## Laser ranging and communications for LISA |

Optics Express, Vol. 18, Issue 20, pp. 20759-20773 (2010)

http://dx.doi.org/10.1364/OE.18.020759

Acrobat PDF (1352 KB)

### Abstract

The Laser Interferometer Space Antenna (LISA) will use Time Delay Interferometry (TDI) to suppress the otherwise dominant laser frequency noise. The technique uses sub-sample interpolation of the recorded optical phase measurements to form a family of interferometric combinations immune to frequency noise. This paper reports on the development of a Pseudo-Random Noise laser ranging system used to measure the sub-sample interpolation time shifts required for TDI operation. The system also includes an optical communication capability that meets the 20 kbps LISA requirement. An experimental demonstration of an integrated LISA phase measurement and ranging system achieved a ≈ 0.19 m rms absolute range error with a 0.5 Hz signal bandwidth, surpassing the 1 m rms LISA specification. The range measurement is limited by mutual interference between the ranging signals exchanged between spacecraft and the interaction of the ranging code with the phase measurement.

© 2010 Optical Society of America

## 1. Introduction

*L*= 5 × 10

^{9}m. Each spacecraft (SC) contains two proof masses that are shielded from external disturbances. Bidirectional laser links are maintained between all SC in the constellation. The relative optical phase between the incoming and outgoing lasers, referenced to their respective test masses, is used to track the changes in separation between the proof masses, allowing detection of gravitational waves. The change in separation of the proof masses Δ

*L*must be monitored with a displacement sensitivity of

1. M. Tinto and J. W. Armstrong, “Cancellation of laser noise in an unequal-arm interferometer detector of gravitational radiation,” Phys. Rev. D **59**(10), 102003 (1999). [CrossRef]

2. M. Tinto, M. Vallisneri, and J. W. Armstrong, “Time-delay interferometric ranging for space-borne gravitationalwave detectors,” Phys. Rev. D , **71**(4), 041101 (2005). [CrossRef]

4. D. A. Shaddock, B. Ware, R. E. Spero, and M. Vallisneri“Postprocessed time-delay interferometry for LISA,” Phys. Rev. D **70**(8), 081101 (2004). [CrossRef]

5. G. de Vine, B. Ware, K. McKenzie, R. E. Spero, W. M. Klipstein, and D. A. Shaddock, “Experimental demonstration of time-delay interferometry for the laser interferometer space antenna,” Phys. Rev. Lett. **104**(21), 211103 (2010). [CrossRef] [PubMed]

6. S. E. Pollack and R. T. Stebbins, “A demonstration of LISA laser communication.”Class. Quantum Grav. **23**, 4201 (2006). [CrossRef]

*τ*=

*t*(

_{Tx}*t*−

*L*/

*c*) −

*t*(

_{Rx}*t*), the difference between the receiving SC clock

*t*and the transmitted time stamp

_{Rx}*t*after propagating at the speed of light

_{Tx}*c*across the arm of length

*L*m. This measurement is sensitive to two fundamental effects;

**Spacecraft separation***L*The drift of the LISA arm lengths result in a changing propagation delay*T*=*L*/*c*between SC and appears as a change in the ranging signal arrival time. Any additional physical delay experienced by the ranging signal, such as group delay or dispersion effects, can be thought of as an increase/decrease in SC separation.**Clock difference***t*−_{Tx}*t*Any differences in spacecraft clocks will cause each SC to generate/measure its ranging signal at a different rate and with a different start time, appearing as an anti-symmetric change in pseudo-range measured between SC._{Rx}

*SC*

_{1}and arriving at

*SC*

_{2}using only combinations of phase measurements.

*ϕ*(

*t*) propagates between

*SC*

_{1}and

*SC*

_{2}. The pulse is recorded by

*SC*

_{1}as

*ϕ*

_{1}(

*t*

_{1}(

*t*)) =

*ϕ*(

*t*) with local clock time

*t*

_{1}(

*t*) and appears at

*SC*

_{2}, with its local clock

*t*

_{2}(

*t*), after propagation delay

*L*/

*c*=

*T*seconds as

*ϕ*

_{2}(

*t*

_{2}(

*t*)) =

*ϕ*(

*t*−

*T*). The pulse of noise will be cancelled by differencing the

*SC*phase measurements at the time the pulse was measured by each

*SC*, giving

*ϕ*

_{1}(

*t*

_{1}(

*t*)) −

*ϕ*

_{2}(

*t*

_{2}(

*t*) =

*t*

_{1}(

*t*) −

*T*) = 0. In other words, TDI attempts to align each data sample recorded on

*SC*

_{2}with clock

*t*

_{2}with an equivalent sample from

*SC*

_{1}, applying sub-sample interpolation where required, such that they a both samples are measuring equivalent phase noise. Determining this alignment requires knowledge of the propagation delay

*T*plus the relationships between the two clocks and is the responsibility of the ranging system.

*T*and is recorded with the same clock as the phase data. Correspondingly, the pseudo-range measured at

*SC*

_{2}is sufficient to identify a) when the sample left

*SC*

_{1}and b) when it was recorded at

*SC*

_{2}. This implicitly compensates for all physical delays (‘

*T*’) and clock deviations. For each phase sample, the pseudo-range represents the delay/advance that must be applied to align the samples between SC using the fractional-delay filtering approach and represents an equivalent physical range estimate

*R*=

*cT*, for speed of light

*c*. The accuracy of the phase data reconstruction and subsequent frequency noise cancellation is limited by the pseudo-range estimate and is the focus of this paper.

*μ*s is proposed to minimise any differential group delay relative to the MHz frequency heterodyne beat notes. To ensure that the DLL is capable of tracking any timing changes, any averaging performed within the DLL must be significantly faster than the expected timing fluctuations. The digital nature of the PRN modulation also lends itself well to the transmission of data between SC and a 20 kbps data communication requirement has also placed upon the ranging system [10]. Initial simulation work has shown the viability of this scheme for use in the LISA environment [9

9. J. J. Esteban, A. F. Garca, J. Eichholz, A. M. Peinado, I. Bykov, G. Heinzel, and K. Danzmann, “Ranging and phase measurement for LISA,” J. Phys.: Conf. Ser. **228**, 012045 (2010). [CrossRef]

## 2. System Description

9. J. J. Esteban, A. F. Garca, J. Eichholz, A. M. Peinado, I. Bykov, G. Heinzel, and K. Danzmann, “Ranging and phase measurement for LISA,” J. Phys.: Conf. Ser. **228**, 012045 (2010). [CrossRef]

*SC*

_{2}and received at

*SC*

_{1}as shown in Fig. 2. PRN ranging codes

*c*

_{1}and

*c*

_{2}are phase modulated onto the respective SC lasers via an Electro-Optic Modulator (EOM). The laser from the far

*SC*

_{2}, with phase

*x*

_{2}(

*t*−

*T*) delayed by the propagation delay

*T*, is interfered with the local laser, with phase

*x*

_{1}, at point

*A*

_{1}on

*SC*

_{1}. The heterodyne beat note, which encodes the difference between the laser phases, is measured at the Detector (

*D*

_{1}) and tracked by the phasemeter. The phasemeter output is passed to the DLL for demodulation of the ranging signal. The outgoing ranging signal is modulated onto the outgoing laser before the interference occurs at

*A*

_{1}.

*c*

_{1}and

*c*

_{2}ie.

*x*

_{2}=

*βc*

_{2}and

*x*

_{1}=

*βc*

_{1}, with modulation depth

*β*= 0.14rad for 1% of the carrier power. Additionally, over the MHz scale PRN frequencies the LISA phase measurement noise will be dominated by carrier shot noise, appearing as an additive phase noise source

*ν*with a white spectrum with a value of

_{PM}*SC*

_{1}clock

*t*

_{1}. The pseudo-range incorporates the propagation delay

*T*between SC and clock differences and can be viewed as a time varying delay

*τ*

_{2}applied to the

*SC*

_{2}laser phase signal. Fortunately, any pseudo-range variations are expected to be slow relative to each pulse period and can be considered static over an averaging period (∼ 50

*μ*s). Any long term phase fluctuations will be tracked by the DLL algorithm introduced in Section 3. Setting the global time relative to the

*SC*

_{1}time

*t*≡

*t*

_{1}gives the shot noise limited phase estimate at the detector as Eq. (1).

### 2.1. Ranging Demodulation

*ĉ*

_{2}(

*t*,

τ ^

_{2}) =

*c*

_{2}(

*t*−

τ ^

_{2}). This local copy, the ‘demodulation’ code, utilises a controllable delay

τ ^

_{2}to estimate the pseudo-range delay

*τ*

_{2}that allows the demodulation code to be aligned with the ranging signal.

τ ^

_{2}must be explicitly acquired. Acquisition is accomplished by correlating the phase estimate against the demodulation code while scanning the delay

τ ^

_{2}, mapping out the auto-correlation of the ranging code. When the misalignment between the demodulation code and ranging signal Δ

*τ*

_{2}=

*τ*

_{2}−

τ ^

_{2}is within 1 chip period

*T*(|Δ

_{P}*τ*

_{2}| <

*T*), a strong correlation peak will be present. The auto-correlation of a PRN sequence evaluated over a period of 50 chips without noise is shown in Fig. 3a, clearly demonstrating the high correlation at 0 delay. While acquisition will not be discussed here, other sources show acquisition within ±1 chip is achievable [7, 9

_{P}9. J. J. Esteban, A. F. Garca, J. Eichholz, A. M. Peinado, I. Bykov, G. Heinzel, and K. Danzmann, “Ranging and phase measurement for LISA,” J. Phys.: Conf. Ser. **228**, 012045 (2010). [CrossRef]

*ĉ*

_{2}and phase estimate

*ϕ*

_{1}can be directly evaluated over an

*N*chip interval for a small acquisition misalignment |Δ

*τ*

_{2}| <

*T*

*. By assuming that each code chip is represented by a pulse*

_{P}*f*(

*t*) of width

*T*and auto-correlation

_{P}*R*(

_{F}*τ*), then each chip of the demodulation code will overlap with at most two chips of both the ranging

*c*

_{2}and interfering

*c*

_{1}codes.

*N*chips weighted by area of overlap. In this instance, the overlap area is defined as

*R*(.) from Eq. (2) evaluated at the delay Δ

_{f}*τ*

_{2}. This dependence is also seen in Fig. 3a, where the M sequence auto-correlation follows that of the Rectangular pulse for small offsets. The second ‘unwanted’ correlation results from coupling in an adjacent chip. For sample

*n*, this contribution is given as the code auto-correlation evaluated at a ±1 chip offset

*C*

_{ĉ2,c2}[

*n*, ±1], which is then weighted by the overlap area

*R*((Δ

_{f}*τ*

_{2}+

*T*)mod

_{P}*T*).

_{P}*c*

_{1}code each contribute a cross-correlation term

*C*

_{ĉ2,c1}[

*n,m*] and

_{i}*C*

_{ĉ2,c1}[

*n,m*+ 1], with weightings

_{i}*R*(

_{f}τ ^

_{2}mod

*T*) and

_{P}*R*((

_{f}τ ^

_{2}+ 1) mod

*T*) respectively. Since the interfering code is generated locally, only the changing demodulation delay will affect the alignment with the demodulation code and the cross-correlation will be calculated with an offset

_{P}*m*= ⌊

_{i}τ ^

_{2}/

*T*⌋, where ⌊

_{P}*x*⌋ is the integer part of

*x*. This determines that the normalised demodulation over

*N*chips (

*τ*=

_{Corr}*NT*), will be given by Eq. (3).

_{P}*τ*

_{2}| ≪

*T*), the output is dominated by the strong, coherent signal from the correlation with the ranging signal, with low level cross-correlation noise from the interfering code. As the misalignment increases, the coherent amplitude will reduce down to the noise levels from the other correlation components. To minimise the interference, the cross-correlation between codes

_{P}*c*

_{1}and

*c*

_{2}should be minimised for all possible delays, however before this effect can be quantified we must discuss the filtering effects introduced by the Phasemeter.

### 2.2. Phase Measurement Effects and Pulse Shaping

*ϕ*

_{1}through controller

*C*. The detector phase signal is mixed with a Local Oscillator (LO) and low pass filtered (LPF) to form an error signal for feedback control of the LO phase. Such a control system locks the LO to be in quadrature with the signal.

_{PLL}*ϕ*

_{1}and the phasemeter Q channel output

*ϕ*can be approximated as

_{Q}*H*(

_{PM}*s*) with Laplace variable

*s*. This transfer function assumes the controller

*C*is dominated by a 1/

_{PLL}*f*slope around the open loop unity gain frequency

*f*≈ 100 kHz. The closed loop transfer function exhibits a high pass behaviour, with a corner frequency

_{u}*f*.

_{u}*ϕ*

_{1}, then the phasemeter Q output

*ϕ*(

_{Q}*t*) will be given by Eq. (4), where the high frequency shot noise is also passed to the output.

*h*. Figure 5a shows the filtering effect upon a Rectangular pulse

_{PM}*f*(

*t*) of width

*T*= 1

_{P}*μ*s, with an unfiltered pulse shown for comparison. The visible decay induced by the filtering causes a measurable delay in the pulse centroid, which will couple into the range estimate.

*f*(

_{M}*t*) defined by Eq. (5). This pulse shape doubles the signal bandwidth by guaranteeing a transition between 1 and −1 occurs every chip. Additionally, the code is now balanced for each chip, with zero DC component and reduced low frequency information as since each chip has 0 DC power. The Manchester pulse also passes through the phasemeter with minimal decay as shown by Fig. 5a.

*T*) for the Manchester pulse shape is small (≪ 1%) compared to other terms, allowing us to simply replace the pulse function with the Manchester pulse

_{P}*f*(

_{M}*t*) and auto-correlation

*R*(

_{M}*τ*). The pulse correlation after the phasemeter is now

*R*(Δ

_{M}*τ*) ★

*h*=

_{PM}*R*(Δ

_{PM}*τ*) and the demodulation correlation can be written as;

### 2.3. Interference Contribution

*C*

_{ĉ2,c2}[

*n*, ±1] =

*ν*

_{ĉ2,c2}[±1], with variance

*C*

_{ĉ2,c1}[

*n,m*] =

_{i}*ν*

_{ĉ2,c1}[

*m*] and

_{i}*C*

_{ĉ2,c1}[

*m*+ 1] =

_{i}*ν*

_{ĉ2,c1}[

*m*+ 1], with variance

_{i}*A*

_{ϕQ,ĉ2}[

*n*] as shown in Eq. (6) below.

τ ^

_{2}will modulate the interference contributions through the pulse auto-correlation. The interference pulse correlation weightings

*R*(

_{f}τ ^

_{2}mod

*T*) and

_{P}*R*((

_{f}*τ*

_{̂}_{2}+ 1)mod

*T*) are each dependent upon the delay, acting as a time varying gain to the correlation noise. On short time scales, performance predictions must be used directly Eq. (6), however on longer averaging periods we can simplify the result.

_{P}*ν*

_{ĉ2,c1}with zero mean and variance

*R*((Δ

_{f}*τ*

_{2}+

*T*)mod

_{P}*T*)

_{P}*ν*

_{ĉ2,c2}[±1] =

*ν*

_{ĉ2,c2}with zero mean and a timing error dependent variance

*ν*, with variance

_{A}*A*

_{ϕQ,ĉ2}[

*n*] can be written as a timing error dependent random variable given by Eq. (7).

*τ*of

_{Corr}*N*= 50 Chips (50

*μ*s or 20 kHz correlation rate). The modulation depth is

*β*= 0.14 for 1% of the carrier power. For perfect demodulation, the correlation noise terms give a noise power

*τ*

_{2}= 0, the Signal to Interference Noise ratio (SINR) is 3

*N*/2. The noise models in the presented in this section were verified by simulation.

## 3. Timing Tracking

*τ*

_{2}be determined to within 3 ns (or 1 m range) and is equivalent to demodulating the ranging signal phase to within 10

^{−3}chips. Changes in this delay will be driven by two dominant timing effects; the LISA arm length fluctuations and the relative clock differences between spacecraft. The delay locked loop derives a timing error signal from an ‘early’ and a ‘late’ correlation to track the resulting changes in code arrival time. The early / late demodulation code is nominally advanced/ delayed half a chip from the ‘on-time’ timing point, however the exact separation may be reduced for better sensitivity at the expense of acquisition reliability. Any change in arrival time will add power to either the early or late correlation, indicating the direction of the change. The difference between the correlation forms a timing error signal proportional to the timing offset Δ

*τ*

_{2}.

*τ*

_{2}and is given by

*τ*

_{2}= ±

*T*/2 as appropriate. This gives the relation in Eq. (8) for the error signal output as a function of timing error. The phasemeter and interference noise contributions have been grouped into noise sources

_{P}*ε*

_{ϕQ,c2}against timing error Δ

*τ*

_{2}. The error signal is approximately linear for errors within | Δ

*τ*

_{2}| ≤

*T*/2, however the error signal zero crossing point occurs at an offset from the ideal timing point Δ

_{P}*τ*

_{2}= 0. This is a direct consequence of the delay introduced by the phasemeter and results in an offset of the zero crossing point Δ

*τ*

_{0}which is dependent upon the phasemeter closed loop decay time 1/

*α*by,

*α*= 2

*π*× 10

^{5}, the timing offset is Δ

*τ*

_{0}≈ 75.2 ns or ≈ 22.6 m with a ∼ 0.82 dB Signal to Noise Ratio (SNR) loss. Importantly, without automatic gain control of the digitized heterodyne beat note, any amplitude deviations will couple into the phasemeter open loop bandwidth

*f*though gain scaling of the 1/

_{u}*f*slope. Consequently the bandwidth

*f*and the error signal zero crossing point will vary with any amplitude variation. As the effective unity gain frequency scales with amplitude noise, this will introduce in an additional source of time varying, timing offset which must be compensated for. Around the target phasemeter operating amplitude

_{u}*A*, the coupling between amplitude noise Δ

_{P}*A*and timing offset is Δ

_{P}*τ*

_{0}≈ 75.2 + 70Δ

*A*/

_{P}*A*ns. Allocating 0.03 ns error to this error source requires knowledge or control of the fractional amplitude fluctuation to within Δ

_{P}*A*/

_{P}*A*≈ 4×10

_{P}^{−4}. This can either be accomplished by automatic gain control or calibrated in offline post-processing using the to the amplitude information recorded by the phasemeter at a fidelity of Δ

*A*/

_{P}*A*≈ 10

_{P}^{−6}.

### 3.1. Closed Loop Timing Analysis

τ ^

_{2}will be updated by the error signal output. Any orbit or clock effects are significantly slower than the ranging system will operate [3], hence a second order Integral, Double Integral (

*II*

^{2}) controller is selected, which maximises low frequency tracking while minimising the sensitivity to high frequency noise. The closed loop system is modelled to predict the effect of phase and interference noise propagating through the timing control loop and an associated noise floor is derived for the timing estimate

τ ^

_{2}. A diagram of the system under consideration is show in Fig. 7.

*ε*

_{ϕQ,c2}(Δ

*τ*

_{2}) =

*βN*/

*τ*(

_{Corr}*R*(Δ

_{PM}*τ*

_{2}+

*T*/2) −

_{P}*R*(Δ

_{PM}*τ*

_{2}−

*T*/2)) ≈ 2

_{P}*βN*/

*τ*Δ

_{Corr}*τ*

_{2}. This linear operation requires the acquisition algorithm to accurately identify the demodulation delay to within |Δ

*τ*

_{2}| <

*T*/2 for this analysis to be valid. From Fig. 6, we can see this assumption is only valid when the timing offset from 0 is accounted for. Within the linear region, the error signal from Eq. (8) can be written as;

_{P}*G*=

*ε*

_{ϕQ,c2}(Δ

*τ*

_{2})/Δ

*τ*

_{2}= 2

*βN*/

*τ*and the

_{Corr}*II*

^{2}controller

*C*

_{II2}(

*s*) =

*k*/

_{I}*s*+ (

*k*

_{I2}/

*s*)

^{2}, with Laplace variable

*s*and controller gains

*k*= 5 × 10

_{I}^{−3}and

*k*

_{I2}= 2.5 × 10

^{−4}selected for a closed loop bandwidth of ≈ 1.4 kHz.

^{−2}ns or ≈ 10 mm rms at 0.5 Hz bandwidth.

## 4. Test bed results

5. G. de Vine, B. Ware, K. McKenzie, R. E. Spero, W. M. Klipstein, and D. A. Shaddock, “Experimental demonstration of time-delay interferometry for the laser interferometer space antenna,” Phys. Rev. Lett. **104**(21), 211103 (2010). [CrossRef] [PubMed]

5. G. de Vine, B. Ware, K. McKenzie, R. E. Spero, W. M. Klipstein, and D. A. Shaddock, “Experimental demonstration of time-delay interferometry for the laser interferometer space antenna,” Phys. Rev. Lett. **104**(21), 211103 (2010). [CrossRef] [PubMed]

**104**(21), 211103 (2010). [CrossRef] [PubMed]

*T*/2 separation between candidate timing points.

_{P}16. W M Folkner, F Hechler, T H Sweetser, M A Vincent, and P L Bender, “LISA orbit selection and stability,” Class. Quantum Grav. **14**, 1405 (1997). [CrossRef]

## 5. Data Modulation and Transmission

**228**, 012045 (2010). [CrossRef]

*τ*

_{2}‖ ≪

*T*/2), all previous analysis remains applicable.

_{P}## 6. Conclusions

## Acknowledgments

## References and links

1. | M. Tinto and J. W. Armstrong, “Cancellation of laser noise in an unequal-arm interferometer detector of gravitational radiation,” Phys. Rev. D |

2. | M. Tinto, M. Vallisneri, and J. W. Armstrong, “Time-delay interferometric ranging for space-borne gravitationalwave detectors,” Phys. Rev. D , |

3. | LISA Frequency control study team, “LISA frequency control white paper,” ESA document LISA-JPL-TN-823 (2009). |

4. | D. A. Shaddock, B. Ware, R. E. Spero, and M. Vallisneri“Postprocessed time-delay interferometry for LISA,” Phys. Rev. D |

5. | G. de Vine, B. Ware, K. McKenzie, R. E. Spero, W. M. Klipstein, and D. A. Shaddock, “Experimental demonstration of time-delay interferometry for the laser interferometer space antenna,” Phys. Rev. Lett. |

6. | S. E. Pollack and R. T. Stebbins, “A demonstration of LISA laser communication.”Class. Quantum Grav. |

7. | E. D. Kaplan, |

8. | W. M. Klipstein, R. E. Spero, and D. A. Shaddock, “Anti-aliasing for LISA photoreceiver signals,” JPL Technical Note, (2006) |

9. | J. J. Esteban, A. F. Garca, J. Eichholz, A. M. Peinado, I. Bykov, G. Heinzel, and K. Danzmann, “Ranging and phase measurement for LISA,” J. Phys.: Conf. Ser. |

10. | D.A. Shaddock, B. Ware, R.E. Spero, K. McKenzie, G de Vine, D. Robison, T. Stecheson, Y. Chong, C. Woodruff, and W. M. Klipstein, “LISA Phasemeter Technology Assessment and Report,” LISA Project Document LIMAS 2009-002 (2009). |

11. | D. A. Shaddock, B. Ware, P. Halverson, R. E. Spero, and W. M. Klipstein, “Overview of the LISA phasemeter,” in AIP Conf. Proc.873, pg. 654–660 (2006). |

12. | Hans-Reiner Schulte “Presentation at LISA Mission Formulation MTR,” (14/15. 4. 2006) |

13. | Vinzenz Wand “Interferometry at low frequencies: Optical phase measurement for LISA and LISA pathfinder”. Ph.D Thesis (4 2007). Leibniz UniversitätHannover |

14. | J. G. Proakis and M. Salehi, |

15. | T. S. Rappaport, |

16. | W M Folkner, F Hechler, T H Sweetser, M A Vincent, and P L Bender, “LISA orbit selection and stability,” Class. Quantum Grav. |

17. | W. Klipstein, P. G. Halverson, R. Peters, R. Cruz, and D. A. Shaddock, “Clock noise removal in LISA,” in AIP Conf. Proc.873, pg. 312–318 (2006). |

**OCIS Codes**

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(280.3400) Remote sensing and sensors : Laser range finder

(350.1270) Other areas of optics : Astronomy and astrophysics

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: August 3, 2010

Revised Manuscript: September 10, 2010

Manuscript Accepted: September 13, 2010

Published: September 15, 2010

**Citation**

Andrew Sutton, Kirk McKenzie, Brent Ware, and Daniel A. Shaddock, "Laser ranging and communications for LISA," Opt. Express **18**, 20759-20773 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-20-20759

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### References

- M. Tinto, and J. W. Armstrong, "Cancellation of laser noise in an unequal-arm interferometer detector of gravitational radiation," Phys. Rev. D Part. Fields 59(10), 102003 (1999). [CrossRef]
- M. Tinto, M. Vallisneri, and J. W. Armstrong, "Time-delay interferometric ranging for space-borne gravitational wave detectors," Phys. Rev. D Part. Fields Gravit. Cosmol. 71(4), 041101 (2005). [CrossRef]
- LISA Frequency control study team, "LISA frequency control white paper," ESA document LISA-JPL-TN-823 (2009).
- D. A. Shaddock, B. Ware, R. E. Spero, and M. Vallisneri, "Postprocessed time-delay interferometry for LISA," Phys. Rev. D Part. Fields Gravit. Cosmol. 70(8), 081101 (2004). [CrossRef]
- G. de Vine, B. Ware, K. McKenzie, R. E. Spero, W. M. Klipstein, and D. A. Shaddock, "Experimental demonstration of time-delay interferometry for the laser interferometer space antenna," Phys. Rev. Lett. 104(21), 211103 (2010). [CrossRef] [PubMed]
- S. E. Pollack, and R. T. Stebbins, "A demonstration of LISA laser communication," Class. Quantum Gravity 23, 4201 (2006). [CrossRef]
- E. D. Kaplan, Understanding GPS Principles and Applications, (Prentice Hall PTR, 1996).
- W. M. Klipstein, R. E. Spero, and D. A. Shaddock, "Anti-aliasing for LISA photoreceiver signals," JPL Technical Note, (2006)
- J. J. Esteban, A. F. Garca, J. Eichholz, A. M. Peinado, I. Bykov, G. Heinzel, and K. Danzmann, "Ranging and phase measurement for LISA," J. Phys.: Conf. Ser. 228, 012045 (2010). [CrossRef]
- D.A. Shaddock, B. Ware, R.E. Spero, K. McKenzie, G. de Vine, D. Robison, T. Stecheson, Y. Chong, C. Woodruff, and W. M. Klipstein, "LISA Phasemeter Technology Assessment and Report," LISA Project Document, LIMAS 2009-002 (2009).
- D. A. Shaddock, B. Ware, P. Halverson, and R. E. Spero, andW. M. Klipstein, "Overview of the LISA phasemeter," in AIP Conf. Proc. 873, pg. 654-660 (2006).
- H.-R. Schulte, Presentation at LISA Mission Formulation MTR 14/15, 4 (2006).
- V. Wand, "Interferometry at low frequencies: Optical phase measurement for LISA and LISA pathfinder," PhD Thesis (4 2007). Leibniz Universität Hannover.
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