1. Introduction

Inspired by the seminal work of Lang and Kobayashi [1] at 1980, self-mixing interference (SMI), threshold laser gain was modulated by optical injection reflected by one rough target to alter the laser intensity, spectrum and slope efficiency, became a non-contact cost-effective sensing method with high sensitivity to phase, easy demodulation and compactness. As a competitive fitting-to-embedded-situation laser technique, tiny laser diode SMIs in-built with photo diodes had applied from general physical sensing (distance, velocity, vibration, bending, displacement, liquid surface, thickness, pressure, refraction index [2], ranging...) to more interdisciplinary studies (mechanics, chemistry, nanoparticle [3], medical biomedical signal [4], vapor [5], laser micro-drilling [6], terahertz imaging [7], material and even nanometer damping of mechanical cantilever [8]...) in last two decades. Recently, more and more excellent SMIs were reported for single target utilizing diodes, such as differential structural SMI [9], phase shifting SMI [10,11], edge-filter enhanced SMI [12,13], and frequency modulated SMI [14]. Nevertheless, as studied by S. Donati [15], diodes subjected to speckle effect [16] from diffusive surfaces would generated pseudo phase and amplitude as a systematic speckle-pattern random error [17]. Hence, diode SMIs in practical application required Aspheric lens for coupling beam to distant object, or performed in short target-to-laser distance (<120μm) [8], or coating treatment on targets. Meanwhile, signal processing needed locked-in method [18], Wavelet [19], or Hilbert transform to prevent accuracy degeneration.

For substantial accuracy improvement, N. Servagent inserted a lithium niobate crystal into external cavity of single-path SMI and obtained resolution of target 10cm away from laser [10]. Subsequently, to further weaken unwanted speckle effect, our team employed a linearly-polarized laser [20] associated with phase-unwrapping algorithm and achieved a better 10nm resolution under weak feedback regime. By phase shifting (PS), a slow interferometric optical phase was segmented into many sub intervals and extracted from each interval, which led to performance of SMI in terms of sensitivity was significantly improved. Unfortunately, one imperfection of our SMI was that only a single target can be mirrored by one laser [21].

Currently, nanotechnology and semiconductor industries had expressed a growing interest in precision simultaneous targets or parameters measurement. As we all know, the widely-used advanced technique, step-and-scan optical lithography [22,23], delivering graphics to surfaces of medium layer or crystals, depended on precision positioning. Another cutting-edge technique, electric circuit layout [24], required two-dimensional geometrical coordinate of components along with x-axis and y-axis. Besides, dual parameters (temperature and stress of storage tanks for nuclear fuel or poisonous materials), two-dimensional profiling, online structural analysis or dual-target measurement are unable to rely on single-path SMIs [21].

In physics, information carried by light is related to amplitude [8], frequency [14], phase [20] and polarization [25]. However, hitherto, single-path SMIs has paid little attention to polarization in literature, polarization is heavily less explored than phase or frequency. To utilize polarization characteristic and satisfy the growing requirement on two-dimensional (2D) measurement [22–24], this paper builds a compact, high-resolution, fast demodulation, dual-target SMI instrument through polarization multiplexing (PM) without optical cross-talk. Unprecedented integration of PM and PS in SMI is studied to outperform the conventional single-path SMIs. Dynamics of dual-mode laser reacting to independent optical injections is investigated with conception-to-proof experiments. The developed SMI allows >1m target-to-laser distance under poor reflection condition and the ameliorated phase algorithm spends less than 20 milliseconds to retrieve two-dimensional phases, which is faster and more sensitive than laser wavelength scanning method using expensive optical spectrum analyzers [26].

In arrangement, we firstly expound the basic conception of integrating two techniques. Then, the observed SMI signal containing multi-harmonics distribution in frequency domain is analyzed to confirm theory. Subsequently, a feasible signal processing is presented with nanometric two-dimensional displacement measurement under laboratory room. At last, experimental comparisons to lock-in technique and a laser Doppler velocity meter (LDV) conclude a one-magnitude resolution improvement than diode SMIs [14] with increased path number, which is applicable for real time simultaneous parameters measurement.

2. Principle

Significant feature of integration of PM and PS is two independent polarization-based optical paths. The schematic in Fig. 1 is mainly implemented on an inexpensive dual longitudinal mode laser, electro-optic modulators (EOM) at perpendicular polarizations and a polarization beam splitter (PBS). Laser source (Atomic He-Ne gas laser, Class-II) driven by constant current (5mA) emits circularly polarized light (632.8nm) to a common polarization beam splitter (PBS) with 45° incident angle, which also receives back-scattered lights for self-mixing. The used PBS is composed by a pair of rectangular prisms coated with multi-media layers, thus, incident circularly polarized light is decomposed into two linearly polarized lights due to fast axis and slow axis of PBS, o-light and e-light, which are rigidly orthogonal in polarization. In experiment, beam waist radii of o and e lights are both less than 1.2mm even at distance of 1m away making impinged areas on targets much narrower than Gauss beam emitted by diodes, therefore, speckle effect is significantly weakened. In optical path, o-light transmits PBS and is back-scattered by moving target_o, e-light is refracted by PBS and back-scattered by target_e. EOMs for PS consisted of magnesium-oxide doped lithium niobate crystals offer a wide spectral-transparency window, broadband range (0~0.25GHz), low insertion losses (<3dB), and large coefficient, which are mechanically fixed at perpendicular direction to keep their intrinsic polarizations consistent with o and e polarizations respectively. Since EOMs slews at different frequencies, induced interferometric phases are transferred to amplitudes of different harmonics [20]. Due to linearly polarization, phase shifting efficiency is improved than [10]. After twice passing through EOMs, reflected phase-shifted o and e lights re-enter into cavity simultaneously to modulate laser intensity. Therefore, independent optical paths share the same cavity, o-path runs: Laser↔PBS_o↔EOM_o↔Target_o, and e-path runs: Laser↔ PBS_e↔EOM_e ↔Target_e. Unlike the dual-beam interferometers, two paths have no reference relation and are able to operate individually. Orthogonality of polarizations prohibits the optical mixing of o and e reflected lights, so optical cross-talk between paths is completely eradicated [25,27].

 

Fig. 1 Schematic of this lensless, phase shifting, polarization multiplexing SMI in absence of sophisticated electronics, which is consisted of a dual-output dual-longitudinal mode He-Ne laser, electro-optic modulators at perpendicular polarizations, polarization beam splitter (PBS) and targets at different directions. Two linearly polarizations by PBS are defined as o and e respectively. VTs (VT1, Thorlabs) denote adjustable resistors for converting photo-currents into voltages. PC denotes computer for signal processing. Generator provides alternating voltages to oscillate target_e (loudspeaker), target_o (precision piezoelectric ceramics transducer, PZT) is driven by a closed-loop controller connected to PC. PD is Si-based photo electric detector (DET36A/M, Thorlabs). DAQ is data acquisition device (USB6361, NI). Because one mode light intensity already mirror status of two targets in following analysis, the dotted PD_e and VT_e can be removed for system simplicity.

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Internal laser cavity for self-mixing works as an optical resonator injected by two reflected lights. Since EOMs slew as optical phase shifters to manipulate the altered phase of laser intensity in sub microwave-photonic band, which satisfies the response time of Si-based photo detector (PD), therefore, the microwave range oscillation of laser intensity is well suited to interrogation by computers visa A/D card. By basic quantum knowledge of multi-mode laser, optical intensities of o and e paths are respectively affected by induced phases from reflected lights, analytical model of integrated SMI is deducted from initial intensities [27] of two modes inside a dual-frequency laser [28]:

After activating EOMs, induced phases are shifted:${\phi }_o\text{'}={\phi }_o+{\alpha }_o\sin ({\omega }_{o}t)$, ${\phi }_e\text{'}={\phi }_e+{\alpha }_e\sin ({\omega }_{e}t)$, where${\alpha }_{o/e}=1.22rad$denote phase shifting depths,${\omega }_{o/e}=2\pi f_{o/e}$denote angular frequencies of slewing rates of EOMs. Due to saturation effect [28,31] of dual-mode gas laser, intensities of two paths fluctuate in duplicated waveform. Told by Eq. (7), if $E=F$, we can get $P_o=-P_e$, therefore, only single photo detector is needed to mirror intensity fluctuations in o and e paths. To analyze harmonics, $P_o$ or $P_e$ expands their multi-harmonics generated by phase shifting [20] in Bessel series:

Harmonics of Eq. (8) contains the first-order and second-order harmonics as below:

In Eqs. (9) and (10), amplitudes of the 1-st and 2-nd harmonics have different magnitudes: $A_{o/e1}=(E/-F)J_1({\alpha }_{o/e})\sin ({\phi }_{o/e})$ and$A_{o/e2}=(E/-F)J_2({\alpha }_{o/e})\cos ({\phi }_{o/e})$. Here we set${\omega }_o\ne {\omega }_e$for sake of filtering four harmonics through band-pass filters. Measured phases are extracted on Eq. (12):

Output of arc-tangent calculation is typical discrete phase values ranging between $-\pi$to$\pi$. In time domain, unwrapping is necessary for converting the intermittent phase information to continuous curves [32]. Then, displacements tracking to phase are obtained in below form:

3. Experimental investigation

Experimental investigation is conducted using the Fig. 1 setup. A dual-output, atom-gas, two-longitudinal mode, He-Ne laser works as optical source [31] lasing at 632.8nm. PBS is placed at 10cm next to emission port of laser for splitting out two polarizations. EOMs are installed at perpendicular platforms to keep their intrinsic polarizations consistent to light polarizations respectively. Sinusoidal driven voltages are applied across the sealed crystal electrodes to shift phase through high-voltage drivers (3211, Newport). When measured targets are 3m away, impinged areas are still highlighted points with 1.5mm beam waist radius. Compared with light spot by using diodes, speckle noises are weakened significantly. Therefore, small diffusive angles (<3mmrad) and desirable signal-to-noise ratio make the remote measurement feasible. After one-hour pre-heating laser tube, o and e lights are periodically manipulated by EOMs to introduce 1.22rad-depth [20] sinusoidal phase shifting. Under weak feedback level, mode competition [30,31,33] and multi-reflection effect can be neglected. Laser cavity resonates with optical injections and outputs a steady-statue sub microwave photonic intensity [34] from end facet of laser to pass PBS1 and interrogated by two photo detectors (PD_o and PD_e).

To check validity of laser model, SMI signals of two paths are observed with micron range displacements in Fig. 2. The relative condition parameters are as follows: target_o moves at sinusoidal motion with 10Hz frequency and 3.5μm amplitude on a nano-position-accuracy piezoelectric transducer; target_e vibrates sinusoidally at 100Hz with amplitude about 5μm on a loudspeaker, EOM_o/e slew at 39.5KHz/98KHz frequencies with sinusoidal shape. Detected photo-currents are I/V converted and differentially sampled at rate of 500KHz/s, 5K data points by double channels of an A/D device (NI, USB-6361) without trigger control.

 

Fig. 2 SMI signals of o-path and e-path detected by PD_o and PD_e respectively in time domain (a) and frequency domain (b). Two insets are zoom-in view of signals. Among multi-harmonics of Fig. 2(b), the narrower-width harmonics with 39.5KHz spacing are o-harmonics, the wider-width harmonics with 98KHz spacing are e-harmonics.

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Figure 2(a) illustrates that, o and e path SMI signals, detected by PD_o and PD_e respectively, are both modulated by phase shifted optical injections, and are consisted of sinusoidal-like, fluctuating amplitude, highly-dense fringes (as shown in inset), density of fringes depends on slewing rates of two EOMs. To analyze spectral characteristics in frequency domain, SMI signals are FFT transformed with Hanning window to obtain visible frequency distribution within 250KHz bandwidth. Interestingly, two spectra displays same frequency distribution. In terms of position, harmonics are both classified into two kinds: o-harmonics (centering on multiple times of 39.5KHz) and e-harmonics (centering at multiple times of 98KHz). O-harmonics have similar widths of 700Hz (seeing Fig. 2(b)) and e-harmonics have widths of 10KHz in both paths, which agrees well with Doppler frequency of target_o/e and proves that SMI signals of two paths are duplicated in frequency domain as well.

To investigate how PS and laser Doppler frequency mutually effect spectral distribution, we perform experiment with varied slewing rates (seeing Figs. 3 and 4). Firstly, we keep the displacements unchanged but gradually increase or decrease the slewing rates, an interesting phenomenon is found, positions of harmonics “move” according to changed slewing rates (seeing Fig. 3). Meanwhile, harmonics bandwidths have no changes by laser Doppler effect. To guarantee that band-pass filters (BPF) extract the full-bandwidth harmonics, $f_o\ne f_e$is a necessary condition for avoiding overlapping between harmonics by providing a sufficient adjacent spectral spacing. Figure 3 also indicates that central frequencies of BPFs need to “move” with slewing rates as well.

 

Fig. 3 One batch of SMI spectra with unchanged displacement, f_o varies 39KHz→74KHz→ 110KHz→100KHz and f_e varies 100KHz→45KHz→36KHz→98KHz.When slewing rates are very close, two kinds of harmonics will overlap and become difficult to distinguish.

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Fig. 4 Another batch of SMI spectra with unchanged slewing rates (f_o = 39.5KHz, f_e = 98KHz) but amplitude of target_e varies 5μm→1.5μm→1μm→12μm, amplitude of target_o varies 3.5μm →10μm→15μm→20μm at 100Hz. Widths of harmonics gradually broaden until cause the overlapping as Fig. 3.

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Secondly, SMI operates with unchanged slewing rates and varied amplitudes of external targets. Seeing Fig. 4, bandwidth of harmonics broadens when amplitude of displacement increases and vice versa. If changing oscillating frequencies of targets, similar phenomenon happens. Conclusively, broadening or narrowing of harmonics bandwidth depends on the Doppler effect, which is useful to estimate maximum velocity of moving targets and alarm possible frequency overlapping. This relationship requires cut-off frequencies of filters to cover full bandwidth harmonics which is a key factor limiting the measured capacities. In Fig. 4, a more diffusive mirror is used as target_o surface leading to that amplitudes of o-harmonics decrease, which means amplitudes of harmonics are affected by optical strength. Although velocity can be estimated from spectral bandwidth of harmonics, but FFT will deteriorate accuracy with few data points and is insensitive to direction change.

4. Signal processing and comparison

A phase-proceeded method is employed to retrieve 2D instantaneous displacement precisely from digital binary data converted by detected SMI photo-current. As above discussed, SMI phases have been time-magnified by high-speed phase-shifting optically, and SMI waveform is now amplitude-magnified at 1000 times digitally. Seeing Fig. 5, the used signal processing are summarized as follows: 1. Acquired SMI signal is low-pass filtered to maintain the useful multi-harmonics less than 250KHz, then, four adaptive BPFs with central frequencies at $f_{o/e}$and$2f_{o/e}$ extract harmonics out. 2. Harmonics are divided by digital arrays:$\cos (2\pi f_{o/e}t)$and$\cos (4\pi f_{o/e}t)$respectively to obtain dynamical amplitudes$A_{o/e1}$and $A_{o/e2}$.3.To suppress random noises, cascaded nonlinear filtering techniques, median filters (MF) [35] and continuous wavelet transform (CWT) [36] are adopted. Subsequently, de-noised harmonic amplitudes are inputted into two arc-tangent calculations to obtain phases. 4. Finally, unwrapped phases are converted into displacements and displayed in software interface.

 

Fig. 5 Signal processing with nonlinear filtering: 1. LPF and amplification. 2. 1-st and 2-nd harmonic extraction. 3. Nonlinear filtering on harmonics amplitudes. 4. Phase unwrapping, where CWT is programmed using VI provided by software toolkit of Labview2014.

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Specially, MF plays the role of digital smoother in processing transient amplitude [37] by averaging operation:

To compare the proposed phase algorithm with lock-in technique [18,25] in terms of resolution, we use a hardware lock-in amplifier solving a simulated frequency-modulated SMI signal on the parameters of target_o as comparison. Simulated SMI signal is generated on retrieved phase of target_o on 38.5KHz frequency carrier. Then simulated signal and reference frequency shifting waveform are digital to analog converted by DAQ and inputted to lock-in amplifier. As illustrated in Fig. 6, this ideal SMI signal is band-passed (30KHz~100KHz) by a hardware filter (SR560, Stanford) and phase-locked by lock-in amplifier (SR830, Stanford). Output phase variation from the lock-in amplifier is displayed in a mixed digital oscilloscope (MSO9064A, Agilent) or sent to PC for unwrapping.

 

Fig. 6 Implementation of lock-in technique on simulated frequency modulated SMI and sinusoidal frequency driven voltages, where the dotted frame denotes the necessary functions inside a lock-in amplifier(notch filters are omitted), which outputs phase information displayed in an digital oscilloscope (OSC) and be sent to computer (PC) for phase unwrapping. PLL denotes a phase lock loop, BPF is a band-pass filter realized by (SR560, Stanford).

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DSP module of lock-in amplifier shifts $\pi /2$phase on reference signal for multiplying with SMI signal to generate quadrature signals in Fig. 6. Therefore, lock-in amplifier actually can only reflect phase or frequency variation around reference frequency leading to lock-in amplifier [18] suffers from a limited bandwidth heavily, even referencing shifting frequency in [25] is only 1.5MHz. As to our hardware lock-in amplifier, frequency of reference signal is up to 102KHz and will be unavailable for higher-frequency shifting, not to mention its high cost by the complicated circuits, DSP logic module and analog boards. The simulated interferometric phase of target_o demodulated out by lock-in technique and by proposed signal processing at same modulation frequency (38.5KHz) are plotted together in Fig. 7, where an average deviation about 3.1nm is found between two methods, showing lock-in may be not obviously more accurate than proposed algorithm. Compared phase-shifting with frequency-shifting, EOM introduces pure phase carrier requiring no additional alignment and never influence laser optical frequency. Since phase-resolve method tracks to laser wavelength in Eq. (13), our algorithm provides a better stability in high-speed measurement without aid of reference path. In addition, acousto-optic modulators (AOM) generate an inclination angle due to optical diffraction [18,25] leading to additional alignment. Meanwhile, resonant or waveguide type EOMs easily cover the radio-frequency, therefore, potentially, phase-shifting technique is feasible for large-amplitude or ultrasonic frequency displacement measurement.

 

Fig. 7 (a). Discrete phase of target_o outputted from arc-tangent calculation on our SMI, the inset is detailed phase points (b). The retrieved displacements of target_o using proposed phase algorithm (blue color) and lock-in technique (red color) respectively, where red-arrow pointed inset on left is detailed phase curves, the other inset on right is the yielded deviation between two results with a maximum error less than 3.5nm.

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To quantitatively evaluate short-term resolution, gradually-increased amplitudes of targets (two precision PZTs) are measured by SMI and a Ploytec-5000 laser Doppler velocity meter (LDV) with unchanged parameters. Maximum short term errors in Fig. 8 are both less than 3nm with relative errors< 0.2%. Since resolution of phase shifting is tunable as expounded by Eq. (14), thus, for some high-accuracy oriented occasions, slewing rates of EOMs can be artificially adjusted according to specific targets. In this system, for low frequency (<200KHz) slewing rates, the phase algorithm theoretically provides resolutions only a little better than 100pm resolution for <10Hz micro displacements. To maintain nanometer resolution, phase shifting frequency (slewing rate of EOM) need be at least 50 times higher than measured Doppler frequency. Based on this rule, millimeter-amplitude or high speed (at acoustic or ultrasonic frequency) displacement requires much radio-frequency phase-shifting. Of course, ultra-slow displacement (frequency < 5Hz) will be retrieved with a better resolution at current experimental conditions.

 

Fig. 8 (a). Amplitudes of target_o increasing from 0.8μm to 0.91μm(PZT step size is 10nm) are measured by two instruments with residuals < 3nm. (b). Amplitudes of target_e increasing from 0.85μm to 0.96μm are measured by two instruments with residuals < 2nm.

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Due to minimum location size (1nm) of precision PZT (P841, PI) and upper frequency of high-voltage EOM drivers, it is difficult to measure pico-meter range displacements in our laboratory. Besides, the stability and repeatability are evaluated by long-term running over ten hours, no accuracy degeneration is concluded. For those cases of no load (targets are powered off), displacement noise floors are less than 1.2nm.

The analytical model for independent external cavities SMI intrinsically reveals how laser gain varies with two orthogonal polarized self-optical-injections. Since optical length can be affected by strain from a micro-cantilever, refraction index, thickness changing of materials, uneven surface of targets, therefore, this model is potential for characterization of parameters simultaneous measurement. In those cases of high feedback, how two feedback factors effects SMI waveform feature need be further investigated.

No limitation exists in displacement waveform. We measure odd waveform shapes of 1m targets-to-laser distance in Fig. 9. Damping and electrocardiogram trajectories from hospital database are digital-to-analog converted by DAQ device and fed to PZTs. Seeing zoom-in view in Fig. 9, SMI automatically discriminate the direction change and amplitude changing. Results of two instrument are plotted together, where results agree well after data-fitting. Error sources underneath discrepancy of SMI and LDV may originate from four kinds: 1. Speckle pattern noise, air refraction index in space, unwanted high feedback and deviated optical path collimation. 2. Mechanical vibrations from devices.3. Electronic jitter of driving voltages to EOMs cause a nonlinear phase error [21], which will get improvement by adoption of a high-resolution signal generator. In observation, noised data point has smaller amplitude than nominal SMI waveform, therefore, nonlinear filtering is necessary to enhance useful feature. 4. If high-reflective materials (aluminum, sliver, high-reflective alloy or multilayer, coated-substrates...) are targeted, the cosine function of Eq. (7) holds on when two variable optical attenuators are used for lowering strength of back-scattered lights.

 

Fig. 9 2D vibrations retrieved by SMI and LDV. (a). Electrocardiogram trajectory (ECG) and damping harmonic vibration lasting 8 seconds. (b). Zoom-in electrocardiogram trajectory of target_e with result of LDV. (c). Zoom-in damping vibration of target_o with LDV result.

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Applications of 2D surface profiling, distance-control, simultaneous parameters sensing, dual-targets tracking and structure analysis are frequently required in the nanotechnology, semiconductor or micro-electro mechanical systems industries, providing a huge space for using this new SMI. This integrated SMI with untreated target surfaces and simple signal processing promises a compact high sensitive 2D instrument well suited for poor feedback condition. Since laser intensity fluctuates within microwave band and is manipulated by tunable phase shifting, therefore, its flexible resolution and measured range can further satisfy requirement of interdisciplinary studies than single path diode SMI or lock-in techniques.

5. Conclusion

The established SMI, running on single laser, single PD for dual independent external targets, has integrated polarization multiplexing and phase shifting techniques, which is significantly improved with a longer target-to-laser distance, low speckle noise and tunable sensitivity. By the orthogonality of polarization, this SMI increases measuring path number and eradicates optical cross-talk. By sinusoidal phase shifting, SMI provides theoretical angstrom resolutions for micron range displacements in each path. Observation in frequency domain also shows good coherence with analytical model deduced from semi-classical laser theory.

Microwave photonic laser intensity contains two kinds of harmonics, which have different positions and widths determined by slewing rates of EOMs and Doppler frequencies induced by targets. Usefully, spectral spacing between adjacent harmonics avoids frequency aliasing and allows signal processing to mirror instantaneous 2D displacements with adjustable ultrahigh resolution. In experiment, this system covers a wider measurable range than lock-in technique and obtains resolutions of nanometers using nonlinear filtering methods in comparison to a LDV. Moreover, narrow linewidth He-Ne laser has a better thermal tolerance than diodes in stability and requires no pumping lasers or focusing lens. Conclusively, we theoretically and experimentally demonstrated a two-path SMI based on integration of PM and PS techniques, two-dimensional displacements are fast and precisely demodulated by pure time-domain phase-resolved algorithm from amplitudes of harmonics.