Abstract

We experimentally validate the vibration suppression capabilities of a weak-value-like protocol. The phase-sensitive heterodyne technique exhibits advantageous characteristics of a weak measurement including anomalous amplification in sensitivity and technical noise suppression. It does not, however, leverage the entanglement between the system and meter to amplify the signal of interest, as is typical in a weak measurement. In this formalism, we demonstrate an amplification in sensitivity to the roll angle of over 700 times. High precision roll experiments anchor numerical simulations to show that the interferometer outperforms standard interferometry by a factor of 500 in terms of peak-to-peak noise amplitude. During the measurement of a rolling stage, technical noise - primarily in the form of vibrations - is substantially attenuated. This is the first demonstration of vibration suppression capabilities that are inherent to the light from a metrology instrument instead of achieved via mechanical damping. The emulation presented in this work also identifies an avenue to achieve anomalous amplification outside of the standard weak measurement protocol.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

The roll angle about the optical axis of a laser has classically been the most challenging degree of freedom to measure using optical methods. In the quest for increased precision, some of the most promising architectures [1–6] are limited by experimental conditions (i.e. vibrations, refractive index, 1/f, laser pointing stability, detector saturation, etc.) as opposed to shot or detector noise. Typical mitigation techniques in published literature can include elaborate environmental control; however, fielded metrology devices are not always supported with the same benefits.

Metrology instruments following the weak-value amplification formalism [7,8] are capable of attenuating the environmental “technical noise” that limits their resolution in practice [8–14]. It is in this construct that many have achieved the measurement of impressive feats [12, 15–19] without the need for exceedingly complex environmental control. Although beneficial for these reasons, the technique includes the disposal of a significant amount of photons during final post-selection of the measurand signal.

A technique has recently emerged to demonstrate anomalous amplification - and the corresponding benefits such as technical noise suppression - outside the standard weak-value protocol [20]. Although it produces a notable increase in sensitivity, the approach does not include a disposal of photons during post-selection of the signal. Instead, the amplification is a result of an imbalance in the complex plane of the signal. The imaginary component is amplified to produce local increases in sensitivity in a similar fashion to weak-value techniques.

In this work, we present an emulation of a weak measurement through the manipulation of the imaginary and real components of a complex signal. Our findings demonstrate the suppression of vibrations which has been touted theoretically as a significant benefit of weak-value amplification [8, 9]. The experimental setup involves a stage rotating about the optical axis of a heterodyne interferometer sensitive to that degree of freedom (the roll angle). During rotation, the stage’s vibrations couple into the natural frequencies of a waveplate mounting fixture. The observation of vibrational content is attenuated while preserving the signal of interest in this context.

Our protocol presents many similarities to a weak-value technique, such as disposing a significant amount of photons and offering rich dynamics in the imaginary component of the signal. However, it is not clear how the interaction exactly mimics a post-selected weak measurement technique [21]. In a weak-value-type interaction entanglement between system and meter proportional to the parameter of interest is induced. In our experiment, the interaction/entanglement comes at the very beginning during the preparation stage, where two frequency modes are entangled to two linear polarization components. The procedure is independent of the roll angle that is eventually estimated, and fails to follow the weak-value recipe. The action of the measured roll comes after such a mixing, affecting only the polarization (system) and being identical for the two states of the meter, i.e. the polarization rotation is frequency-independent.

Our work builds on previous studies of a similar architecture [5, 6] through analysis in the context of a weak value and experiments in the presence of vibrations. The applications of vibration attenuation in a weak-value-emulated interferometer extend to machine tool metrology and calibration [22], particle and virus detection [23], refractometry [24], and gravitational wave sensing [25–27].

2. Theory

The heterodyne roll interferometer is shown in Fig. 1. The heterodyne frequency in this experiment is created via two acousto-optic modulators (AOMs) [28]. Frequency stabilized light at 633 nm is split and input to two AOMs driven at slightly different frequencies (80 MHz and 80.07 MHz to create a heterodyne frequency of nominally 70 kHz upon interference, the additive 160.07 MHz frequency is removed via filtering). The output first-order diffracted light is collected at the Bragg angle after frequency upshifting. Both beams with slightly different frequencies are rotated into orthogonal linear polarizations using a half-wave plate (HWP). The beams are then recombined using a polarizing beamsplitter to be collinear and input into the interferometer. A small portion of the heterodyne input beam (nominally 5%) is separated and passed through a 45° polarizer to be used for pre-selected reference interference. The remaining portion of the beam passes through a quarter-wave plate (QWP) and half-wave plate (HWP) where the HWP is mounted to the roll rotation stage. Finally, light is passed through a final polarizer to create post-selected measurement signal. The reference and measurement detectors, PDr and PDm, are fed into the reference and measurement channels of a lock-in amplifier used to extract phase and, after post-processing, roll. The QWP/HWP combination can be tuned to nearly align the fast axis of each waveplate. In doing so, the waveplates create a measurand that is coupled to the propagating light. The system and the meter are well defined: The system is the degree of freedom used for the pre- and the post-selection procedure, i.e. polarization, and the two-level meter is composed by the two center frequencies or modes of the electric field, where a heterodyne phase detection is performed.

 

Fig. 1 Full schematic of the measurement setup used for interferometric roll sensing; BS, beamsplitter, AOM, acousto-optic modulator, HWP, halfwave plate, PBS, polarizing beamsplitter, pol., polarizer, PDr and PDm, reference and measurement photodetectors, QWP, quarterwave plate. A Renishaw interferometer measures pitch for calibration from an orthogonal measurement position.

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Each element in this setup can be described by its corresponding Jones matrices,

P=[1000],H(α)=[cos(2α)sin(2α)sin(2α)cos(2α)],Q=[100i]R(θ)=[cos(θ)sin(θ)sin(θ)cos(θ)],andE0=[A1eiω1tA2eiω2t],
where E0 is the input orthogonally polarized heterodyne beam with amplitudes, A1 and A2, and angular frequencies, ω1 and ω2, R(θ) is a rotation matrix to simplify the math for the QWP and is therefore a function of QWP fast-axis angle θ, Q is the QWP, H(α) is the HWP and is a function of roll angle α, and P is a polarizer. The input light is propagated through the setup according to
Es=PH(α)R(θ)QR(θ)E0,
where Es is the output electric field 2×1 matrix. After performing the multiplication, the output is
Es=[A1eiω1t{isin(θ)[sin(2α+θ)]+cos(θ)[cos(2α+θ)]}+A2eiω2t{icos(θ)[sin(2α+θ)]sin(θ)[cos(2α+θ)]}][10].
The resulting irradiance is the modulus squared of this electric field which can be shown to be equal to
I=|Es|2=C12A12+C22A22+2C1C2A1A2cos[(ω1ω2)t+ϕm].
In Eq. (4), ϕm is the phase difference accrued between the pre-selected and post-selected states of the interferometer and C1 and C2 are amplitude terms. These terms are equal to
tan(ϕm)=tan(4α+2θ)sin(2θ),C1=122+cos[4α]+cos[4(α+θ)],andC2=122cos[4α]cos[4(α+θ)].

The results of Eq. (5) can be seen in Fig. 2 after the DC terms, C12A12+C22A22, are removed with a high-pass filter. Amplitude (2C1C2A1A2) and phase (ϕm) of the AC signal is then simulated as a function of roll angle, α, after setting A1 and A2 equal to unity. It can be seen that in π/4 intervals, the system undergoes a significant increase in phase sensitivity. Figure 2(a) on its own is encouraging, but it comes at a cost. Figure 2(b) shows that these same π/4 intervals produce drops in signal amplitude due to post-selection. Therefore, increased phase sensitivity is compensated with decreased signal amplitude. Figure 2(c) is the derivative of Fig. 2(a). It can be seen that the phase sensitivity is amplified by over 700X for a QWP plate angle of 0.160°. This QWP angle is later experimentally validated in Fig. 3 in addition to other small QWP angles with high amplification. It is important to note that the amplification takes a distinct form in the complex plane. Imaginary-valued weak measurements are known to be favorable for metrological tasks [9, 29, 30] and the current setup is no exception. The amplification is primarily a result of an imbalance in the phasor representation of the interferometer. Although Fig. 2(c) shows impressive phase sensitivity as a function of roll, the attenuation of the real part of the complex signal results in signal instability.

 

Fig. 2 Simulations of the weak-value-emulated amplification in the heterodyne roll interferometer. (a) The phase sensitivity between pre- and post-selection as a function of QWP angle, θ, and HWP ‘roll’ angle, α. (b) The AC amplitude of the interferometer drops to near zero when the phase sensitivity is amplified. (c) The derivative of phase as a function of roll angle showing that the phase sensitivity is amplified by over 700 times. The QWP angle of 0.160° is experimentally validated later in Fig. 3. (d) Amplification manifests mainly in the imaginary part of the complex phasor signal.

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3. Experimental results

The validation of this roll interferometer is performed in comparison to an industry-standardized Renishaw pitch measurement of the same moving stage. The Renishaw retroreflector target is rigidly attached to the stage with epoxy to ensure maximum stiffness. The HWP measurand of the weak-value-emulated interferometer is mounted into a rotating optical carriage and then connected to the stage using a 1/2 inch post. Therefore, the HWP has added compliance (in the form of 1/4′-20 mounting screw to stage, #8–32 mounting screw to rotating optical carriage, and the rotating optic carriage itself) compared to the Renishaw measurement. This results in lower-frequency vibrations that can be observed in the weak-value-emulated setup but not the Renishaw measurement.

To perform a measurement, a roll experiment is performed at an arbitrary QWP angle, where the sensitivity is enhanced by iteratively rotating the QWP and HWP to decrease AC signal amplitude (as in Fig. 2(b) and observed on an oscilloscope). In this fashion, as the amplitude decreases the phase sensitivity is enhanced. After a measurement is taken, the QWP angle is calibrated to agree with the orthogonal Renishaw pitch calibration as shown in the setup in Fig. 1.

Roll is extracted from the phase term (ϕm) in Eq. (5) according to

α=θ214arctan[tan(ϕm)sin(2θ)]
where θ is the fast-axis angle of the QWP, and ϕm is output from a lock-in-amplifier (LIA). The time constant of the LIA was set to 10 μs and all recorded phase data was subjected to a final 135 Hz low-pass filter.

The rolling and vibrating stage was a precision NewFocusTM Motorized Five-Axis Tilt Aligner by Newport. The time-domain and frequency-domain results presented in Fig. 3 indicate that as the QWP fast axis approached the fast axis of the HWP, technical noise - mainly in the form of vibrations - is directly suppressed. In Fig. 3, the QWP angles of 1.719°, 0.974°, 0.630°, 0.378°, and 0.160° correspond to amplifications in roll sensitivity of 66.7 rad/rad, 117.7 rad/rad, 181.9 rad/rad, 303.2 rad/rad, and 716.2 rad/rad, respectively.

 

Fig. 3 The effects of increased phase sensitivity in the interferometer in the (a) time domain and (b) frequency domain. As the fast axis of the QWP, θ, approaches the fast axis of the HWP, α, technical noise - mainly in the form of vibrations - is suppressed.

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The input laser for all experiments in this work was a 1.2 mW stabilized HeNe laser. Unfortunately, that level of optical power can only achieve a roll resolution of approximately 80 μrad due to the severe drop in AC amplitude predicted in the simulation in Fig. 2(b). The irradiance of the post-selected light at the measurement detector for all experimental results presented in this paper was approximately 54 μW and did not change considerably through the QWP angles shown in Fig. 3.

It is instructive to observe the statistical levels of noise created in the interferometer as a function of the power level stated above. If the measured signal irradiance is high enough, the noise is entirely due to quantum statistics, i.e. shot noise. This contribution in terms of phase noise is given by [31,32]

ϕs=1λ2hcλBηP,
where λ is the laser wavelength, h is Planck’s constant, c is the speed of light, B is the bandwidth of the system (135 Hz after our final low-pass filter), η is the detector’s quantum efficiency, and P is power. The measurement detector used for the measurements in this paper (Thorlabs PDA10A) has a quantum efficiency of 0.75 as derived using its responsivity value of 0.382 at 633 nm wavelength. If we ignore the quantum fluctuations of the light itself and only consider classical noise from the detector, the phase resolution is inversely proportional to the signal-to-noise ratio as given by [31]
ϕd=NEPBP,
where NEP is the noise equivalent power of the detector (42.7pW/Hz for Thorlabs PDA10A at 633 nm). The term NEPB is the amount of light necessary to produce a DC signal change equal to the RMS phase noise of the system.

The above expressions can be evaluated as a function of optical power as seen in Fig. 4(a). For the nominal power levels shown in this paper, ∼54 μW, our results are inherently limited by the detector noise. The results are further re-worked according to Eq. (6) in Fig. 4(b). The phase noise cannot be directly input to that formula to evaluate roll noise; rather, the change in roll as a function of the phase noise needs to be evaluated in the same fashion that the sensitivity was calculated for Fig. 2(c). Although the results in this work demonstrate the attenuation of vibrations, they are not completely eliminated and still limit our results above the detector and shot noise limits. If measurements of this device are performed in ideal circumstances without substantial technical noise present, the resolution can theoretically reach sub-μrad resolution - even with our current ∼54μW of optical power after post-selection of the signal.

 

Fig. 4 Statistical limitations of shot noise and detector noise as outlined in Eqs. (7)(8). In the absence of all forms of technical noise, our setup is fundamentally limited by detector noise due to the discarding of photons during post-selection.

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As a final validation of the theoretical construct of this work, the results shown in Fig. 3 are reworked in terms of peak-to-peak (pk-pk) noise level in Fig. 5. The pk-pk error bars are the result of 5 measurement partitions per experimental run to generate 95% confidence intervals. The theoretical line is traced to indicate the trend of Eq. (6). It is no surprise that the vibration suppression results follow Eq. (6). In fact, the derived post-processing formula does not exclusively attenuate vibrations, rather, technical noise as a whole. To demonstrate this, numerical simulations are also imposed with white phase noise with the same nominal pk-pk amplitude as observed in the experiments in this paper (pk-pk phase amplitude 0.2 rad with standard deviation of 0.03 rad).

 

Fig. 5 Experimental peak-to-peak noise results with vibration content as shown in Fig. 3. The numerical simulation assumes white noise with the same nominal amplitude as the raw phase signals recorded in this paper followed by post-processing from Eq. (6) - the theoretical line.

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4. Discussion and conclusions

While the results presented in this work demonstrate high precision roll-angle measurements, it should be noted that they do not overcome major feats in the optical metrology community. In fact, the roll axis of the same stage used in this research was measured by means of a HWP with better resolution, decreased cost, reduced complexity, and better long-term signal stability [1]. To do so, the HWP was rigidly epoxied to the stage to increase stiffness compared to the weak-value-emulated measurements in this paper. Therefore, no vibrations were observed under 100 Hz.

However, if one were to consider the amplification compared to standard interferometry (defined as QWP = 45° yielding a linear roll-to-phase response in Fig. 2), it is clearly evident that weak-value-emulated amplification vastly outperforms its standard interferometric counterpart in this setup. That conclusion on its own complements the results of the theoretical exploration by Brunner and Simon [29] which predict imaginary-valued weak-value interferometry vastly outperforms standard interferometers.

There are instances where one may find the need to damp vibrations during measurement without the need for complex vibration control. Weak-value-emulated (and standard weak value) amplification can prove beneficial in these instances. Alternatively, in the case of optical measurements performed without substantial technical noise present, these types of measurements may be rendered unnecessary.

5. Funding

National Science Foundation under awards CMMI:1265824 and STTR:1417032 and the US Department of Commerce, National Institute of Standards and Technology under Award No. 70NANB12H186. Steven Gillmer is currently an MIT Lincoln Laboratory employee. No laboratory funding or resources were used to produce the result/findings reported in this publication. Opinions, interpretations, conclusions, and recommendations are those of the authors, and do not necessarily represent the view of the United States Government.

6. Acknowledgments

The authors acknowledge valuable discussions with Mohammad Mirhosseini, Thomas Brown, and Miguel Alonso in the development of this work.

References

1. S. R. Gillmer, X. Yu, C. Wang, and J. D. Ellis, “Robust high-dynamic-range optical roll sensing,” Opt. Lett. 40, 2497–2500 (2015). [CrossRef]   [PubMed]  

2. S. Li, C. Yang, E. Zhang, and G. Jin, “Compact optical roll-angle sensor with large measurement range and high sensitivity,” Opt. Lett. 30, 242–244 (2005). [CrossRef]   [PubMed]  

3. W. Chen, S. Zhang, and X. Long, “Angle measurement with laser feedback instrument,” Opt. Express 21, 8044–8050 (2013). [CrossRef]   [PubMed]  

4. Y. Le, W. Hou, K. Hu, and K. Shi, “High-sensitivity roll-angle interferometer,” Opt. Lett. 38, 3600–3603 (2013). [CrossRef]   [PubMed]  

5. C.-M. Wu and Y.-T. Chuang, “Roll angular displacement measurement system with microradian accuracy,” Sensors Actuators A: Phys. 116, 145–149 (2004). [CrossRef]  

6. Z. Liu, D. Lin, H. Jiang, and C. Yin, “Roll angle interferometer by means of waveplates,” Sensors Actuators A: Phys. 104, 127–131 (2003). [CrossRef]  

7. Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351–1354 (1988). [CrossRef]   [PubMed]  

8. J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, “Colloquium: Understanding quantum weak values: Basics and applications,” Rev. Mod. Phys. 86, 307 (2014). [CrossRef]  

9. A. N. Jordan, J. Martínez-Rincón, and J. C. Howell, “Technical advantages for weak-value amplification: when less is more,” Phys. Rev. X 4, 011031 (2014).

10. J. Harris, R. W. Boyd, and J. S. Lundeen, “Weak value amplification can outperform conventional measurement in the presence of detector saturation,” Phys. Rev. Lett. 118, 070802 (2017). [CrossRef]   [PubMed]  

11. A. N. J. David, J. Starling, P. Ben Dixon, and J. C. Howell, “Optimizing the signal-to-noise ratio of a beam-deflection measurement with interferometric weak values,” Phys. Rev. A 80, 041803 (2009). [CrossRef]  

12. O. Hosten and P. Kwiat, “Observation of the spin hall effect of light via weak measurements,” Science 319, 5864 (2008). [CrossRef]  

13. G. I. Viza, J. Martínez-Rincón, G. B. Alves, A. N. Jordan, and J. C. Howell, “Experimentally quantifying the advantages of weak-value-based metrology,” Phys. Rev. A 92, 032127 (2015). [CrossRef]  

14. J. P. Torres and L. J. Salazar-Serrano, “Weak value amplification: a view from quantum estimation theory that highlights what it is and what isn?t,” Sci. Reports 6, 19702 (2016). [CrossRef]  

15. P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, “Ultrasensitive beam deflection measurement via interferometric weak value amplification,” Phys. Rev. Lett. 102, 173601 (2009). [CrossRef]   [PubMed]  

16. J. Martínez-Rincón, C. A. Mullarkey, G. I. Viza, W.-T. Liu, and J. C. Howell, “Ultrasensitive inverse weak-value tilt meter,” Opt. Lett. 42, 13 (2017). [CrossRef]  

17. O. S. Magaa-Loaiza, M. Mirhosseini, B. Rodenburg, and R. W. Boyd, “Amplification of angular rotations using weak measurements,” Phys. Rev. Lett. 112, 200401 (2014). [CrossRef]  

18. X. Qiu, L. Xie, X. Liu, L. Luo, Z. Li, Z. Zhang, and J. Du, “Precision phase estimation based on weak-value amplification,” Appl. Phys. Lett. 110, 071105 (2017). [CrossRef]  

19. D. J. Starling, P. B. Dixon, A. N. Jordan, and J. C. Howell, “Precision frequency measurements with interferometric weak values,” Phys. Rev. A 82, 063822 (2010). [CrossRef]  

20. J. Martínez-Rincón, W.-T. Liu, G. I. Viza, and J. C. Howell, “Can anomalous amplification be attained without postselection?” Phys. Rev. Lett. 116, 100803 (2016). [CrossRef]   [PubMed]  

21. J. Martínez-Rincón, “Weak-values metrological techniques for parameter estimation,” Ph.D. thesis (2017).

22. A. H. Slocum, Precision Machine Design (Society of Manufacturing Engineers, 1992).

23. A. Mitra, B. Deutsch, F. Ignatovich, C. Dykes, and L. Novotny, “Nano-optofluidic detection of single viruses and nanoparticles,” ACS Nano 4, 3 (2010). [CrossRef]  

24. G. Wu, M. Takahashi, K. Arai, H. Inaba, and K. Minoshima, “Extremely high-accuracy correction of air refractive index using two-colour optical frequency combs,” Sci. Reports 3, 1894 (2013). [CrossRef]  

25. T. Schuldt, M. Gohlke, D. Weise, U. Johann, A. Peters, and C. Braxmaier, “Picometer and nanoradian optical heterodyne interferometry for translation and tilt metrology of the lisa gravitational reference sensor,” Class. Quantum Gravity 26, 8 (2009). [CrossRef]  

26. M.-J. Hu and Y.-S. Zhang, “Gravitational wave detection via weak measurements amplification,” arXiv (2017).

27. B. Abbott et al., “Observation of gravitational waves from a binary black hole merger,” Phys. Rev. Lett. 116, 061102 (2016). [CrossRef]   [PubMed]  

28. J. D. Ellis, Field Guide to Displacement Measuring Interferometry (SPIE, 2014). [CrossRef]  

29. N. Brunner and C. Simon, “Measuring small longitudinal phase shifts: Weak measurements or standard interferometry?” Phys. Rev. Lett. 105, 010405 (2010). [CrossRef]   [PubMed]  

30. J. Dressel and A. N. Jordan, “Significance of the imaginary part of the weak value,” Phys. Rev. A 85, 012107 (2012). [CrossRef]  

31. J. Lawall and E. Kessler, “Michelson interferometry with 10 pm accuracy,” Rev. Sci. Instruments 71, 7 (2000). [CrossRef]  

32. A. Stevenson, M. Gray, H.-A. Bachor, and D. McClelland, “Quantum-noise-limited interferometric phase measurements,” Appl. Opt. 32, 19 (1993). [CrossRef]  

References

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  1. S. R. Gillmer, X. Yu, C. Wang, and J. D. Ellis, “Robust high-dynamic-range optical roll sensing,” Opt. Lett. 40, 2497–2500 (2015).
    [Crossref] [PubMed]
  2. S. Li, C. Yang, E. Zhang, and G. Jin, “Compact optical roll-angle sensor with large measurement range and high sensitivity,” Opt. Lett. 30, 242–244 (2005).
    [Crossref] [PubMed]
  3. W. Chen, S. Zhang, and X. Long, “Angle measurement with laser feedback instrument,” Opt. Express 21, 8044–8050 (2013).
    [Crossref] [PubMed]
  4. Y. Le, W. Hou, K. Hu, and K. Shi, “High-sensitivity roll-angle interferometer,” Opt. Lett. 38, 3600–3603 (2013).
    [Crossref] [PubMed]
  5. C.-M. Wu and Y.-T. Chuang, “Roll angular displacement measurement system with microradian accuracy,” Sensors Actuators A: Phys. 116, 145–149 (2004).
    [Crossref]
  6. Z. Liu, D. Lin, H. Jiang, and C. Yin, “Roll angle interferometer by means of waveplates,” Sensors Actuators A: Phys. 104, 127–131 (2003).
    [Crossref]
  7. Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351–1354 (1988).
    [Crossref] [PubMed]
  8. J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, “Colloquium: Understanding quantum weak values: Basics and applications,” Rev. Mod. Phys. 86, 307 (2014).
    [Crossref]
  9. A. N. Jordan, J. Martínez-Rincón, and J. C. Howell, “Technical advantages for weak-value amplification: when less is more,” Phys. Rev. X 4, 011031 (2014).
  10. J. Harris, R. W. Boyd, and J. S. Lundeen, “Weak value amplification can outperform conventional measurement in the presence of detector saturation,” Phys. Rev. Lett. 118, 070802 (2017).
    [Crossref] [PubMed]
  11. A. N. J. David, J. Starling, P. Ben Dixon, and J. C. Howell, “Optimizing the signal-to-noise ratio of a beam-deflection measurement with interferometric weak values,” Phys. Rev. A 80, 041803 (2009).
    [Crossref]
  12. O. Hosten and P. Kwiat, “Observation of the spin hall effect of light via weak measurements,” Science 319, 5864 (2008).
    [Crossref]
  13. G. I. Viza, J. Martínez-Rincón, G. B. Alves, A. N. Jordan, and J. C. Howell, “Experimentally quantifying the advantages of weak-value-based metrology,” Phys. Rev. A 92, 032127 (2015).
    [Crossref]
  14. J. P. Torres and L. J. Salazar-Serrano, “Weak value amplification: a view from quantum estimation theory that highlights what it is and what isn?t,” Sci. Reports 6, 19702 (2016).
    [Crossref]
  15. P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, “Ultrasensitive beam deflection measurement via interferometric weak value amplification,” Phys. Rev. Lett. 102, 173601 (2009).
    [Crossref] [PubMed]
  16. J. Martínez-Rincón, C. A. Mullarkey, G. I. Viza, W.-T. Liu, and J. C. Howell, “Ultrasensitive inverse weak-value tilt meter,” Opt. Lett. 42, 13 (2017).
    [Crossref]
  17. O. S. Magaa-Loaiza, M. Mirhosseini, B. Rodenburg, and R. W. Boyd, “Amplification of angular rotations using weak measurements,” Phys. Rev. Lett. 112, 200401 (2014).
    [Crossref]
  18. X. Qiu, L. Xie, X. Liu, L. Luo, Z. Li, Z. Zhang, and J. Du, “Precision phase estimation based on weak-value amplification,” Appl. Phys. Lett. 110, 071105 (2017).
    [Crossref]
  19. D. J. Starling, P. B. Dixon, A. N. Jordan, and J. C. Howell, “Precision frequency measurements with interferometric weak values,” Phys. Rev. A 82, 063822 (2010).
    [Crossref]
  20. J. Martínez-Rincón, W.-T. Liu, G. I. Viza, and J. C. Howell, “Can anomalous amplification be attained without postselection?” Phys. Rev. Lett. 116, 100803 (2016).
    [Crossref] [PubMed]
  21. J. Martínez-Rincón, “Weak-values metrological techniques for parameter estimation,” Ph.D. thesis (2017).
  22. A. H. Slocum, Precision Machine Design (Society of Manufacturing Engineers, 1992).
  23. A. Mitra, B. Deutsch, F. Ignatovich, C. Dykes, and L. Novotny, “Nano-optofluidic detection of single viruses and nanoparticles,” ACS Nano 4, 3 (2010).
    [Crossref]
  24. G. Wu, M. Takahashi, K. Arai, H. Inaba, and K. Minoshima, “Extremely high-accuracy correction of air refractive index using two-colour optical frequency combs,” Sci. Reports 3, 1894 (2013).
    [Crossref]
  25. T. Schuldt, M. Gohlke, D. Weise, U. Johann, A. Peters, and C. Braxmaier, “Picometer and nanoradian optical heterodyne interferometry for translation and tilt metrology of the lisa gravitational reference sensor,” Class. Quantum Gravity 26, 8 (2009).
    [Crossref]
  26. M.-J. Hu and Y.-S. Zhang, “Gravitational wave detection via weak measurements amplification,” arXiv (2017).
  27. B. Abbott and et al., “Observation of gravitational waves from a binary black hole merger,” Phys. Rev. Lett. 116, 061102 (2016).
    [Crossref] [PubMed]
  28. J. D. Ellis, Field Guide to Displacement Measuring Interferometry (SPIE, 2014).
    [Crossref]
  29. N. Brunner and C. Simon, “Measuring small longitudinal phase shifts: Weak measurements or standard interferometry?” Phys. Rev. Lett. 105, 010405 (2010).
    [Crossref] [PubMed]
  30. J. Dressel and A. N. Jordan, “Significance of the imaginary part of the weak value,” Phys. Rev. A 85, 012107 (2012).
    [Crossref]
  31. J. Lawall and E. Kessler, “Michelson interferometry with 10 pm accuracy,” Rev. Sci. Instruments 71, 7 (2000).
    [Crossref]
  32. A. Stevenson, M. Gray, H.-A. Bachor, and D. McClelland, “Quantum-noise-limited interferometric phase measurements,” Appl. Opt. 32, 19 (1993).
    [Crossref]

2017 (3)

J. Harris, R. W. Boyd, and J. S. Lundeen, “Weak value amplification can outperform conventional measurement in the presence of detector saturation,” Phys. Rev. Lett. 118, 070802 (2017).
[Crossref] [PubMed]

J. Martínez-Rincón, C. A. Mullarkey, G. I. Viza, W.-T. Liu, and J. C. Howell, “Ultrasensitive inverse weak-value tilt meter,” Opt. Lett. 42, 13 (2017).
[Crossref]

X. Qiu, L. Xie, X. Liu, L. Luo, Z. Li, Z. Zhang, and J. Du, “Precision phase estimation based on weak-value amplification,” Appl. Phys. Lett. 110, 071105 (2017).
[Crossref]

2016 (3)

J. Martínez-Rincón, W.-T. Liu, G. I. Viza, and J. C. Howell, “Can anomalous amplification be attained without postselection?” Phys. Rev. Lett. 116, 100803 (2016).
[Crossref] [PubMed]

B. Abbott and et al., “Observation of gravitational waves from a binary black hole merger,” Phys. Rev. Lett. 116, 061102 (2016).
[Crossref] [PubMed]

J. P. Torres and L. J. Salazar-Serrano, “Weak value amplification: a view from quantum estimation theory that highlights what it is and what isn?t,” Sci. Reports 6, 19702 (2016).
[Crossref]

2015 (2)

G. I. Viza, J. Martínez-Rincón, G. B. Alves, A. N. Jordan, and J. C. Howell, “Experimentally quantifying the advantages of weak-value-based metrology,” Phys. Rev. A 92, 032127 (2015).
[Crossref]

S. R. Gillmer, X. Yu, C. Wang, and J. D. Ellis, “Robust high-dynamic-range optical roll sensing,” Opt. Lett. 40, 2497–2500 (2015).
[Crossref] [PubMed]

2014 (3)

A. N. Jordan, J. Martínez-Rincón, and J. C. Howell, “Technical advantages for weak-value amplification: when less is more,” Phys. Rev. X 4, 011031 (2014).

J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, “Colloquium: Understanding quantum weak values: Basics and applications,” Rev. Mod. Phys. 86, 307 (2014).
[Crossref]

O. S. Magaa-Loaiza, M. Mirhosseini, B. Rodenburg, and R. W. Boyd, “Amplification of angular rotations using weak measurements,” Phys. Rev. Lett. 112, 200401 (2014).
[Crossref]

2013 (3)

Y. Le, W. Hou, K. Hu, and K. Shi, “High-sensitivity roll-angle interferometer,” Opt. Lett. 38, 3600–3603 (2013).
[Crossref] [PubMed]

W. Chen, S. Zhang, and X. Long, “Angle measurement with laser feedback instrument,” Opt. Express 21, 8044–8050 (2013).
[Crossref] [PubMed]

G. Wu, M. Takahashi, K. Arai, H. Inaba, and K. Minoshima, “Extremely high-accuracy correction of air refractive index using two-colour optical frequency combs,” Sci. Reports 3, 1894 (2013).
[Crossref]

2012 (1)

J. Dressel and A. N. Jordan, “Significance of the imaginary part of the weak value,” Phys. Rev. A 85, 012107 (2012).
[Crossref]

2010 (3)

A. Mitra, B. Deutsch, F. Ignatovich, C. Dykes, and L. Novotny, “Nano-optofluidic detection of single viruses and nanoparticles,” ACS Nano 4, 3 (2010).
[Crossref]

N. Brunner and C. Simon, “Measuring small longitudinal phase shifts: Weak measurements or standard interferometry?” Phys. Rev. Lett. 105, 010405 (2010).
[Crossref] [PubMed]

D. J. Starling, P. B. Dixon, A. N. Jordan, and J. C. Howell, “Precision frequency measurements with interferometric weak values,” Phys. Rev. A 82, 063822 (2010).
[Crossref]

2009 (3)

P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, “Ultrasensitive beam deflection measurement via interferometric weak value amplification,” Phys. Rev. Lett. 102, 173601 (2009).
[Crossref] [PubMed]

T. Schuldt, M. Gohlke, D. Weise, U. Johann, A. Peters, and C. Braxmaier, “Picometer and nanoradian optical heterodyne interferometry for translation and tilt metrology of the lisa gravitational reference sensor,” Class. Quantum Gravity 26, 8 (2009).
[Crossref]

A. N. J. David, J. Starling, P. Ben Dixon, and J. C. Howell, “Optimizing the signal-to-noise ratio of a beam-deflection measurement with interferometric weak values,” Phys. Rev. A 80, 041803 (2009).
[Crossref]

2008 (1)

O. Hosten and P. Kwiat, “Observation of the spin hall effect of light via weak measurements,” Science 319, 5864 (2008).
[Crossref]

2005 (1)

2004 (1)

C.-M. Wu and Y.-T. Chuang, “Roll angular displacement measurement system with microradian accuracy,” Sensors Actuators A: Phys. 116, 145–149 (2004).
[Crossref]

2003 (1)

Z. Liu, D. Lin, H. Jiang, and C. Yin, “Roll angle interferometer by means of waveplates,” Sensors Actuators A: Phys. 104, 127–131 (2003).
[Crossref]

2000 (1)

J. Lawall and E. Kessler, “Michelson interferometry with 10 pm accuracy,” Rev. Sci. Instruments 71, 7 (2000).
[Crossref]

1993 (1)

A. Stevenson, M. Gray, H.-A. Bachor, and D. McClelland, “Quantum-noise-limited interferometric phase measurements,” Appl. Opt. 32, 19 (1993).
[Crossref]

1988 (1)

Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351–1354 (1988).
[Crossref] [PubMed]

Abbott, B.

B. Abbott and et al., “Observation of gravitational waves from a binary black hole merger,” Phys. Rev. Lett. 116, 061102 (2016).
[Crossref] [PubMed]

Aharonov, Y.

Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351–1354 (1988).
[Crossref] [PubMed]

Albert, D. Z.

Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351–1354 (1988).
[Crossref] [PubMed]

Alves, G. B.

G. I. Viza, J. Martínez-Rincón, G. B. Alves, A. N. Jordan, and J. C. Howell, “Experimentally quantifying the advantages of weak-value-based metrology,” Phys. Rev. A 92, 032127 (2015).
[Crossref]

Arai, K.

G. Wu, M. Takahashi, K. Arai, H. Inaba, and K. Minoshima, “Extremely high-accuracy correction of air refractive index using two-colour optical frequency combs,” Sci. Reports 3, 1894 (2013).
[Crossref]

Bachor, H.-A.

A. Stevenson, M. Gray, H.-A. Bachor, and D. McClelland, “Quantum-noise-limited interferometric phase measurements,” Appl. Opt. 32, 19 (1993).
[Crossref]

Ben Dixon, P.

A. N. J. David, J. Starling, P. Ben Dixon, and J. C. Howell, “Optimizing the signal-to-noise ratio of a beam-deflection measurement with interferometric weak values,” Phys. Rev. A 80, 041803 (2009).
[Crossref]

Boyd, R. W.

J. Harris, R. W. Boyd, and J. S. Lundeen, “Weak value amplification can outperform conventional measurement in the presence of detector saturation,” Phys. Rev. Lett. 118, 070802 (2017).
[Crossref] [PubMed]

J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, “Colloquium: Understanding quantum weak values: Basics and applications,” Rev. Mod. Phys. 86, 307 (2014).
[Crossref]

O. S. Magaa-Loaiza, M. Mirhosseini, B. Rodenburg, and R. W. Boyd, “Amplification of angular rotations using weak measurements,” Phys. Rev. Lett. 112, 200401 (2014).
[Crossref]

Braxmaier, C.

T. Schuldt, M. Gohlke, D. Weise, U. Johann, A. Peters, and C. Braxmaier, “Picometer and nanoradian optical heterodyne interferometry for translation and tilt metrology of the lisa gravitational reference sensor,” Class. Quantum Gravity 26, 8 (2009).
[Crossref]

Brunner, N.

N. Brunner and C. Simon, “Measuring small longitudinal phase shifts: Weak measurements or standard interferometry?” Phys. Rev. Lett. 105, 010405 (2010).
[Crossref] [PubMed]

Chen, W.

Chuang, Y.-T.

C.-M. Wu and Y.-T. Chuang, “Roll angular displacement measurement system with microradian accuracy,” Sensors Actuators A: Phys. 116, 145–149 (2004).
[Crossref]

David, A. N. J.

A. N. J. David, J. Starling, P. Ben Dixon, and J. C. Howell, “Optimizing the signal-to-noise ratio of a beam-deflection measurement with interferometric weak values,” Phys. Rev. A 80, 041803 (2009).
[Crossref]

Deutsch, B.

A. Mitra, B. Deutsch, F. Ignatovich, C. Dykes, and L. Novotny, “Nano-optofluidic detection of single viruses and nanoparticles,” ACS Nano 4, 3 (2010).
[Crossref]

Dixon, P. B.

D. J. Starling, P. B. Dixon, A. N. Jordan, and J. C. Howell, “Precision frequency measurements with interferometric weak values,” Phys. Rev. A 82, 063822 (2010).
[Crossref]

P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, “Ultrasensitive beam deflection measurement via interferometric weak value amplification,” Phys. Rev. Lett. 102, 173601 (2009).
[Crossref] [PubMed]

Dressel, J.

J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, “Colloquium: Understanding quantum weak values: Basics and applications,” Rev. Mod. Phys. 86, 307 (2014).
[Crossref]

J. Dressel and A. N. Jordan, “Significance of the imaginary part of the weak value,” Phys. Rev. A 85, 012107 (2012).
[Crossref]

Du, J.

X. Qiu, L. Xie, X. Liu, L. Luo, Z. Li, Z. Zhang, and J. Du, “Precision phase estimation based on weak-value amplification,” Appl. Phys. Lett. 110, 071105 (2017).
[Crossref]

Dykes, C.

A. Mitra, B. Deutsch, F. Ignatovich, C. Dykes, and L. Novotny, “Nano-optofluidic detection of single viruses and nanoparticles,” ACS Nano 4, 3 (2010).
[Crossref]

Ellis, J. D.

Gillmer, S. R.

Gohlke, M.

T. Schuldt, M. Gohlke, D. Weise, U. Johann, A. Peters, and C. Braxmaier, “Picometer and nanoradian optical heterodyne interferometry for translation and tilt metrology of the lisa gravitational reference sensor,” Class. Quantum Gravity 26, 8 (2009).
[Crossref]

Gray, M.

A. Stevenson, M. Gray, H.-A. Bachor, and D. McClelland, “Quantum-noise-limited interferometric phase measurements,” Appl. Opt. 32, 19 (1993).
[Crossref]

Harris, J.

J. Harris, R. W. Boyd, and J. S. Lundeen, “Weak value amplification can outperform conventional measurement in the presence of detector saturation,” Phys. Rev. Lett. 118, 070802 (2017).
[Crossref] [PubMed]

Hosten, O.

O. Hosten and P. Kwiat, “Observation of the spin hall effect of light via weak measurements,” Science 319, 5864 (2008).
[Crossref]

Hou, W.

Howell, J. C.

J. Martínez-Rincón, C. A. Mullarkey, G. I. Viza, W.-T. Liu, and J. C. Howell, “Ultrasensitive inverse weak-value tilt meter,” Opt. Lett. 42, 13 (2017).
[Crossref]

J. Martínez-Rincón, W.-T. Liu, G. I. Viza, and J. C. Howell, “Can anomalous amplification be attained without postselection?” Phys. Rev. Lett. 116, 100803 (2016).
[Crossref] [PubMed]

G. I. Viza, J. Martínez-Rincón, G. B. Alves, A. N. Jordan, and J. C. Howell, “Experimentally quantifying the advantages of weak-value-based metrology,” Phys. Rev. A 92, 032127 (2015).
[Crossref]

A. N. Jordan, J. Martínez-Rincón, and J. C. Howell, “Technical advantages for weak-value amplification: when less is more,” Phys. Rev. X 4, 011031 (2014).

D. J. Starling, P. B. Dixon, A. N. Jordan, and J. C. Howell, “Precision frequency measurements with interferometric weak values,” Phys. Rev. A 82, 063822 (2010).
[Crossref]

P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, “Ultrasensitive beam deflection measurement via interferometric weak value amplification,” Phys. Rev. Lett. 102, 173601 (2009).
[Crossref] [PubMed]

A. N. J. David, J. Starling, P. Ben Dixon, and J. C. Howell, “Optimizing the signal-to-noise ratio of a beam-deflection measurement with interferometric weak values,” Phys. Rev. A 80, 041803 (2009).
[Crossref]

Hu, K.

Hu, M.-J.

M.-J. Hu and Y.-S. Zhang, “Gravitational wave detection via weak measurements amplification,” arXiv (2017).

Ignatovich, F.

A. Mitra, B. Deutsch, F. Ignatovich, C. Dykes, and L. Novotny, “Nano-optofluidic detection of single viruses and nanoparticles,” ACS Nano 4, 3 (2010).
[Crossref]

Inaba, H.

G. Wu, M. Takahashi, K. Arai, H. Inaba, and K. Minoshima, “Extremely high-accuracy correction of air refractive index using two-colour optical frequency combs,” Sci. Reports 3, 1894 (2013).
[Crossref]

Jiang, H.

Z. Liu, D. Lin, H. Jiang, and C. Yin, “Roll angle interferometer by means of waveplates,” Sensors Actuators A: Phys. 104, 127–131 (2003).
[Crossref]

Jin, G.

Johann, U.

T. Schuldt, M. Gohlke, D. Weise, U. Johann, A. Peters, and C. Braxmaier, “Picometer and nanoradian optical heterodyne interferometry for translation and tilt metrology of the lisa gravitational reference sensor,” Class. Quantum Gravity 26, 8 (2009).
[Crossref]

Jordan, A. N.

G. I. Viza, J. Martínez-Rincón, G. B. Alves, A. N. Jordan, and J. C. Howell, “Experimentally quantifying the advantages of weak-value-based metrology,” Phys. Rev. A 92, 032127 (2015).
[Crossref]

A. N. Jordan, J. Martínez-Rincón, and J. C. Howell, “Technical advantages for weak-value amplification: when less is more,” Phys. Rev. X 4, 011031 (2014).

J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, “Colloquium: Understanding quantum weak values: Basics and applications,” Rev. Mod. Phys. 86, 307 (2014).
[Crossref]

J. Dressel and A. N. Jordan, “Significance of the imaginary part of the weak value,” Phys. Rev. A 85, 012107 (2012).
[Crossref]

D. J. Starling, P. B. Dixon, A. N. Jordan, and J. C. Howell, “Precision frequency measurements with interferometric weak values,” Phys. Rev. A 82, 063822 (2010).
[Crossref]

P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, “Ultrasensitive beam deflection measurement via interferometric weak value amplification,” Phys. Rev. Lett. 102, 173601 (2009).
[Crossref] [PubMed]

Kessler, E.

J. Lawall and E. Kessler, “Michelson interferometry with 10 pm accuracy,” Rev. Sci. Instruments 71, 7 (2000).
[Crossref]

Kwiat, P.

O. Hosten and P. Kwiat, “Observation of the spin hall effect of light via weak measurements,” Science 319, 5864 (2008).
[Crossref]

Lawall, J.

J. Lawall and E. Kessler, “Michelson interferometry with 10 pm accuracy,” Rev. Sci. Instruments 71, 7 (2000).
[Crossref]

Le, Y.

Li, S.

Li, Z.

X. Qiu, L. Xie, X. Liu, L. Luo, Z. Li, Z. Zhang, and J. Du, “Precision phase estimation based on weak-value amplification,” Appl. Phys. Lett. 110, 071105 (2017).
[Crossref]

Lin, D.

Z. Liu, D. Lin, H. Jiang, and C. Yin, “Roll angle interferometer by means of waveplates,” Sensors Actuators A: Phys. 104, 127–131 (2003).
[Crossref]

Liu, W.-T.

J. Martínez-Rincón, C. A. Mullarkey, G. I. Viza, W.-T. Liu, and J. C. Howell, “Ultrasensitive inverse weak-value tilt meter,” Opt. Lett. 42, 13 (2017).
[Crossref]

J. Martínez-Rincón, W.-T. Liu, G. I. Viza, and J. C. Howell, “Can anomalous amplification be attained without postselection?” Phys. Rev. Lett. 116, 100803 (2016).
[Crossref] [PubMed]

Liu, X.

X. Qiu, L. Xie, X. Liu, L. Luo, Z. Li, Z. Zhang, and J. Du, “Precision phase estimation based on weak-value amplification,” Appl. Phys. Lett. 110, 071105 (2017).
[Crossref]

Liu, Z.

Z. Liu, D. Lin, H. Jiang, and C. Yin, “Roll angle interferometer by means of waveplates,” Sensors Actuators A: Phys. 104, 127–131 (2003).
[Crossref]

Long, X.

Lundeen, J. S.

J. Harris, R. W. Boyd, and J. S. Lundeen, “Weak value amplification can outperform conventional measurement in the presence of detector saturation,” Phys. Rev. Lett. 118, 070802 (2017).
[Crossref] [PubMed]

Luo, L.

X. Qiu, L. Xie, X. Liu, L. Luo, Z. Li, Z. Zhang, and J. Du, “Precision phase estimation based on weak-value amplification,” Appl. Phys. Lett. 110, 071105 (2017).
[Crossref]

Magaa-Loaiza, O. S.

O. S. Magaa-Loaiza, M. Mirhosseini, B. Rodenburg, and R. W. Boyd, “Amplification of angular rotations using weak measurements,” Phys. Rev. Lett. 112, 200401 (2014).
[Crossref]

Malik, M.

J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, “Colloquium: Understanding quantum weak values: Basics and applications,” Rev. Mod. Phys. 86, 307 (2014).
[Crossref]

Martínez-Rincón, J.

J. Martínez-Rincón, C. A. Mullarkey, G. I. Viza, W.-T. Liu, and J. C. Howell, “Ultrasensitive inverse weak-value tilt meter,” Opt. Lett. 42, 13 (2017).
[Crossref]

J. Martínez-Rincón, W.-T. Liu, G. I. Viza, and J. C. Howell, “Can anomalous amplification be attained without postselection?” Phys. Rev. Lett. 116, 100803 (2016).
[Crossref] [PubMed]

G. I. Viza, J. Martínez-Rincón, G. B. Alves, A. N. Jordan, and J. C. Howell, “Experimentally quantifying the advantages of weak-value-based metrology,” Phys. Rev. A 92, 032127 (2015).
[Crossref]

A. N. Jordan, J. Martínez-Rincón, and J. C. Howell, “Technical advantages for weak-value amplification: when less is more,” Phys. Rev. X 4, 011031 (2014).

J. Martínez-Rincón, “Weak-values metrological techniques for parameter estimation,” Ph.D. thesis (2017).

McClelland, D.

A. Stevenson, M. Gray, H.-A. Bachor, and D. McClelland, “Quantum-noise-limited interferometric phase measurements,” Appl. Opt. 32, 19 (1993).
[Crossref]

Miatto, F. M.

J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, “Colloquium: Understanding quantum weak values: Basics and applications,” Rev. Mod. Phys. 86, 307 (2014).
[Crossref]

Minoshima, K.

G. Wu, M. Takahashi, K. Arai, H. Inaba, and K. Minoshima, “Extremely high-accuracy correction of air refractive index using two-colour optical frequency combs,” Sci. Reports 3, 1894 (2013).
[Crossref]

Mirhosseini, M.

O. S. Magaa-Loaiza, M. Mirhosseini, B. Rodenburg, and R. W. Boyd, “Amplification of angular rotations using weak measurements,” Phys. Rev. Lett. 112, 200401 (2014).
[Crossref]

Mitra, A.

A. Mitra, B. Deutsch, F. Ignatovich, C. Dykes, and L. Novotny, “Nano-optofluidic detection of single viruses and nanoparticles,” ACS Nano 4, 3 (2010).
[Crossref]

Mullarkey, C. A.

Novotny, L.

A. Mitra, B. Deutsch, F. Ignatovich, C. Dykes, and L. Novotny, “Nano-optofluidic detection of single viruses and nanoparticles,” ACS Nano 4, 3 (2010).
[Crossref]

Peters, A.

T. Schuldt, M. Gohlke, D. Weise, U. Johann, A. Peters, and C. Braxmaier, “Picometer and nanoradian optical heterodyne interferometry for translation and tilt metrology of the lisa gravitational reference sensor,” Class. Quantum Gravity 26, 8 (2009).
[Crossref]

Qiu, X.

X. Qiu, L. Xie, X. Liu, L. Luo, Z. Li, Z. Zhang, and J. Du, “Precision phase estimation based on weak-value amplification,” Appl. Phys. Lett. 110, 071105 (2017).
[Crossref]

Rodenburg, B.

O. S. Magaa-Loaiza, M. Mirhosseini, B. Rodenburg, and R. W. Boyd, “Amplification of angular rotations using weak measurements,” Phys. Rev. Lett. 112, 200401 (2014).
[Crossref]

Salazar-Serrano, L. J.

J. P. Torres and L. J. Salazar-Serrano, “Weak value amplification: a view from quantum estimation theory that highlights what it is and what isn?t,” Sci. Reports 6, 19702 (2016).
[Crossref]

Schuldt, T.

T. Schuldt, M. Gohlke, D. Weise, U. Johann, A. Peters, and C. Braxmaier, “Picometer and nanoradian optical heterodyne interferometry for translation and tilt metrology of the lisa gravitational reference sensor,” Class. Quantum Gravity 26, 8 (2009).
[Crossref]

Shi, K.

Simon, C.

N. Brunner and C. Simon, “Measuring small longitudinal phase shifts: Weak measurements or standard interferometry?” Phys. Rev. Lett. 105, 010405 (2010).
[Crossref] [PubMed]

Slocum, A. H.

A. H. Slocum, Precision Machine Design (Society of Manufacturing Engineers, 1992).

Starling, D. J.

D. J. Starling, P. B. Dixon, A. N. Jordan, and J. C. Howell, “Precision frequency measurements with interferometric weak values,” Phys. Rev. A 82, 063822 (2010).
[Crossref]

P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, “Ultrasensitive beam deflection measurement via interferometric weak value amplification,” Phys. Rev. Lett. 102, 173601 (2009).
[Crossref] [PubMed]

Starling, J.

A. N. J. David, J. Starling, P. Ben Dixon, and J. C. Howell, “Optimizing the signal-to-noise ratio of a beam-deflection measurement with interferometric weak values,” Phys. Rev. A 80, 041803 (2009).
[Crossref]

Stevenson, A.

A. Stevenson, M. Gray, H.-A. Bachor, and D. McClelland, “Quantum-noise-limited interferometric phase measurements,” Appl. Opt. 32, 19 (1993).
[Crossref]

Takahashi, M.

G. Wu, M. Takahashi, K. Arai, H. Inaba, and K. Minoshima, “Extremely high-accuracy correction of air refractive index using two-colour optical frequency combs,” Sci. Reports 3, 1894 (2013).
[Crossref]

Torres, J. P.

J. P. Torres and L. J. Salazar-Serrano, “Weak value amplification: a view from quantum estimation theory that highlights what it is and what isn?t,” Sci. Reports 6, 19702 (2016).
[Crossref]

Vaidman, L.

Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351–1354 (1988).
[Crossref] [PubMed]

Viza, G. I.

J. Martínez-Rincón, C. A. Mullarkey, G. I. Viza, W.-T. Liu, and J. C. Howell, “Ultrasensitive inverse weak-value tilt meter,” Opt. Lett. 42, 13 (2017).
[Crossref]

J. Martínez-Rincón, W.-T. Liu, G. I. Viza, and J. C. Howell, “Can anomalous amplification be attained without postselection?” Phys. Rev. Lett. 116, 100803 (2016).
[Crossref] [PubMed]

G. I. Viza, J. Martínez-Rincón, G. B. Alves, A. N. Jordan, and J. C. Howell, “Experimentally quantifying the advantages of weak-value-based metrology,” Phys. Rev. A 92, 032127 (2015).
[Crossref]

Wang, C.

Weise, D.

T. Schuldt, M. Gohlke, D. Weise, U. Johann, A. Peters, and C. Braxmaier, “Picometer and nanoradian optical heterodyne interferometry for translation and tilt metrology of the lisa gravitational reference sensor,” Class. Quantum Gravity 26, 8 (2009).
[Crossref]

Wu, C.-M.

C.-M. Wu and Y.-T. Chuang, “Roll angular displacement measurement system with microradian accuracy,” Sensors Actuators A: Phys. 116, 145–149 (2004).
[Crossref]

Wu, G.

G. Wu, M. Takahashi, K. Arai, H. Inaba, and K. Minoshima, “Extremely high-accuracy correction of air refractive index using two-colour optical frequency combs,” Sci. Reports 3, 1894 (2013).
[Crossref]

Xie, L.

X. Qiu, L. Xie, X. Liu, L. Luo, Z. Li, Z. Zhang, and J. Du, “Precision phase estimation based on weak-value amplification,” Appl. Phys. Lett. 110, 071105 (2017).
[Crossref]

Yang, C.

Yin, C.

Z. Liu, D. Lin, H. Jiang, and C. Yin, “Roll angle interferometer by means of waveplates,” Sensors Actuators A: Phys. 104, 127–131 (2003).
[Crossref]

Yu, X.

Zhang, E.

Zhang, S.

Zhang, Y.-S.

M.-J. Hu and Y.-S. Zhang, “Gravitational wave detection via weak measurements amplification,” arXiv (2017).

Zhang, Z.

X. Qiu, L. Xie, X. Liu, L. Luo, Z. Li, Z. Zhang, and J. Du, “Precision phase estimation based on weak-value amplification,” Appl. Phys. Lett. 110, 071105 (2017).
[Crossref]

ACS Nano (1)

A. Mitra, B. Deutsch, F. Ignatovich, C. Dykes, and L. Novotny, “Nano-optofluidic detection of single viruses and nanoparticles,” ACS Nano 4, 3 (2010).
[Crossref]

Appl. Opt. (1)

A. Stevenson, M. Gray, H.-A. Bachor, and D. McClelland, “Quantum-noise-limited interferometric phase measurements,” Appl. Opt. 32, 19 (1993).
[Crossref]

Appl. Phys. Lett. (1)

X. Qiu, L. Xie, X. Liu, L. Luo, Z. Li, Z. Zhang, and J. Du, “Precision phase estimation based on weak-value amplification,” Appl. Phys. Lett. 110, 071105 (2017).
[Crossref]

Class. Quantum Gravity (1)

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Opt. Express (1)

Opt. Lett. (4)

Phys. Rev. A (4)

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D. J. Starling, P. B. Dixon, A. N. Jordan, and J. C. Howell, “Precision frequency measurements with interferometric weak values,” Phys. Rev. A 82, 063822 (2010).
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J. Dressel and A. N. Jordan, “Significance of the imaginary part of the weak value,” Phys. Rev. A 85, 012107 (2012).
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A. N. Jordan, J. Martínez-Rincón, and J. C. Howell, “Technical advantages for weak-value amplification: when less is more,” Phys. Rev. X 4, 011031 (2014).

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[Crossref]

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Figures (5)

Fig. 1
Fig. 1 Full schematic of the measurement setup used for interferometric roll sensing; BS, beamsplitter, AOM, acousto-optic modulator, HWP, halfwave plate, PBS, polarizing beamsplitter, pol., polarizer, PDr and PDm, reference and measurement photodetectors, QWP, quarterwave plate. A Renishaw interferometer measures pitch for calibration from an orthogonal measurement position.
Fig. 2
Fig. 2 Simulations of the weak-value-emulated amplification in the heterodyne roll interferometer. (a) The phase sensitivity between pre- and post-selection as a function of QWP angle, θ, and HWP ‘roll’ angle, α. (b) The AC amplitude of the interferometer drops to near zero when the phase sensitivity is amplified. (c) The derivative of phase as a function of roll angle showing that the phase sensitivity is amplified by over 700 times. The QWP angle of 0.160° is experimentally validated later in Fig. 3. (d) Amplification manifests mainly in the imaginary part of the complex phasor signal.
Fig. 3
Fig. 3 The effects of increased phase sensitivity in the interferometer in the (a) time domain and (b) frequency domain. As the fast axis of the QWP, θ, approaches the fast axis of the HWP, α, technical noise - mainly in the form of vibrations - is suppressed.
Fig. 4
Fig. 4 Statistical limitations of shot noise and detector noise as outlined in Eqs. (7)(8). In the absence of all forms of technical noise, our setup is fundamentally limited by detector noise due to the discarding of photons during post-selection.
Fig. 5
Fig. 5 Experimental peak-to-peak noise results with vibration content as shown in Fig. 3. The numerical simulation assumes white noise with the same nominal amplitude as the raw phase signals recorded in this paper followed by post-processing from Eq. (6) - the theoretical line.

Equations (8)

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P = [ 1 0 0 0 ] , H ( α ) = [ cos ( 2 α ) sin ( 2 α ) sin ( 2 α ) cos ( 2 α ) ] , Q = [ 1 0 0 i ] R ( θ ) = [ cos ( θ ) sin ( θ ) sin ( θ ) cos ( θ ) ] , and E 0 = [ A 1 e i ω 1 t A 2 e i ω 2 t ] ,
E s = PH ( α ) R ( θ ) QR ( θ ) E 0 ,
E s = [ A 1 e i ω 1 t { i sin ( θ ) [ sin ( 2 α + θ ) ] + cos ( θ ) [ cos ( 2 α + θ ) ] } + A 2 e i ω 2 t { i cos ( θ ) [ sin ( 2 α + θ ) ] sin ( θ ) [ cos ( 2 α + θ ) ] } ] [ 1 0 ] .
I = | E s | 2 = C 1 2 A 1 2 + C 2 2 A 2 2 + 2 C 1 C 2 A 1 A 2 cos [ ( ω 1 ω 2 ) t + ϕ m ] .
tan ( ϕ m ) = tan ( 4 α + 2 θ ) sin ( 2 θ ) , C 1 = 1 2 2 + cos [ 4 α ] + cos [ 4 ( α + θ ) ] , and C 2 = 1 2 2 cos [ 4 α ] cos [ 4 ( α + θ ) ] .
α = θ 2 1 4 arctan [ tan ( ϕ m ) sin ( 2 θ ) ]
ϕ s = 1 λ 2 hc λ B η P ,
ϕ d = NEP B P ,

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