Sinusoidal Phase Modulation Research Articles

Integrating-bucket phase stabilization method for fiber-optic fringe projector

Ac phase tracking technique (PTAC) is an established and effective method for phase stabilization of fiber-optic fringe projectors (FoFP), but it may reduce the fringe visibility owing to the introduction of an extra sinusoidal phase modulation. In this paper we propose an integrating-bucket method to stabilize the phase of FoFPs which is applicable to both linear phase shifting (LPS) profilometry and sinusoidal phase shifting (SiPS) profilometry. The method requires no extra modulation of the interference phase; instead, it divides the interference signal period into N (N > 2) episodes with equal time intervals and directly integrates these episodes to obtain N discrete data. By manipulating these data the interference phase can be calculated and accordingly phase disturbance can be eliminated by the feedback loop. The proposed method is as effective as the PTAC technique but at the same time more computationally efficient because the operations involved are merely add, subtraction and arctangent solution. We built a digital phase controller with the proposed method, and managed to restrict the maximum phase drift of an FoFP to within 15 mrad in 5 min for both LPS and SiPS, which is similar to the result obtained with the PTAC technique. What’s more, we measured the mean value variation of the recovered phase map of a stabilized FoFP profilometer, and results are 30 mrad for LPS and 50 mrad for SiPS in 5 min.

Extraction of Carrier Phase Delay for Nonlinear Errors Compensation of PGC Demodulation in an SPM Interferometer

Traditional phase generated carrier (PGC) schemes require that the carrier phase delay be kept zero and the phase modulation depth be maintained at a certain value; otherwise, serious nonlinear errors will be introduced into the phase demodulation of a sinusoidal phase modulation interferometer. Proposed herein is a robust carrier phase delay extraction method for nonlinear errors compensation of PGC demodulation that mitigates the effect of the interference phase shift. The carrier phase delay is directly extracted by adopting the first-, second-, and fourth-order in-phase and quadrature harmonic components generated by quadrature multi-harmonics frequency down conversion. Further, with the extracted phase delay, real-time compensations for the phase delay and phase demodulation depth are realized simultaneously in PGC demodulation. Theoretical analysis and derivation of the proposed method are given in detail. Simulation and displacement measurement experiments with different phase delays and phase modulation depths are performed to verify the effectiveness of the proposed method. The experimental results demonstrate that the proposed method is able to precisely extract the phase delay and efficiently eliminate the effects of the phase delay and modulation depth instability.

Arbitrary phase shifting method for fiber-optic fringe projection profilometry based on temporal sinusoidal phase modulation

A new arbitrary phase shifting method is proposed in this paper for fiber-optic fringe projection profilometry. By introducing a small temporal sinusoidal phase modulation to the phase shift and employing the phase-generated carrier (PGC) method, the time-varying interference phase α(t)can be demodulated from the Michelson interference signal, which is reflected by the fiber end faces and recorded concurrently with the recording of the fringe pattern. Integrate sin [α(t)] and cos [α(t)] respectively, and then the phase shifts between frames can be determined and consequently used to solve the phase map with least-squares PS algorithm. The noise model of this method is deduced from which the optimum modulation amplitude can be selected. In the experiment, four fringe patterns with uncalibrated phase shifts were recorded under strong environmental disturbance, and were analyzed by utilizing the Carre algorithm, the proposed method and the modified Fourier transform (MFT) method respectively. A visual criterion is used to evaluate the performances of these methods. Experimental results indicate that the proposed method is much more robust to phase shift miscalibration than the Carre algorithm, and its performance is comparable to that of the MFT method while its computational complexity is substantially lower. Therefore, the proposed method is suitable for real-time phase shift calibration.

Artifact-free robust single-shot background subtraction for optical coherence tomography

A proof-of-principle demonstration of a novel method of background subtraction for optical coherence tomography is presented using a full-field time-domain set-up. This single-shot method is based on time-averaged sampling of a sinusoidal phase modulation in the reference arm. It can significantly suppress motion artifacts as well as entirely eliminate phase-noise induced background subtraction errors that corrupt the images obtained using other methods. When used with Fourier domain set-ups, this technique can also eliminate arbitrarily strong autocorrelation artifacts.

Interferometric delay tracking for low-noise Mach-Zehnder-type scanning measurements.

Precise delay control is of paramount importance in optical pump-probe measurements. Here, we report on a high-precision delay tracking technique for mechanical scanning measurements in a Mach-Zehnder interferometer configuration. The setup employs a 1.55-µm continuous-wave laser beam propagating along the interferometer arms. Sinusoidal phase modulation at 30 MHz, and demodulation of the interference signal at the fundamental frequency and its second harmonic, enables delay tracking with sampling rates of up to 10 MHz. At an interferometer arm length of 1 m, root-mean-square error values of the relative delay tracking below 10 attoseconds for both stationary and mechanically scanned (0.2 mm/s) operation are demonstrated. By averaging several scans, a precision of the delay determination better than 1 as is reached. We demonstrate this performance with a mechanical chopper periodically interrupting one of the interferometer arms, which opens the door to the combination of high-sensitivity lock-in detection with (sub-)attosecond-precision relative delay determination.

Laser diode frequency stabilization bases on Iodine saturated absorption and sinusoidal phase modulation with suppression of residual amplitude modulation

Laser diode frequency stabilization bases on Iodine saturated absorption and sinusoidal phase modulation with suppression of residual amplitude modulation

Impact of a temporal sinusoidal phase modulation on the optical spectrum

We discuss the effects of imparting a temporal sinusoidal phase modulation to a continuous wave on the frequency spectrum. While a practical analytical solution to this problem already exists, we present here a physical interpretation based on interference processes. This simple model will help students better understand the origin of the oscillatory structure that can be observed in the resulting spectrum and that is characteristic of Bessel functions of the first kind. We illustrate our approach with an example from the field of optics.

Iodine-frequency-stabilized laser diode and displacement-measuring interferometer based on sinusoidal phase modulation

We propose a sinusoidal phase modulation method to achieve both the frequency stabilization of an external-cavity laser diode (ECLD) to an 127I2 saturated absorption transition near 633 nm and displacement measurement using a Mach–Zehnder interferometer. First, the frequency of the ECLD is stabilized to the b21 hyperfine component of the P(33) 6-3 transition of 127I2 by combining sinusoidal phase modulation by an electro-optic modulator and frequency modulation spectroscopy by chopping the pump beam using an acousto-optic modulator. Even though a small modulation index of m = 3.768 rad is utilized, a relative frequency stability of 10−11 order is obtained over a sampling time of 400 s. Secondly, the frequency-stabilized ECLD is applied as a light source to a Mach–Zehnder interferometer. From the two consecutive modulation harmonics (second and third orders) involved in the interferometer signal, the displacement of the moving mirror is determined for four optical path differences (L0 = 100, 200, 500, and 1000 mm). The measured modulation indexes for the four optical path differences coincide with the designated value (3.768 rad) within 0.5%. Compared with the sinusoidal frequency modulation Michelson interferometer (Vu et al 2016 Meas. Sci. Technol. 27 105201) which was demonstrated by some of the same authors of this paper, the phase modulation Mach–Zhender interferometer could fix the modulation index to a constant value for the four optical path differences. In this report, we discuss the measurement principle, experimental system, and results.

High-accuracy vibration sensor based on a Fabry-Perot interferometer with active phase-tracking technology.

A novel position-sensitive Fabry-Perot interferometer was constructed with direct phase modulation by a built-in electro-optic modulator. Pure sinusoidal phase modulation of the light was produced, and the first harmonic of the interference signal was extracted to dynamically maintain the interferometer phase to the most sensitive point of the interferogram. Therefore, the minute vibration of the object was coded on the variation of the interference signal and could be directly retrieved by the output voltage of a photodetector. The operating principle and the signal processing method for active feedback control of the interference phase have been demonstrated in detail. The developed vibration sensor was calibrated through a high-precision piezo-electric transducer and tested by a nano-positioning stage under a vibration magnitude of 60nm and a frequency of 300Hz. The active phase-tracking method of the system provides high immunity against environmental disturbances. Experimental results show that the proposed interferometer can effectively reconstruct tiny vibration waveforms with subnanometer resolution, paving the way for high-accuracy vibration sensing, especially for micro-electro-mechanical systems/nano-electro-mechanical systems and ultrasonic devices.

Impact of initial pulse characteristics on the mitigation of self-phase modulation by sinusoidally time varying phase

A simple and efficient approach to suppress undesirable self-phase modulation (SPM) of optical pulses propagating in fiber-optic systems is based on imposing a sinusoidal temporal phase modulation on the pulses to offset the chirp generated by SPM (Audo et al. Opt Lett 42(15):2902–2905, 2017). Here, we present a detailed analysis of this method. We derive an exact formula for the reduction of the SPM-induced rms spectrum broadening of an initially Gaussian pulse enabled by the sinusoidal compensation, and we assess the effects of the initial pulse shape and duration on the effectiveness of the technique by means of numerical simulation. The differences between pre- and post-propagation compensation schemes are also discussed.

Optical fiber Fabry-Pérot micro-displacement sensor for MEMS in-plane motion stage

Fabry-Pérot interferometer sensors have been widely used in Micro-Electro-Mechanical-Systems (MEMS) due to high displacement accuracy and immunity to electromagnetic noises, but they are still limited by micro scale measurement range. In this paper, a Fabry-Pérot interferometer in-plane displacement sensor is proposed for measuring the displacement of MEMS devices utilizing a polished optical fiber and a modulated laser source. The polished optical fiber and a sidewall of a MEMS device form an optical cavity for the proposed sensor. The sinusoidal phase modulation with extreme point search algorithm enables the proposed sensor to measure displacements larger than the wavelengths of the laser light in real time. The experimental results show that the proposed displacement sensor has a capability to measure displacements larger than 3μm and it shows the measurement accuracy less than 35nm. The proposed displacement sensor is then embedded on a single degree-of-freedom MEMS motion stage and tested to monitor its displacement in real time.

Extinction ratio enhancement by sinusoidal phase modulation and nonlinear spectral focusing

We propose a new approach to enhance the extinction ratio of a pulse impaired by a coherent continuous-wave background. Taking advantage of a sinusoidal phase modulation followed by a nonlinear spectral focusing and an optical bandpass filtering, we are able to significantly decrease the level of the unwanted background. The principle of the method is experimentally validated at telecommunication wavelengths.

Theory and Experiment Study on Laser Self-Mixing Interference with Multiway Feedback External Cavity

The self-mixing interference (SMI) effect with multiway feedback is analyzed in this paper. The basic theory of SMI with multiway feedback external cavity (MFEC) can be derived from the theory of interference as well as the theory of laser self-mixing interference (LSMI). Based on the external cavity structure with MFEC, a SMI model with two feedback external cavity structure is proposed. The expressions of the phase and output power in the MFEC are deduced and some simulation analyses are made. It is found that the output signal of the SMI system with two external cavity is sinusoidal or sawtooth waves, which is similar to the signal of SMI with a low frequency sinusoidal phase modulation. The density of interference signal fringes is uneven in one period and the shape of the interference wave is determined by the intensity of the external feedback. Furthermore, a SMI with two feedback external cavity experiment system is considered, and the experiment verifies the simulations.

Nonlinear spectrum broadening cancellation by sinusoidal phase modulation.

We propose and experimentally demonstrate a new approach to dramatically reduce the spectral broadening induced by self-phase modulation occurring in a Kerr medium. By using a temporal sinusoidal phase modulation, we efficiently cancel to a large extent the chirp induced by the nonlinear effect. Experimental validation carried out in a passive or amplifying fiber confirms the interest of the technique for the mitigation of the spectral expansion of long pulses.

Controlled longitudinal emittance blow-up using band-limited phase noise in CERN PSB

Controlled longitudinal emittance blow-up (from 1 eVs to 1.4 eVs) for LHC beams in the CERN PS Booster is currently achievied using sinusoidal phase modulation of a dedicated high-harmonic RF system. In 2021, after the LHC injectors upgrade, 3 eVs should be extracted to the PS. Even if the current method may satisfy the new requirements, it relies on low-power level RF improvements. In this paper another method of blow-up was considered, that is the injection of band-limited phase noise in the main RF system (h=1), never tried in PSB but already used in CERN SPS and LHC, under different conditions (longer cycles). This technique, which lowers the peak line density and therefore the impact of intensity effects in the PSB and the PS, can also be complementary to the present method. The longitudinal space charge, dominant in the PSB, causes significant synchrotron frequency shifts with intensity, and its effect should be taken into account. Another complication arises from the interaction of the phase loop with the injected noise, since both act on the RF phase. All these elements were studied in simulations of the PSB cycle with the BLonD code, and the required blow-up was achieved.

Sinusoidal phase-modulating interferometer with ellipse fitting and a correction method.

In a sinusoidal phase-modulating interferometer, sinusoidal modulation of the phase of the laser or the reference wave is necessary. However, modulation of the phase also involves an intensity modulation of the light, which leads to a measurement error if conventional signal processing is used. In addition, the error of modulation depth and the phase delay of demodulation also increase the measurement error. A novel signal processing, with ellipse fitting and a correction method, is proposed. Numerical simulation results and experimental results prove that the novel signal processing can compensate for the measurement error caused by the intensity modulation, the error of modulation depth, and the phase delay of demodulation.

Linear sampling and magnification technique based on phase modulators and dispersive elements: The temporal lenticular lens

In this work, we exploit the space/time duality in optics to implement a temporal lenticular lens allowing to simultaneously sample and magnify an arbitrary-shaped optical signal. More specifically, by applying a sinusoidal phase-modulation, the signal under test is propagated through a discrete dispersive element that samples and magnifies its initial waveform. Thanks to this temporal lenticular lens, optical sampling associated to an intensity magnification factor of 3.6 is experimentally demonstrated at a repetition rate of 10 GHz.

Co-axis digital holography based on sinusoidal phase modulation using generalized lock-in detection.

Sinusoidal phase modulating (SPM) interferometers are used to precisely measure complex light fields with simple interferometric setups. Recently, a generalized lock-in technique has been proposed for optimizing the signal extraction in phase-modulated interferometers. This article shows its applicability in digital holography as well as digital holographic interferometry. A compact homemade setup based on the Michelson interferometer was used to observe the scattering sample surface with a spatial resolution of 32 LP/mm; out-of-plane rotation was measured with a theoretical detection limit of 0.004°. The comparison between the generalized lock-in detection and traditional SPM interferometer method was done theoretically and experimentally.

A simplified algorithm for digital fringe analysis in two-wave interferometry with sinusoidal phase modulation

A compact simplified algorithm for digital detection of the amplitude and phase of the interferometric signal delivered by a two-wave interferometer with sinusoidal phase modulation is presented. The algorithm consists of simple mathematical combinations of four frames obtained by integration by a camera of the time-varying intensity in an interference pattern during the four successive quarters of the modulation period. The algorithm is invariant by circular permutation of the four image frames. Any set of four consecutive frames can be used for the calculations, which simplifies the practical implementation of the method. A numerical simulation has been carried out to evaluate the efficiency of the algorithm for fringe envelope extraction in low-coherence interferometry. A theoretical analysis of the effect of noise in phase map calculation is conducted. A comparison with the conventional sinusoidal phase-shifting algorithm is made.

Real-time phase delay compensation of PGC demodulation in sinusoidal phase-modulation interferometer for nanometer displacement measurement.

As the phase delay between the carrier component of the detected interference signal and the carrier has adverse effect for phase generated carrier (PGC) demodulation, it is essential to compensate the phase delay to improve the accuracy of precision displacement measurement in sinusoidal phase-modulation interferometer (SPMI). In this paper, a real-time phase delay compensation method is proposed by regulating a compensating phase introduced to the carrier to maximize the output of the low pass filter so as to make the carrier synchronize with the interference signal. The influence of phase delay for PGC demodulation is analyzed and the method for real-time phase delay compensation is described in detail. The simulation of the method was performed to verify the validity of the phase delay compensation algorithm. A SPMI using an EOM was constructed and several comparative experiments were carried out to demonstrate the feasibility of the proposed method. The experimental results show that the phase delay can be compensated accurately in real time, and nanometer accuracy is achieved for precision displacement measurement.