6.2 Thermal co-alignment
7.1.1 Experimental setup for multiplexed signals
The simplified experimental setup is shown in Figure 7.1. We connected three polarization- maintaining (PM) optical fibers with ultra polished connector (UPC) ends in series: length
L1 = 20 m, L2 = 10 m, L3 = 2.5 m. When two connector end-faces are joined, a small
portion of light (∼1% of the power for UPCs) is back-reflected at these junctions. We used DI to independently measure the phase of reflections from connectors R1, R2, R3,
and R4 shown in Figure 7.1.
Two Nd:YAG nonplanar ring oscillator lasers (Lightwave 126, Innolight Mephisto) operating at 1064 nm were offset phase-locked to provide a 14 MHz heterodyne beat note frequency. Light from the carrier laser was coupled into a PM fiber and directed through a Photline Technologies (NIR-LN series) waveguide electro-optic phase modulator (EOM). The EOM was driven with a PRN code 232−1 chips long, producing the 0-π rad phase
modulation with anfchip= 80 MHz chipping frequency. We used a long PRN code so that
errors from the already low auto-correlation of code segments would be random, allowing further improvement by averaging.
The back-reflections from the fiber connectors were combined with light from the local oscillator and the resulting signal was detected at the photodetector (New Focus 1611- fc). This photodetector output was digitized using a 16-bit Analog-to-Digital Converter sampling at 80 MHz. The PRN decoding and phase extraction was performed in real-time using a field-programmable gate array (FPGA). The FPGA implemented four parallel processing channels with decoding delays matched to the unique propagation delays of the back-reflections fromR1,R2,R3, andR4. The signal due to reflectionR1, for example, was
§7.1 Digitally enhanced interferometry 97 Fiber Coupler EOM PRN code 20 m 10 m 2.5 m Fiber Coupler Photodetector Laser Laser Signal Generators Fiber Connectors
R1
R2
R3
R4
L1
L2
L3
Local Oscillator Carrier 50/50 50/50 Offset Phase-Locked by 14 MHzFigure 7.1: Digital interferometry setup for monitoring length changes of L1,L2, and L3
by tracking reflections from R1, R2, R3, and R4. Signal generators are used to inject
test signals for performance characterization. EOM: electro-optic modulator. PRN code: pseudo-random noise code.
isolated by multiplying the photodetector signal by a version of the PRN code delayed by the total electronic and optical group delay of the signal including generation, modulation, propagation via R1 and measurement at the FPGA. After decoding, the phase of each
channel was extracted from the decoded signals using a digital phase-locked loop on the same FPGA.
The optical path length change along any fiber length could be found by subtracting the phase measurements from the connector reflections at each end of the fiber. For example, length fluctuations of L1 were measured by taking the difference of the phase
measurements from reflectionsR1 and R2,φR1 andφR2:
δL1 =
λ
4πn(φR2−φR1), (7.1)
whereλis the laser wavelength,nis the index of refraction of silica, and the phasesφare in radians.
Lay et al. [86] predicted the rms displacement resolution for monitoring one length due to the finite isolation of the PRN code. We expand on this model to predict the sensitivity of a system detecting N reflections. IfN signals with different PRN delays are present at the photodetector, the phase noise Root Power Spectral Density (RPSD) due to interference from other signals would be
˜ φRi= v u u t 1 fchip PN k=1Pk Pi − 1 ! [rad/√Hz], (7.2)
where Pk is the power of the kth signal and Pi is the power of the desired (correctly
decoded) signal. 7.1.2 Results
Piezo fiber stretchers were inserted into each fiber segment allowing the lengths to be independently modulated to characterize the signal response. Displacement signals were injected via these fiber stretchers: a 1 Hz square wave (L1), a 3 Hz triangle wave (L2),
and a 5 Hz sine wave (L3). Figure 7.2 shows simultaneous displacement measurements for
each length and demonstrates clear recovery of the injected displacement signals.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 −1.5 −1 −0.5 0 0.5 1 Time [s] D isp la ce me n t [µ m] Length 1 Length 3 Length 2
Figure 7.2: Simultaneous displacement measurements of fiber lengths when stretched with different waveforms. Modulating L1 with a 1 Hz square wave, L2 with a 3 Hz triangle
wave, and L3 with a 5 Hz sine wave. DI clearly recovers the individual displacement
signals.
To measure the crosstalk between signals, we modulated L2 with a 5 Hz sine wave
and measured the cross-coupling into the displacement measurements of the adjacent, unmodulatedL1andL3lengths. A portion of the time domain data is shown in Figure 7.3,
where the sinusoidal 5 Hz signal is noticeably absent from theδL1 andδL3 measurements.
The crosstalk can be quantified in the frequency domain by examining the RPSD of the displacement data, shown in Figure 7.4. The amplitude of the 5 Hz signal is 2.1µm/√Hz inδL2, 5.3 nm/
√
Hz inδL3, and indistinguishable from the noise floor of∼200 pm/
√
Hz in
δL1. This corresponds to a crosstalk of<2.6×10−3 (-52 dB) intoL3 and<10−4(-80 dB)
into L1. Electronic tests indicate that the crosstalk between lengths is limited by errors
in the PRN modulation and particularly the finite bandwidth of the analog electronics used to amplify the code. The tail of the input electronics’ impulse response results in a single reflection being partially decoded at several subsequent delays. This bandwidth effect would only corrupt reflections downstream and explains the crosstalk asymmetry shown in Figure 7.4 (i.e.,L2 signals appear inL3 measurements but do not couple intoL1
measurements). We note that our PRN scheme is susceptible to polarization crosstalk as in conventional interferometry. However, this crosstalk was not our limiting noise source; the difference in path length fluctuations between the two polarizations is small compared to the total fluctuations.
The limiting noise source at frequencies below 1 Hz was laser frequency noise. Recall from §5.2.2 that laser frequency noise δν corrupts each displacement measurement in
§7.1 Digitally enhanced interferometry 99 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 Length 3 Length 1 Length 2 Time [s] D isp la ce me n t [µ m]
Figure 7.3: Time domain data showing low crosstalk between signals when modulatingL2
with a 5 Hz sine wave.
proportion to the interferometer arm length Li:
δLi ≈δL0i+
δν
ν Li, (7.3)
whereδL0i denotes the true length change. In Figure 7.3 the error due to the slow drift in laser frequency is largest in the displacement of the longest fiber, L1 (20 m), yet barely
visible in the shortest length, L3 (2.5 m). The low frequency noise in Figure 7.4 appears
highly correlated between channels but is scaled by the length.
Immunity from laser frequency noise is commonly achieved by using an interferometer with equal length arms [87, 88]; laser frequency noise affects the phase of the light in each arm by the same amount and cancels when interfered. With DI multiplexing, we are freed from the constraint of equal arm lengths because length measurements can be independently scaled prior to subtraction. This independent length knowledge enables us to form a differential length measurement that is free from frequency noise. For example,
L2is half the length ofL1and should be half as sensitive to changes in the laser’s frequency.
The frequency noise inδL2can be removed by forming a combination with a scaled version
of δL1.
Figure 7.5 shows measurement ofL2 before and after the subtraction of laser frequency
noise. To demonstrate a genuine signal-to-noise ratio improvement we added a sinusoidal displacement signal in L2 at 0.4 Hz, where the system was frequency noise limited. The
measurement noise is substantially reduced at frequencies below 1 Hz while the 0.4 Hz displacement signal remains. We modulated the laser frequency by 500 kHz at 5 Hz to measure the magnitude of frequency noise suppression. This frequency modulation signal is suppressed by a factor of 40 (33 dB). To measure each length in frequency noise-free mode, one length must act as a frequency noise reference; each measurement will be made relative to this reference length. Any other noise in the reference length will couple into
10−2 10−1 100 101 10−11 10−10 10−9 10−8 10−7 10−6 Length 2 Displacement Signal Crosstalk Length 3 Length 1 Frequency [Hz] R PSD [ m/ √ H z]
Figure 7.4: RPSDs ofδL1,δL2, andδL3 when modulatingL2 with a 5 Hz sine wave. The
crosstalk between length measurements is determined by the ratio of the amplitudes of the 5 Hz peak in each spectrum.
the subtraction, thus using the longest length in a quiet environment will produce the best results.
7.1.3 Discussion
In DI systems, a PRN code modulated at 80 MHz (as in this experiment) can isolate reflections with a minimum spacing of 2.5 m in fiber, i.e. with separation greater than one code chip. With currently available technology, it is possible to modulate at frequencies exceeding 10 GHz, which could reduce the minimum separation between targets to∼2 cm. In this experiment, we focused on demonstrating DI’s multiplexing capability, targeted in a fiber implementation. The 200 pm/√Hz code noise floor in this setup would suffice for use in a measurement such as GRACE Follow-On, but not for LISA. In a different DI experiment also using a heterodyne readout, our measured noise floor was 5 pm/√Hz at 1 Hz [89], so our noise floor from Figure 7.5 is not a fundamental limit to the technique.
Recently, a more advanced DI modulation scheme [90] has demonstrated sub-picometer sensitivity levels compatible with LISA requirements. Instead of the two-level (0 /π) PRN code, the new scheme utilizes a four-level code which removes the need for a frequency- shifted local oscillator. This DI implementation is thus in a homodyne configuration, while still allowing the phase to be precisely measured from the in-phase and quadrature compo- nents. This further reduces the hardware needed for the interferometric measurement – as now it eliminates the need for a dedicated, frequency-shifted local oscillator beam (which is typically accomplished in heterodyne systems using a second laser or an acousto-optic modulator).
§7.2 Looking forward 101