Displacement measurement using an optoelectronic oscillator with an intra-loop Michelson interferometer

Jehyun Lee; Sooyoung Park; Dae Han Seo; Sin Hyuk Yim; Seokchan Yoon; D. Cho

doi:10.1364/OE.24.021910

1. Introduction

From an apparatus to calibrate a gauge block to a gravitational-wave detector [1], a Michelson interferometer is indispensable in displacement measurements. In its simplest realization, one counts interference fringes by measuring output power from the interferometer as one of the mirrors moves. This is known as homodyne detection. In 1970, engineers at Hewlett-Packard invented a new detection scheme that employed two frequencies [2]. Each frequency component traverses orthogonal arms of a Michelson interferometer and the beat signals before and after the interferometer are compared. Displacement shows up as a phase shift between them. This is known as heterodyne detection. It is an improvement over the homodyne detection in that it is immune to a slow drift of laser power; moreover, its sensitivity does not depend on the slope of the fringe. Heterodyne detection is widely used to align patterns in semiconductor lithography with sub-nanometer precision [3].

In this article, we report a new displacement measurement scheme using an optoelectronic oscillator (OEO) with an intra-loop Michelson interferometer. In this scheme, which we will call OEO detection, a displacement produces a shift in oscillation frequency of the OEO. Although the three methods - homodyne, heterodyne and OEO - have the same fundamental detection limit (as will be discussed in Section 3), in practice OEO detection provides certain advantages over the two other methods because frequency is usually easiest to measure.

The OEO was first proposed in 1978 [4] and many realizations followed in the early 1980s. See references in [5]. As shown in Fig. 1, a modulated laser beam is detected by a photodetector to produce a beat signal at the modulation frequency f_m, which is amplified and fed back to the modulator. With sufficient gain, it oscillates spontaneously. It was originally developed as a stable oscillator at a frequency of tens of GHz [6]. Recently, there has been growing interest in the use of OEO as a sensor for various physical quantities, for example, refractive index [7], force [8], strain [9], and distance [10–12]. When used as a sensor, the OEO loop includes a part, whose optical path length depends on a physical quantity of interest, so that a change in the quantity causes a shift ΔF in the OEO frequency F_OEO.

Fig. 1 Optoelectronic oscillator. $f_{c}$ is the carrier frequency of the laser and $f_{s}$ is the frequency of the sideband produced by the modulator.

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Our OEO detection method also measures ΔF due to a displacement and it appears similar to the sensors in [7–12]. However, one major difference is that by incorporating an optical interferometer inside the loop, effect of a displacement is measured with respect to an optical wavelength and ΔF in the OEO detection is proportional to f_c of the laser. In comparison, ΔF of the aforementioned sensors is proportional to F_OEO, which is in the RF region. In a typical case, the ratio f_c/F_OEO is larger than a million and the interferometer-based OEO detection scheme is extremely more sensitive.

There were previous proposals to integrate an interferometer with an OEO for a sensitive detection of either displacement or rotation. In 1991 a Soviet group constructed an OEO in a Mach-Zehnder configuration using an acousto-optic modulator (AOM) as a beam splitter [13]. They pointed out that its oscillation frequency was highly sensitive to the difference in the interferometer arm lengths and carried out a proof-of-principle experiment. Their main interest, however, was tuning the oscillator by changing one of the arm lengths. When the result is interpreted as a displacement measurement, its precision was 360 nm, mainly limited by the OEO frequency instability. There was also an interesting proposal by Konopsky to use an OEO in the form of a Sagnac interferometer with a counter-propagating pair of a carrier and a sideband as an optical gyroscope [14].

2. Operational principles of heterodyne and OEO detection

A heterodyne detection scheme is shown in Fig. 2. A carrier at frequency f_c from a laser is modulated to produce a sideband at f_s = f_c + f_m whose plane of polarization is perpendicular to that of the carrier. Part of the modulator output is picked off by a non-polarizing beam splitter (NPBS) and detected by the first photodetector (PD1). The part transmitting the NPBS is split into the carrier and the sideband by a polarizing beam splitter (PBS). They traverse orthogonal arms of a Michelson interferometer. Output from the interferometer is detected by the second photodetector (PD2). In front of the photodetectors, there are linear polarizers (not shown in Fig. 2) with their transmission axes rotated by 45 $^{°}$ so that the carrier and the sideband interfere to produce beat signals. The beat signals from PD1 and PD2 are compared to extract phase difference $ϕ$ . When mirror A is fixed and B is displaced by Δl, $ϕ$ changes by $Δ ϕ = 2 \bar{k} Δ l .$ Here $\bar{k} = (k_{c} + k_{s}) / 2$ is the average wavenumber and we neglected $Δ k = k_{c} - k_{s}$ compared with $\bar{k} .$

Fig. 2 Heterodyne detection scheme. NPBS: non-polarizing beam splitter; PBS: polarizing beam splitter; PD: photodetector.

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In the OEO detection scheme in Fig. 3, there is no reference oscillator to drive the modulator. Instead, output from PD2 is amplified and fed back to the modulator to form a loop. The boundary condition of the interferometric OEO for the q-th mode is

K_{q} (L_{s} + l_{a}) + \bar{k} l_{d} = 2 π q,

where K_q is a wavenumber for the RF OEO oscillation whereas

\bar{k}

is for the optical field. L_s is sum of the optical path shared by f_c and f_s and the electronic path from PD2 to the modulator. It is shown by the red dashed line in Fig. 3.

l_{a} = l_{A} + l_{B}

and

l_{d} = 2 (l_{A} - l_{B})

are, respectively, average and difference of path lengths traversed by f_c and f_s through the interferometer.

l_{A}

and

l_{B}

are distances from the PBS to mirrors A and B, respectively. From the boundary condition, F_OEO of the q-th mode is

F_{q} = \frac{c}{L} q - \frac{l_{d}}{L} \bar{f},

where L = L_s + l_a and

\bar{f} = (f_{c} + f_{s}) / 2.

If the mirror B is displaced by Δl, then

Fig. 3 OEO detection scheme.

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Δ F_{q} = \frac{2 Δ l}{L} \bar{f} .

We note that for an OEO sensor without an interferometer, a path length change ΔL produces

Δ F_{q} = \frac{Δ L}{L} F_{q} .

For example, consider

\bar{f} =

500 THz, F_OEO = 100 MHz, and L = 10 m. Using OEO detection, 2Δl = 1 nm produces ΔF = 50 kHz; a simple sensor with ΔL = 1 nm results in ΔF = 0.01 Hz.

3. Comparison of measurement uncertainties

3.1 Fundamental limits

In homodyne detection, if a laser beam with power P_o and wavenumber k (wavelength = λ) is used, output power from a Michelson interferometer is:

P (Δ z) = P_{o} (1 + \cos k Δ z),

where Δz = 2Δl. The displacement measurement is most sensitive at kΔz = π/2, where the slope is maximum. When P(Δz) is measured during τ, the measurement uncertainty due to statistical fluctuation is:

δ (Δ z) = \frac{λ}{2 π} \sqrt{\frac{ℏ ω}{P_{o} τ}} .

In heterodyne detection, if the total power P_o is evenly split between the carrier and the sideband, output power at PD2 from the Michelson interferometer in Fig. 2 is:

P (t) = P_{o} (1 + \cos (ω_{m} t + Δ ϕ)),

where ω_m = 2π f_m and

Δ ϕ = \bar{k} Δ z .

We neglect the vector nature of the light fields in this analysis; the polarization effect would have reduced P(t) by a factor of 4. Δϕ with respect to the reference signal at PD1 can be measured from a time lag between zero crossings of the two signals. The minimum uncertainty in Δϕ can be obtained by the chi-squared fit of

{\bar{P}}_{j}

measured from jΔt to (j + 1)Δt,

{\bar{P}}_{j} = \frac{1}{Δ t} \int_{j Δ t}^{(j + 1) Δ t} P (t) d t,

for j = 1, 2, .. n, to Eq. (7). Here

Δ t < < T < < n Δ t

and T is the period of modulation 1/f_m. The uncertainty in Δϕ from such measurements can be formally expressed as

\frac{1}{{(δ (Δ ϕ))}^{2}} = \sum_{j = 1}^{n} \frac{1}{{(\frac{\partial (Δ ϕ)}{\partial P_{j}} δ P_{j})}^{2}},

where P_j = P(jΔt) in Eq. (7) and δP_j =

\sqrt{P_{j}} ℏ ω / Δ t

is the standard deviation of

{\bar{P}}_{j}

from P_j. When the total measurement time

τ = n Δ t

is sufficiently longer than T, Eq. (9) reduces to

\begin{array}{l} δ (Δ ϕ) = \sqrt{\frac{ℏ ω}{P_{o} τ} .} \end{array}

From

Δ ϕ = k Δ z

, it leads to the same uncertainty as that of the homodyne detection in Eq. (6).

In the OEO detection, if a laser beam with power P_o is used, the full width at half maximum (FWHM) of the OEO output spectrum in an ideal case is

\begin{array}{l} δ F_{O E O} = \frac{1}{2 π} \frac{ℏ ω}{P_{o} t_{r}^{2}}, \end{array}

where t_r = L/c is the round-trip time for a photon in the OEO loop [14]. This is analogous to the Schawlow-Townes limit on laser linewidth [15] with the photon lifetime due to spontaneous emission replaced by t_r. If we make a frequency measurement during τ, the uncertainty due to the finite linewidth is

δ (Δ F) = \sqrt{\frac{δ F_{O E O}}{2 π τ}} .

From Eq. (3), δ(ΔF) leads to the same uncertainty in Δz as that in Eq. (6).

We conclude that all three detection methods based on a Michelson interferometer have the same fundamental uncertainty given in Eq. (6). This limit is very small. For example, if P_o = 1 mW, λ = 600 nm and τ = 1 ms, then δ(Δz) = 55 fm.

3.2 Practical limits

In practice, there are a few technical noise sources that degrade the measurements: (i) power or frequency noise of a laser, (ii) environmental perturbations like acoustic noise or air turbulence, and (iii) timing jitter or electronic noise of a detector that limits the precision of a phase or frequency measurement.

(i) While the power noise of a laser directly affects homodyne detection, its effect on heterodyne or OEO detection is of the second order. Because a modulation technique is used in those detections, effect of the laser frequency instability is also limited. It is important, however, to operate the laser in a single longitudinal mode to obtain the stable beat signal.
(ii) External perturbations to the interferometer produce spurious displacements that cannot be distinguished from Δl to be measured. They affect the three methods in the same way, and the interferometer should be carefully isolated. By comparing Fig. 2 and Fig. 3, we note that while the heterodyne detection is sensitive to perturbations downstream of the NPBS only, the OEO detection is susceptible to those throughout the loop including the modulator.
(iii) When Δϕ is measured from a time lag, precision of phase detection is limited by finite timing resolution δt. If Δϕ is measured τ/T times during τ as in Fig. 4(a), each measurement has a typical error 2π (δt/T) and the uncertainty in displacement measurement over τ is: $\begin{array}{l} δ (Δ z) = \frac{λ δ t}{\sqrt{τ} T} . \end{array}$

For an OEO detection, suppose that ΔF with respect to a reference oscillator is measured from the phase shift Δϕ(τ) accumulated during τ as in Fig. 4(b). From the relation ΔF = Δϕ(τ)/2πτ, the uncertainty in ΔF due to δt is δ(ΔF) = δt/τT. This leads to the uncertainty in Δz,

δ (Δ z) = L \frac{F_{q} δ t}{f_{0} τ} .

We note that while δ(Δz) of the heterodyne detection scales as

1 / \sqrt{τ}

, that of the OEO detection scales as 1/τ. This is an advantage of the frequency measurement.

Fig. 4 (a) Measurement of phase shift in heterodyne detection and (b) measurement of frequency shift in OEO detection.

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4. Apparatus

We built an apparatus that could be used for both heterodyne and OEO detections to compare their performances. We use an extended-cavity diode laser at λ = 660 nm as a light source. Its output goes through an optical isolator and a single-mode fiber for mode filtering. Its spectrum is monitored by using a confocal spectrum analyzer to guarantee single-frequency operation, and the frequency is stabilized to one of the cavity modes. Polarization of the fiber output is defined by a linear polarizer with a horizontal transmission axis. Transmitted power through the linear polarizer is actively stabilized by using an upstream AOM. In this way, all aspects of the laser beam are tightly controlled.

The laser beam, which serves as a carrier, impinges on a modulator that produces a sideband with vertical polarization [16]. The sideband at f_s = f_c + f_m is produced by an AOM operating at 80 MHz and it is combined with a carrier at a PBS placed 30 cm downstream. See Fig. 5. In a commercial version [17], they use a special AOM with birefringent crystals, which are arranged in such a way that a sideband with orthogonal polarization emerges along the carrier.

Fig. 5 Modulator to produce an orthogonally polarized sideband. AOM: acousto-optic modulator; M: mirror; HWP: half-wave plate. Polarization states of the carrier (red) and the sideband (blue) are shown.

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A schematic of the apparatus configured for OEO detection is shown in Fig. 6. The combined beam produced by the modulator goes to a Michelson interferometer via an NPBS. Mirrors A and B are attached to piezo transducers (PZT: PI Ceramic Model PD080.31). The NPBS directs part of the beams to 150-MHz bandwidth photodetectors (Thorlabs Model PDA10A) PD1 and PD2. In order to achieve sub-nanometer precision, the apparatus should be isolated from air turbulence and ambient vibrations. The critical parts surrounded by a box in Fig. 6 are placed in a vacuum chamber, which sits on an active vibration-isolation stage. In addition, to eliminate power-line noise we use batteries wherever possible.

Fig. 6 Apparatus configured for OEO detection in a phase-locked loop (PLL) scheme. DC: directional coupler; DPD: digital phase detector.

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For a heterodyne detection, the apparatus is configured as in Fig. 2. The beat signals from PD1 and PD2 are compared to extract a phase difference using a digital phase detector (DPD) built around a 200-MHz phase discriminator (Analog Devices Model AD9901). For an OEO detection, the apparatus is configured to form a loop by feeding output from PD2 to the modulator. It is problematic, however, to use a frequency shift ΔF as a direct measure of a displacement Δl due to the nonlinear relation between Δl and ΔF. Suppose that mirror B is displaced by Δl to produce ΔF. The shift then changes AOM deflection angle of the sideband, thereby its subsequent path, and eventually Δl. In order to linearize the measurement, we phase lock the OEO to a reference oscillator and use the correction signal to PZT A as a measure of Δl of mirror B. By using the phase-locked loop (PLL), we keep the OEO running at the same frequency regardless of Δl. Instead of the PZT, an analog phase shifter between PD2 and the RF amplifier could be used as a transducer. We note that a loop filter comprising the PLL can limit the measurement bandwidth.

Although the PLL scheme appears to be a phase-shift measurement, we want to emphasize that it is a frequency-shift measurement just as a PLL in a modern FM radio measures frequency modulation imprinted on a carrier. If uncorrected, phase error detected by the DPD grows linearly in time as in Fig. 4 (b).

5. Measurements and results

When the loop is closed, the OEO oscillates at around 81 MHz. In addition to the fixed amplifier gain of 35 dB, we adjust the laser power to control the loop gain continuously. We have a stable oscillation at 3 mW. A typical output spectrum is shown in Fig. 7 (a). Although a few other modes are observed, they are negligible: the nearest mode at 1.8-MHz offset is −75 dB smaller than the main one. The near single-mode operation is possible owing to resonant response of the AOM tuned to 80 MHz and its frequency-dependent deflection angle. Because the OEO is aligned for the main mode with a very tight tolerance defined by small effective area of the photodetectors, downstream misalignment of the sidebands provides strong discrimination. The free spectral range of 1.8 MHz corresponds to the round-trip time t_r = 0.56 μs, most of which arises from propagation time of the acoustic wave through the AOM [18]. Figure 7 (b) shows the spectra for both free-running and phase-locked OEO. FWHM of the free-running case is 900 Hz measured at the resolution bandwidth (RBW) of 300 Hz. FWHM for the PLL case is 100 Hz at 30-Hz RBW. The peaks at the offset of $\pm 30$ kHz represent the bandwidth of the PLL system.

Fig. 7 (a) Output spectrum of the OEO. Free spectral range is 1.8 MHz. (b) Comparison of OEO spectra before and after the phase lock.

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We compare precision of the two detection methods by measuring baseline noise. For the heterodyne detection, we measure fluctuation of the DPD output V(DPD) while mirrors A and B are held fixed. For the OEO detection we measure that of the correction voltage V(PZT) to the PZT A while mirror B is fixed. After subtracting dc offset, each output is amplified by a factor of 100 and fed to a data acquisition system to measure the standard deviation over measurement time τ from 40 μs to 500 ms. The results converted to displacement are presented in Fig. 8. The DPD and the PZT are calibrated by independent measurements: ΔV(DPD)/Δϕ = 1 V/π and Δl/ΔV(PZT) = 11.5 nm/V.

Fig. 8 Standard deviation of the error signal for heterodyne detection and the correction signal for OEO detection. Each results are average values of 50 measurements.

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Noise of both methods is not larger than 100 pm for τ longer than 100 μs. At τ = 100 ms, it reaches 2 pm and 10 pm, respectively, for heterodyne and OEO detections. We note that the noise of OEO detection is consistently larger than that of heterodyne detection. As discussed in Section 3, while heterodyne detection is sensitive to perturbations downstream of the NPBS only, OEO detection is susceptible to those throughout the loop including the modulator. Particularly, given that our modulator has a long arm, it is prone to external perturbations.

Most of the noise is coming from acoustic and mechanical perturbations of environment in spite of the vacuum seal and the active vibration isolation of the apparatus. Figure 9 shows the Fourier transform of V(DPD) and V(PZT) expressed in unit of $pm/ \sqrt{Hz}$ . They show noise peaks between 50 Hz and 1 kHz, which coincide with resonance frequencies of various optical mounts. When we give a small kick to the apparatus, those noise peaks increase abruptly. Other than a small 60-Hz (and its high harmonic) noise, independent measurement of RF devices shows that electronic noise is negligible compared with mechanical noise.

Fig. 9 Displacement noise of heterodyne and OEO detections.

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While measurement bandwidth of the heterodyne method is limited only by detection electronics such as a phase detector, that of the OEO method is additionally limited by a PLL system with an actuator inside the OEO loop. We use a PZT as an actuator, which exhibits resonance at around 300 kHz in driving the mirror. Servo bandwidth of the PLL is further reduced to 30 kHz. These limits are apparent in the Bode plots of V(DPD) in Fig. 10 and V(PZT A) in Fig. 11 for heterodyne and OEO methods, respectively. The Bode plots are taken by driving PZT B harmonically. Heterodyne detection has a flat response up to 100 kHz, before the PZT resonance shows up. The OEO detection has a resonant response at 30 kHz from the PLL system. This situation can be improved by using an electronic or electro-optic phase shifter instead of the mechanical one with inertia.

Fig. 10 Bode plot of heterodyne detection.

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Fig. 11 Bode plot of OEO detection.

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In summary, for the case of our apparatus the OEO detection was somewhat inferior to the heterodyne detection in specifications of noise immunity and bandwidth. We note, however, that when the reference and interferometric signals cannot be well separated as in a gyroscope application or when detection electronics instead of environmental perturbation are a limiting factor as discussed in Section 3, the OEO method can be either complimentary to or better than the heterodyne method. The bandwidth limit can also be significantly reduced by using an electronic or electro-optic phase shifter instead of the PZT-mirror system with inertia.

In our experiment we use an AOM-based modulator. Owing to a long arm, it is susceptible to vibration noise. It also couples modulation frequency with a deflection angle, and thereby makes it necessary to use a phase-lock loop. With a better designed modulator, advantage of the frequency-shift measurement of the OEO method for various physical quantities would be more pronounced.

Funding

Agency for Defense Development (Grant to the Atom Optic Sensor Laboratory for National Defense).

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Displacement measurement using an optoelectronic oscillator with an intra-loop Michelson interferometer

Abstract

1. Introduction

2. Operational principles of heterodyne and OEO detection

3. Comparison of measurement uncertainties

3.1 Fundamental limits

3.2 Practical limits

4. Apparatus

5. Measurements and results

Funding

References and links

Cited By

Figures (11)

Equations (14)

Optics Express