To this paper we discuss that the phase readout in low noise laser interferometers can significantly deviate from the underlying optical pathlength difference (OPD). The cross coupling of beam tilt to the interferometric phase readout is compared to the OPD. For such a system it is shown that the amount of tilt to phase readout coupling depends strongly on the involved beams and their parameters, as well as on the detector properties and the precise definition of the phase. The unique single element photodiode phase is therefore compared to three common phase definitions for quadrant diodes. It is shown that neither phase definition globally shows the least amount of cross coupling of angular itDeutsches Zentrum für Luft- und Raumfahrt (DLR)/No 50 OQ 130

Heinzel, Gerhard

Schuster, Sönke

Tröbs, Michael

Wanner, Gudrun

Journal of Physics : Conference Series

Institutionelles Repositorium der Leibniz Universität Hannover

This content has been downloaded from IOPscience. Please scroll down to see the full text.Download details:IP Address: 194.95.157.141This content was downloaded on 08/08/2016 at 12:21Please note that terms and conditions apply.A brief comparison of optical pathlength difference and various definitions for theinterferometric phaseView the table of contents for this issue, or go to the journal homepage for more2015 J. Phys.: Conf. Ser. 610 012043(http://iopscience.iop.org/1742-6596/610/1/012043)Home Search Collections Journals About Contact us My IOPscienceA brief comparison of optical pathlength differenceand various definitions for the interferometric phaseGudrun Wanner1, So¨nke Schuster1, Michael Tro¨bs1 and GerhardHeinzel11 Max-Planck-Institute for Gravitational Physics (Albert-Einstein-Institute) , Callinstr. 38,D-30167 Hannover, GermanyandLeibniz University of Hannover, Hannover, GermanyE-mail: gudrun.wanner@aei.mpg.deAbstract. In this paper we discuss that the phase readout in low noise laser interferometerscan significantly deviate from the underlying optical pathlength difference (OPD). The crosscoupling of beam tilt to the interferometric phase readout is compared to the OPD. For sucha system it is shown that the amount of tilt to phase readout coupling depends strongly onthe involved beams and their parameters, as well as on the detector properties and the precisedefinition of the phase. The unique single element photodiode phase is therefore compared tothree common phase definitions for quadrant diodes. It is shown that neither phase definitionglobally shows the least amount of cross coupling of angular jitter.1. IntroductionA frequent use of laser interferometers is to sense distance variations. Assume a very simpleand perfect Mach-Zehnder interferometer where a mirror in one beam path is rotated aroundthe beam reflection point. For simulations and interpretation the interferometer can then bereduced to the setup shown in Fig. 1. Here, the rays labeled zMB and zp indicate the propagationaxis of measurement and reference beam respectively. In this paper, we discuss this very simplesetup and its resulting phase readout. We highlight the fact that the resulting phase readoutcan deviate significantly from what might be expected from the change in the optical path, evenin absence of standard noise sources such as laser frequency noise, electronic readout noise orthermal noise. This significant deviation originates from the phase- and intensity profiles oflaser beams which cannot be sufficiently represented by plane waves or rays and secondly fromincomplete beam detection and beam clipping on finite photodiodes as well as on the actualdefinition of the interferometric phase in the case of quadrant diodes.photodiodexpivotFigure 1: Effective beam paths in a simple laserinterferometer, if a mirror in the measurement beam pathrotates around the beam reflection point (thus coincidingwith the pivot) by α/2. For α = 0 both beams impingeorthogonally on the center of the photodiode.10th International LISA Symposium (LISAX) IOP PublishingJournal of Physics: Conference Series 610 (2015) 012043 doi:10.1088/1742-6596/610/1/012043Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distributionof this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.Published under licence by IOP Publishing Ltd 12. Interferometric phase vs optical pathlength difference (OPD)The difference between OPD and interferometric phase can be seen for instance by looking atthe tilt to length (TTL) coupling in the given simplified setup.With no medium involved, the OPD can be computed geometrically: it is the differencebetween the distances zMB and zp the measurement and reference beam propagate betweenpivot and diode (see Fig. 1):OPD = zMB − zp =(1cos(α)− 1)zp ≈ α22zp . (1)Assuming rotation angles in the order of 100µrad and a distance zp of about 20 cm, the OPD isin the order of nanometers. For picometer interferometers such as LISA and LISA Pathfinder,it can be seen from this very simple and general example, that the coupling of tilt (i.e. angularjitter) to the OPD can be a significant effect. However, the OPD is not what is measured inthese missions - or generally in interferometers, but the interferometric phase. We define herethe interferometric phase as the phase that can be extracted from the photocurrent resultingfrom the two interfering beams on a photodiode (see [1] for a mathematical description of theinterferometric phase).To show that the interferometric phase can deviate significantly from the OPD, we will showtwo special cases.Example case 1: We assume that both beams are circular fundamental Gaussian beams. Letthe waist of both beams be located halfway between pivot and photodiode and the distancezp between pivot and photodiode is chosen to be twice the Rayleigh range of the beams. Thewavefronts of both beams on the detector surface can then be approximated by a section of asphere which has its center in the pivot of the rotation. This means, that a mirror tilt maps thewavefront onto itself. For small rotations we therefore expect negligible cross coupling from tiltto the interferometric phase. We have shown previously [1] with a numerical simulation, thatfor this special case the coupling is well below pm-level.Example case 2: Assume the same case as before, but let the center of rotation be an arbitrarypoint. If the detector is a large single element detector which does not clip either beam, and ifthe parameters of the interfering beams are matched, the coupling of tilt into the interferometricphase is negligible, as we will show below.We have shown before [2, eq (34)] that the resulting phase is then φ ≈ α2z/(4z0), where z0 isthe Rayleigh range of both beams and z the distance from their waist. In order to comparethe interferometric phase to the OPD, it needs to be converted to a length, which we callthe longitudinal pathlength signal LPS = φ/k. For typical wavenumbers k in the order of2pi/(1µm) ≈ 107/m, tilt angles in the order up to a few hundred µrad, and assuming thatRayleigh range z0 and distance z from the waist are of comparable order, we see that the tilt tolength coupling is negligible:LPS ≈ α2kz4z0≈ (10−4)2107m  1 pm . (2)In both examples the OPD lies in the order of nanometers while the phase readout is neg-ligible. In example case 1 this is due to the very specific choice of beam parameters. Thisshows that the nature of the beams (e.g. circular fundamental Gaussian beams) needs to beaccounted for when predicting the interferometric phase. In example case 2 the vanishing crosscoupling results from the cancellation of two effects [3], which holds for arbitrary but matchedfundamental Gaussian beams that are detected without clipping by a single element photodiode.10th International LISA Symposium (LISAX) IOP PublishingJournal of Physics: Conference Series 610 (2015) 012043 doi:10.1088/1742-6596/610/1/0120432If either beam is clipped for instance by the slits of a quadrant photodiode (QPD) or becausethe diode is insufficiently large (/ 3 times spot size on the detector), a significant coupling canoccur (see Fig. 2(a)). This shows that clipping effects and therefore the detector shape havesignificant impact on the phase readout.Interferometric phase of a quadrant diode Quadrant photodiodes (QPDs) are often used forphase detection in interferometers, because their use allows to locate the centroid as well as therelative angle of the incident beams[4, 5]. Any quadrant diode naturally provides not one but fourphotocurrents, which need to be combined in order to generate one phase signal. Consequently,there are several options to define a phase signal for quadrant diodes. One possibility to describethese signals is via the complex amplitude C of the photocurrent, which we define as the integralover the detector surface of the product of the electric field of the measurement beam andcomplex conjugated electric field of the reference beam and which is given by:C :=∫dSEmE∗r = P¯ c exp(iφ) =: A exp(iφ) . (3)Here, P¯ is the time averaged detected power and c the measured contrast of the interference.The complex amplitude C can be determined experimentally for instance by a single bin discreteFourier transform, as done in LISA Pathfinder [6]. The interferometric phase φ is then simplythe argument of the complex amplitude C. If a large single element detector is assumed, S can beset to the detector plane. If a quadrant detector is assumed, there are four finite integrals to beperformed, resulting in four complex amplitudes Cn and consequently four interferometric phasesφn, and four contrasts cn (n=1,..,4). The sum of the complex amplitudes plus the hypotheticalcomplex amplitudes of the slits would exactly be equal to the complex amplitude of a singleelement detector of identical diameter. Therefore, the definition of a QPD phase closest to thephase of a single element photodiode is:φLPF = arg(4∑n=1Cn)= arg(4∑n=1An exp(iφn)). (4)This phase definition was chosen in LISA Pathfinder (LPF). Alternatively the arithmetic meanof the quadrant phases φn could be computed:φAP =144∑n=1φn =14arg(4∏n=1Cn). (5)Here, the product notation on the right hand side is a better choice for implementation, since itis more robust to phase jumps than the arithmetic mean on the left hand side.While φAP might be an intuitive choice, it has clear disadvantages compared with the LPFdefinition: in the case of poorly centered beams the phases of all quadrants contribute equallyto φAP, even though the phase of one or several quadrants might originate from a very lownumber of photons, while the phases of other quadrants might originate from strong signals.This case is naturally compensated in φLPF, because the summation of the complex amplitudesis effectively a vector addition.The power imbalance problem in the averaged phase can be accounted for, by weighing thequadrant phases for instance by the amplitudes An := P¯ncn:φWAP =A1φ1 +A2φ2 +A3φ3 +A4φ4A1 +A2 +A3 +A4. (6)10th International LISA Symposium (LISAX) IOP PublishingJournal of Physics: Conference Series 610 (2015) 012043 doi:10.1088/1742-6596/610/1/0120433-0.4-0.3-0.2-0.1 0 0.1-400 -300 -200 -100  0  100  200  300  400LPS = Phase/k [nm]Measurement beam tilt angle _ [µrad]   OPD  SEPD   LPF   AP   WAP(a) Example 2.0-1.6-1.2-0.8-0.4 0-400 -300 -200 -100  0  100  200  300  400LPS = Phase/k [nm]Measurement beam tilt angle _ [µrad]   OPD  SEPD   LPF   AP   WAP(b) Example 2.1-0.4-0.3-0.2-0.1 0 0.1-400 -300 -200 -100  0  100  200  300  400LPS = Phase/k [nm]Measurement beam tilt angle _ [µrad]   OPD  SEPD   LPF   AP   WAP(c) Example 2.2-0.4-0.3-0.2-0.1 0-400 -300 -200 -100  0  100  200  300  400LPS = Phase/k [nm]Measurement beam tilt angle _ [µrad]   OPD  SEPD   LPF   AP   WAP(d) Example 2.3Figure 2: Longitudinal pathlength signal (LPS) for the setup of example case 2 with the settingslisted in Tab. 1. These graphs were generated with the numerical software tool IfoCAD[7].Subfig. (a) is the matched beam parameter case described in example case 2. Subfig.(b)-(d)show the same setup for different choices of mismatched beam parameters.Parameter description value Parameter description valuediameter of circular QPD 10 mm measurement beam power 1.2 mWphotodiode gap width 70µm reference beam power 0.7 mWλ: wavelength laser beams 1064 nm zp: distance from pivot to photodiode 5 mmEx 2.0 Ex 2.1 Ex 2.2 Ex 2.3distance RB from waist to QPD [m] -0.4 -0.4 0 0distance MB from waist to QPD [m] -0.4 -0.6 0 0RB Rayleigh range [m] 0.5 3.137 2.5 2.0MB Rayleigh range [m] 0.5 4.124 2.0 2.5Table 1: Parameter settings for Figure 2. RB: reference beam, MB: measurement beamThe φWAP is a first order approximation of φLPF, since the first series expansion of the complexamplitude is C ≈ A+ iAφ:φLPF ≈ arg(4∑n=1(An + iAnφn))≈ arctan(A1φ1 +A2φ2 +A3φ3 +A4φ4A1 +A2 +A3 +A4)(7)≈ A1φ1 +A2φ2 +A3φ3 +A4φ4A1 +A2 +A3 +A4= φWAP . (8)There are of course numerous further options to define the phase readout for quadrant detectors.The definitions φLPF, φAP and φWAP are those we regard as most intuitive and physically closestto the phase readout of a single element detector. A direct comparison of these signals is shown10th International LISA Symposium (LISAX) IOP PublishingJournal of Physics: Conference Series 610 (2015) 012043 doi:10.1088/1742-6596/610/1/0120434in Fig. 2 for variations of example case 2. These numerical simulations were generated withthe software tool IfoCAD[7], the given phase definitions and the settings listed in Tab. 1. Moredetailed information on the IfoCAD algorithms to compute interferometric phases are given in[1, 8]. Subfig. 2(a)) shows the case of matched beam parameters. As discussed above, the LPSof a single element diode (SEPD) does not sense the measurement beam tilt, provided that thedetector is sufficiently large that the beams are fully detected - which is the case in this example.However, all here defined phase signals of a quadrant diode show a significant amount of TTLcross coupling. Since we assume identically shaped SEPDs and QPDs, φLPF equals φSEPD in thehypothetical limit that the gap width goes to zero. Therefore one can say that any differenceoccurring between φSEPD and φLPF results fully from the clipping at the insensitive gaps betweenthe quadrants of the QPD. Therefore, Subfig. 2(a) shows the vanishing TTL coupling on SEPDsas described above as well as the TTL cross coupling on QPDs originating from incompletebeam detection.For mismatched beam parameters, the amount of TTL cross coupling of each phase signal de-pends on the exact setup and the defined beam parameters, as shown in Fig 2(b)-2(d). It can beseen that either of the three QPD LPS signals can show the least amount of TTL cross coupling.In other words, neither signal is globally an optimal definition for a phase readout. The beamparameters in the examples were chosen to highlight this fact, but are otherwise arbitrary. Adetailed comparison of the phase signals is work in progress and will be discussed in a separatepaper.3. ConclusionsFor high performance interferometers, such as LISA Pathfinder, and the Laser RangingInstrument (LRI) of GRACE Follow-On, it is necessary to distinguish carefully the opticalpathlength difference from the interferometric phase, since these parameters could deviatesignificantly, depending on beam parameters and setup. For future high performanceinterferometers, such as in space based gravitational wave detectors like LISA and DECIGO,and in future interferometric geodesy missions beyond GRACE Follow-On, the fact that thecross coupling of angular jitter to phase noise depends on the phase definition could be usedto reduce the resulting phase noise. For this however, dedicated analyses need to be performedwhich also include effects such as the robustness of the phase signal to phase jumps, coupling ofother optical noise types or the robustness to quadrant failure.AcknowledgmentsWe gratefully acknowledge support by Deutsches Zentrum fu¨r Luft- und Raumfahrt (DLR)with funding of the Bundesministerium fu¨r Wirtschaft und Technologie with a decision of theDeutschen Bundestag (DLR project reference No 50 OQ 1301).4. References[1] Wanner G et al 2012 Optics Communications 285 4831–4839[2] Wanner G, Heinzel G 2014 Applied Optics 53, No. 14[3] Schuster S, Wanner G, Tro¨bs M, Heinzel G: “Vanishing tilt-to-length coupling for a singular case in two-beamlaser interferometers with Gaussian beams” submitted to Applied Optics[4] Morrison E et al 1994 Automatic alignment of optical interferometers Appl. Opt. 33, No. 22[5] Morrison E et al 1994 Experimental demonstration of an automatic alignment system for opticalinterferometers Appl. Opt. 33, No. 22[6] Heinzel G et al Class. Quantum Grav. 21, (2004) 581-587[7] http://www.lisa.aei-hannover.de/ifocad/[8] Kochkina E et al ASP Conference Series, 46710th International LISA Symposium (LISAX) IOP PublishingJournal of Physics: Conference Series 610 (2015) 012043 doi:10.1088/1742-6596/610/1/0120435

A brief comparison of optical pathlength difference and various definitions for the interferometric phase

Wanner, G.

Schuster, S.

Tröbs, M.

Heinzel, G.

MPG.PuRe

This content has been downloaded from IOPscience. Please scroll down to see the full text.Download details:IP Address: 194.94.224.254This content was downloaded on 03/06/2015 at 08:48Please note that terms and conditions apply.A brief comparison of optical pathlength difference and various definitions for theinterferometric phaseView the table of contents for this issue, or go to the journal homepage for more2015 J. Phys.: Conf. Ser. 610 012043(http://iopscience.iop.org/1742-6596/610/1/012043)Home Search Collections Journals About Contact us My IOPscienceA brief comparison of optical pathlength differenceand various definitions for the interferometric phaseGudrun Wanner1, So¨nke Schuster1, Michael Tro¨bs1 and GerhardHeinzel11 Max-Planck-Institute for Gravitational Physics (Albert-Einstein-Institute) , Callinstr. 38,D-30167 Hannover, GermanyandLeibniz University of Hannover, Hannover, GermanyE-mail: gudrun.wanner@aei.mpg.deAbstract. In this paper we discuss that the phase readout in low noise laser interferometerscan significantly deviate from the underlying optical pathlength difference (OPD). The crosscoupling of beam tilt to the interferometric phase readout is compared to the OPD. For sucha system it is shown that the amount of tilt to phase readout coupling depends strongly onthe involved beams and their parameters, as well as on the detector properties and the precisedefinition of the phase. The unique single element photodiode phase is therefore compared tothree common phase definitions for quadrant diodes. It is shown that neither phase definitionglobally shows the least amount of cross coupling of angular jitter.1. IntroductionA frequent use of laser interferometers is to sense distance variations. Assume a very simpleand perfect Mach-Zehnder interferometer where a mirror in one beam path is rotated aroundthe beam reflection point. For simulations and interpretation the interferometer can then bereduced to the setup shown in Fig. 1. Here, the rays labeled zMB and zp indicate the propagationaxis of measurement and reference beam respectively. In this paper, we discuss this very simplesetup and its resulting phase readout. We highlight the fact that the resulting phase readoutcan deviate significantly from what might be expected from the change in the optical path, evenin absence of standard noise sources such as laser frequency noise, electronic readout noise orthermal noise. This significant deviation originates from the phase- and intensity profiles oflaser beams which cannot be sufficiently represented by plane waves or rays and secondly fromincomplete beam detection and beam clipping on finite photodiodes as well as on the actualdefinition of the interferometric phase in the case of quadrant diodes.photodiodexpivotFigure 1: Effective beam paths in a simple laserinterferometer, if a mirror in the measurement beam pathrotates around the beam reflection point (thus coincidingwith the pivot) by α/2. For α = 0 both beams impingeorthogonally on the center of the photodiode.10th International LISA Symposium (LISAX) IOP PublishingJournal of Physics: Conference Series 610 (2015) 012043 doi:10.1088/1742-6596/610/1/012043Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distributionof this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.Published under licence by IOP Publishing Ltd 12. Interferometric phase vs optical pathlength difference (OPD)The difference between OPD and interferometric phase can be seen for instance by looking atthe tilt to length (TTL) coupling in the given simplified setup.With no medium involved, the OPD can be computed geometrically: it is the differencebetween the distances zMB and zp the measurement and reference beam propagate betweenpivot and diode (see Fig. 1):OPD = zMB − zp =(1cos(α)− 1)zp ≈ α22zp . (1)Assuming rotation angles in the order of 100µrad and a distance zp of about 20 cm, the OPD isin the order of nanometers. For picometer interferometers such as LISA and LISA Pathfinder,it can be seen from this very simple and general example, that the coupling of tilt (i.e. angularjitter) to the OPD can be a significant effect. However, the OPD is not what is measured inthese missions - or generally in interferometers, but the interferometric phase. We define herethe interferometric phase as the phase that can be extracted from the photocurrent resultingfrom the two interfering beams on a photodiode (see [1] for a mathematical description of theinterferometric phase).To show that the interferometric phase can deviate significantly from the OPD, we will showtwo special cases.Example case 1: We assume that both beams are circular fundamental Gaussian beams. Letthe waist of both beams be located halfway between pivot and photodiode and the distancezp between pivot and photodiode is chosen to be twice the Rayleigh range of the beams. Thewavefronts of both beams on the detector surface can then be approximated by a section of asphere which has its center in the pivot of the rotation. This means, that a mirror tilt maps thewavefront onto itself. For small rotations we therefore expect negligible cross coupling from tiltto the interferometric phase. We have shown previously [1] with a numerical simulation, thatfor this special case the coupling is well below pm-level.Example case 2: Assume the same case as before, but let the center of rotation be an arbitrarypoint. If the detector is a large single element detector which does not clip either beam, and ifthe parameters of the interfering beams are matched, the coupling of tilt into the interferometricphase is negligible, as we will show below.We have shown before [2, eq (34)] that the resulting phase is then φ ≈ α2z/(4z0), where z0 isthe Rayleigh range of both beams and z the distance from their waist. In order to comparethe interferometric phase to the OPD, it needs to be converted to a length, which we callthe longitudinal pathlength signal LPS = φ/k. For typical wavenumbers k in the order of2pi/(1µm) ≈ 107/m, tilt angles in the order up to a few hundred µrad, and assuming thatRayleigh range z0 and distance z from the waist are of comparable order, we see that the tilt tolength coupling is negligible:LPS ≈ α2kz4z0≈ (10−4)2107m  1 pm . (2)In both examples the OPD lies in the order of nanometers while the phase readout is neg-ligible. In example case 1 this is due to the very specific choice of beam parameters. Thisshows that the nature of the beams (e.g. circular fundamental Gaussian beams) needs to beaccounted for when predicting the interferometric phase. In example case 2 the vanishing crosscoupling results from the cancellation of two effects [3], which holds for arbitrary but matchedfundamental Gaussian beams that are detected without clipping by a single element photodiode.10th International LISA Symposium (LISAX) IOP PublishingJournal of Physics: Conference Series 610 (2015) 012043 doi:10.1088/1742-6596/610/1/0120432If either beam is clipped for instance by the slits of a quadrant photodiode (QPD) or becausethe diode is insufficiently large (/ 3 times spot size on the detector), a significant coupling canoccur (see Fig. 2(a)). This shows that clipping effects and therefore the detector shape havesignificant impact on the phase readout.Interferometric phase of a quadrant diode Quadrant photodiodes (QPDs) are often used forphase detection in interferometers, because their use allows to locate the centroid as well as therelative angle of the incident beams[4, 5]. Any quadrant diode naturally provides not one but fourphotocurrents, which need to be combined in order to generate one phase signal. Consequently,there are several options to define a phase signal for quadrant diodes. One possibility to describethese signals is via the complex amplitude C of the photocurrent, which we define as the integralover the detector surface of the product of the electric field of the measurement beam andcomplex conjugated electric field of the reference beam and which is given by:C :=∫dSEmE∗r = P¯ c exp(iφ) =: A exp(iφ) . (3)Here, P¯ is the time averaged detected power and c the measured contrast of the interference.The complex amplitude C can be determined experimentally for instance by a single bin discreteFourier transform, as done in LISA Pathfinder [6]. The interferometric phase φ is then simplythe argument of the complex amplitude C. If a large single element detector is assumed, S can beset to the detector plane. If a quadrant detector is assumed, there are four finite integrals to beperformed, resulting in four complex amplitudes Cn and consequently four interferometric phasesφn, and four contrasts cn (n=1,..,4). The sum of the complex amplitudes plus the hypotheticalcomplex amplitudes of the slits would exactly be equal to the complex amplitude of a singleelement detector of identical diameter. Therefore, the definition of a QPD phase closest to thephase of a single element photodiode is:φLPF = arg(4∑n=1Cn)= arg(4∑n=1An exp(iφn)). (4)This phase definition was chosen in LISA Pathfinder (LPF). Alternatively the arithmetic meanof the quadrant phases φn could be computed:φAP =144∑n=1φn =14arg(4∏n=1Cn). (5)Here, the product notation on the right hand side is a better choice for implementation, since itis more robust to phase jumps than the arithmetic mean on the left hand side.While φAP might be an intuitive choice, it has clear disadvantages compared with the LPFdefinition: in the case of poorly centered beams the phases of all quadrants contribute equallyto φAP, even though the phase of one or several quadrants might originate from a very lownumber of photons, while the phases of other quadrants might originate from strong signals.This case is naturally compensated in φLPF, because the summation of the complex amplitudesis effectively a vector addition.The power imbalance problem in the averaged phase can be accounted for, by weighing thequadrant phases for instance by the amplitudes An := P¯ncn:φWAP =A1φ1 +A2φ2 +A3φ3 +A4φ4A1 +A2 +A3 +A4. (6)10th International LISA Symposium (LISAX) IOP PublishingJournal of Physics: Conference Series 610 (2015) 012043 doi:10.1088/1742-6596/610/1/0120433-0.4-0.3-0.2-0.1 0 0.1-400 -300 -200 -100  0  100  200  300  400LPS = Phase/k [nm]Measurement beam tilt angle _ [µrad]   OPD  SEPD   LPF   AP   WAP(a) Example 2.0-1.6-1.2-0.8-0.4 0-400 -300 -200 -100  0  100  200  300  400LPS = Phase/k [nm]Measurement beam tilt angle _ [µrad]   OPD  SEPD   LPF   AP   WAP(b) Example 2.1-0.4-0.3-0.2-0.1 0 0.1-400 -300 -200 -100  0  100  200  300  400LPS = Phase/k [nm]Measurement beam tilt angle _ [µrad]   OPD  SEPD   LPF   AP   WAP(c) Example 2.2-0.4-0.3-0.2-0.1 0-400 -300 -200 -100  0  100  200  300  400LPS = Phase/k [nm]Measurement beam tilt angle _ [µrad]   OPD  SEPD   LPF   AP   WAP(d) Example 2.3Figure 2: Longitudinal pathlength signal (LPS) for the setup of example case 2 with the settingslisted in Tab. 1. These graphs were generated with the numerical software tool IfoCAD[7].Subfig. (a) is the matched beam parameter case described in example case 2. Subfig.(b)-(d)show the same setup for different choices of mismatched beam parameters.Parameter description value Parameter description valuediameter of circular QPD 10 mm measurement beam power 1.2 mWphotodiode gap width 70µm reference beam power 0.7 mWλ: wavelength laser beams 1064 nm zp: distance from pivot to photodiode 5 mmEx 2.0 Ex 2.1 Ex 2.2 Ex 2.3distance RB from waist to QPD [m] -0.4 -0.4 0 0distance MB from waist to QPD [m] -0.4 -0.6 0 0RB Rayleigh range [m] 0.5 3.137 2.5 2.0MB Rayleigh range [m] 0.5 4.124 2.0 2.5Table 1: Parameter settings for Figure 2. RB: reference beam, MB: measurement beamThe φWAP is a first order approximation of φLPF, since the first series expansion of the complexamplitude is C ≈ A+ iAφ:φLPF ≈ arg(4∑n=1(An + iAnφn))≈ arctan(A1φ1 +A2φ2 +A3φ3 +A4φ4A1 +A2 +A3 +A4)(7)≈ A1φ1 +A2φ2 +A3φ3 +A4φ4A1 +A2 +A3 +A4= φWAP . (8)There are of course numerous further options to define the phase readout for quadrant detectors.The definitions φLPF, φAP and φWAP are those we regard as most intuitive and physically closestto the phase readout of a single element detector. A direct comparison of these signals is shown10th International LISA Symposium (LISAX) IOP PublishingJournal of Physics: Conference Series 610 (2015) 012043 doi:10.1088/1742-6596/610/1/0120434in Fig. 2 for variations of example case 2. These numerical simulations were generated withthe software tool IfoCAD[7], the given phase definitions and the settings listed in Tab. 1. Moredetailed information on the IfoCAD algorithms to compute interferometric phases are given in[1, 8]. Subfig. 2(a)) shows the case of matched beam parameters. As discussed above, the LPSof a single element diode (SEPD) does not sense the measurement beam tilt, provided that thedetector is sufficiently large that the beams are fully detected - which is the case in this example.However, all here defined phase signals of a quadrant diode show a significant amount of TTLcross coupling. Since we assume identically shaped SEPDs and QPDs, φLPF equals φSEPD in thehypothetical limit that the gap width goes to zero. Therefore one can say that any differenceoccurring between φSEPD and φLPF results fully from the clipping at the insensitive gaps betweenthe quadrants of the QPD. Therefore, Subfig. 2(a) shows the vanishing TTL coupling on SEPDsas described above as well as the TTL cross coupling on QPDs originating from incompletebeam detection.For mismatched beam parameters, the amount of TTL cross coupling of each phase signal de-pends on the exact setup and the defined beam parameters, as shown in Fig 2(b)-2(d). It can beseen that either of the three QPD LPS signals can show the least amount of TTL cross coupling.In other words, neither signal is globally an optimal definition for a phase readout. The beamparameters in the examples were chosen to highlight this fact, but are otherwise arbitrary. Adetailed comparison of the phase signals is work in progress and will be discussed in a separatepaper.3. ConclusionsFor high performance interferometers, such as LISA Pathfinder, and the Laser RangingInstrument (LRI) of GRACE Follow-On, it is necessary to distinguish carefully the opticalpathlength difference from the interferometric phase, since these parameters could deviatesignificantly, depending on beam parameters and setup. For future high performanceinterferometers, such as in space based gravitational wave detectors like LISA and DECIGO,and in future interferometric geodesy missions beyond GRACE Follow-On, the fact that thecross coupling of angular jitter to phase noise depends on the phase definition could be usedto reduce the resulting phase noise. For this however, dedicated analyses need to be performedwhich also include effects such as the robustness of the phase signal to phase jumps, coupling ofother optical noise types or the robustness to quadrant failure.AcknowledgmentsWe gratefully acknowledge support by Deutsches Zentrum fu¨r Luft- und Raumfahrt (DLR)with funding of the Bundesministerium fu¨r Wirtschaft und Technologie with a decision of theDeutschen Bundestag (DLR project reference No 50 OQ 1301).4. References[1] Wanner G et al 2012 Optics Communications 285 4831–4839[2] Wanner G, Heinzel G 2014 Applied Optics 53, No. 14[3] Schuster S, Wanner G, Tro¨bs M, Heinzel G: “Vanishing tilt-to-length coupling for a singular case in two-beamlaser interferometers with Gaussian beams” submitted to Applied Optics[4] Morrison E et al 1994 Automatic alignment of optical interferometers Appl. Opt. 33, No. 22[5] Morrison E et al 1994 Experimental demonstration of an automatic alignment system for opticalinterferometers Appl. Opt. 33, No. 22[6] Heinzel G et al Class. Quantum Grav. 21, (2004) 581-587[7] http://www.lisa.aei-hannover.de/ifocad/[8] Kochkina E et al ASP Conference Series, 46710th International LISA Symposium (LISAX) IOP PublishingJournal of Physics: Conference Series 610 (2015) 012043 doi:10.1088/1742-6596/610/1/0120435

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A brief comparison of optical pathlength difference and various definitions for the interferometric phase

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