Calibration of optical tweezers based on an autoregressive model Zi-Qiang Wang,1 Jin-Hua Zhou,1 Min-Cheng Zhong,1 Di Li,1 and Yin-Mei Li1,2,* 1
Department of Optics and Optical Engineering, University of Science and Technology of China, Hefei, Anhui, 230026, China 2 Hefei National Laboratory for Physical Sciences at the Microscale, Hefei, Anhui, 230026, China * [email protected]
Abstract: The power spectrum density (PSD) has long been explored for calibrating optical tweezers stiffness. Fast Fourier transform (FFT) based spectral estimator is typically used. This approach requires a relatively longer data acquisition time to achieve adequate spectral resolution. In this paper, an autoregressive (AR) model is proposed to obtain the spectrum density using a limited number of samples. According to our method, the arithmetic model has been established with burg arithmetic, and the final prediction error criterion has been used to select the most appropriate order of the AR model, the power spectrum density has been estimated based the AR model. Then, the optical tweezers stiffness has been determined with the simple calculation from the power spectrum. Since only a small number of samples are used, the data acquisition time is significantly reduced and real-time stiffness calibration becomes feasible. To test this calibration method, we study the variation of the trap stiffness as a function of the parameters of the data length and the trapping depth. Both of the simulation and experiment results have showed that the presented method returns precise results and outperforms the conventional FFT method when using a limited number of samples. ©2014 Optical Society of America OCIS codes: (350.4855) Optical tweezers or optical manipulation; (170.4520) Optical confinement and manipulation; (120.4640) Optical instruments.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
M. C. Zhong, X. B. Wei, J. H. Zhou, Z. Q. Wang, and Y. M. Li, “Trapping red blood cells in living animals using optical tweezers,” Nat Commun 4, 1768 (2013). R. Reyes-Lamothe, D. J. Sherratt, and M. C. Leake, “Stoichiometry and Architecture of Active DNA Replication Machinery in Escherichia coli,” Science 328(5977), 498–501 (2010). F. M. Fazal and S. M. Block, “Optical tweezers study life under tension,” Nat. Photonics 5(6), 318–321 (2011). T. L. Min, P. J. Mears, L. M. Chubiz, C. V. Rao, I. Golding, and Y. R. Chemla, “High-resolution, long-term characterization of bacterial motility using optical tweezers,” Nat. Methods 6(11), 831–835 (2009). M. Capitanio, G. Romano, R. Ballerini, M. Giuntini, F. S. Pavone, D. Dunlap, and L. Finzi, “Calibration of optical tweezers with differential interference contrast signals,” Rev. Sci. Instrum. 73(4), 1687–1696 (2002). K. Visscher, S. P. Gross, and S. M. Block, “Construction of multiple-beam optical traps with nanometerresolution position sensing,” IEEE J. Sel. Top. Quantum Electron. 2(4), 1066–1076 (1996). K. Berg- Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75(3), 594–612 (2004). W. P. Wong and K. Halvorsen, “The effect of integration time on fluctuation measurements: calibrating an optical trap in the presence of motion blur,” Opt. Express 14(25), 12517–12531 (2006). B. M. Lansdorp and O. A. Saleh, “Power spectrum and Allan variance methods for calibrating single-molecule video-tracking instruments,” Rev. Sci. Instrum. 83(2), 025115 (2012). K. D. Wulff, D. G. Cole, and R. L. Clark, “An adaptive system identification approach to optical trap calibration,” Opt. Express 16(7), 4420–4425 (2008). C.-K. Yeh and P.-C. Li, “Doppler angle estimation using AR modeling,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 49(6), 683–692 (2002). S. Nassar, K.-P. Schwarz, N. El-Sheimy, and A. Noureldin, “Modeling inertial sensor errors using autoregressive (AR) models,” Navigation 51, 259–268 (2004). Z. Li, J. Shen, X. Sun, and Y. Wang, “Nanoparticle size measurement from dynamic light scattering data based on an autoregressive model,” Laser Phys. Lett. 10(9), 095701 (2013).
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Received 3 Mar 2014; revised 23 May 2014; accepted 13 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016956 | OPTICS EXPRESS 16956
14. E. D. Übeylı and İ. Güler, “Comparison of eigenvector methods with classical and model-based methods in analysis of internal carotid arterial Doppler signals,” Comput. Biol. Med. 33(6), 473–493 (2003). 15. Z. Q. Wang, “calibration OT based AR spectrum,” MatlabCentral (2014) http://www.mathworks.cn/matlabcentral/fileexchange/47059. 16. F. Czerwinski, A. C. Richardson, and L. B. Oddershede, “Quantifying Noise in Optical Tweezers by Allan Variance,” Opt. Express 17(15), 13255–13269 (2009). 17. K. Svoboda and S. M. Block, “Biological Applications of Optical Forces,” Annu. Rev. Biophys. Biomol. Struct. 23(1), 247–285 (1994). 18. H. Felgner, O. Müller, and M. Schliwa, “Calibration of Light Forces in Optical Tweezers,” Appl. Opt. 34(6), 977–982 (1995). 19. L. P. Ghislain, N. A. Switz, and W. W. Webb, “Measurement of Small Forces Using an Optical Trap,” Rev. Sci. Instrum. 65(9), 2762–2768 (1994). 20. K. C. Vermeulen, G. J. L. Wuite, G. J. M. Stienen, and C. F. Schmidt, “Optical trap stiffness in the presence and absence of spherical aberrations,” Appl. Opt. 45(8), 1812–1819 (2006). 21. K. C. Neuman, E. A. Abbondanzieri, and S. M. Block, “Measurement of the effective focal shift in an optical trap,” Opt. Lett. 30(11), 1318–1320 (2005).
1. Introduction Currently, optical tweezers are an important tool in the areas of biomedical and biophysical researches. The relatively small magnitudes and noninvasive nature of the forces exerted by an optical trap are especially useful to manipulate cells [1] and motor proteins [2, 3]. Besides optical manipulation, optical tweezers have also been used to measure weak forces involved in biological system such as to arrest swimming bacteria [4] or to investigate the nature of forces involved in linear motor proteins [3]. The optical forces are rarely measured directly. Instead, the stiffness of the trap is first determined, then used in conjunction with the measured displacement from the center of the trap to supply the force on an object through Hook’s law ( F = −kx ), where F is the force, k is the trapping stiffness, x is the displacement from the trapping center. Accordingly, to enable quantitative force measurements, the optical stiffness has to be determined firstly. There were some common methods for calibrating the trapping stiffness, such as the drag method [5], the equipartition theorem method [6], and the PSD method [7]. The drag method calibrates the stiffness by creating a fluid flow around the particle and observing the displacement. The stiffness can be calculated by k = γν / x , where ν is the velocity, and γ is Stoke’s drag coefficient ( γ = 6πη r with η fluid viscosity). This method requires an actuator to move the particle with respect to the fluid, e.g., a moving stage, pump, or the trap itself. The equipartition method uses the equipartition theorem in which the mean-squared displacement and thermal energy of the system are related to stiffness according to k = k BT / x 2 , where k B is Boltzmann’s constant and T is the absolute temperature. Because this method is affect by any added noise, drift and filter, it is usually the least accurate of all. The PSD method [8, 9], quite accurate when the particle’s viscous drag coefficient, γ , is known, is based on the determining the power spectrum of the position of a trapped object. With this method, the power spectral is fitted with a Lorentzian, yielding the corner frequency fc, which is proportional to the trap stiffness ( k = 2πγ f c ).The PSD method needs a fast detection system while the drag method does not need one. Thus either a fast camera or a quadrant photodiode interferometric detection system is required for PSD method and the experiments with PSD method are simpler and quicker than that with drag method. So the PSD method is used more widely than others. The PSD of the position of the trapping bead is often estimated with fast Fourier transform (FFT) method. This FFT-based power spectrum estimation method is known as classical methods and has been widely studied in many kinds of signals. This method is also the most commonly used signal processing technique for calibrating the optical tweezers. However, for most PSD calibrations, 2 s of measurement time and 10 kHz of sampling frequency are required [5, 10]. Thus, the FFT based method is not suitable for fast calibration due to the requirement of a relative longer data acquisition time. Recently, model-based methods (AR model is commonly used) are put forward to estimate the PSD. This method has been used in processing different signals [11–13], such as
#207561 - $15.00 USD (C) 2014 OSA
Received 3 Mar 2014; revised 23 May 2014; accepted 13 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016956 | OPTICS EXPRESS 16957
Doppler signals and the dynamic light scattering signals. In the Doppler signals research [11], the research has found the AR method outperformed the FFT method at smaller Doppler angles. The experimental data for Doppler angles, ranging from 33° to 72°, showed that the AR method using only eight flow samples had an average estimation error of 3.6 °,which compare favorably to the average error of 4.7 ° for the FFT method using 64 flow samples. In the dynamic light scattering signals research [13], Standard polystyrene particles of 50, 90 and 100 nm were measured. The experimental results indicated that the AR method had estimation error of 2%, which compare favorably to the error of 12% for the autocorrelation method using 4096 samples. In this letter, we show how to use AR spectrum with Burg algorithm to calibrating the optical tweezers stiffness. This method is very suitable for rapid measurement applications. Moreover, the spectrum base AR method is flat, therefore, the method does not require curve fitting which will decrease the estimating error. AR model is the common model. 2. AR method estimation and the stiffness calibration 2.1 Power spectrum estimation based on Burg arithmetic An autoregressive (AR) model is a possible method of modeling a stationary stochastic process. This model specifies that the output variable depends linearly on its own previous values. The autoregressive model can be written as p
x ( n ) = − ai x ( n − i ) + ω ( n )
0 ≤ n ≤ N −1
(1)
i =1
Where N is the length of the sampling data, p is the order of the AR model and ai (i = 1, 2, ⋅⋅⋅, p ) are the AR Process parameters, ω ( n ) denotes the white noise sequence with
a mean value of zero and variance of σ 2 . The power spectrum of x ( n ) can be expressed as P (e jω ) =
σ p
1 + ai e
2
(2)
2
− jω i
z
−i
.
i =1
Thus, the problem has come down to estimate the parameters ai of the AR model and the variance of the white noise sequence for the given sampling values. There are three kinds of arithmetic [11, 12, 14] for extracting these parameters: the autocorrelation arithmetic, the Burg arithmetic and the covariance arithmetic. The autocorrelation arithmetic is the simplest one, but its resolution is very low. The covariance arithmetic is the most complex one. The Burg algorithm is a more commonly used method because of its simplicity, acceptable accuracy, and smaller amounts of data. Moreover, the method has a fast algorithm. The Burg algorithm needs not directly estimate the parameters ai , instead, it estimates reflection coefficient uˆm . The coefficient uˆm is estimated directly with a recursive algorithm. In each recursion step, a single reflection coefficient is estimated. To estimate pth reflection coefficient an AR (P) model is fitted to the data with the first p − 1 reflection coefficients u1 , ⋅⋅⋅, u p −1 fixed to the value found in the previous steps. The parameters aˆi of step p are related to the parameters of the previous step by the Levinson–Durbin algorithm [12]. The new reflection coefficient u p is determined by minimizing the sum of the forward and backward residuals. An essential characteristic of the Burg algorithm is that the number of residuals decreases with each recursion step. The criterion to be minimized with respect to u p is the sum of absolute squares of the forward residuals and the backward residuals (See appendix).
#207561 - $15.00 USD (C) 2014 OSA
Received 3 Mar 2014; revised 23 May 2014; accepted 13 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016956 | OPTICS EXPRESS 16958
2.2 The order of the AR model The accuracy of the AR model depends greatly on the order of p. We use the final prediction error (FPE) to compute p, and the FPE is described as FPE ( i ) = ρi2
N + i +1 , N − i −1
(3)
Where N is the length of the sampling data, and ρi is the power estimation of the p order prediction error. When the order of the AR model increases, ρi will decrease, but N + i +1 will increase. Therefore, when the order p reaches some point, FPE ( i ) will have its N − i −1 minimal value, and that would be the best order p for the AR model.
2.3. The stiffness of optical trap measurement The theoretical calibration of the optical trap based on power spectrum can be expressed as [7] 2
Pi =< xi / Tmsr >=
k BT / 2π 2γ A = 2 . 2 2 f c + fi fc + fi 2
(4)
There has been an adopted FFT method. Here, we estimate the power spectrum density with A AR method. According to (4), Pmax = 2 when f i = 0 , then, there is relationship of f i = f c fc when Pi = Pmax / 2 . So f c can be obtained. Thus, according k = 2πγ f c , the stiffness of optical trap can be determined. The principle is show in the Fig. 1. This stiffness calibration method is implemented using a custom-made MATLAB program [15].
Fig. 1. AR spectrum of position signal for a 1 μm radius polystyrene bead immersed in water solution at room temperature. The laser power is 0.4 W, and the AR spectrum has been normalized.
3. Numerical studies In order to verify the AR method, we simulate the signal of the bead in optical trap with Monte-Carlo method [16]. The simulation conditions are as follows: the viscosity coefficient #207561 - $15.00 USD (C) 2014 OSA
Received 3 Mar 2014; revised 23 May 2014; accepted 13 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016956 | OPTICS EXPRESS 16959
of water is 0.89 × 10−3 N·s·m−2. The radius of the sphere is 0.5 μm, the temperature is 25°C, the sample frequency is 20 kHz, the stiffness and the data length N could be varied. At first, the different stiffness signals are simulated for the data length is 1 × 106. Each stiffness signal is simulated with 10 times. Then the stiffness of these signals is calibrated signal with AR method and FFT method particularly. The mean value stand deviation and the error of each stiffness are calculated and shown in the Table 1. It shows that both the AR method and FFT method can obtain the correct result. So, it also verifies the correction of the Monte-Carlo simulation method. With comparison of the error, we conclude that the result with AR method is more accurate than classic spectrum. Table 1. A comparison of the calibration methods performed on different stiffness simulation signal (data length is 1 × 106, each stiffness signal is simulated with 10 times) Simulation stiffness kx (pN/μm)
10 20 30 40 50 a
Error b(%)
Stiffness with AR method kx(pN/μm) Mean
STDa
10.04 20.12 30.12 40.09 50.04
0.12 0.11 0.12 0.16 0.36
Stands for Standard Deviation; b Error=
0.35 0.61 0.39 0.22 0.07
Stiffness with FFT method kx(pN/μm) Mean
STD
10.17 20.13 30.16 40.16 50.34
0.26 0.28 0.54 0.33 0.53
Error(%)
1.65 0.66 0.53 0.40 0.68
k x -Mean × 100% kx
Then, the signals of different data length are simulated for the fixed stiffness of 20pN/μm. Each length data signal is simulated with 10 times. We calculated the mean value and stand deviation and the error for each length data signal as shown in Table 2. The results show that the shorter data length is, the greater the fluctuation of the stiffness is. When N = 2000, the stiffness and the error with AR method and FFT method are 19.19 pN/μm, 4.02% and 22.24 pN/μm and 11.2% respectively; when N = 1000000, the stiffness and the error with AR method and with FFT method are 20.12 pN/μm, 0.61% and 20.13 pN/μm, 0.67% respectively. With comparison of the error in Table 2, we also conclude the calibration result with AR method is more precise than that with FFT method when the sample data length is shorter. Table 2. A Comparison of the calibration methods performed on different data length based on Monte-Carlo simulation signal (kx = 20 pN/μm, each length data signal is simulated with 10 times) Sample Data Length(N)
2000 3000 5000 10000 20000 1000000
AR method calibration kx (pN/nm) Mean
STD
19.19 20.09 20.30 19.88 19.81 20.12
1.76 3.09 2.19 0.83 0.71 0.11
Error(%)
4.02 0.45 1.48 0.61 0.97 0.61
FFT method calibration kx (pN/μm) Mean
STD
22.24 21. 98 21.64 20.39 20.25 20.13
5.06 6.83 4.88 2.88 2.92 0.28
Error(%)
11.20 9.90 8.20 1.97 1.23 0.67
4. Experimental verification
4.1 Instrument description The experimental set-up is schematically shown in Fig. 2. The optical tweezers was implemented in an inverted microscope (Olympus IX71) where the laser beam (AFL-106440-R-CL, Amonics Limited, 1064 nm) was tightly focused by an oil immersion objective. The laser was switched on at least an hour prior to experiment for optimal pointing stability.
#207561 - $15.00 USD (C) 2014 OSA
Received 3 Mar 2014; revised 23 May 2014; accepted 13 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016956 | OPTICS EXPRESS 16960
The chamber consists of two glass cover slips separated by one layer of double-sticky tape, and it was filled with a dilute solution of polystyrene spheres in millipore water. The experiments were carried out at room temperature (25°C). The scattered laser light was focused through a lens onto a photodiode after the sample chamber. Data was collected by a Quadrant photodiode (Hamamatsu, JAPAN, s1557). The recorded voltage signal was amplified, before streamed through an acquisition card (National Instruments, USA, PCI6251) onto a hard drive. Our custom-made data acquiring software was programmed in LABVIEW (National Instruments, USA). In addition, a video image of the trapped bead was recorded by a CCD camera (Coolsnap cf,USA). It ensured that only a single bead was in the trap both at the beginning and end of an experiment.
Fig. 2. The OT setup.
4.2 The experimental results and discussion To demonstrate the calibration method, we study the variation of the trap stiffness as a function of the data length and the trapping depth respectively. The experiment conditions are as follows: the laser power is 200 mW; the sample frequency is 20 kHz and data length is 20000; the diameter is 1.0 μm; the trap depth is 10, 15, 20 and 25 μm respectively. Every trap depth is measured with seven times. 1) The stiffness as a function of data length We calibrate the optical stiffness with AR method and FFT method for different lengths of data. The trap depth is 15 μm. The calibration results are shown in Fig. 3. It shows that the calibration result tends to a stable value with the increase of the data length and the calibration results with two methods are consistent. And the error bar shows that calibration result with AR method is more precise than that with FFT method. For example, when N = 2000, the relative error with AR method and with FFT method are 5.1% and 14.2% respectively.
#207561 - $15.00 USD (C) 2014 OSA
Received 3 Mar 2014; revised 23 May 2014; accepted 13 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016956 | OPTICS EXPRESS 16961
60
FFT Method AR Method
50
stiffness(pN/um)
40
30
20
10
0 0
5000
10000
15000
20000
data length
Fig. 3. Calibration of the optical tweezers at different data lengths where the polystyrene bead radius is 0.5μm and the trap depth is 15μm. These error bars are determined by calculating the standard deviation from a set of seven different measurements.
2) The stiffness as a function of trapping depth Trapping capability diminishes with the depth in the flow chamber. This is due to the optical aberration when using oil-immersion objectives [17]. We calibrate the stiffness with AR method and FFT method at different trapping depth of 10, 15, 20, 25 μm. The results with the two methods are shown in Fig. 4. 50
FFT Method AR Method
45
Stiffness(pN/um)
40 35 30 25 20 15 10 5 0 8
10
12
14
16
18
20
Trap depth(um)
22
24
26
Fig. 4. Calibration of the optical tweezers at different trapping depth. The polystyrene bead radius is 1 μm. These error bars are determined by calculating the standard deviation from a set of seven different measurements.
Figure 4 shows that the stiffness is decreasing when the trapping depth is increasing, which is in agreement with reports in the literature [18–21]. The two calibration results are consistent. Based on the error bar of Fig. 4, the results show that calibration from the AR method is more stable than the result obtained by FFT method. Those results also show that when a trap is to be used at a depth of 20 um, the aberrations are too big and the water immersion objectives should be used to avoid aberrations. The experiment results show that AR method successfully identify the corner frequency of the spherical particles confined in an optical trap. For short time series, the accuracy is
#207561 - $15.00 USD (C) 2014 OSA
Received 3 Mar 2014; revised 23 May 2014; accepted 13 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016956 | OPTICS EXPRESS 16962
improved. For example, in a 2000 data point series we found a reduction of 60% in trap stiffness estimation error. 5. Conclusion In this paper, we introduce a new calibration method based on AR spectrum. To validate this method, the trap stiffness at different trap depth and with different data length have been researched numerically and experimentally. All of these results validate that AR method provides the accurate results as the results obtained from the FFT method. Moreover, for short time series, the accuracy of calibration with this method is improved, for example in a 2000 data point series we found a reduction of 60% in trap stiffness estimation error. Accordingly, the data acquisition time is significantly reduced and real-time stiffness calibration has become feasible. This study may enable some chemical or biological research in when the experiment duration is short. Appendix The Burg arithmetic is a method of autoregressive power spectrum estimation in which the energy sum of forward and backward prediction errors is at its lowest under the Levinson– Durbin recursion constraint. Let the forward and backward prediction errors of the linearity prediction AR model be e pf and ebp , where f and b denote forward prediction and backward prediction, respectively, p
eˆ pf ( n ) = x ( n ) + aˆ p ( i ) x ( n − i ),
n = p + 1,..., N ,
(5)
i =1 p
eˆbp ( n ) = x ( n − p ) + aˆ *p ( i )x ( n − p + i ) ,
n = p + 1,..., N .
(6)
i =1
where N is the length of the sampling data. Let the power sum of forward and backward prediction errors is 1 2
ρ pfb = [ ρ pf + ρ bp ] =
2 2 1 N −1 f 1 N −1 b ep ( n ) + ep ( n ) N − p n= p N − p n= p
(7)
In the Eq. (7), when the order is from 1 to p, the prediction errors satisfy the following recursive-in-order expressions, emf ( n ) = emf −1 ( n ) + um emb −1 ( n − 1) emb ( n ) = emb −1 ( n ) + um* emf −1 ( n − 1) , e0f ( n ) = e0b ( n ) = x ( n )
m = 1, 2, , p
(8)
Thus in the Eq. (7), ρ pfb is only a function of reflection coefficient um , m = 1, 2, ⋅⋅⋅, p . As the order is m, let ρ mfb is the minimum value, then the reflection coefficient uˆm can be estimated. Substituting Eq. (8) into (7), let ∂ρ fb / ∂um = 0 , yields −2 uˆm =
N
n = m +1 N
n = m +1
f m −1
e
eˆmf −1 ( n )emb*−1 ( n − 1)
(n)
2
+
N
n = m +1
emb −1 ( n − 1)
2
.
(9)
Dual to Eq. (9), uˆm is obtained.
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Received 3 Mar 2014; revised 23 May 2014; accepted 13 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016956 | OPTICS EXPRESS 16963
Then, the parameters of the m order AR model can be computed by the Levinson–Durbin recursion formulas aˆm −1 ( i ) + uˆm am −1 ( m − i ) , aˆm ( i ) = uˆ p ,
i = 1,..., p − 1, i = p.
ρ mfb = (1 − um2 ) ρ mfb−1 .
(10) (11)
The preceding procedure can be summarized as below: a) The initial condition is e0f ( n ) = x ( n ) ,
e0b ( n ) = x ( n ) .
Substituting it into Eq. (9) we can obtain the uˆ1 . 2 1 N −1 x ( n ) , we obtain the AR model parameter a1 (1) = u1 , and N n=0 the power sum of forward prediction and backward prediction error ρ1fb ( 0 ) = (1 − u12 ) rx ( 0 ) .
b) According to rx ( 0 ) =
c) According to Eq. (8), we obtain forward prediction error e1f ( n ) and backward
prediction error e1b ( n ) . Then according Eq. (5), we estimate the Reflection coefficient.
d) According to Levinsion recurrence relation of Eq. (10) and Eq. (11), we obtain AR model parameter a2 (1) and a2 ( 2 ) , and ρ 2fb as m = 2 . e) Repeat the process, until m = p , and then we obtain all the AR model parameters. f) With the AR model parameters, we obtain the AR spectrum. g) Normalizing the AR spectrum, then let Pi = 1 / 2 , we obtain the corner frequency f c = fi . According to k = 2πγ f c , we obtained the optical trap stiffness. Acknowledgments
This work was financially supported by the National Natural Science Foundation of China (Grant Nos. 11374292, 21073174 and 11302220) and the National Basic Research Program of China (Grant No. 2011CB910402).
#207561 - $15.00 USD (C) 2014 OSA
Received 3 Mar 2014; revised 23 May 2014; accepted 13 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016956 | OPTICS EXPRESS 16964
Department of Optics and Optical Engineering, University of Science and Technology of China, Hefei, Anhui, 230026, China 2 Hefei National Laboratory for Physical Sciences at the Microscale, Hefei, Anhui, 230026, China * [email protected]
Abstract: The power spectrum density (PSD) has long been explored for calibrating optical tweezers stiffness. Fast Fourier transform (FFT) based spectral estimator is typically used. This approach requires a relatively longer data acquisition time to achieve adequate spectral resolution. In this paper, an autoregressive (AR) model is proposed to obtain the spectrum density using a limited number of samples. According to our method, the arithmetic model has been established with burg arithmetic, and the final prediction error criterion has been used to select the most appropriate order of the AR model, the power spectrum density has been estimated based the AR model. Then, the optical tweezers stiffness has been determined with the simple calculation from the power spectrum. Since only a small number of samples are used, the data acquisition time is significantly reduced and real-time stiffness calibration becomes feasible. To test this calibration method, we study the variation of the trap stiffness as a function of the parameters of the data length and the trapping depth. Both of the simulation and experiment results have showed that the presented method returns precise results and outperforms the conventional FFT method when using a limited number of samples. ©2014 Optical Society of America OCIS codes: (350.4855) Optical tweezers or optical manipulation; (170.4520) Optical confinement and manipulation; (120.4640) Optical instruments.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
M. C. Zhong, X. B. Wei, J. H. Zhou, Z. Q. Wang, and Y. M. Li, “Trapping red blood cells in living animals using optical tweezers,” Nat Commun 4, 1768 (2013). R. Reyes-Lamothe, D. J. Sherratt, and M. C. Leake, “Stoichiometry and Architecture of Active DNA Replication Machinery in Escherichia coli,” Science 328(5977), 498–501 (2010). F. M. Fazal and S. M. Block, “Optical tweezers study life under tension,” Nat. Photonics 5(6), 318–321 (2011). T. L. Min, P. J. Mears, L. M. Chubiz, C. V. Rao, I. Golding, and Y. R. Chemla, “High-resolution, long-term characterization of bacterial motility using optical tweezers,” Nat. Methods 6(11), 831–835 (2009). M. Capitanio, G. Romano, R. Ballerini, M. Giuntini, F. S. Pavone, D. Dunlap, and L. Finzi, “Calibration of optical tweezers with differential interference contrast signals,” Rev. Sci. Instrum. 73(4), 1687–1696 (2002). K. Visscher, S. P. Gross, and S. M. Block, “Construction of multiple-beam optical traps with nanometerresolution position sensing,” IEEE J. Sel. Top. Quantum Electron. 2(4), 1066–1076 (1996). K. Berg- Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75(3), 594–612 (2004). W. P. Wong and K. Halvorsen, “The effect of integration time on fluctuation measurements: calibrating an optical trap in the presence of motion blur,” Opt. Express 14(25), 12517–12531 (2006). B. M. Lansdorp and O. A. Saleh, “Power spectrum and Allan variance methods for calibrating single-molecule video-tracking instruments,” Rev. Sci. Instrum. 83(2), 025115 (2012). K. D. Wulff, D. G. Cole, and R. L. Clark, “An adaptive system identification approach to optical trap calibration,” Opt. Express 16(7), 4420–4425 (2008). C.-K. Yeh and P.-C. Li, “Doppler angle estimation using AR modeling,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 49(6), 683–692 (2002). S. Nassar, K.-P. Schwarz, N. El-Sheimy, and A. Noureldin, “Modeling inertial sensor errors using autoregressive (AR) models,” Navigation 51, 259–268 (2004). Z. Li, J. Shen, X. Sun, and Y. Wang, “Nanoparticle size measurement from dynamic light scattering data based on an autoregressive model,” Laser Phys. Lett. 10(9), 095701 (2013).
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Received 3 Mar 2014; revised 23 May 2014; accepted 13 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016956 | OPTICS EXPRESS 16956
14. E. D. Übeylı and İ. Güler, “Comparison of eigenvector methods with classical and model-based methods in analysis of internal carotid arterial Doppler signals,” Comput. Biol. Med. 33(6), 473–493 (2003). 15. Z. Q. Wang, “calibration OT based AR spectrum,” MatlabCentral (2014) http://www.mathworks.cn/matlabcentral/fileexchange/47059. 16. F. Czerwinski, A. C. Richardson, and L. B. Oddershede, “Quantifying Noise in Optical Tweezers by Allan Variance,” Opt. Express 17(15), 13255–13269 (2009). 17. K. Svoboda and S. M. Block, “Biological Applications of Optical Forces,” Annu. Rev. Biophys. Biomol. Struct. 23(1), 247–285 (1994). 18. H. Felgner, O. Müller, and M. Schliwa, “Calibration of Light Forces in Optical Tweezers,” Appl. Opt. 34(6), 977–982 (1995). 19. L. P. Ghislain, N. A. Switz, and W. W. Webb, “Measurement of Small Forces Using an Optical Trap,” Rev. Sci. Instrum. 65(9), 2762–2768 (1994). 20. K. C. Vermeulen, G. J. L. Wuite, G. J. M. Stienen, and C. F. Schmidt, “Optical trap stiffness in the presence and absence of spherical aberrations,” Appl. Opt. 45(8), 1812–1819 (2006). 21. K. C. Neuman, E. A. Abbondanzieri, and S. M. Block, “Measurement of the effective focal shift in an optical trap,” Opt. Lett. 30(11), 1318–1320 (2005).
1. Introduction Currently, optical tweezers are an important tool in the areas of biomedical and biophysical researches. The relatively small magnitudes and noninvasive nature of the forces exerted by an optical trap are especially useful to manipulate cells [1] and motor proteins [2, 3]. Besides optical manipulation, optical tweezers have also been used to measure weak forces involved in biological system such as to arrest swimming bacteria [4] or to investigate the nature of forces involved in linear motor proteins [3]. The optical forces are rarely measured directly. Instead, the stiffness of the trap is first determined, then used in conjunction with the measured displacement from the center of the trap to supply the force on an object through Hook’s law ( F = −kx ), where F is the force, k is the trapping stiffness, x is the displacement from the trapping center. Accordingly, to enable quantitative force measurements, the optical stiffness has to be determined firstly. There were some common methods for calibrating the trapping stiffness, such as the drag method [5], the equipartition theorem method [6], and the PSD method [7]. The drag method calibrates the stiffness by creating a fluid flow around the particle and observing the displacement. The stiffness can be calculated by k = γν / x , where ν is the velocity, and γ is Stoke’s drag coefficient ( γ = 6πη r with η fluid viscosity). This method requires an actuator to move the particle with respect to the fluid, e.g., a moving stage, pump, or the trap itself. The equipartition method uses the equipartition theorem in which the mean-squared displacement and thermal energy of the system are related to stiffness according to k = k BT / x 2 , where k B is Boltzmann’s constant and T is the absolute temperature. Because this method is affect by any added noise, drift and filter, it is usually the least accurate of all. The PSD method [8, 9], quite accurate when the particle’s viscous drag coefficient, γ , is known, is based on the determining the power spectrum of the position of a trapped object. With this method, the power spectral is fitted with a Lorentzian, yielding the corner frequency fc, which is proportional to the trap stiffness ( k = 2πγ f c ).The PSD method needs a fast detection system while the drag method does not need one. Thus either a fast camera or a quadrant photodiode interferometric detection system is required for PSD method and the experiments with PSD method are simpler and quicker than that with drag method. So the PSD method is used more widely than others. The PSD of the position of the trapping bead is often estimated with fast Fourier transform (FFT) method. This FFT-based power spectrum estimation method is known as classical methods and has been widely studied in many kinds of signals. This method is also the most commonly used signal processing technique for calibrating the optical tweezers. However, for most PSD calibrations, 2 s of measurement time and 10 kHz of sampling frequency are required [5, 10]. Thus, the FFT based method is not suitable for fast calibration due to the requirement of a relative longer data acquisition time. Recently, model-based methods (AR model is commonly used) are put forward to estimate the PSD. This method has been used in processing different signals [11–13], such as
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Received 3 Mar 2014; revised 23 May 2014; accepted 13 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016956 | OPTICS EXPRESS 16957
Doppler signals and the dynamic light scattering signals. In the Doppler signals research [11], the research has found the AR method outperformed the FFT method at smaller Doppler angles. The experimental data for Doppler angles, ranging from 33° to 72°, showed that the AR method using only eight flow samples had an average estimation error of 3.6 °,which compare favorably to the average error of 4.7 ° for the FFT method using 64 flow samples. In the dynamic light scattering signals research [13], Standard polystyrene particles of 50, 90 and 100 nm were measured. The experimental results indicated that the AR method had estimation error of 2%, which compare favorably to the error of 12% for the autocorrelation method using 4096 samples. In this letter, we show how to use AR spectrum with Burg algorithm to calibrating the optical tweezers stiffness. This method is very suitable for rapid measurement applications. Moreover, the spectrum base AR method is flat, therefore, the method does not require curve fitting which will decrease the estimating error. AR model is the common model. 2. AR method estimation and the stiffness calibration 2.1 Power spectrum estimation based on Burg arithmetic An autoregressive (AR) model is a possible method of modeling a stationary stochastic process. This model specifies that the output variable depends linearly on its own previous values. The autoregressive model can be written as p
x ( n ) = − ai x ( n − i ) + ω ( n )
0 ≤ n ≤ N −1
(1)
i =1
Where N is the length of the sampling data, p is the order of the AR model and ai (i = 1, 2, ⋅⋅⋅, p ) are the AR Process parameters, ω ( n ) denotes the white noise sequence with
a mean value of zero and variance of σ 2 . The power spectrum of x ( n ) can be expressed as P (e jω ) =
σ p
1 + ai e
2
(2)
2
− jω i
z
−i
.
i =1
Thus, the problem has come down to estimate the parameters ai of the AR model and the variance of the white noise sequence for the given sampling values. There are three kinds of arithmetic [11, 12, 14] for extracting these parameters: the autocorrelation arithmetic, the Burg arithmetic and the covariance arithmetic. The autocorrelation arithmetic is the simplest one, but its resolution is very low. The covariance arithmetic is the most complex one. The Burg algorithm is a more commonly used method because of its simplicity, acceptable accuracy, and smaller amounts of data. Moreover, the method has a fast algorithm. The Burg algorithm needs not directly estimate the parameters ai , instead, it estimates reflection coefficient uˆm . The coefficient uˆm is estimated directly with a recursive algorithm. In each recursion step, a single reflection coefficient is estimated. To estimate pth reflection coefficient an AR (P) model is fitted to the data with the first p − 1 reflection coefficients u1 , ⋅⋅⋅, u p −1 fixed to the value found in the previous steps. The parameters aˆi of step p are related to the parameters of the previous step by the Levinson–Durbin algorithm [12]. The new reflection coefficient u p is determined by minimizing the sum of the forward and backward residuals. An essential characteristic of the Burg algorithm is that the number of residuals decreases with each recursion step. The criterion to be minimized with respect to u p is the sum of absolute squares of the forward residuals and the backward residuals (See appendix).
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Received 3 Mar 2014; revised 23 May 2014; accepted 13 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016956 | OPTICS EXPRESS 16958
2.2 The order of the AR model The accuracy of the AR model depends greatly on the order of p. We use the final prediction error (FPE) to compute p, and the FPE is described as FPE ( i ) = ρi2
N + i +1 , N − i −1
(3)
Where N is the length of the sampling data, and ρi is the power estimation of the p order prediction error. When the order of the AR model increases, ρi will decrease, but N + i +1 will increase. Therefore, when the order p reaches some point, FPE ( i ) will have its N − i −1 minimal value, and that would be the best order p for the AR model.
2.3. The stiffness of optical trap measurement The theoretical calibration of the optical trap based on power spectrum can be expressed as [7] 2
Pi =< xi / Tmsr >=
k BT / 2π 2γ A = 2 . 2 2 f c + fi fc + fi 2
(4)
There has been an adopted FFT method. Here, we estimate the power spectrum density with A AR method. According to (4), Pmax = 2 when f i = 0 , then, there is relationship of f i = f c fc when Pi = Pmax / 2 . So f c can be obtained. Thus, according k = 2πγ f c , the stiffness of optical trap can be determined. The principle is show in the Fig. 1. This stiffness calibration method is implemented using a custom-made MATLAB program [15].
Fig. 1. AR spectrum of position signal for a 1 μm radius polystyrene bead immersed in water solution at room temperature. The laser power is 0.4 W, and the AR spectrum has been normalized.
3. Numerical studies In order to verify the AR method, we simulate the signal of the bead in optical trap with Monte-Carlo method [16]. The simulation conditions are as follows: the viscosity coefficient #207561 - $15.00 USD (C) 2014 OSA
Received 3 Mar 2014; revised 23 May 2014; accepted 13 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016956 | OPTICS EXPRESS 16959
of water is 0.89 × 10−3 N·s·m−2. The radius of the sphere is 0.5 μm, the temperature is 25°C, the sample frequency is 20 kHz, the stiffness and the data length N could be varied. At first, the different stiffness signals are simulated for the data length is 1 × 106. Each stiffness signal is simulated with 10 times. Then the stiffness of these signals is calibrated signal with AR method and FFT method particularly. The mean value stand deviation and the error of each stiffness are calculated and shown in the Table 1. It shows that both the AR method and FFT method can obtain the correct result. So, it also verifies the correction of the Monte-Carlo simulation method. With comparison of the error, we conclude that the result with AR method is more accurate than classic spectrum. Table 1. A comparison of the calibration methods performed on different stiffness simulation signal (data length is 1 × 106, each stiffness signal is simulated with 10 times) Simulation stiffness kx (pN/μm)
10 20 30 40 50 a
Error b(%)
Stiffness with AR method kx(pN/μm) Mean
STDa
10.04 20.12 30.12 40.09 50.04
0.12 0.11 0.12 0.16 0.36
Stands for Standard Deviation; b Error=
0.35 0.61 0.39 0.22 0.07
Stiffness with FFT method kx(pN/μm) Mean
STD
10.17 20.13 30.16 40.16 50.34
0.26 0.28 0.54 0.33 0.53
Error(%)
1.65 0.66 0.53 0.40 0.68
k x -Mean × 100% kx
Then, the signals of different data length are simulated for the fixed stiffness of 20pN/μm. Each length data signal is simulated with 10 times. We calculated the mean value and stand deviation and the error for each length data signal as shown in Table 2. The results show that the shorter data length is, the greater the fluctuation of the stiffness is. When N = 2000, the stiffness and the error with AR method and FFT method are 19.19 pN/μm, 4.02% and 22.24 pN/μm and 11.2% respectively; when N = 1000000, the stiffness and the error with AR method and with FFT method are 20.12 pN/μm, 0.61% and 20.13 pN/μm, 0.67% respectively. With comparison of the error in Table 2, we also conclude the calibration result with AR method is more precise than that with FFT method when the sample data length is shorter. Table 2. A Comparison of the calibration methods performed on different data length based on Monte-Carlo simulation signal (kx = 20 pN/μm, each length data signal is simulated with 10 times) Sample Data Length(N)
2000 3000 5000 10000 20000 1000000
AR method calibration kx (pN/nm) Mean
STD
19.19 20.09 20.30 19.88 19.81 20.12
1.76 3.09 2.19 0.83 0.71 0.11
Error(%)
4.02 0.45 1.48 0.61 0.97 0.61
FFT method calibration kx (pN/μm) Mean
STD
22.24 21. 98 21.64 20.39 20.25 20.13
5.06 6.83 4.88 2.88 2.92 0.28
Error(%)
11.20 9.90 8.20 1.97 1.23 0.67
4. Experimental verification
4.1 Instrument description The experimental set-up is schematically shown in Fig. 2. The optical tweezers was implemented in an inverted microscope (Olympus IX71) where the laser beam (AFL-106440-R-CL, Amonics Limited, 1064 nm) was tightly focused by an oil immersion objective. The laser was switched on at least an hour prior to experiment for optimal pointing stability.
#207561 - $15.00 USD (C) 2014 OSA
Received 3 Mar 2014; revised 23 May 2014; accepted 13 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016956 | OPTICS EXPRESS 16960
The chamber consists of two glass cover slips separated by one layer of double-sticky tape, and it was filled with a dilute solution of polystyrene spheres in millipore water. The experiments were carried out at room temperature (25°C). The scattered laser light was focused through a lens onto a photodiode after the sample chamber. Data was collected by a Quadrant photodiode (Hamamatsu, JAPAN, s1557). The recorded voltage signal was amplified, before streamed through an acquisition card (National Instruments, USA, PCI6251) onto a hard drive. Our custom-made data acquiring software was programmed in LABVIEW (National Instruments, USA). In addition, a video image of the trapped bead was recorded by a CCD camera (Coolsnap cf,USA). It ensured that only a single bead was in the trap both at the beginning and end of an experiment.
Fig. 2. The OT setup.
4.2 The experimental results and discussion To demonstrate the calibration method, we study the variation of the trap stiffness as a function of the data length and the trapping depth respectively. The experiment conditions are as follows: the laser power is 200 mW; the sample frequency is 20 kHz and data length is 20000; the diameter is 1.0 μm; the trap depth is 10, 15, 20 and 25 μm respectively. Every trap depth is measured with seven times. 1) The stiffness as a function of data length We calibrate the optical stiffness with AR method and FFT method for different lengths of data. The trap depth is 15 μm. The calibration results are shown in Fig. 3. It shows that the calibration result tends to a stable value with the increase of the data length and the calibration results with two methods are consistent. And the error bar shows that calibration result with AR method is more precise than that with FFT method. For example, when N = 2000, the relative error with AR method and with FFT method are 5.1% and 14.2% respectively.
#207561 - $15.00 USD (C) 2014 OSA
Received 3 Mar 2014; revised 23 May 2014; accepted 13 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016956 | OPTICS EXPRESS 16961
60
FFT Method AR Method
50
stiffness(pN/um)
40
30
20
10
0 0
5000
10000
15000
20000
data length
Fig. 3. Calibration of the optical tweezers at different data lengths where the polystyrene bead radius is 0.5μm and the trap depth is 15μm. These error bars are determined by calculating the standard deviation from a set of seven different measurements.
2) The stiffness as a function of trapping depth Trapping capability diminishes with the depth in the flow chamber. This is due to the optical aberration when using oil-immersion objectives [17]. We calibrate the stiffness with AR method and FFT method at different trapping depth of 10, 15, 20, 25 μm. The results with the two methods are shown in Fig. 4. 50
FFT Method AR Method
45
Stiffness(pN/um)
40 35 30 25 20 15 10 5 0 8
10
12
14
16
18
20
Trap depth(um)
22
24
26
Fig. 4. Calibration of the optical tweezers at different trapping depth. The polystyrene bead radius is 1 μm. These error bars are determined by calculating the standard deviation from a set of seven different measurements.
Figure 4 shows that the stiffness is decreasing when the trapping depth is increasing, which is in agreement with reports in the literature [18–21]. The two calibration results are consistent. Based on the error bar of Fig. 4, the results show that calibration from the AR method is more stable than the result obtained by FFT method. Those results also show that when a trap is to be used at a depth of 20 um, the aberrations are too big and the water immersion objectives should be used to avoid aberrations. The experiment results show that AR method successfully identify the corner frequency of the spherical particles confined in an optical trap. For short time series, the accuracy is
#207561 - $15.00 USD (C) 2014 OSA
Received 3 Mar 2014; revised 23 May 2014; accepted 13 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016956 | OPTICS EXPRESS 16962
improved. For example, in a 2000 data point series we found a reduction of 60% in trap stiffness estimation error. 5. Conclusion In this paper, we introduce a new calibration method based on AR spectrum. To validate this method, the trap stiffness at different trap depth and with different data length have been researched numerically and experimentally. All of these results validate that AR method provides the accurate results as the results obtained from the FFT method. Moreover, for short time series, the accuracy of calibration with this method is improved, for example in a 2000 data point series we found a reduction of 60% in trap stiffness estimation error. Accordingly, the data acquisition time is significantly reduced and real-time stiffness calibration has become feasible. This study may enable some chemical or biological research in when the experiment duration is short. Appendix The Burg arithmetic is a method of autoregressive power spectrum estimation in which the energy sum of forward and backward prediction errors is at its lowest under the Levinson– Durbin recursion constraint. Let the forward and backward prediction errors of the linearity prediction AR model be e pf and ebp , where f and b denote forward prediction and backward prediction, respectively, p
eˆ pf ( n ) = x ( n ) + aˆ p ( i ) x ( n − i ),
n = p + 1,..., N ,
(5)
i =1 p
eˆbp ( n ) = x ( n − p ) + aˆ *p ( i )x ( n − p + i ) ,
n = p + 1,..., N .
(6)
i =1
where N is the length of the sampling data. Let the power sum of forward and backward prediction errors is 1 2
ρ pfb = [ ρ pf + ρ bp ] =
2 2 1 N −1 f 1 N −1 b ep ( n ) + ep ( n ) N − p n= p N − p n= p
(7)
In the Eq. (7), when the order is from 1 to p, the prediction errors satisfy the following recursive-in-order expressions, emf ( n ) = emf −1 ( n ) + um emb −1 ( n − 1) emb ( n ) = emb −1 ( n ) + um* emf −1 ( n − 1) , e0f ( n ) = e0b ( n ) = x ( n )
m = 1, 2, , p
(8)
Thus in the Eq. (7), ρ pfb is only a function of reflection coefficient um , m = 1, 2, ⋅⋅⋅, p . As the order is m, let ρ mfb is the minimum value, then the reflection coefficient uˆm can be estimated. Substituting Eq. (8) into (7), let ∂ρ fb / ∂um = 0 , yields −2 uˆm =
N
n = m +1 N
n = m +1
f m −1
e
eˆmf −1 ( n )emb*−1 ( n − 1)
(n)
2
+
N
n = m +1
emb −1 ( n − 1)
2
.
(9)
Dual to Eq. (9), uˆm is obtained.
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Received 3 Mar 2014; revised 23 May 2014; accepted 13 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016956 | OPTICS EXPRESS 16963
Then, the parameters of the m order AR model can be computed by the Levinson–Durbin recursion formulas aˆm −1 ( i ) + uˆm am −1 ( m − i ) , aˆm ( i ) = uˆ p ,
i = 1,..., p − 1, i = p.
ρ mfb = (1 − um2 ) ρ mfb−1 .
(10) (11)
The preceding procedure can be summarized as below: a) The initial condition is e0f ( n ) = x ( n ) ,
e0b ( n ) = x ( n ) .
Substituting it into Eq. (9) we can obtain the uˆ1 . 2 1 N −1 x ( n ) , we obtain the AR model parameter a1 (1) = u1 , and N n=0 the power sum of forward prediction and backward prediction error ρ1fb ( 0 ) = (1 − u12 ) rx ( 0 ) .
b) According to rx ( 0 ) =
c) According to Eq. (8), we obtain forward prediction error e1f ( n ) and backward
prediction error e1b ( n ) . Then according Eq. (5), we estimate the Reflection coefficient.
d) According to Levinsion recurrence relation of Eq. (10) and Eq. (11), we obtain AR model parameter a2 (1) and a2 ( 2 ) , and ρ 2fb as m = 2 . e) Repeat the process, until m = p , and then we obtain all the AR model parameters. f) With the AR model parameters, we obtain the AR spectrum. g) Normalizing the AR spectrum, then let Pi = 1 / 2 , we obtain the corner frequency f c = fi . According to k = 2πγ f c , we obtained the optical trap stiffness. Acknowledgments
This work was financially supported by the National Natural Science Foundation of China (Grant Nos. 11374292, 21073174 and 11302220) and the National Basic Research Program of China (Grant No. 2011CB910402).
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Received 3 Mar 2014; revised 23 May 2014; accepted 13 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016956 | OPTICS EXPRESS 16964
