1. Introduction

Sub-micron interferometric displacement sensors (IDS) based on homodyne interferometers have been widely used as surface profile measurement devices, velocity meters, and all-optical vibration transducers [1–3]. The most common homodyne interferometric displacement sensor is implemented using a Michelson interferometer. The Michelson interferometric displacement sensor converts displacement-induced variation on the phase of a laser into power variation allowing for sub-micron displacement measurement. The power variation with displacement in a Michelson IDS is given by P(D) = P^max cos² (2πD/λ +ϕ₀) where λ is the laser wavelength, D is the displacement, and ϕ₀ is the initial phase at D = 0 [4]. The sensitivity (σ) of a Michelson IDS, defined as the maximum variation-rate of P/P^max with D, is given by σ = 2π/λ which is limited by the wavelength λ of the laser.

Heterodyne interferometers, such as the two-frequency IDS [5], have also been widely used for sub-micron displacement measurement. A two-frequency IDS converts displacement-induced variation on the phase of a laser into a corresponding variation of the phase of a sinusoidally-modulated optical signal (SMOS). The power of the SMOS is given by P(t,D) = P^max cos² [π f_st + 2πD/λ − ϕ₀] where f_s is the power modulation frequency [5]. Existing implementations of two-frequency IDS convert the SMOS into a sinusoidal electrical signal using a photo-detector and then retrieve the displacement value using analog or digital phase demodulation techniques. Utilization of advanced phase demodulation techniques in a two-frequency IDS opens the way for sensitivity enhancement beyond the wavelength limit that is inherent in a Michelson IDS.

Recently, we have demonstrated a novel non-interferometric displacement sensor with micron-level resolution based on a Kerr phase-interrogator [6]. In this approach, displacement induces variation on the phase of a SMOS by changing the length of the path through which the SMOS propagates. Then, a Kerr phase-interrogator converts the resulting phase variation into power variation using all-optical signal processing by the nonlinear Kerr effect [6]. The power as a function of displacement is given by P(D) = P^max cos² (2πD/λ_s +ϕ₀), where λ_s = v_g/f_s is the wavelength of the SMOS with a typical value of several millimeters, and v_g is the group velocity. The sensitivity of this displacement sensor is given by σ = 2π/λ_s which is much lower than that of a Michelson IDS because λ_s is typically three orders of magnitude longer than the wavelength λ of the laser of the Michelson IDS.

Sensitivity higher than that of a Michelson IDS can be achieved by utilization of a hybrid system composed of a Kerr phase-interrogator and a two-frequency IDS. In this hybrid displacement sensor (HDS), a two-frequency IDS converts the displacement-induced laser phase variation into SMOS phase variation using optical interference, and then, a Kerr phase-interrogator converts the SMOS phase variation into power variation using the nonlinear Kerr effect. Similar to a Michelson IDS, the HDS all-optically converts laser phase variation into power variation; however, the HDS has higher sensitivity than a Michelson IDS because the advanced demodulation scheme of the Kerr phase-interrogator eliminates the wavelength limitation on sensitivity.

In this paper, we propose and demonstrate a HDS for sub-micron displacement measurement and sensitivity enhancement beyond the wavelength limit of a Michelson IDS. A novel experimental setup combines a Kerr phase-interrogator and a two-frequency IDS to embody the proposed HDS. Theoretical analysis shows that displacement induces phase variation on one of the spectral components of a SMOS leading to a corresponding phase variation on the SMOS. A Kerr phase-interrogator is utilized to convert phase-variation of the SMOS into power-variation allowing for sub-micron displacement measurement. Investigation of the Kerr phase-interrogator when the peak of the nonlinear Kerr-induced phase modulation ϕ_SPM is greater than 10 radians reveals that the sensitivity of a HDS is σ = 0.36ϕ_SPM × 2π/λ, which is higher than the sensitivity of a Michelson IDS by a factor of 0.36ϕ_SPM. Finally, experimental results demonstrate displacement sensing at sub-micron resolution and sensitivity enhancement in agreement with the theoretical model.

2. Experimental setup

Figure 1(a) presents a schematic of the sub-wavelength displacement measurement setup comprised of a Kerr phase-interrogator and a two-frequency IDS. A continuous-wave (CW) laser (RIO) operating at a wavelength λ₀ is amplitude-modulated using a sinusoidal electrical signal generator (HP 83752A) oscillating at f_m = 10 GHz to obtain a SMOS with power oscillating at a frequency f_s = 2f_m. The optical spectrum of the SMOS is composed of two distinct peaks at λ₁ and λ₂ separated by $\mathrm{\Delta }\lambda ={\lambda }_0^2{f}_s/c$ with c being the speed of light in vacuum, as illustrated in Fig. 1. A fiber-coupled polarization beam splitter (from General Photonics) divides the power of the SMOS into a sensor path and a reference path.

 

Fig. 1 Schematic of the sub-micron displacement measurement setup based on a Kerr phase-interrogator and a two-frequency IDS. RF: radio-frequency; CW: continuous-wave; EOM: electro-optic modulator; PSD: power spectral density; PC: polarization controller; FPS: fiber polarization splitter; FPC: fiber polarization combiner; PM: polarization maintaining; PBS: polarization beam splitter; M: mirror; and EDFA: Erbium-doped fiber amplifier.

Download Full Size | PDF

The SMOS in the sensor path propagates through a polarization-maintaining circulator (from General Photonics) and is launched with linear polarization at 45° from the principal axis of a 10 m long polarization-maintaining fiber with a beat-length of 2 mm. Due to the wavelength dependence of birefringence in the polarization-maintaining fiber, the two wavelength components of the SMOS exit the birefringent fiber at port (b) with orthogonal polarizations [7]. A free-space polarization-beam-splitter separates the two orthogonally polarized spectral components, as illustrated in Fig. 1. Each spectral components E_i at λ_i travels a separate path with length L_i and is reflected by a mirror M_i. Mirror M₁ is attached to a piezo-electric actuator (Thorlabs PK2FQP2) and displacement of M₁ is induced by variation of the voltage applied to the piezo-electric actuator from a voltage source (HP E3631A). The displacement of M₁ leads to variation in the phase of E₁ given by ϕ₁ = k₁L₁ +ϕ_1,0 = 2k₁D +ϕ_1,0, where k₁ = 2π/λ₁, D is the displacement distance, and ϕ_1,0 is a constant. Mirror M₂ is fixed and the phase of E₂ is given by ϕ₂ = ϕ_2,0, where ϕ_2,0 is a constant.

After reflection from mirrors M_i, the spectral components travel back through the polarization-maintaining fiber and exit with parallel polarizations at port (a) to reconstruct the SMOS. The SMOS travels back through the polarization-maintaining circulator and a fiber-coupled polarization beam combiner (from General Photonics) recombines the signals from the sensor and the reference paths. The combined signal at the output of the fiber polarization combiner is amplified using an Erbium-doped fiber amplifier (Amonics AEDFA-33-B-FA) and is launched into a nonlinear Kerr medium comprised of a fiber with a length of L_kerr = 5.7 km, a loss coefficient of α_dB = 0.47 dB/km measured using the cut-back method, a coupling-loss of 2 dB at each fiber end, a waveguide nonlinearity of γ = 4.2 W⁻¹km⁻¹ measured as described in [8], and a chromatic-dispersion of D_c = 3 ps/nm-km measured using the modulation phase-shift method [9, 10]. The electric field amplitudes of the SMOSs from the sensor and the reference paths are given by

3. Sensitivity enhancement

Sensitivity enhancement is achieved when the Kerr phase-interrogator is operated under the condition ϕ_SPM ≥ 0.5 where the power of the first-order side-band is given by [6]

 

Fig. 2 Normalized power of the first order-sideband as a function of phase calculated using Eq. (5) for ϕ_SPM = 0.5, 5, 10, and 50.

Download Full Size | PDF

 

Fig. 3 Exact and approximate values of P₁ as a function of ϕ calculated using Eq. (5) and Eq. (6) with a) ϕ_SPM = 5, b) ϕ_SPM = 10, and c) ϕ_SPM = 1000.

Download Full Size | PDF

4. Experimental results and discussion

To demonstrate displacement measurement under the condition ϕ_SPM < 0.5, the combined signal at the output of the polarization beam combiner is amplified to 40 mW and then launched into the Kerr medium. The power of the first order side-band is recorded as the voltage of the piezo-electric actuator is increased in steps of 0.1 V corresponding to a displacement step of 28.5 nm. Figure 4 presents the measured values of P₁ as a function of displacement showing sinusoidal dependence of the side-band power with displacement as predicted by Eq. (3). Also presented in Fig. 4 is the theoretical value of P₁ (D) obtained from Eq. (3) with λ₁ = 1548 nm showing close agreement between theory and experiment. The measured sensitivity from Fig. 4 is σ = 3.86×10⁶ m⁻¹ in close agreement with the the theoretically calculated value σ = 4.06 × 10⁶ m⁻¹ obtained from Eq. (4) with λ₁ = 1548 nm.

 

Fig. 4 Experimentally measured and theoretically calculated side-band power as a function of displacement for ϕ_SPM < 0.5.

Download Full Size | PDF

To demonstrate sensitivity enhancement, the power at the input of the Kerr medium is further amplified to 450 mW corresponding to ϕ_SPM = 5 and the value of P₁ is measured as the mirror M₁ is displaced in steps of 28.5 nm. Figure 5 presents the measured values of P₁ as a function of displacement along with the theoretical value calculated using Eq. (5) with λ₁ = 1548 nm and ϕ_SPM = 5 showing close agreement between theory and experiment. A sensitivity of σ = 6.76 × 10⁶ m⁻¹ is measured from Fig. 5 indicating a sensitivity enhancement by a factor of 1.75 in comparison with the sensitivity at ϕ_SPM < 0.5. Figure 3(a) shows that the approximate value of P₁ from Eq. (6) for ϕ in the range determined by the condition −π/4 < 0.36×ϕ_SPM ×(ϕ −π/2 − mπ) < π/4 correponding to $P_1/P_1^{\max }$ < 0.5 closely matches the exact value of P₁ from Eq. (5); therefore, Eq. (7) provides a close estimate of the sensitivity at ϕ_SPM = 5. Using Eq. (7), the theoretical estimate of the sensitivity enhancement is 0.36ϕ_SPM = 1.8 in close agreement with the experimentally measured value.

 

Fig. 5 Experimental measurements of power as a function of displacement for ϕ_SPM = 5.

Download Full Size | PDF

Optimal displacement sensing is achieved around operating points where the power variation with displacement is maximum. For ϕ_SPM < 0.5, the operating points are located at the quadrature points around D_op such that ϕ = 2πD_op/λ₁ + (ϕ_1,0 −ϕ_2,0)/2−ϕ_ref = π/4 + qπ/2 with q being an integer. For ϕ_SPM > 10, the operating points from Eq. (6) are located at D_op where ϕ satisfies the condition 0.36 ×ϕ_SPM × (ϕ−π/2−mπ) = ±π/4. The operating point can be shifted by using a variable delay-line in the reference arm to vary t_⊥ and change ϕ_ref = π f_st_⊥ [6].

The displacement resolution under the condition ϕ_SPM < 0.5 is calculated from the minimum detectable phase-change Δϕ = 2πΔD/λ₁ = α leading to ΔD = αλ₁/2π. Differentiation of Eq. (3) around the operating points leads to $\left|\delta \varphi \right|=|\delta P_1/P_1^{\max }|$, where δϕ represents the fluctuations of phase-shift and δP₁ represent the noise-induced power fluctuations. With a maximum power fluctuation of 1%, the minimum resolvable differential phase-shift is α = max {|δϕ|} = 10⁻² and the displacement resolution is ΔD = 2.46 nm for λ₁ = 1548 nm. Under the condition ϕ_SPM > 10, the displacement resolution is calculated from 0.36ϕ_SPM × Δϕ = 0.36ϕ_SPM ×2πΔD/λ₁ =α which leads to ΔD =αλ₁/(0.72πϕ_SPM) indicating that the displacement resolution is refined by a factor of 0.36ϕ_SPM.

The dynamic-range D_DR is defined as the quasi-linear range over which P₁ varies from 0.2P_max to 0.8P_max around the operating points to obtain a one-to-one correspondence between P₁ and D [13]. For ϕ_SPM < 0.5, P₁ = 0.2P_max at 2πD_min/λ₁ + ϕ₀ = 0.35π and P₁ = 0.8P_max at 2πD_max/λ₁ +ϕ₀ = 0.15π leading to a dynamic range D_DR = |D_min − D_max| = 0.1λ₁. For λ₁ = 1548 nm, the dynamic-range is D_DR = 154.8nm and the number of resolvable points within this range is the rounded ratio between the dynamic-range and the displacement resolution ⌊D_DR/ΔD⌋ = ⌊0.2π/α⌋ = 62. Similarly, for ϕ_SPM > 10, P₁ = 0.2P_max at 0.36 × ϕ_SPM × (2πD_min/λ₁ +ϕ₀ −π/2 − mπ) = 0.15π and P₁ = 0.8P_max at 0.36 × ϕ_SPM × (2πD_max/λ₁ +ϕ₀ −π/2 − mπ) = 0.35π leading to a dynamic-range D_DR = 0.1λ₁/(0.36ϕ_SPM) which corresponds to a reduction by a factor of 0.36ϕ_SPM when compared with D_DR for ϕ_SPM < 0.5. The number of resolvable points when ϕ_SPM > 10 is given by ⌊D_DR/ΔD⌋ = ⌊0.2π/α⌋ which is identical to the number of resolvable points when ϕ_SPM < 0.5.

5. Comparison with Michelson interferometric displacement sensors

The power as a function of displacement for a Michelson IDS is given by P(ϕ) = P^max cos² (ϕ) with ϕ = 2πD/λ +ϕ₀ [4] which is identical to Eq. (3). Therefore, the sensitivity of a Michelson IDS is the same as the sensitivity of a HDS when the Kerr phase-interrogator operates under the condition ϕ_SPM < 0.5. However, when the Kerr phase-interrogator operates under the condition ϕ_SPM > 10, the power as a function of displacement for the HDS is given by Eq. (6). In this case, the sensitivity of the HDS given by Eq. (7) is higher than that of the Michelson IDS by a factor 0.36ϕ_SPM indicating sensitivity enhancement beyond the wavelength limit of a Michelson IDS. This sensitivity enhancement can be adapted for a variety of other sensing applications that utilize Michelson interferometers such as vibration monitoring, gravitational wave detection, temperature/strain measurement, and refractive-index sensing.

6. Conclusion

A novel hybrid displacement sensor for sub-micron displacement measurement is obtained by combining a Kerr phase-interrogator and a two-frequency interferometric displacement sensor. The advanced demodulation scheme of the Kerr phase-interrogator enhances the sensitivity of the hybrid displacement sensor beyond the wavelength limited sensitivity of a Michelson interferometric displacement sensor. A sub-micron displacement sensor with a sensitivity enhancement factor of 1.75 is experimentally demonstrated. Future work will focus on utilizing this sensitivity enhancement approach in novel devices for vibration monitoring and refractive-index sensing.