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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 4 — Feb. 14, 2011
  • pp: 2874–2885
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Active beam position stabilization of pulsed lasers for long-distance ion profile diagnostics at the Spallation Neutron Source (SNS)

Robert A. Hardin, Yun Liu, Cary Long, Alexander Aleksandrov, and Willem Blokland  »View Author Affiliations


Optics Express, Vol. 19, Issue 4, pp. 2874-2885 (2011)
http://dx.doi.org/10.1364/OE.19.002874


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Abstract

A high peak-power Q-switched laser has been used to monitor the ion beam profiles in the superconducting linac at the Spallation Neutron Source (SNS). The laser beam suffers from position drift due to movement, vibration, or thermal effects on the optical components in the 250-meter long laser beam transport line. We have designed, bench-tested, and implemented a beam position stabilization system by using an Ethernet CMOS camera, computer image processing and analysis, and a piezo-driven mirror platform. The system can respond at frequencies up to 30 Hz with a high position detection accuracy. With the beam stabilization system, we have achieved a laser beam pointing stability within a range of 2 μrad (horizontal) to 4 μrad (vertical), corresponding to beam drifts of only 0.5 mm × 1 mm at the furthest measurement station located 250 meters away from the light source.

© 2011 OSA

1. Introduction

Variations in laser beam position can be a serious problem in experiments employing laser-based diagnostics and maintaining a stable beam position is a crucial element in many laser-based applications [1

1. A. Stalmashonak, N. Zhavoronkov, I. V. Hertel, S. Vetrov, and K. Schmid, “Spatial control of femtosecond laser system output with submicroradian accuracy,” Appl. Opt. 45(6), 1271–1274 (2006). [CrossRef] [PubMed]

4

4. J. Liu, Y. Kida, T. Teramoto, and T. Kobayashi, “Generation of stable sub-10 fs pulses at 400 nm in a hollow fiber for UV pump-probe experiment,” Opt. Express 18(5), 4664–4672 (2010). [CrossRef] [PubMed]

]. The laser wire system at the Spallation Neutron Source (SNS) located at Oak Ridge National Laboratory (ORNL) is used to measure ion beam density profiles [5

5. Y. Liu, A. Aleksandrov, S. Assadi, W. Blokland, C. Deibele, W. Grice, C. Long, T. Pelaia, and A. Webster, “Laser wire beam profile monitor in the spallation neutron source (SNS) superconducting linac,” Nucl. Instr. Meth. A 612(2), 241–253 (2010). [CrossRef]

,6

6. W. Blokland, A. Barker, and W. Grice, “Drift Compensation for the SNS Laserwire,” Proceedings of ICALEPCS 2007, Knoxville, Tennessee, USA (2007).

] in a noninvasive manner. Using a single light source to perform the profile measurements along the superconducting linac (SCL) is operationally and economically more efficient than multiple sources. Additionally, due to radiation in the accelerator beam line the light source is placed in a separate building and a laser transport line (LTL) is used to deliver the beam to the measurement stations. Since the total beam path length is up to 250 meters and the laser and transport line are at different building foundations, even a small drift can cause the laser beam to miss the downstream optics. Therefore, stabilization of the laser beam with a feedback system is essential to ensure the laser beam is transported to the furthest measurement station.

Figure 1
Fig. 1 (Top) Measured beam position as a function of time for the SNS laser wire system and the (Bottom) frequency spectrum of those positions.
shows the beam position drift over a 30 minute period and the frequency spectrum of those drifts. The spectrum shows that the beam drift appears to be a low frequency noise with major frequency components lower than 1 to 2 Hz. This indicates that the typical movement the feedback system must respond to is the slow drift of the beam. A high peak power Q-switched Nd:YAG laser was used as the laser wire light source, capable of producing a maximum pulse energy of 1.5 J with a 7 ns pulse width and a 30 Hz pulse rate. The high peak power is needed to achieve a sufficient signal-to-noise ratio for ion beam profile measurements, while the pulsed nature of the laser requires a discrete feedback algorithm. To avoid divergence, due to the long distance the beam must travel, the beam size has to be large (~1 inch diameter). Additionally, due to the large beam size, a conventional position sensor cannot be used for the beam position detection due to limited space in the measurement stations as well as limited dynamic range and the potential for damage. A Prosilica GE640 camera was used in conjunction with analysis software that calculates the centroid of the beam position. With a large laser beam, the feedback mirror mount must also be capable of driving a large two-inch diameter mirror. A more detailed description of the laser wire system is given in Ref. 5

5. Y. Liu, A. Aleksandrov, S. Assadi, W. Blokland, C. Deibele, W. Grice, C. Long, T. Pelaia, and A. Webster, “Laser wire beam profile monitor in the spallation neutron source (SNS) superconducting linac,” Nucl. Instr. Meth. A 612(2), 241–253 (2010). [CrossRef]

.

In this work we propose a simple modified feedback algorithm to use in conjunction with a new piezomotor driven mirror mount. We analyze the algorithm to determine its capabilities and limitations and then implement and bench test the algorithm and piezomotor mount before installing them in the SNS laser wire system for testing. Both the initial bench and laser wire tests show a significant improvement in beam position stabilization using the feedback algorithm and piezomotor mirror mount. By using the piezoelectric driven mount and optimizing the overall image acquisition and processing speed, the entire feedback system is able to respond at the laser repetition rate. As a result, the new feedback system effectively stabilizes the laser beam position on a pulse to pulse basis over distances of 250 meters to the current observational limits of our system. The implemented feedback system can easily be used in many other general beam position stabilization applications.

2. Modeling and bench test experiment of feedback scheme

The algorithm used for the feedback routine is given by Eq. (1), where y(t) is the observed position of the laser beam, x(t) is the external perturbation input which represents a modulation signal in the simulation/bench test experiment or an external noise in the actual system, V(t) is the output voltage from the piezomotor controller, δ(t) is the difference between the position set point SP and the observed position in pixels y(t), γ is the feedback gain, and Vpp (Volt per pixel) is the coefficient between the controller voltage and the beam position. Vpp is measured by manually adjusting the output voltages and observing the number of pixels the beam moves for each axis (horizontal and vertical). Obviously, Vpp is both an axis and a (image sensor) location dependent parameter. td is the time difference between two consecutive measurements and in our experiments using pulsed laser, td = 1/fd where fd is the repetition rate of the laser pulse.

y(t)=x(t)+V(t)/Vpp,
(1a)
V(t)=V(ttd)+γδ(ttd),
(1b)
δ(t)=Vpp[SPy(t)].
(1c)

Making a Fourier transform over the above equations, we can have a transfer function in the frequency domain as

T(ω)=Y(ω)X(ω)=1exp(iωtd)1+(γ1)exp(iωtd).
(2)

The amplitude of the transfer function, defined as the (noise) reduction ratio, is then expressed by Eq. (3) as a function of frequency and gain factor

g(f)=|Y(f)X(f)|=11γ+(γ/2sin(πftd))2.
(3)

A plot of the reduction ratio spectrum at a number of gain factors is shown in Fig. 3
Fig. 3 Nyquist stability diagram for gain factors of 0.1, 0.5, 1, 1.5, 2, and 3 as a function of frequency for f d = 20 Hz.
. Here we assumed the laser repetition rate fd=20 Hz to match the bench test parameter.

Figure 2
Fig. 2 Transfer function magnitudes for gain factors of 0.1, 0.5, 1, 1.5, 2, and 3 as a function of frequency for f d=20 Hz.
illustrates that as the gain factor increases, the frequency bandwidth producing an effective reduction (i.e., reduction ratios less than 1) also increases. This would indicate that the larger the gain factor is, except around γ=2 where the system shows a resonant response, the better reduction ratio the feedback system will achieve. However, this does not take into consideration the stability.

To determine the stability for the gain factors we employ the Nyquist stability criterion by examining the (γ-1)exp(-iωtd) factor in the denominator, which we define as D(). The Nyquist stability diagram is a graphical method of applying the stability criterion by plotting the real and imaginary parts of D() in the complex plane, and determining if the point (−1,0) is enclosed by the plot D() [11

11. D. A. Neamen, Electronic Circuit Analysis and Design (McGraw-Hill, 2001).

,12

12. J. DiStefano, A. Stubberud, and I. Williams, Schaum’s Outlines: Feedback and Control Systems, 2nd Edition. (McGraw-Hill, 1990).

]. If the Nyquist plot loop encircles or passes through the point (−1,0) the system is unstable, otherwise it is stable.

The Nyquist stability diagrams for the same gain factors in Fig. 2 are shown in Fig. 3. The stability criterion point of (−1,0) is denoted as a diamond. While a γ of 0.1 is close to being unstable, as the gain is increased to 2, the Nyquist plots first shrink in to the origin at a gain factor of 1 (square), and then expands again until at encircling the point (−1,0) at a gain factor of 2 and becoming unstable. Note that the plot of γ= 0.5 overlaps with that of γ= 1.5. Thus the Nyquist stability diagrams indicate the range of stability for the gain factors to be 0<γ<2. Furthermore, for the gain factors within the range of stability, the overall stability does not depend on frequency.

A schematic of the experimental setup for bench testing the feedback scheme is shown in Fig. 4
Fig. 4 Bench test experimental setup. FBM is the feedback mirror, DM is the driving mirror, and VS is the view screen.
. Two laser sources were used in the bench test experiments: a simple cw laser and a 20 Hz pulsed Q-switched laser. The cw laser is used to measure the frequency response of the mirror mount, while the pulsed laser is used to test the feedback algorithm. The beam path from the laser source consists of the feedback mirror (FBM), a driving mirror (DM), a few sets of beam relay mirrors (not shown), and a view screen (VS). The beam path length from the FBM to the target is approximately 12 meters. The FBM is a 2” mirror mounted on a piezo tilt platform (Physik Instrumente GmbH S330.2SD). The DM is used to generate external perturbation to examine the feedback routine. The view screen is a 2” diameter plate for use with the Prosilica GE640 camera for beam position imaging. The camera sends the images to the Camera PC for position (horizontal and vertical) analysis, and then the position information is transferred over an Ethernet connection to the Feedback PC. The laser beam position is determined by calculating a weighted centroid calculated from a large number of pixels of the image sensor, the thermal noise of the image sensor has very little influence on the calculation of the beam position, assuming that the spatial beam profile of the laser does not change significantly on a shot-to-shot basis. Other possible noise limitation in the position sensing system is the saturation of some pixels on the image sensor which can cause a significant error in the centroid calculation. In the actual system, the gain level and the aperture size of the camera are adjusted so that the pixels do not saturate. Therefore, the noise limitation in the position sensing system is negligible. The Feedback PC running the feedback routine outputs voltages using a NI 6052E card through the Physik Instrumente GmbH E616.SS0 Piezo controller to the FBM.

The first test of the FBM is to investigate the frequency response of the piezo tilt platform with the 2” mirror. We drive the FBM with a sinusoidal signal generated from the Feedback PC while measuring a cw laser beam position using a position sensitive detector as the target. Both the driving signal and the position signal are recorded from an oscilloscope for analysis. The position variation amplitudes, normalized to the peak, are shown in Fig. 5
Fig. 5 Mirror response to sinusoidal driving as a function of frequency.
. The magnitude decays to approximately −3 dB around 30 Hz, which is the repetition rate of the laser used in the laser wire system. Thus, the mirror mount has no significant issues responding to the laser pulse rates of the laser wire system in addition to allowing us the ability to ignore the frequency dependence of the mirror based on the frequency range of motion observed in Fig. 1.

To verify whether the feedback system can reduce the external perturbation, we drive the DM (See Fig. 4) with a sinusoidal input and measure the laser beam position with the feedback system off and on. The experiment has been conducted for different oscillation frequencies (of the driving signal) and various gain factors γ. A comparison of the analytical (Eq. (3)) and experimental reduction ratios (feedback ON/feedback OFF magnitudes) for several gain factors, as a function of driving frequency, is shown in Fig. 6
Fig. 6 Reduction Ratio experimental data and model comparison. Marks are the bench test data and lines are modeling results.
. The reduction ratios are determined by calculating the standard deviation on the position differences from the set point over 10 oscillation periods in both feedback off and on phases. A set of vertical experimental reduction ratios as a function of frequency are indicated by the data points (circle, square, etc.) while the analytical reduction ratios are shown as lines. As the driving frequency is reduced from 3 Hz, the reduction ratios for all gain factors approach the limit of a perfect feedback system of zero. Assuming a single pixel to be the minimum amplitude, including any errors, the reduction ratio minimum would be approximately 0.043 in the horizontal and 0.037 in the vertical. Additionally, for all gain factors tested, reduction ratios at or below 2 Hz are below 1. At 3 Hz and the gain factors of 0.5 and 0.8, the beam movement is no longer being reduced, but rather being enhanced. This demonstrates that the piezoelectric driven mirror and feedback routine examined here should easily compensate beam motion (reduction ratio < 1) up to frequencies of 3 Hz, or roughly 15% of the laser pulse rate (20 Hz), for an appropriate gain factor.

Experimental data up to 2 Hz fit the analytical result quite well, with errors typically within 5% of the model, while at 3 Hz the error typically runs around 13%. To achieve reduction ratios of less than 0.5, Fig. 6 suggests that gain factors in the range of 1.2 to 1.5 in conjunction with oscillation frequencies below 2 Hz are needed. Based on these data, the piezomotor mirror and feedback algorithm tests were complete and installed in the SNS laser wire system for full operational tests.

3. Application to SNS laser wire system

Since the beam movement is no longer a driven sinusoid, the analysis does not include the reduction ratios as a function of frequency, but has changed to monitoring the horizontal and vertical positions over time with the feedback system on or off. This allows for determining a spatial distribution function map of the beam positions by counting the number of times the beam is at a specific x and y coordinate over a particular amount of time. The standard deviation for each measurement is calculated to determine a consistent range of spatial variation. This provides the means for determining the radial stability calculations for each camera and camera axis, based on the values given in Table 1.

Shown in Fig. 8
Fig. 8 Beam position standard deviations for two unique position set points with the feedback on for cameras A and B as a function of gain factors.
are feedback measurements made with each camera for two unique set points and three gain factors (0.5, 1, and 1.5). These data were taken over a two minute span with a 30 Hz laser pulse rate and had 3600 points each. The standard deviations were calculated and clearly for a gain factor of 1 the standard deviations for both cameras, and both axes, are maintained at or below 0.75 mm (1 pixel) of deviation. While for gain factors of 0.5 and 1.5, the standard deviations vary significantly relative to those with a gain factor of 1. Thus while all three gain factors produce stabilization of the beam, a gain factor of 1 produces the best result for stabilizing the beam position in the SNS laser wire system. This can be explained based on the linear stability analysis of the control model. Applying a small perturbation to the steady state of Eq. (1), one can obtain the Lyapunov exponent that is proportional to ln(1-γ). Negative (real part of) Lyapunov exponents (corresponding to 0<γ<2) are necessary to achieve a stable feedback control. At γ=1, the Lyapunov exponent has a value of negative infinity, which indicates the feedback system is most stable at the gain factor of unity [13

13. E. Ott, Chaos in Dynamical System (Cambridge University Press, 2002).

]. Fig. 8 also illustrates that the location of the set point can play a role in how well the feedback system stabilizes the beam. For example, the data for camera B with a gain factor of 0.5 shows a standard deviation of ~2.2 mm (red open circle) for one set point and a deviation of ~0.6 mm (black closed circle). While not investigated in this work, a more thorough examination of the set point effect on the beam stabilization would be beneficial.

To illustrate the effectiveness of the feedback system a spatial distribution map for both cameras with the feedback off and on is shown in Fig. 9
Fig. 9 Spatial distribution maps for Camera B (a) feedback off, (b) feedback on, and Camera A (c) feedback off, (d) feedback on. The feedback on plots (b) and (d) have gain factors of 1.0.
. The spatial distribution maps have been shifted relative to their respective set points to show the relative differences about a common point. Figures 9(a) and (b) show Camera B with the feedback off and on, while (c) and (d) is Camera A with the feedback off and on. Both Figs. 9(b) and (d) have gain factors of 1.0. While (a) and (c) show the beam moving over a large area without the feedback, when the feedback is on, both locations show a compact range of positions. A calculation of the standard deviations, reduction ratios, and angular stability for the data shown in Fig. 9 is given in Table 2

Table 2. Laser wire feedback system standard deviations, reduction ratios, and angular stability with the feedback on for a gain factor of 1.

table-icon
View This Table
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.

For these data the standard deviations with the feedback off range from 1.8 to 5.9 pixels, while the range of standard deviations with the feedback on only range from 0.61 to 1.0 pixel, producing reduction ratios of 0.15 to 0.36. The important point is that while the range of movement with the feedback off may vary over a large range, the standard deviation of the movement with the feedback on is significantly reduced to less than a single pixel about the set point. Thus, based on the calculated angular resolution of the cameras in Table 1, the angular stability is shown to range from 2.1 to 4.2 μrad.

4. Conclusions

An improved piezoelectric mirror based feedback system has been designed and implemented in the SNS laser wire system to stabilize the beam position of a 30 Hz Q-switched laser. The piezomotor driven mirror is able to respond at the laser pulse repetition rate. The feedback scheme introduced in this work shows stability for a gain factor range of 0 < γ< 2, and produces successful noise reduction effect for a bandwidth of up to 3 Hz. Bench test data agree with the analytical prediction quite well. After the installation of the feedback system in the SNS laser wire diagnostic system, the standard deviation of the laser beam position was reduced to less than one pixel, corresponding to angular stabilities within a range of 2.1 to 4.2 μrad over a distance of 225 meters. The laser beam pointing stabilization has significantly improved the laser wire measurement performance.

While we have shown the ability to stabilize the beam position to within a single pixel with our current measurement capabilities, there is still a room for improvement. The first would be to address the set point position sensitivity of the feedback. While for a gain factor of 1 the position does not have the sensitivity to the set point as with the other gain factors, the ability to remove or account for this effect would allow for the possibility of improving the system. Improving the measurement resolution would be beneficial in reducing the overall angular stability of the system, possibly even approaching the sub-microradian range.

Acknowledgements

References and links

1.

A. Stalmashonak, N. Zhavoronkov, I. V. Hertel, S. Vetrov, and K. Schmid, “Spatial control of femtosecond laser system output with submicroradian accuracy,” Appl. Opt. 45(6), 1271–1274 (2006). [CrossRef] [PubMed]

2.

F. Lindau, O. Lundh, A. Persson, K. Cassou, S. Kazamias, D. Ros, F. Plé, G. Jamelot, A. Klisnick, S. de Rossi, D. Joyeux, B. Zielbauer, D. Ursescu, T. Kühl, and C. G. Wahlström, “Quantitative study of 10 Hz operation of a soft x-ray laser-energy stability and target considerations,” Opt. Express 15(15), 9486–9493 (2007). [CrossRef] [PubMed]

3.

T. Kanai, A. Suda, S. Bohman, M. Kaku, S. Yamaguchi, and K. Midorikawa, “Pointing stabilization of a high-repetition-rate high-power femtosecond laser for intense few-cycle pulse generation,” Appl. Phys. Lett. 92(6), 061106 (2008). [CrossRef]

4.

J. Liu, Y. Kida, T. Teramoto, and T. Kobayashi, “Generation of stable sub-10 fs pulses at 400 nm in a hollow fiber for UV pump-probe experiment,” Opt. Express 18(5), 4664–4672 (2010). [CrossRef] [PubMed]

5.

Y. Liu, A. Aleksandrov, S. Assadi, W. Blokland, C. Deibele, W. Grice, C. Long, T. Pelaia, and A. Webster, “Laser wire beam profile monitor in the spallation neutron source (SNS) superconducting linac,” Nucl. Instr. Meth. A 612(2), 241–253 (2010). [CrossRef]

6.

W. Blokland, A. Barker, and W. Grice, “Drift Compensation for the SNS Laserwire,” Proceedings of ICALEPCS 2007, Knoxville, Tennessee, USA (2007).

7.

P. M. Sinclair, J. R. Drummond, and A. D. May, “Digital divider circuit for a laser-beam-pointing stabilization system,” Appl. Opt. 34(3), 408–409 (1995). [CrossRef] [PubMed]

8.

F. Breitling, R. S. Weigel, M. C. Downer, and T. Tajima, “Laser pointing stabilization and control in the submicroradian regime with neural networks,” Rev. Sci. Instrum. 72(2), 1339–1342 (2001). [CrossRef]

9.

Y. Wu, D. French, and I. Jovanovic, “Passive beam pointing stabilization,” Opt. Lett. 35(2), 250–252 (2010). [CrossRef] [PubMed]

10.

P. Fritschel, N. Mavalvala, D. Shoemaker, D. Sigg, M. Zucker, and G. González, “Alignment of an interferometric gravitational wave detector,” Appl. Opt. 37(28), 6734–6747 (1998). [CrossRef]

11.

D. A. Neamen, Electronic Circuit Analysis and Design (McGraw-Hill, 2001).

12.

J. DiStefano, A. Stubberud, and I. Williams, Schaum’s Outlines: Feedback and Control Systems, 2nd Edition. (McGraw-Hill, 1990).

13.

E. Ott, Chaos in Dynamical System (Cambridge University Press, 2002).

OCIS Codes
(000.2170) General : Equipment and techniques
(000.3110) General : Instruments, apparatus, and components common to the sciences
(120.4820) Instrumentation, measurement, and metrology : Optical systems
(260.5210) Physical optics : Photoionization
(140.3425) Lasers and laser optics : Laser stabilization

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: December 20, 2010
Revised Manuscript: January 14, 2011
Manuscript Accepted: January 17, 2011
Published: January 31, 2011

Citation
Robert A. Hardin, Yun Liu, Cary Long, Alexander Aleksandrov, and Willem Blokland, "Active beam position stabilization of pulsed lasers for long-distance ion profile diagnostics at the Spallation Neutron Source (SNS)," Opt. Express 19, 2874-2885 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-4-2874


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References

  1. A. Stalmashonak, N. Zhavoronkov, I. V. Hertel, S. Vetrov, and K. Schmid, “Spatial control of femtosecond laser system output with submicroradian accuracy,” Appl. Opt. 45(6), 1271–1274 (2006). [CrossRef] [PubMed]
  2. F. Lindau, O. Lundh, A. Persson, K. Cassou, S. Kazamias, D. Ros, F. Plé, G. Jamelot, A. Klisnick, S. de Rossi, D. Joyeux, B. Zielbauer, D. Ursescu, T. Kühl, and C. G. Wahlström, “Quantitative study of 10 Hz operation of a soft x-ray laser-energy stability and target considerations,” Opt. Express 15(15), 9486–9493 (2007). [CrossRef] [PubMed]
  3. T. Kanai, A. Suda, S. Bohman, M. Kaku, S. Yamaguchi, and K. Midorikawa, “Pointing stabilization of a high-repetition-rate high-power femtosecond laser for intense few-cycle pulse generation,” Appl. Phys. Lett. 92(6), 061106 (2008). [CrossRef]
  4. J. Liu, Y. Kida, T. Teramoto, and T. Kobayashi, “Generation of stable sub-10 fs pulses at 400 nm in a hollow fiber for UV pump-probe experiment,” Opt. Express 18(5), 4664–4672 (2010). [CrossRef] [PubMed]
  5. Y. Liu, A. Aleksandrov, S. Assadi, W. Blokland, C. Deibele, W. Grice, C. Long, T. Pelaia, and A. Webster, “Laser wire beam profile monitor in the spallation neutron source (SNS) superconducting linac,” Nucl. Instr. Meth. A 612(2), 241–253 (2010). [CrossRef]
  6. W. Blokland, A. Barker, and W. Grice, “Drift Compensation for the SNS Laserwire,” Proceedings of ICALEPCS 2007, Knoxville, Tennessee, USA (2007).
  7. P. M. Sinclair, J. R. Drummond, and A. D. May, “Digital divider circuit for a laser-beam-pointing stabilization system,” Appl. Opt. 34(3), 408–409 (1995). [CrossRef] [PubMed]
  8. F. Breitling, R. S. Weigel, M. C. Downer, and T. Tajima, “Laser pointing stabilization and control in the submicroradian regime with neural networks,” Rev. Sci. Instrum. 72(2), 1339–1342 (2001). [CrossRef]
  9. Y. Wu, D. French, and I. Jovanovic, “Passive beam pointing stabilization,” Opt. Lett. 35(2), 250–252 (2010). [CrossRef] [PubMed]
  10. P. Fritschel, N. Mavalvala, D. Shoemaker, D. Sigg, M. Zucker, and G. González, “Alignment of an interferometric gravitational wave detector,” Appl. Opt. 37(28), 6734–6747 (1998). [CrossRef]
  11. D. A. Neamen, Electronic Circuit Analysis and Design (McGraw-Hill, 2001).
  12. J. DiStefano, A. Stubberud, and I. Williams, Schaum’s Outlines: Feedback and Control Systems, 2nd Edition. (McGraw-Hill, 1990).
  13. E. Ott, Chaos in Dynamical System (Cambridge University Press, 2002).

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