## Abstract

We propose and demonstrate sub-micron displacement sensing and sensitivity enhancement using a two-frequency interferometer and a Kerr phase-interrogator. Displacement induces phase variation on a sinusoidally modulated optical signal by changing the length of the path that either of the signal’s two spectral components propagates through. A Kerr phase-interrogator converts the resulting phase variation into power variation allowing for sub-micron displacement sensing. The sensitivity of this novel displacement sensor is enhanced beyond the wavelength-limited sensitivity of the widely used Michelson interferometric displacement sensor. The proposed approach for sensitivity enhancement creates a whole new class of sensors with ultra-high sensitivity.

© 2015 Optical Society of America

## 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}* cos

^{2}(2

*πD/λ*+

*ϕ*

_{0}) where

*λ*is the laser wavelength,

*D*is the displacement, and

*ϕ*

_{0}is the initial phase at

*D*= 0 [4]. The sensitivity (

*σ*) of a Michelson IDS, defined as the maximum variation-rate of

*P/P*with

^{max}*D*, is given by

*σ*= 2

*π/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}* cos

^{2}[

*π f*+ 2

_{s}t*πD/λ − ϕ*

_{0}] where

*f*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.

_{s}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}* cos

^{2}(2

*πD/λ*+

_{s}*ϕ*

_{0}), where

*λ*=

_{s}*v*is the wavelength of the SMOS with a typical value of several millimeters, and

_{g}/f_{s}*v*is the group velocity. The sensitivity of this displacement sensor is given by

_{g}*σ*= 2

*π/λ*which is much lower than that of a Michelson IDS because

_{s}*λ*is typically three orders of magnitude longer than the wavelength

_{s}*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*= 0.36

*ϕ*2

_{SPM}×*π/λ*, which is higher than the sensitivity of a Michelson IDS by a factor of 0.36

*ϕ*. Finally, experimental results demonstrate displacement sensing at sub-micron resolution and sensitivity enhancement in agreement with the theoretical model.

_{SPM}## 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 *λ*_{0} 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*= 2

_{s}*f*. The optical spectrum of the SMOS is composed of two distinct peaks at

_{m}*λ*

_{1}and

*λ*

_{2}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.

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*travels a separate path with length

_{i}*L*and is reflected by a mirror M

_{i}*. Mirror M*

_{i}_{1}is attached to a piezo-electric actuator (Thorlabs PK2FQP2) and displacement of M

_{1}is induced by variation of the voltage applied to the piezo-electric actuator from a voltage source (HP E3631A). The displacement of M

_{1}leads to variation in the phase of

*E*

_{1}given by

*ϕ*

_{1}=

*k*

_{1}

*L*

_{1}+

*ϕ*

_{1,0}= 2

*k*

_{1}

*D*+

*ϕ*

_{1,0}, where

*k*

_{1}= 2

*π/λ*

_{1},

*D*is the displacement distance, and

*ϕ*

_{1,0}is a constant. Mirror M

_{2}is fixed and the phase of

*E*

_{2}is given by

*ϕ*

_{2}=

*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*= 5.7 km, a loss coefficient of

_{kerr}*α*= 0.47 dB/km measured using the cut-back method, a coupling-loss of 2 dB at each fiber end, a waveguide nonlinearity of

_{dB}*γ*= 4.2 W

^{−1}km

^{−1}measured as described in [8], and a chromatic-dispersion of

*D*= 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

_{c}*ϕ*

_{‖}=

*k*

_{1}

*D*+ (

*ϕ*

_{1,0}

*−ϕ*

_{2,0})/2,

*ϕ*=

_{⊥}*ϕ*with

_{ref}*ϕ*being a constant,

_{ref}*D*is the displacement of M

_{1}, and

*P*/2 is the peak power of each SMOS. Self-induced phase-modulation in the Kerr medium leads to the formation of distinct sidebands

_{p}*P*with

_{i}*i*= 1,2,3,… [6, 8, 10–12]. The power of the first-order side-band

*P*

_{1}as a function of

*D*when the maximum nonlinear phase-shift accumulated in the Kerr medium

*ϕ*=

_{SPM}*γP*with

_{p}L_{eff}*L*being the effective-length of the Kerr medium

_{eff}*L*= [1

_{eff}*−*exp(

*−αL*)]

_{kerr}*/α*satisfies the condition

*ϕ*< 0.5 is given by [6] where

_{SPM}*ϕ*=

*ϕ*

_{‖}

*− ϕ*= 2

_{⊥}*πD/λ*

_{1}+

*ϕ*

_{0}with

*ϕ*

_{0}= (

*ϕ*

_{1,0}

*−ϕ*

_{2,0})/2

*− ϕ*, and the displacement measurement sensitivity is

_{ref}## 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]

*J*(

_{n}*x*) is the Bessel function of the first kind. Figure 2 presents the calculated value of

*P*

_{1}as a function of

*ϕ*for

*ϕ*= 0.5, 5, 10, and 50 illustrating that the variation of

_{SPM}*P*

_{1}with

*ϕ*becomes faster around

*ϕ*= (0.5 +

*m*)

*π*with

*m*being an integer. The value of

*P*

_{1}as a function of

*ϕ*around

*ϕ*= (0.5 +

*m*)

*π*is approximated by

*ϕ*that satisfy

*−π/*2 < 0.36

*×ϕ*(

_{SPM}×*ϕ−π/*2

*−mπ*) <

*π*/2. Figure 3 shows that Eq. (6) is valid for

*ϕ*> 10 and has been verified for

_{SPM}*ϕ*as large as 10

_{SPM}^{4}and beyond. Using Eq. (6), the displacement measurement sensitivity is

*ϕ*< 0.5 in Eq. (4) indicates an enhancement by a factor of 0.36

_{SPM}*ϕ*.

_{SPM}## 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

*P*

_{1}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

*P*

_{1}(

*D*) obtained from Eq. (3) with

*λ*

_{1}= 1548 nm showing close agreement between theory and experiment. The measured sensitivity from Fig. 4 is

*σ*= 3.86

*×*10

^{6}m

^{−1}in close agreement with the the theoretically calculated value

*σ*= 4.06

*×*10

^{6}m

^{−1}obtained from Eq. (4) with

*λ*

_{1}= 1548 nm.

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

*P*

_{1}is measured as the mirror M

_{1}is displaced in steps of 28.5 nm. Figure 5 presents the measured values of

*P*

_{1}as a function of displacement along with the theoretical value calculated using Eq. (5) with

*λ*

_{1}= 1548 nm and

*ϕ*= 5 showing close agreement between theory and experiment. A sensitivity of

_{SPM}*σ*= 6.76

*×*10

^{6}m

^{−1}is measured from Fig. 5 indicating a sensitivity enhancement by a factor of 1.75 in comparison with the sensitivity at

*ϕ*< 0.5. Figure 3(a) shows that the approximate value of

_{SPM}*P*

_{1}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 ${P}_{1}/{P}_{1}^{\mathrm{max}}$ < 0.5 closely matches the exact value of

*P*

_{1}from Eq. (5); therefore, Eq. (7) provides a close estimate of the sensitivity at

*ϕ*= 5. Using Eq. (7), the theoretical estimate of the sensitivity enhancement is 0.36

_{SPM}*ϕ*= 1.8 in close agreement with the experimentally measured value.

_{SPM}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*such that

_{op}*ϕ*= 2

*πD*

_{op}/λ_{1}+ (

*ϕ*

_{1,0}

*−ϕ*

_{2,0})/2

*−ϕ*=

_{ref}*π/*4 +

*qπ/*2 with

*q*being an integer. For

*ϕ*> 10, the operating points from Eq. (6) are located at

_{SPM}*D*where

_{op}*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*

_{s}t_{⊥}[6].

The displacement resolution under the condition *ϕ _{SPM}* < 0.5 is calculated from the minimum detectable phase-change Δ

*ϕ*= 2

*π*Δ

*D/λ*

_{1}=

*α*leading to Δ

*D*=

*αλ*

_{1}/2

*π*. Differentiation of Eq. (3) around the operating points leads to $\left|\delta \varphi \right|=|\delta {P}_{1}/{P}_{1}^{\mathrm{max}}|$, where

*δϕ*represents the fluctuations of phase-shift and

*δP*

_{1}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 Δ

*D*= 2.46 nm for

*λ*

_{1}= 1548 nm. Under the condition

*ϕ*> 10, the displacement resolution is calculated from 0.36

_{SPM}*ϕ*Δ

_{SPM}×*ϕ*= 0.36

*ϕ*2

_{SPM}×*π*Δ

*D/λ*

_{1}=

*α*which leads to Δ

*D*=

*αλ*

_{1}/(0.72

*πϕ*) indicating that the displacement resolution is refined by a factor of 0.36

_{SPM}*ϕ*.

_{SPM}The dynamic-range *D _{DR}* is defined as the quasi-linear range over which

*P*

_{1}varies from 0.2

*P*to 0.8

_{max}*P*around the operating points to obtain a one-to-one correspondence between

_{max}*P*

_{1}and

*D*[13]. For

*ϕ*< 0.5,

_{SPM}*P*

_{1}= 0.2

*P*at 2

_{max}*πD*

_{min}/λ_{1}+

*ϕ*

_{0}= 0.35

*π*and

*P*

_{1}= 0.8

*P*at 2

_{max}*πD*

_{max}/λ_{1}+

*ϕ*

_{0}= 0.15

*π*leading to a dynamic range

*D*= |

_{DR}*D*| = 0.1

_{min}− D_{max}*λ*

_{1}. For

*λ*

_{1}= 1548 nm, the dynamic-range is

*D*= 154.8nm and the number of resolvable points within this range is the rounded ratio between the dynamic-range and the displacement resolution ⌊

_{DR}*D*/Δ

_{DR}*D*⌋ = ⌊0.2

*π/α*⌋ = 62. Similarly, for

*ϕ*> 10,

_{SPM}*P*

_{1}= 0.2

*P*at 0.36

_{max}*× ϕ*(2

_{SPM}×*πD*

_{min}/λ_{1}+

*ϕ*

_{0}

*−π/*2

*− mπ*) = 0.15

*π*and

*P*

_{1}= 0.8

*P*at 0.36

_{max}*× ϕ*(2

_{SPM}×*πD*

_{max}/λ_{1}+

*ϕ*

_{0}

*−π/*2

*− mπ*) = 0.35

*π*leading to a dynamic-range

*D*= 0.1

_{DR}*λ*

_{1}/(0.36

*ϕ*) which corresponds to a reduction by a factor of 0.36

_{SPM}*ϕ*when compared with

_{SPM}*D*for

_{DR}*ϕ*< 0.5. The number of resolvable points when

_{SPM}*ϕ*> 10 is given by ⌊

_{SPM}*D*/Δ

_{DR}*D*⌋ = ⌊0.2

*π/α*⌋ which is identical to the number of resolvable points when

*ϕ*< 0.5.

_{SPM}## 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}* cos

^{2}(

*ϕ*) with

*ϕ*= 2

*πD/λ*+

*ϕ*

_{0}[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*< 0.5. However, when the Kerr phase-interrogator operates under the condition

_{SPM}*ϕ*> 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 0.36

_{SPM}*ϕ*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.

_{SPM}## 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.

## Acknowledgments

The authors are thankful to the NSERC Discovery Grant and Canada Research Chair Program (CRC in Fiber Optics and Photonics).

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