to the switch-like microactuator, the enlarged bi-direction OT deflection has been realized. The OT expansion and deflection that can reach micron scale is generally large enough for microsystems. Moreover, the advantages of these kinds of OT microactuators include: they provide a novel method of microactuation for MEMS or micro/nano-technology; they enable wireless and remote control; easier integration or miniaturization becomes practical and feasible; no electric or electronic interference between the actuators and MEMS devices would be caused; the actuators can be simply made of single layer materials and a variety of conductive or non-conductive materials can be employed. Therefore, the OT microactuators would be widely utilized in the fields of MEMS and micro/nano-technology.

Acknowledgments

This work has been supported by the National High Technology Research and Development Program of China under Grant No. 2006AA04Z237, and by the National Natural Science Foundation of China under Grant No. 50775205.

References and links

1.

J. Kyokane, K. Tsujimoto, Y. Yanagisawa, T. Ueda, and M. Fukuma, “Actuator using electrostriction effect of fullerenol-doped polyurethane elastomer (PUE) films,” IEICE Transactions on Electronics E 87C, 136–141 (2004).

2.

J. L. Yeh, C. Y. Hui, and N. C. Tien, “Electrostatic model for an asymmetric comb drive,” Journal of Microelectromechanical Systems 9, 126–130 (2000). [CrossRef]

3.

D. L. Devoe and A. P. Pisano, “Modeling and optimal design of piezoelectric cantilever microactuators,” Journal of Microelectromechanical Systems 6, 266–270(1997). [CrossRef]

4.

X. Huang and H. Shen, “Nonlinear free and forced vibration of simply supported shear deformable laminated plates with piezoelectric actuators,” International Journal of Mechanical Sciences 47, 187–208 (2005). [CrossRef]

5.

Y. W. Park and D. Y. Kim, “Development of a magnetostrictive microactuator,” Journal of Magnetism and Magnetic Materials E 1765–1766, 272–276 (2004).

6.

C. N. Saikrishna, K. V. Ramaiah, and S. K. Bhaumik, “Effects of thermo-mechanical cycling on the strain response of Ni-Ti-Cu shape memory alloy wire actuator,” Materials Science and Engineering A-Structural Materials Properties Microstructure and Processing 428, 217–224 (2006). [CrossRef]

7.

T. Lalinský, M. Držík, J. Chlpík, M. Krnáč, Š Haščík, Ž. Mozolová, and I. Kostič, “Thermo-mechanical characterization of micromachined GaAs-based thermal converter using contactless optical methods,” Sensors and Actuators A 123–124, 99–105 (2005). [CrossRef]

8.

L. J. Li and D. Uttamchandani, “Modified asymmetric microelectrothermal actuator: analysis and experimentation,” J. Micromech. Microeng. 14, 1734–1741 (2004). [CrossRef]

9.

M. F. Pai and N. C. Tien, “Low voltage electrothermal vibromotor for silicon optical bench applications,” Sensors Actuators A 83, 237–243 (2000). [CrossRef]

10.

J. K. Luo, A. J. Flewitt, S. M. Spearing, N. A. Fleck, and W. I. Milne, “Comparison of microtweezers based on three lateral thermal actuator configurations,” J. Micromech. Microeng. 15, 1294–1302 (2005). [CrossRef]

11.

S. C. Chen, C. P. Grigoropoulos, H. K. Park, P. Kerstens, and A. C. Tam, “Photothermal displacement measurement of transient melting and surface deformation during pulsed laser heating,” Appl. Phy. Lett. 73, 2093–2095 (1998). [CrossRef]

12.

Y. L. He, H. J. Zhang, and D. X. Zhang, “Theoretical and experimental study of photo-thermal expansion using an atomic force microscope,” J. Micromech. Microeng. 15, 1637–1640 (2005). [CrossRef]

13.

Jeffrey T. Butler, Victor M. Bright, and William D. Cowan, “Average power control and positioning of polysilicon thermal actuators,” Sensors and Actuators 72, 88–97(1999). [CrossRef]

14.

Lijie Li and Deepak Uttamchandani, “Modified asymmetric micro-electrothermal actuator,” J. Micromech. Microeng. 14, 1734–1741(2004). [CrossRef]

OCIS Codes
(120.6780) Instrumentation, measurement, and metrology : Temperature
(120.6810) Instrumentation, measurement, and metrology : Thermal effects
(120.7000) Instrumentation, measurement, and metrology : Transmission
(230.4000) Optical devices : Microstructure fabrication
(350.5340) Other areas of optics : Photothermal effects

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: July 3, 2008
Revised Manuscript: August 4, 2008
Manuscript Accepted: August 5, 2008
Published: August 15, 2008

Citation
Dongxian Zhang, Haijun Zhang, Chao Liu, and Jianzhong Jiang, "Theoretical and experimental study of optothermal expansion and optothermal microactuator," Opt. Express 16, 13476-13485 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-17-13476


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References

  1. J. Kyokane, K. Tsujimoto, Y. Yanagisawa, T. Ueda, and M. Fukuma, "Actuator using electrostriction effect of fullerenol-doped polyurethane elastomer (PUE) films," IEICE Transactions on Electronics E 87C, 136-141 (2004).
  2. J. L. Yeh, C. Y. Hui, and N. C. Tien, "Electrostatic model for an asymmetric comb drive," Journal of Microelectromechanical Systems 9,126-130 (2000). [CrossRef]
  3. D. L. Devoe and A. P. Pisano, "Modeling and optimal design of piezoelectric cantilever microactuators," Journal of Microelectromechanical Systems 6,266-270(1997). [CrossRef]
  4. X. Huang and H. Shen, "Nonlinear free and forced vibration of simply supported shear deformable laminated plates with piezoelectric actuators," International Journal of Mechanical Sciences 47,187-208 (2005). [CrossRef]
  5. Y. W. Park and D. Y. Kim, "Development of a magnetostrictive microactuator," Journal of Magnetism and Magnetic Materials E 1765-1766,272-276 (2004).
  6. C. N. Saikrishna, K. V. Ramaiah, and S. K. Bhaumik, "Effects of thermo-mechanical cycling on the strain response of Ni-Ti-Cu shape memory alloy wire actuator," Materials Science and Engineering A-Structural Materials Properties Microstructure and Processing 428,217-224 (2006). [CrossRef]
  7. T. Lalinský, M. Držík, J. Chlpík, M. Krná�?, Š. Haš�?ík, Ž. Mozolová, and I. Kosti�?, "Thermo-mechanical characterization of micromachined GaAs-based thermal converter using contactless optical methods," Sensors and Actuators A 123-124,99-105 (2005). [CrossRef]
  8. L. J. Li and D. Uttamchandani, "Modified asymmetric microelectrothermal actuator: analysis and experimentation," J. Micromech. Microeng. 14, 1734-1741 (2004). [CrossRef]
  9. M. F. Pai, and N. C. Tien, "Low voltage electrothermal vibromotor for silicon optical bench applications," Sensors Actuators A 83, 237-243 (2000). [CrossRef]
  10. J. K. Luo, A. J. Flewitt, S. M. Spearing, N. A. Fleck, and W. I. Milne, "Comparison of microtweezers based on three lateral thermal actuator configurations," J. Micromech. Microeng. 15,1294-1302 (2005). [CrossRef]
  11. S. C. Chen, C. P. Grigoropoulos, H. K. Park, P. Kerstens, and A. C. Tam, "Photothermal displacement measurement of transient melting and surface deformation during pulsed laser heating," Appl. Phy. Lett. 73,2093-2095 (1998). [CrossRef]
  12. Y. L. He, H. J. Zhang, and D. X. Zhang, "Theoretical and experimental study of photo-thermal expansion using an atomic force microscope," J. Micromech. Microeng. 15,1637-1640 (2005). [CrossRef]
  13. J. T. Butler, V. M. Bright, and W. D. Cowan, "Average power control and positioning of polysilicon thermal actuators," Sensors and Actuators 72, 88-97 (1999). [CrossRef]
  14. L. Li and D. Uttamchandani, "Modified asymmetric micro-electrothermal actuator," J. Micromech. Microeng. 14, 1734-1741(2004). [CrossRef]

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