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, PD_r and PD_m, 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, PD_r and PD_m, 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,

The results of Eq. (5) can be seen in Fig. 2 after the DC terms, $C_1^2{A}_1^2+C_2^2{A}_2^2$, are removed with a high-pass filter. Amplitude (2C₁C₂A₁A₂) and phase (ϕ_m) 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 (ϕ_m) in Eq. (5) according to

The rolling and vibrating stage was a precision NewFocus^TM 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]

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.