Laser interferometry with precision and multi-fringe dynamic range at low frequencies is a core technology to measure the motion of various objects (test masses) in space and ground based experiments for gravitational wave detection and geodesy. Even though available interferometer schemes are well understood, their construction remains complex, often involving, for example, the need to build quasi-monolithic optical benches with dozens of components. In recent years techniques have been investigated that aim to reduce this complexity by combining phase modulation techniques with sophisticated digital readout algorithms. This article presents a new scheme that uses strong laser frequency modulations in combination with the deep phase modulation readout algorithm to construct simpler and easily scalable interferometers.
© 2015 Optical Society of America
High dynamic range laser interferometry with precision at mHz frequencies to read out the motion of test masses has been achieved using heterodyne detection schemes, for example with kHz heterodyne beat note frequencies for LISA pathfinder  and MHz frequencies for LISA . Similar techniques are also applied in ground based experiments, like suspension platform interferometers in gravitational wave detectors . The complexity and cost of the light preparation, the interferometer construction and the readout schemes of such interferometers have lead to an expanding investigation into alternative techniques that can be applied in future experiments. Ideally, such interferometers can, at some point, sense all degrees of freedom of multiple objects with minimal complexity.
Deep phase modulation (DPM) [4, 5, 6] is one of these techniques. It uses a strong sinusoidal phase modulation in one interferometer arm, with an amplitude larger than one interference fringe, to generate a comb of beat notes. By determining the amplitudes of these beats and using a non-linear fit algorithm, the interferometric phase can be reconstructed. This method greatly simplifies the complex light preparation schemes used, for example, in LISA Pathfinder, to generate AC readout signals in the kHz range.
Digitally enhanced interferometry (DeI)  is another scheme that allows one to multiplex multiple optical signals through a single optical and analog detection path, which leads to totally new interferometer topologies. This is achieved by phase-modulating pseudo-random noise (PRN) codes onto the laser light. The PRN chip rate determines the minimal delay and arm length difference between two separable signals. Experiments using DeI have shown performance levels in the order of at low frequencies , however, such precisions have not yet been demonstrated with chip rates suitable for simplifying interferometers with sizes of a few cm . A homodyne variant of DeI has also demonstrated levels of , but only at frequencies above 1 Hz . This technique, however, has additionally been used to demonstrate self-homodyning of the signals, which can further simplify optical layouts.
Another variant of self-homodyning has been used in multi-fringe readout techniques of low-finesse fiber micro-cavities . In contrast to the DeI scheme by Sutton et al.  the self-homodyning is here not achieved by a fast phase modulation, but by a correspondingly strong sinusoidal frequency modulation of the laser source. Similar schemes have also been investigated in unequal arm length interferometer set-ups and are sometimes referred to as frequency-modulated continuous-wave (FMCW) ranging techniques [12, 13]. Some of these schemes have been used to demonstrate phase readout noise levels at frequencies around 1 kHz , using a simple Bessel function analysis that relies on a constant modulation depth. Multiplexed variants using a time-gated analysis of the waveforms have also been investigated . Just recently one of these schemes, using smooth window functions , has achieved phase readout noise levels of above 1 Hz ( corresponds to for λ0 = 1064 nm). None of the above mentioned schemes has so far been evaluated for long-term phase tracking of a moving test mass with performance goals of noise at 1 mHz.
In this article a new scheme is presented that combines a strong laser frequency modulation with self-homodyning and the readout technique developed for DPM. This approach, which is in the following referred to as deep frequency modulation (DFM), promises to simplify the light preparation, as well as the interferometer set-ups themselves. In the following, the modulation scheme and its ties to DPM are presented. A readout scheme using DFM is described, together with the expected major noise sources and mitigation strategies. The potential of DFM to perform absolute ranging and its combination with a DeI multiplexed readout scheme are briefly discussed. This is followed by the presentation of first results, obtained with a real-time simulation.
2. Deep frequency modulation
We assume an unequal arm length Mach-Zehnder interferometer with an effective length difference of ΔL, as shown in Fig. 1. The frequency of the laser source is modulated with the DFM:Fig. 1. It is equivalent to the one used in DPM [4, 5], with the exception that the effective modulation depth m is given as m = 2πΔfτ, meaning that it scales linearly with the arm length difference and with the frequency modulation depth. To apply the DPM readout algorithm the output power Pout is first converted into a voltage vout via a photo diode and a trans-impedance amplifier. The signal is then demodulated with sine and cosine tones at N harmonics of the modulation frequency and low-pass filtered. The result of the demodulation for the n-th harmonic (for n > 1) are the quadrature Qn and in-phase In components of its complex amplitude [4, 17] : 4], which estimates the four signal parameters k, m, φ and ψ. This is done by minimising a least-square expression χ2, which, in simple terms, compares the measured complex amplitudes with a set of amplitudes derived from the analytic expressions in Equation 7. It was shown that this readout scheme is able to achieve phase measurement performance levels corresponding to between 1 Hz and 1 mHz . Modulation depth values sufficient for the DPM readout algorithms are on the order of m ≈ 9. Assuming one uses a rapidly tunable laser source with a modulation frequency of 1 kHz and a depth of 9 GHz, a delay of 160 ps is required to apply this readout, which corresponds to an arm length difference in vacuum of ΔL ≈ 4.8 cm.
The potential advantages of using DFM are described by an optical set-up shown in Fig. 2. The laser light is split into various fibers that supply individual interferometers. Each one of them acts as an optical head (OH), that consist only of a fiber output coupler, a beam splitter, a mirror and a quadrant photo diode or receiver with modest bandwidth requirements (≈ 100 kHz). Using the phase information determined from all four quadrants, displacement and tip and tilt of the TM can be determined, by using the DPM readout algorithms and differential wavefront sensing . The self-homodyning of DFM ensures that any fiber length noise is strongly suppressed, leaving only laser frequency noise as predicted dominant noise source, the mitigation of which is discussed in the next paragraph. The optical layout for reading out the motion of two test masses requires only two OHs that are placed in a distance of more than ΔL/2 ≈ 2.4 cm, making it much simpler than classic Mach-Zehnder interferometer layouts. Furthermore, a large number of OHs can be connected to a single laser, making DFM easily scalable for the readout of multiple test masses and/or degrees of freedom.
The unequal arm length required to operate a DFM interferometer inherently couples laser frequency noise into the measurement. For achieving a TM displacement sensitivity of , a frequency noise f̃ at low frequencies of better than is required (assuming a laser center wavelength of λ0 = 1064 nm).
Various frequency stabilisation schemes can be employed to reduce the low frequency noise of rapidly tunable lasers required for DFM. The here proposed method is to feed the light into a stable, unequal arm length reference interferometer (RI in Fig. 2). Thereby, excess laser frequency noise is measured as reference phase φr, which can either be subtracted from the displacement signals φs, or, in case of dynamic range limitations, be used to actively stabilise the laser at low frequencies. Since the DPM readout also provides access to the effective modulation depth m and to the modulation phase ψm, they can as well be actively stabilised to desired values, if measured in the RI.
The coupling of other classic interferometer noise sources into the measurement precision has to be considered. Laser amplitude fluctuations, present at the modulation frequency and its higher harmonics, will spoil the readout performance by influencing the value of the corresponding complex amplitudes fed into the fit algorithm. Classic amplitude stabilisation schemes can be included to reduce excess amplitude noise. For levels above feasible stabilisation bandwidth limits, which becomes more relevant for higher modulation frequencies, a correction can probably be implemented. This would be done by determining the complex amplitudes of the amplitude signal separately and by using these values in a feed-forward loop to correct the amplitudes used for the phase determination. Any limitations due to shot noise are not considered limiting, since even with modest interference contrast and photo diode responsitivity an input power of about 200 μW in an OH is more than sufficient to achieve pm performance. Noise in the photo receivers and in the phase measurement chain  is well understood at kHz frequencies and can be made sufficiently small by design.
So far perfect sinusoidal frequency modulations were assumed. Light sources applied in similar studies have shown non-linear modulation characteristics . For DFM any presence of higher harmonics of fm in the frequency modulation signal will add beat signals at the critical demodulation frequencies. This changes the measured complex amplitudes, which introduces noise and non-linear couplings, similar to the effect introduced by non-flat photo receiver transfer functions, discussed in the original DPM study . One way to mitigate this effect is to measure the higher order modulations [19, 16] and to either actively correct the modulation waveform, or to apply a correction of the complex signal amplitudes.
One additional feature of DFM, which is usually explored in FMCW techniques, is that the effective modulation depth m encodes the light travel time τ and, therefore, also the absolute position of the TM, considering the geometry of the OH. Since the DPM readout recovers this information as well, one can potentially combine the ranging and the phase readout. This requires, however, that either the delay in the reference interferometer used to stabilise m is well known, or, that the frequency modulation depth Δf is measured separately with the desired accuracy. For a distance to the TM of 2.4 cm, a relative readout accuracy of m of 4 × 10−5 leads to an absolute length accuracy in the order of a single interference fringe (1064 nm), potentially closing the gap between interferometer and ranging signal. The two signals can be combined with an appropriate filter to increase the dynamic range and to improve the very low frequency performance of the readout.
DFM can in principle be combined with DeI to implement a multiplexed readout scheme. Figure 3 shows a possible implementation. A phase modulation of pseudo random noise codes (c(t)) onto the light distributed to the OHs can be used to generate effective amplitude modulations of the interference signals .10, 9] might limit the performance. The optical detection through a back reflection into the fiber can not be combined with differential wavefront sensing, making this a set-up only for displacement sensing and potentially absolute ranging, either by using the effective modulation depth m, by using PRN ranging algorithms [21, 22], or by a combination of the two.
The OHs shown in Fig. 3 can also be simplified further to a cavity-like set-up, by either placing a single mirror into the beam bath, or by using a reflection of the fiber end itself. This technique is somewhat equivalent to the one presented by Nowakowski et al. , however, the demodulation of only the first two reflections from the cavity results, in case of sufficiently fast modulations or long arm length mismatch, in a fully sinusoidal response to phase changes, avoiding periodic errors due to multiple reflections. The ranging potential and the straight forward optical set-up make this scheme an interesting alternative to study.
To investigate the achievable performance levels of DFM a numerical experiment has been implemented on a personal computer, using the C programming language. A sketch of the simulation is shown in Fig. 4. Time series equivalent to sampled photo receiver signals were generated for two separate interferometers, one with a stable optical path length difference ΔLr = 4.8cm, acting as reference interferometer, and one with a varying optical path length difference ΔLi ≈ 4.7cm, the signal of interest, representing, for example, the output of an OH that interrogates the motion of a test mass. The initial data was generated with a rate of 250 kHz and the signal properties are described in the following.
The average path length differences for both interferometers differ by 1 mm, allowing both readouts to operate close to a desired effective modulation depth of about m ≈ 9. Larger differences in the path length have not yet been analysed. Laser frequency noise couples into both interferometers and it was simulated with a conservative noise level of at 1 Hz. The coupling was continually scaled with the delays corresponding to each optical path length difference. Additive noise, which includes effects such as shot noise, electronic noise and quantisation noise, was simulated with Gaussian noise generators and added to each signal. An additive noise level of was chosen, which corresponds to a signal-to-noise ratio that is commonly achieved with existing phase measurement systems. A frequency modulation was chosen with a modulation depth of Δf = 9GHz and a frequency of fm = 1kHz. The signal simulated for the reference interferometer is4]. The fit estimates four output parameters for each channel, the AC signal amplitude ax,e, the modulation depth mx,e, the estimated interferometric phase φx,e, and the modulation phase ψx,e. The varying parameters injected into the signal generation are also averaged and decimated, making them available for direct comparison with the estimated values. Such comparisons are well suited to investigate non-linear behaviours, because they represent absolute measurements that are not subject to most common mode suppressions. Time series were generated that correspond to measurement times of 4000 s. Figure 5 shows the results for a simulation with a constant amplitude v0 = 1V and a perfect, sinusoidal frequency modulation.
For the non-negligible laser frequency noise levels assumed here the estimated interferometric phase of the measurement channel needs to be corrected with the reference interferometer readout. Since the frequency noise scales differently for each interferometer, the determined modulation depths have to be included to get the correct results.Fig. 5. The residuals are dominated by a white noise with a level of about . This level was found to scale linear with the chosen additive white noise n̂x, similar to the coupling in simple heterodyne detection schemes, but increased by a factor due to the inclusion of a second noise term from the reference interferometer during the frequency noise correction. Some small residual peaks are visible around 10 mHz, which are probably induced by the non-sufficient suppression of the implemented averaging filter, as described by Schwarze et. al. . The standard deviation of the residuals for is on the order of 1.6 · 10−6 radian. By calculating the standard deviation of the difference between the estimated path length difference Ls,e and the simulation input Ls, a relative uncertainty of less than 1.2 · 10−6 was estimated. For a length of 4.7 cm this is equivalent to a length uncertainty of less than 60 nm for the absolute ranging (assuming the frequency modulation depth is known with arbitrary accuracy).
Non-sinusoidal frequency modulation
To investigate the influence of higher harmonic components of the laser frequency modulation on the readout a modified modulation signal was implemented.Fig. 6. The introduction of the higher harmonic frequency modulation has a significant impact on the performance. The ratio between both modulation depths is almost exactly equal to the linear phase readout range. This result indicates that high dynamic range readout systems using DFM will either require a very clean sinusoidal laser modulation scheme, an active stabilisation of the parasitic tones, or a further extension of the current readout scheme by post processing corrections or modification of the fit algorithm.
4. Summary and conclusion
Deep frequency modulation interferometry is a new technique with the potential to perform low frequency, multi-fringe laser interferometry with precision. It uses the self-homodyning of two beams interfered in an unequal arm length interferometer to generate a beat note that can be read out using the algorithms developed for deep phase modulation. The scheme is highly scalable, it greatly simplifies optical set-ups and the projected footprint, cost, mass and power consumption make it a promising candidate for the readout of multiple test masses and degrees of freedom in future space missions [23, 24], fundamental research , and metrology applications. Attractive extensions of DFM include absolute ranging and multiplexed readout schemes.
The conceptual design of DFM can easily be adapted to interferometers with longer arm length, where laser frequency noise mitigation will then become the critical parameter for the achievable performance.
The values used in this article for the modulation parameters Δf and fm are based on commercially available external cavity lasers. Other light sources, for example current modulated laser diodes , can potentially improve on these parameters, which can then, in turn, enable set-ups with further miniaturisation and/or higher readout rates. Experimental studies will be necessary to determine which combination of light source, modulation scheme and stabilisation scheme will result in the desired levels of residual amplitude modulations and purest sinusoidal frequency modulation.
Experimental verification and extended studies of its individual components (laser source, stabilisations, optical head, phasemeter, fit algorithm) are the next steps to qualify DFM for deployment.
The author would like to thank Felipe Guzmán Cervantes, Katharina-Sophie Isleif, Moritz Mehmet and Gerhard Heinzel for useful discussions and proof reading.
References and links
1. G. Heinzel, V. Wand, A. García, O. Jennrich, C. Braxmaier, D. Robertson, K. Middleton, D. Hoyland, A. Rüdiger, R. Schilling, U. Johann, and K. Danzmann, “The LTP interferometer and phasemeter,” Class. Quantum Grav. 21, S581 (2004). [CrossRef]
2. R. Spero, B. Bachman, G. de Vine, J. Dickson, W. M. Klipstein, T. Ozawa, K. McKenzie, D. A. Shaddock, D. Robison, A. Sutton, and B. Ware, “Progress in Interferometry for LISA at JPL,” Class. Quantum Grav. 28, 094007 (2011). [CrossRef]
3. K. Dahl, G. Heinzel, B. Willke, K. A. Strain, S. Goßler, and K. Danzmann, “Suspension platform interferometer for the AEI 10 m prototype: concept, design and optical layout,” Class. Quantum Grav. 29, 095024 (2012). [CrossRef]
5. T. S. Schwarze, O. Gerberding, F. G. Cervantes, G. Heinzel, and K. Danzmann, “Advanced phasemeter for deep phase modulation interferometry,” Opt. Express 22, 18214–18223 (2014). [CrossRef] [PubMed]
6. M. Terán, V. Martín, L. Gesa, I. Mateos, F. Gibert, N. Karnesis, J. Ramos-Castro, T. Schwarze, O. Gerberding, G. Heinzel, F. Guzmán, and M. Nofrarias, “Towards a fpga-controlled deep phase modulation interferometer,” arXiv preprint arXiv:1411.6910 (2014).
8. G. de Vine, D. S. Rabeling, B. J. Slagmolen, T. T. Lam, S. Chua, D. M. Wuchenich, D. E. McClelland, and D. A. Shaddock, “Picometer level displacement metrology with digitally enhanced heterodyne interferometry,” Opt. Express 17, 828–837 (2009). [CrossRef] [PubMed]
9. K.-S. Isleif, O. Gerberding, S. Köhlenbeck, A. Sutton, B. Sheard, S. Goßler, D. Shaddock, G. Heinzel, and K. Danzmann, “Highspeed multiplexed heterodyne interferometry,” Opt. Express 22, 24689–24696 (2014). [CrossRef] [PubMed]
11. B. K. Nowakowski, D. T. Smith, and S. T. Smith, “Development of a miniature, multichannel, extended fabryperot fiber-optic laser interferometer system for low frequency si-traceable displacement measurement,” in “Proceedings of the 29th ASPE Annual Meeting,” (2014).
13. J. Zheng, Optical frequency-modulated continuous-wave (FMCW) interferometry, vol. 107 (Springer Science & Business Media, 2005).
14. A. Dandridge, A. B. Tveten, and T. G. Giallorenzi, “Homodyne demodulation scheme for fiber optic sensors using phase generated carrier,” IEEE Trans. Microw. Theory Techn. 30, 1635–1641 (1982). [CrossRef]
15. I. Sakai, R. Youngquist, and G. Parry, “Multiplexing of optical fiber sensors using a frequency-modulated source and gated output,” J. Lightwave Technol. 5, 932–940 (1987). [CrossRef]
16. T. Kissinger, T. O. Charrett, and R. P. Tatam, “Range-resolved interferometric signal processing using sinusoidal optical frequency modulation,” Opt. Express 23, 9415–9431 (2015). [CrossRef]
17. M. Abramowitz and I. A. Stegun et al., Handbook of mathematical functions, vol. 1 (Dover New York, 1972).
18. T. Schuldt, M. Gohlke, D. Weise, U. Johann, A. Peters, and C. Braxmaier, “Picometer and nanoradian optical heterodyne interferometry for translation and tilt metrology of the LISA gravitational reference sensor,” Class. Quantum Grav. 26, 085008 (2009). [CrossRef]
19. T. Kissinger, T. O. H. Charrett, and R. P. Tatam, “Fibre segment interferometry using code-division multiplexed optical signal processing for strain sensing applications,” Meas. Sci. Technol. 24, 094011 (2013). [CrossRef]
20. D. Shaddock, B. Ware, P. Halverson, R. E. Spero, and B. Klipstein, “Overview of the LISA Phasemeter,” AIP Conf. Proc. 873, 689–696 (2006). [CrossRef]
21. J. J. Esteban, A. F. Garcia Marin, S. Barke, A. M. Peinado, F. Guzman Cervantes, I. Bykov, G. Heinzel, and K. Danzmann, “Experimental demonstration of weak-light laser ranging and data communication for LISA,” Opt. Express 19, 15937 (2011). [CrossRef] [PubMed]
22. A. J. Sutton, K. McKenzie, B. Ware, G. de Vine, R. E. Spero, W. Klipstein, and D. A. Shaddock, “Improved optical ranging for space based gravitational wave detection,” Class. Quantum Grav. 30, 075008 (2013). [CrossRef]
23. K. Danzmann and A. Rüdiger, “LISA technology - concept, status, prospects,” Class. Quantum Grav. 20, S1 (2003). [CrossRef]
24. J. Luo, F. Gao, Y.-Z. Bai, C.-G. Shao, and Z.-B. Zhou, “Test of the equivalence principle with optical readout in space,” Laser 2, f1 (2008).