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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 3 — Jan. 30, 2012
  • pp: 3261–3267
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Stable optical phase modulation with micromirrors

Caleb Knoernschild, Taehyun Kim, Peter Maunz, Stephen G. Crain, and Jungsang Kim  »View Author Affiliations


Optics Express, Vol. 20, Issue 3, pp. 3261-3267 (2012)
http://dx.doi.org/10.1364/OE.20.003261


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Abstract

We measure the motional fluctuations of a micromechanical mirror using a Michelson interferometer, and demonstrate its interferometric stability. The position stability of the micromirror is dominated by the thermal mechanical noise of the structure. With this level of stability, we utilize the micromirror to realize an optical phase modulator by simply reflecting light off the mirror and modulating its position. The resonant frequency of the modulator can be tuned by applying a voltage between the mirror and an underlying electrode. Full modulation depth of ±π is achieved when the mirror resonantly excited with a sinusoidal voltage at an amplitude of 11V.

© 2012 OSA

1. Introduction

Micromirrors fabricated using micro-electromechanical systems (MEMS) technology are well suited to steer a potentially large number of laser beams. MEMS systems are used in all-optical switching [1

1. D. T. Neilson, R. Frahm, P. Kolodner, C. A. Bolle, R. Ryf, J. Kim, A. R. Papazian, C. J. Nuzman, A. Gasparyan, N. R. Basavanhally, V. A. Aksyuk, and J. V. Gates, “256 × 256 port optical cross-connect subsystem,” J. Lightwave Technol. 22, 1499–1509 (2004). [CrossRef]

, 2

2. J. Kim, C. J. Nuzman, B. Kumar, D. F. Lieuwen, J. S. Kraus, A. Weiss, C. P. Lichtenwalner, A. R. Papazian, R. E. Frahm, N. R. Basavanhally, D. A. Ramsey, V. A. Aksyuk, F. Pardo, M. E. Simon, V. Lifton, H. B. Chan, M. Haueis, A. Gasparyan, H. R. Shea, S. Arney, C. A. Bolle, P. R. Kolodner, R. Ryf, D. T. Neilson, and J. V. Gates, “1100 × 1100 port MEMS-based optical crossconnect with 4-dB maximum loss,” IEEE Photon. Technol. Lett. 15, 1537–1539 (2003). [CrossRef]

], scanning projection displays [3

3. R. A. Conant, P. M. Hagelin, U. Krishnamoorthy, M. Hart, O. Solgaard, K. Y. Lau, and R. S. Muller, “A raster-scanning full-motion video display using polysilicon micromachined mirrors,” Sens. Actuators A 83, 291–296 (2000). [CrossRef]

, 4

4. K. Castelino, V. Milanovic, and D. McCormick, “MEMS-based high-speed low-power vector display,” International Conference on Optical MEMS and Their Applications IEEE/LEOS (IEEE, 2005), pp. 127 –128. [CrossRef]

], and optical addressing of quantum information processing systems [5

5. D. L. Moehring, M. J. Madsen, K. C. Younge, R. N. Kohn Jr., P. Maunz, L.-M. Duan, C. Monroe, and B. B. Blinov, “Quantum networking with photons and trapped atoms,” J. Opt. Soc. Am. B 24, 300–315 (2007). [CrossRef]

8

8. C. Knoernschild, X. L. Zhang, L. Isenhower, A. T. Gill, F. P. Lu, M. Saffman, and J. Kim, “Independent individual addressing of multiple neutral atom qubits with a micromirror-based beam steering system,” Appl. Phys. Lett. 97, 134101 (2010). [CrossRef]

]. Compared to traditional beam steering techniques such as acousto-optic (AO) and electro-optic (EO) deflectors [9

9. M. Gottlieb, C. L. M. Ireland, and J. M. Ley, Electro-Optic and Acousto-Optic Scanning and Deflection (M. Dekker, 1983).

11

11. S. Kim, R. R. Mcleod, M. Saffman, and K. H. Wagner, “Doppler-free, multiwavelength acousto-optic deflector for two-photon addressing arrays of Rb atoms in a quantum information processor,” Appl. Opt. 47, 1816–1831 (2008). [CrossRef] [PubMed]

], MEMS-based approaches typically consume much less actuation energy and device volume per beam. These advantages make scalable systems capable of simultaneously steering a large number of beams possible. In many applications in quantum information processing with atoms, the optical phases of the multiple beams addressing the atomic qubits have to be maintained [12

12. S. X. Wang, J. Labaziewicz, Y. Ge, R. Shewmon, and I. L. Chuang, “Demonstration of a quantum logic gate in a cryogenic surface-electrode ion trap,” Phys. Rev. A 81, 062332 (2010). [CrossRef]

].

In this paper, we present a characterization of the pointing and phase stability of MEMS mirrors. Using a Michelson interferometer, we demonstrate interferometric stability of carefully designed micromirrors. These micromirrors are shown to have motional stability down to thermal noise levels. Taking advantage of the interferometric stability, scalability, and low actuation requirements of MEMS mirrors, we introduce a method of MEMS-based optical phase modulation. Resonant vibrations of the MEMS mirrors, which are frequency tunable with bias voltage, are shown to enable controllable optical phase modulation at radio frequencies (RF). With this modulation scheme, phase modulation on multiple laser systems can be performed in a single, high optical density device.

2. MEMS mirror design

We consider a micromirror consisting of a circular polysilicon mirror plate tethered to anchors with two torsional springs (Fig. 1(a)) [13

13. C. Kim, C. Knoernschild, B. Liu, and J. Kim, “Design and characterization of MEMS micromirrors for ion-trap quantum computation,” IEEE J. Sel. Top. Quantum Electron. 13, 322–329 (2007). [CrossRef]

]. The MEMS structure is fabricated using Sandia’s SUMMiT V MEMS foundry process, where alternating layers of polysilicon (structural and routing layers) and silicon dioxide (SiO2: sacrificial layers) are used to form the devices. At the end of the fabrication process, the SiO2 is selectively removed to enable free motion of the polysilicon structural layers. The released mirror is electrostatically actuated by applying a voltage between the grounded mirror plate and the underlying electrodes. The mirror structure features several mechanical modes of vibration with the lowest resonant frequencies corresponding to the tilting and sagging modes. The behavior of these modes can be captured using a damped harmonic oscillator model:
Iθ¨+Dθθ˙+2κθ=12C(θ,z)θV2+Fθ,
(1)
Mz¨+Dzz˙+2Kz=12C(θ,z)zV2+Fz,
(2)
where θ and z are the variables describing the tilt angle and vertical sag of the mirror plate, respectively. Variables I (M), Dθ (Dz), κ (K), C(θ,z), V and Fθ(z) denote the rotational inertia (mass) of the mirror plate, damping coefficient for the tilting (sagging) mode, torsional stiffness (spring constant) of the torsional springs, capacitance between the mirror plate and the underlying actuation electrode, applied voltage between the electrode and the mirror plate, and other driving torque (force) for the tilting (sagging) mode, respectively. The mirror features a tilting (sagging) resonance in the θ direction at ωθ=2κ/I (z direction at ωz=2K/M). Modes with higher resonant frequencies exist that involve higher-order distortion of the mirror plate. The dominant damping mechanism for the mirror motion is squeeze film damping by the air between the mirror plate and the substrate [14

14. M. Andrews, I. Harris, and G. Turner, “A comparison of squeeze-film theory with measurements on a microstructure,” Sens. Actuators A 36, 79–87 (1993). [CrossRef]

]. For optimal beam steering performance, the mirror is designed to feature near critical damping for the tilt motion at atmospheric pressures [13

13. C. Kim, C. Knoernschild, B. Liu, and J. Kim, “Design and characterization of MEMS micromirrors for ion-trap quantum computation,” IEEE J. Sel. Top. Quantum Electron. 13, 322–329 (2007). [CrossRef]

, 15

15. C. Knoernschild, C. Kim, B. Liu, F. P. Lu, and J. Kim, “MEMS-based optical beam steering system for quantum information processing in two-dimensional atomic systems,” Opt. Lett. 33, 273–275 (2008). [CrossRef] [PubMed]

]. The damping can be reduced by either placing the mirror in vacuum, or by modifying the device design to provide smaller mirror radius and larger gap between the mirror and the substrate.

Fig. 1 Schematic of the measurement setup. (a) Scanning electron micrograph of a MEMS mirror. (b) MEMS mirror stability measurement setup using a Michelson interferometer. (c) Shifting the reflection point on the MEMS mirror for sensitivity in the tilting mode.

The spectral response of the primary tilting mode to a driving torque in the absence of applied voltage (V = 0) can be calculated by solving Eq. (1) in Fourier domain:
|θ˜(ω)|2=|F˜θ(ω)|2(2κ)21(1ω2/ωθ2)2+ω2/(Q2ωθ2),
(3)
where Q=2κI/Dθ is the quality factor of the tilting mode, and θ̃(ω) and θ(ω) are the Fourier transform of the tilt angle and the angular driving torque, respectively.

3. Stability measurement

We measured the pointing stability of this MEMS mirror in a Michelson interferometer, as shown in Fig. 1(b). A single mode, λ = 780 nm wavelength laser diode is directed to a 50/50 beam splitter. Half of the light is sent toward the MEMS device in the sample arm of the interferometer and the other half onto a standard mirror in the reference arm. The light in the sample arm is focused by a 50 mm focal length lens to a beam waist of 15 μm at the MEMS mirror. The MEMS mirror has a radius of R = 130 μm and resonant frequency of ωθ/2π = 135 kHz for the tiling mode (ωz/2π = 258 kHz for sagging mode). It is housed inside a vacuum chamber to control the damping. Light in the reference arm of the interferometer travels through a second 50 mm focal length lens before reflecting off the standard flat mirror. The standard mirror is mounted on a kinematic mount with a piezoelectric actuator to tune the optical path length difference of the interferometer. Light from the two arms recombines on the beam splitter before a final lens focuses it onto a photodetector. The photocurrent measured by the photodetector, assuming no losses, follows the standard interference equation and is given by Idet = Iph[1+cos(2πdopt/λ + ϕi)]/2, where Iph is the photocurrent proportional to the optical power used in the interferometer, dopt is the optical path length introduced by the micromirror motion, and ϕi is the bias point of the interferometer determined by the position of the standard mirror [16

16. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley-Interscience, 2007).

]. The interferometer is actively stabilized at ϕi = π/2 to maximize sensitivity.

The sagging mode of the mirror changes the path length in the sample arm by slightly shifting the reflection plane along the direction of propagation. The tilting motion induces a tilt in the sample arm beam path, and does not lead to an optical path length change in the interferometer. A path length change using the tilting mode is realized by aligning the beam to reflect off the mirror a distance rl ≈ 0.75R away from the axis of rotation (Fig. 1(c)), leading to dopt = 2rlθ.

Figure 2 shows the noise spectrum of the interferometer output, corresponding to the phase noise arising from the motion of the micromirror. The two curves correspond to operating the mirror in a background pressure of ∼ 5 and 760 Torr with a corresponding quality factor Q ≈ 45 and 2 for the primary tilting mode, respectively. The photocurrent from the detector is converted to a voltage signal using a transimpedance amplifier with tranimpedance gain of Rf = 2 kΩ. The detected photocurrent of Iph = 0.6mA from 1.5mW of optical power received at the detector corresponds to an optical shot noise level of 2qIphRf28nV/Hz, where q is the absolute value of the electron charge (dashed line). This shot noise value describes the fundamental sensitivity of the interferometer’s ability to detect a phase shift with a noise level of 5×108rad/Hz. The source laser diode used in our experiment has inherently low intensity noise, featuring shot noise level above 400kHz, and only a modest increase (by a factor of two) for frequencies down to 30 kHz without any additional stabilization technique.

Fig. 2 Interferometer output representing the position noise spectrum of a micromirror at a pressure of 5 Torr (Q ≈ 45) and 760 Torr (Q ≈ 2) with 0V bias voltage. The dotted line shows shot noise level corresponding to laser intensity fluctuations. Simulations of the mode shapes corresponding to the five peaks (A – E) are shown by a color plot of maximum displacement. The inset shows tuning of tiling mode resonant frequency as a function of bias voltage. Dots indicate measured data while the solid line shows the theoretical prediction.

When a DC bias voltage is applied between the mirror and underlying electrode, the resonance frequency of the mechanical oscillator shifts to a lower value. The nonlinearity inherent in capacitive actuation (right hand side of Eqs. (1) and (2) produces greater electrostatic forces for a given voltage at higher angles of tilt. The inset in Fig. 2 illustrates this effect known as electrostatic softening for the tilting mode. Data points are plotted against the theoretical behavior which is calculated based on measured values of the mirror’s mechanical constants and simulated capacitance values. This mechanism can be used to tune the resonant frequency of a given mechanical oscillator.

4. Phase modulation

The demonstrated stability of the mirror suggests that the resonant modes of the micromirrors can be utilized to modulate the phase of an optical field at RF with low additional phase noise. The depth of modulation for a micromirror driven at mechanical resonance is amplified by the quality factor Q, enabling significant modulation with only modest driving voltages. Furthermore, the resonant frequency of the micromirror can be tuned using electrostatic softening. We characterized the optical phase modulation properties with the Michelson interferometer (Fig. 1(b)) using the sagging mode at 258kHz and reflecting the laser beam at the center of the micromirror. When the optical path length difference is modulated by (βλ/2π) sin ωt, the detected current Idet is given by
Idet1+cos(ϕi+βsinωt)=1+J0(β)cosϕi+2{J2(β)cos2ωt+J4(β)cos4ωt+}cosϕi2{J1(β)sinωt+J3(β)sin3ωt+}sinϕi.
(4)
In the experiment, we set ϕi = π/2 to suppress all even order modulation sidebands. Due to quadratic dependence on the applied voltage shown in Eq. (2), we use a RF voltage at frequency f/2 to induce a phase modulation at frequency f. Depth of the optical path length modulation is proportional to the square of the amplitude of the driving RF voltage vd, so we let βsvd2 to define the proportionality constant s. Figure 3(a) shows a spectrum of the interferometer output when the micromirror is driven with RF voltage amplitude of vd = 10V. The even order modulation sidebands are suppressed, and the relative peak heights of the odd order modulation sidebands can be determined. By fitting the peak heights of first order modulation sideband to AJ1(svd2) as a function of vd, we obtained s = 0.025 ± 0.0005/V2.

Fig. 3 (a) Spectrum of interferometer output for a given modulation amplitude of a sagging mode at f =258 kHz. (b) Normalized spectrum peak heights as a function of RF voltage amplitude. Bessel functions of different orders Jn(svd2) are plotted using a single fitting parameter s = 0.025/V2 (lines), while the measured DC values (circles) and peak heights are shown with squares (f), diamonds (3f), and triangles (5f).

We plot the peak heights of each odd order modulation sideband as a function of the RF voltage amplitude in Fig. 3(b). The DC components corresponding to J0(svd2) are measured by scanning ϕi from 0 to π/2 for each RF amplitude, and fit using the s value obtained from first order modulation sidebands. The amplitude of J0 values have to be normalized since the DC and AC current are measured using different transimpedance gain in our circuit. The data fit well to Bessel functions with a single scaling parameter s up to fifth order. A DC voltage of 135V is necessary to induce a phase shift of π for the 780 nm laser for this mirror, but a smaller RF voltage amplitude of vd ≈ 11 V is sufficient to generate phase oscillation between ±π when a mechanical resonance is utilized. The reduction in driving voltage is consistent with the quality factor Qs ∼ 300 of the sagging mode under the operating pressure. The contribution of mechanical thermal noise of the mirrors on the RMS phase noise of the modulator is independent of the quality factor of the mechanical resonance used, and was estimated to be less than 10−4 rad in Section 3.

Phase modulation based on reflection off a micromirror works over a broad wavelength range, and is insensitive to polarization, power of the incoming light and potential thermal drifts from driving RF power. Using micromirrors with higher resonant frequencies, higher modulation frequencies can be achieved than by using macroscopic mirrors [18

18. J. E. Debs, N. P. Robins, A. Lance, M. B. Kruger, and J. D. Close, “Piezo-locking a diode laser with saturated absorption spectroscopy,” Appl. Opt. 47, 5163–5166 (2008). [CrossRef] [PubMed]

]. A large array of such micromirror-based phase modulators can be fabricated in a single chip in the case where a scalable solution is desired.

5. Summary

In this work, we showed that micromirrors can be used to construct beam steering devices with interferometric phase stability. This stability was shown to be limited by thermal noise associated with the mechanical structure of the mirror. The resonant modes excited by the thermal noise of the MEMS device were identified and frequency controlled with a DC bias voltage. Leveraging the stability, tunability, scalability, and low actuation requirements associated with MEMS mirrors, we propose a MEMS-based optical phase modulating system and demonstrate ±π phase oscillations at 258 kHz driven by an RF amplitude of 11 V. Micromirrors designed for higher modulation frequencies could easily provide MHz phase oscillations. This MEMS-based phase modulation system could enable phase control of multiple laser systems across a range of wavelengths using only a single device.

Acknowledgments

The authors would like to thank Felix P. Lu at Applied Quantum Technologies in Durham, North Carolina for help with the micromirrors. This work was supported by ARO, office of the Director of National Intelligence, and Intelligence Advanced Research Projects Activity.

References and links

1.

D. T. Neilson, R. Frahm, P. Kolodner, C. A. Bolle, R. Ryf, J. Kim, A. R. Papazian, C. J. Nuzman, A. Gasparyan, N. R. Basavanhally, V. A. Aksyuk, and J. V. Gates, “256 × 256 port optical cross-connect subsystem,” J. Lightwave Technol. 22, 1499–1509 (2004). [CrossRef]

2.

J. Kim, C. J. Nuzman, B. Kumar, D. F. Lieuwen, J. S. Kraus, A. Weiss, C. P. Lichtenwalner, A. R. Papazian, R. E. Frahm, N. R. Basavanhally, D. A. Ramsey, V. A. Aksyuk, F. Pardo, M. E. Simon, V. Lifton, H. B. Chan, M. Haueis, A. Gasparyan, H. R. Shea, S. Arney, C. A. Bolle, P. R. Kolodner, R. Ryf, D. T. Neilson, and J. V. Gates, “1100 × 1100 port MEMS-based optical crossconnect with 4-dB maximum loss,” IEEE Photon. Technol. Lett. 15, 1537–1539 (2003). [CrossRef]

3.

R. A. Conant, P. M. Hagelin, U. Krishnamoorthy, M. Hart, O. Solgaard, K. Y. Lau, and R. S. Muller, “A raster-scanning full-motion video display using polysilicon micromachined mirrors,” Sens. Actuators A 83, 291–296 (2000). [CrossRef]

4.

K. Castelino, V. Milanovic, and D. McCormick, “MEMS-based high-speed low-power vector display,” International Conference on Optical MEMS and Their Applications IEEE/LEOS (IEEE, 2005), pp. 127 –128. [CrossRef]

5.

D. L. Moehring, M. J. Madsen, K. C. Younge, R. N. Kohn Jr., P. Maunz, L.-M. Duan, C. Monroe, and B. B. Blinov, “Quantum networking with photons and trapped atoms,” J. Opt. Soc. Am. B 24, 300–315 (2007). [CrossRef]

6.

L. Luo, D. Hayes, T. Manning, D. Matsukevich, P. Maunz, S. Olmschenk, J. Sterk, and C. Monroe, “Protocols and techniques for a scalable atom-photon quantum network,” Fortschr. Phys. 57, 1133–1152 (2009). [CrossRef]

7.

C. Knoernschild, C. Kim, F. P. Lu, and J. Kim, “Multiplexed broadband beam steering system utilizing high speed MEMS mirrors,” Opt. Express 17, 7233–7244 (2009). [CrossRef] [PubMed]

8.

C. Knoernschild, X. L. Zhang, L. Isenhower, A. T. Gill, F. P. Lu, M. Saffman, and J. Kim, “Independent individual addressing of multiple neutral atom qubits with a micromirror-based beam steering system,” Appl. Phys. Lett. 97, 134101 (2010). [CrossRef]

9.

M. Gottlieb, C. L. M. Ireland, and J. M. Ley, Electro-Optic and Acousto-Optic Scanning and Deflection (M. Dekker, 1983).

10.

F. Schmidt-Kaler, H. Häffner, S. Gulde, M. Riebe, G. Lancaster, T. Deuschle, C. Becher, W. Hänsel, J. Eschner, C. Roos, and R. Blatt, “How to realize a universal quantum gate with trapped ions,” Appl. Phys. B 77, 789–796 (2003). [CrossRef]

11.

S. Kim, R. R. Mcleod, M. Saffman, and K. H. Wagner, “Doppler-free, multiwavelength acousto-optic deflector for two-photon addressing arrays of Rb atoms in a quantum information processor,” Appl. Opt. 47, 1816–1831 (2008). [CrossRef] [PubMed]

12.

S. X. Wang, J. Labaziewicz, Y. Ge, R. Shewmon, and I. L. Chuang, “Demonstration of a quantum logic gate in a cryogenic surface-electrode ion trap,” Phys. Rev. A 81, 062332 (2010). [CrossRef]

13.

C. Kim, C. Knoernschild, B. Liu, and J. Kim, “Design and characterization of MEMS micromirrors for ion-trap quantum computation,” IEEE J. Sel. Top. Quantum Electron. 13, 322–329 (2007). [CrossRef]

14.

M. Andrews, I. Harris, and G. Turner, “A comparison of squeeze-film theory with measurements on a microstructure,” Sens. Actuators A 36, 79–87 (1993). [CrossRef]

15.

C. Knoernschild, C. Kim, B. Liu, F. P. Lu, and J. Kim, “MEMS-based optical beam steering system for quantum information processing in two-dimensional atomic systems,” Opt. Lett. 33, 273–275 (2008). [CrossRef] [PubMed]

16.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley-Interscience, 2007).

17.

Z. Djuric, “Mechanisms of noise sources in microelectromechanical systems,” Microelectron. Reliab. 40, 919–932 (2000). [CrossRef]

18.

J. E. Debs, N. P. Robins, A. Lance, M. B. Kruger, and J. D. Close, “Piezo-locking a diode laser with saturated absorption spectroscopy,” Appl. Opt. 47, 5163–5166 (2008). [CrossRef] [PubMed]

OCIS Codes
(120.5060) Instrumentation, measurement, and metrology : Phase modulation
(230.4040) Optical devices : Mirrors
(230.4685) Optical devices : Optical microelectromechanical devices

ToC Category:
Optical Devices

History
Original Manuscript: November 9, 2011
Revised Manuscript: January 20, 2012
Manuscript Accepted: January 22, 2012
Published: January 27, 2012

Citation
Caleb Knoernschild, Taehyun Kim, Peter Maunz, Stephen G. Crain, and Jungsang Kim, "Stable optical phase modulation with micromirrors," Opt. Express 20, 3261-3267 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-3-3261


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References

  1. D. T. Neilson, R. Frahm, P. Kolodner, C. A. Bolle, R. Ryf, J. Kim, A. R. Papazian, C. J. Nuzman, A. Gasparyan, N. R. Basavanhally, V. A. Aksyuk, and J. V. Gates, “256 × 256 port optical cross-connect subsystem,” J. Lightwave Technol.22, 1499–1509 (2004). [CrossRef]
  2. J. Kim, C. J. Nuzman, B. Kumar, D. F. Lieuwen, J. S. Kraus, A. Weiss, C. P. Lichtenwalner, A. R. Papazian, R. E. Frahm, N. R. Basavanhally, D. A. Ramsey, V. A. Aksyuk, F. Pardo, M. E. Simon, V. Lifton, H. B. Chan, M. Haueis, A. Gasparyan, H. R. Shea, S. Arney, C. A. Bolle, P. R. Kolodner, R. Ryf, D. T. Neilson, and J. V. Gates, “1100 × 1100 port MEMS-based optical crossconnect with 4-dB maximum loss,” IEEE Photon. Technol. Lett.15, 1537–1539 (2003). [CrossRef]
  3. R. A. Conant, P. M. Hagelin, U. Krishnamoorthy, M. Hart, O. Solgaard, K. Y. Lau, and R. S. Muller, “A raster-scanning full-motion video display using polysilicon micromachined mirrors,” Sens. Actuators A83, 291–296 (2000). [CrossRef]
  4. K. Castelino, V. Milanovic, and D. McCormick, “MEMS-based high-speed low-power vector display,” International Conference on Optical MEMS and Their Applications IEEE/LEOS (IEEE, 2005), pp. 127 –128. [CrossRef]
  5. D. L. Moehring, M. J. Madsen, K. C. Younge, R. N. Kohn, P. Maunz, L.-M. Duan, C. Monroe, and B. B. Blinov, “Quantum networking with photons and trapped atoms,” J. Opt. Soc. Am. B24, 300–315 (2007). [CrossRef]
  6. L. Luo, D. Hayes, T. Manning, D. Matsukevich, P. Maunz, S. Olmschenk, J. Sterk, and C. Monroe, “Protocols and techniques for a scalable atom-photon quantum network,” Fortschr. Phys.57, 1133–1152 (2009). [CrossRef]
  7. C. Knoernschild, C. Kim, F. P. Lu, and J. Kim, “Multiplexed broadband beam steering system utilizing high speed MEMS mirrors,” Opt. Express17, 7233–7244 (2009). [CrossRef] [PubMed]
  8. C. Knoernschild, X. L. Zhang, L. Isenhower, A. T. Gill, F. P. Lu, M. Saffman, and J. Kim, “Independent individual addressing of multiple neutral atom qubits with a micromirror-based beam steering system,” Appl. Phys. Lett.97, 134101 (2010). [CrossRef]
  9. M. Gottlieb, C. L. M. Ireland, and J. M. Ley, Electro-Optic and Acousto-Optic Scanning and Deflection (M. Dekker, 1983).
  10. F. Schmidt-Kaler, H. Häffner, S. Gulde, M. Riebe, G. Lancaster, T. Deuschle, C. Becher, W. Hänsel, J. Eschner, C. Roos, and R. Blatt, “How to realize a universal quantum gate with trapped ions,” Appl. Phys. B77, 789–796 (2003). [CrossRef]
  11. S. Kim, R. R. Mcleod, M. Saffman, and K. H. Wagner, “Doppler-free, multiwavelength acousto-optic deflector for two-photon addressing arrays of Rb atoms in a quantum information processor,” Appl. Opt.47, 1816–1831 (2008). [CrossRef] [PubMed]
  12. S. X. Wang, J. Labaziewicz, Y. Ge, R. Shewmon, and I. L. Chuang, “Demonstration of a quantum logic gate in a cryogenic surface-electrode ion trap,” Phys. Rev. A81, 062332 (2010). [CrossRef]
  13. C. Kim, C. Knoernschild, B. Liu, and J. Kim, “Design and characterization of MEMS micromirrors for ion-trap quantum computation,” IEEE J. Sel. Top. Quantum Electron.13, 322–329 (2007). [CrossRef]
  14. M. Andrews, I. Harris, and G. Turner, “A comparison of squeeze-film theory with measurements on a microstructure,” Sens. Actuators A36, 79–87 (1993). [CrossRef]
  15. C. Knoernschild, C. Kim, B. Liu, F. P. Lu, and J. Kim, “MEMS-based optical beam steering system for quantum information processing in two-dimensional atomic systems,” Opt. Lett.33, 273–275 (2008). [CrossRef] [PubMed]
  16. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley-Interscience, 2007).
  17. Z. Djuric, “Mechanisms of noise sources in microelectromechanical systems,” Microelectron. Reliab.40, 919–932 (2000). [CrossRef]
  18. J. E. Debs, N. P. Robins, A. Lance, M. B. Kruger, and J. D. Close, “Piezo-locking a diode laser with saturated absorption spectroscopy,” Appl. Opt.47, 5163–5166 (2008). [CrossRef] [PubMed]

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