Critically coupled resonant optical cavities are often used as mode cleaners
in optical systems to improve the signal to noise ratio (SNR) of a signal that
is encoded as an amplitude modulation of a laser beam. Achieving the best SNR
requires maintaining the alignment of the mode cleaner relative to the laser
beam on which the signal is encoded. An automatic alignment system which is
primarily sensitive to the carrier field component of the beam will not, in
general, provide optimal SNR. We present an approach that modifies traditional
dither alignment sensing by applying a large amplitude modulation on the signal
field, thereby producing error signals that are sensitive to the signal
sideband field alignment. When used in conjunction with alignment actuators,
this approach can improve the detected SNR; we demonstrate a factor of 3
improvement in the SNR of a kilometer-scale detector of the Laser
Interferometer Gravitational-wave Observatory. This approach can be generalized
to other types of alignment sensors

Anderson

Breitenbach

Grote

K. Kawabe

Kawabe

Kippenberg

M. Evans

N. Mavalvala

N. Smith-Lefebvre

S. Ballmer

S. Waldman

Sigg

V. Frolov

Optics Letters

English

arXiv

Crossref

Optimal alignment sensing of a readout mode cleaner cavity

Critically coupled resonant optical cavities are often used as mode cleaners in optical systems to improve the signal-to-noise ratio (SNR) of a signal that is encoded as an amplitude modulation of a laser beam. Achieving the best SNR requires maintaining the alignment of the mode cleaner relative to the laser beam on which the signal is encoded. An automatic alignment system that is primarily sensitive to the carrier field component of the beam will not, in general, provide optimal SNR. We present an approach that modifies traditional dither alignment sensing by applying a large amplitude modulation on the signal field, thereby producing error signals that are sensitive to the signal sideband field alignment. When used in conjunction with alignment actuators, this approach can improve the detected SNR; we demonstrate a factor of 3 improvement in the SNR of a kilometer-scale detector of the Laser Interferometer Gravitational-Wave Observatory. This approach can be generalized to other types of alignment sensors.National Science Foundation (U.S.) (Cooperative Agreement PHY-0757058

Mavalvala, Nergis

Kawabe, K.

Frolov, V. V.

Smith, Nicolas de Mateo

Evans, Matthew J

Waldman, Samuel J.

DSpace@MIT

arXiv:1110.4122v1  [physics.optics]  18 Oct 2011Optimal Alignment Sensing of a Readout Mode Cleaner CavityN. Smith-Lefebvre,1 S. Ballmer,2 M. Evans,1 S. Waldman,1 K. Kawabe,3 V. Frolov,4 N. Mavalvala11LIGO Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139.2Syracuse University, Syracuse, New York 12344.3LIGO Hanford Observatory, PO Box 159, Richland, WA 99352-0159.4LIGO Livingston Observatory, PO Box 940, Livingston, LA 70754-0940.Compiled October 20, 2011Critically coupled resonant optical cavities are often used as mode cleaners in optical systems to improvethe signal to noise ratio (SNR) of a signal that is encoded as an amplitude modulation of a laser beam.Achieving the best SNR requires maintaining the alignment of the mode cleaner relative to the laser beamon which the signal is encoded. An automatic alignment system which is primarily sensitive to the carrierfield component of the beam will not, in general, provide optimal SNR. We present an approach that modifiestraditional dither alignment sensing by applying a large amplitude modulation on the signal field, therebyproducing error signals that are sensitive to the signal sideband field alignment. When used in conjunctionwith alignment actuators, this approach can improve the detected SNR; we demonstrate a factor of 3improvement in the SNR of a kilometer-scale detector of the Laser Interferometer Gravitational-wave Observa-tory. This approach can be generalized to other types of alignment sensors. c© 2011 Optical Society of AmericaOCIS codes: 220.1140, 140.4780In any quantum noise limited photodetection processwhere some physical quantity is encoded as an amplitudemodulation on a laser beam, higher-order modes (HOM)of the transverse electromagnetic (TEM) optical fieldpresent a problem. Often the HOM carry no signal infor-mation, but nevertheless contribute to the noise. Opticalsensing systems where HOMs can compromise the signalto noise ratio (SNR) include laser interferometer gravi-tational wave detectors [1–3], cavity optomechanics [4]experiments, and quantum optics experiments [5].A critically coupled optical cavity can attenuate theHOM content of a laser beam. Such cavities also act astemporal filters for reducing laser amplitude and phasenoise for frequencies above the cavity linewidth. Whensuch mode cleaner filter cavities are placed at the output(readout) port of an optical system, deriving error signalsto control the length and alignment of the Output ModeCleaner (OMC) cavity poses a particular challenge. Thesignal-rich optical field may be weak compared to theHOM components.In this article we describe a solution to the problemof aligning the OMC cavity used at the output port ofthe 4 kilometer long laser interferometers of the LaserInterferometer Gravitational-wave Observatory (LIGO)[1]. Though we consider the LIGO optical readout here,our scheme applies to any optical system where a signalis encoded as an amplitude modulation of a laser field,and the spatial mode of the signal-induced modulationsideband differs from that of the DC carrier field. In thecase of LIGO, the signal is a gravitational wave inducedmodulation of the optical field, typically at frequencies10 Hz to 10 kHz shifted from the carrier.Fig. 1 shows a schematic representation of the read-out system we consider. The laser beam incident on theOMC comprises a carrier field and signal sideband fieldswhich must be aligned to the OMC cavity. In the ab-sence of technical noise sources, an automatic alignmentsystem should maintain the cavity alignment that max-imizes the SNR of the detected signal with respect tothe photon shot noise. In general the carrier and side-band fields do not occupy the same spatial modes andtheir transmission through the OMC varies differently asa function of the input alignment.As a simple example, consider a beam reflected froman over-coupled Fabry-Perot cavity (OC). The OC isheld slightly offset from resonance such that periodiclength excitations cause amplitude modulation that canbe detected on a photodetector. Consider the case of astatic misalignment of the input carrier field. The re-flected carrier field will also be misaligned, while the sig-nal field, being generated inside the OC, will be in thesame spatial mode as the OC. Thus the carrier and signalfields will have a relative misalignment. In this example,the cause of the alignment mismatch is a misalignment ofthe input field to the OC. However, in more complicatedcases, such as in the case of a LIGO detector composedof multiple, coupled cavities, the source of extra modalcontent may not be so easily removed.Consider a carrier field with transmitted amplitude cand amplitude modulated upper and lower signal side-bands fields with transmitted amplitudes s. The shotnoise limited SNR of the signal when measured on aphotodetector is proportional toSNR ∝ cs√c2 + 2s2≈ s. (1)Here we have made the following simplifying assump-tions: the carrier field amplitude is much greater thanthat of the signal field, the signal fields are pure am-plitude modulation, and that there is perfect spatialoverlap of the carrier and the sidebands after transmis-sion through the OMC. We see that the alignment sys-1PhotodetectorOMCCarrier BeamSignal BeamDithered Steering MirrorsFig. 1. A signal is encoded by interferometry (or anyother means) as an amplitude modulation of a laserbeam. The beam is then steered by two mirrors which aredithered in angle before passing though a mode cleanercavity. Alignment signals can be derived by demodula-tion of the transmitted photocurrent. The misalignmenthas been grossly exaggerated in the figure.tem that maximizes optical SNR of fields detected af-ter transmission through an OMC simply maximizes theamplitude transmission of the signal field through theOMC. We do not address contributions to readout noisedue to alignment jitter of the beam incident on the OMCin this article.To leading order, typical alignment schemes sense theTEM01 and TEM10 modes of the carrier field measuredin the the OMC eigenmode basis [6]. The presence ofthese modes is interpreted as a misalignment. In thisarticle we will show how to modify a common alignmenttechnique to be sensitive to signal field misalignments,and thus allow one to maximize the SNR of the signal.Consider a single alignment degree of freedom wheremisalignments of the carrier and the signal fields arerepresented by one set of misalignment angles θc andθs. Dither alignment sensing is commonly achieved bymechano-optical modulation of an angular degree of free-dom, e.g by driving steering mirrors that direct the beaminto the OMC, as shown in Fig. 1. Each mirror is ditheredin angle with constant amplitude at a given frequency tomodulate the input pointing into the cavity.Figure 2 shows the frequency content of the field trans-mitted through the OMC. The carrier field is separatedin frequency from two amplitude modulated signal side-band fields at ±fb. The dithering steering mirror pro-duce two sidebands at ±fd, with a field amplitude pro-portional to θc (alternatively, the TEM01 mode ampli-tude [7]). The signal field also has dither sidebands pro-portional to θs. The demodulated signal is made of prod-ucts of field amplitudes separated by the demodulationfrequency.We will use the notation P (f) to represent the OMCtransmitted photocurrent demodulated at the frequencyf . We define the standard dither alignment signal asSstandard = P (fd) ∝ 2c2θc + 4s2θs ≈ 2c2θc, (2)where c is the carrier field amplitude, s is the signalfield amplitude. For the case where the carrier power ismuch greater than the signal field power (c2 ≫ s2), theFig. 2. Arrow diagram showing electric fields after trans-mission through the OMC. In the figure, fd is the angulardither frequency, fb is the beacon modulation frequency.demodulated alignment signal is sensitive only to mis-alignments of the carrier field. In this scheme the errorsignal is nulled at an alignment where the total transmit-ted cavity power is maximized. We will refer to this asthe standard dither scheme.To better sense the alignment of the signal field onemay create a large amplitude modulation of the signal,say by modulating the length of the signal cavity. Thismodulation creates a frequency tag of the spatial modeof the signal field. We refer to this as a “beacon” modu-lation, and fb is the modulation frequency.As in the standard dither scheme, steering mirrors areused to dither the angle of the beam. The desired errorsignal in this scheme is produced by demodulating thetransmitted power at fb + fd or fd − fb, or the sum ofthese signals. We define the beacon alignment signal asSbeacon = P (fd + fb) ∝ 2scθc + 2csθs. (3)When this error signal is nulled, the SNR is improvedin comparison to standard dither, but sensitivity to thecarrier field remains, so the SNR will not be maximized.Because the beacon modulation is small, Sbeacon willhave a higher relative noise level than the standard ditherapproach.The beacon alignment scheme was compared directlyto a standard dither scheme using the OMC cavity atthe output of the 4 km LIGO Interferometer at Hanford(H1). The readout was arranged as in Fig. 1. The angulardither frequencies were between 1.5 and 2.5 kHz, whilethe beacon modulation was a 10 Hz excitation of thedifferential arm length.The differential arm length of the interferometer issensed as an amplitude modulation at the output port.The static carrier at the antisymmetric port is generatedby an offset of the differential arm length. The OMC fil-ters the spatial and frequency content of the beam beforethe beam is split equally and detected on two photode-tectors. The LIGO interferometers are examples of opti-cal systems where a beacon based alignment system per-forms significantly better than a standard dither scheme.Fig. 3 shows the amplitude spectral density of thetransmission of the OMC of the H1 interferometer. Theplot is centered on an injected calibration signal at 1144Hz and the curves are normalized to the peak height ofthis line. The calibration line is surrounded by primarilyshot noise. The dashed green curve (color online) shows2900 950 1000 1050 1100 1150 1200 1250 1300 1350 140010−210−1100frequency (Hz)NormalizedTransmittedASD(Hz−1/2)Beacon alignmentDither alignmentFig. 3. The noise amplitude spectral density of the LIGOH1 detector using two types of alignment schemes. Thecurves are normalized to a calibration line at 1144 Hz.The small line structures are resonances of the suspen-sion wires supporting the mirrors. The beacon schemeshows an SNR improvement of about a factor of 3.the performance using dither sensing, while the solid redcurve shows beacon sensing. The SNR of the calibrationline is improved by about a factor of 3.1 by using a bea-con scheme. A factor of 2.4 is due to an increase of signalstrength and the remaining factor of 1.3 is due to a re-duction in the total transmitted power, and hence theshot noise. We propose that the poor performance of thestandard technique is due to excess HOMs in additionto the TEM01 mode associated with misalignment.It is possible to combine the standard dither signalwith the beacon signal to produce a signal which is sen-sitive to signal field misalignments only. We also makeuse of the DC transmitted power, PDC = P (0) ≈ c2,and the beacon modulation amplitude, P (fb) ≈ 2cs. Anoptimal alignment signal can be constructed as follows:Soptimal = Sbeacon − P (fb)2PDC Sstandard (4)∝ 2csθs + 2scθc − 2cs2c2 (2c2θc)∝ 2csθs.An alternative (though mathematically equivalent)technique would be to just make a dither servo whichmaximizes the optical SNR directly. This may be a moredesirable approach depending on how signals are readout and if digital processing is possible. In this approach,the SNR should be calculated in real time and demodu-lated at the dither frequency to provide the error signal.As above, if the beacon amplitude on the photodetec-tor is P (fb) and the DC power is PDC , the optical SNRisSNR = P (fb)√PDC. (5)Any dither sensing scheme is essentially measuring thepartial derivative of a signal with respect to the dithereddegree of freedom [8]. Thus, a dither servo which maxi-mizes the SNR will measure a signal proportional to∂∂θ(P (fb)√PDC)= 1√PDC(∂P (fb)∂θ− P (fb)2PDC∂PDC∂θ), (6)which is equivalent to (4) up to constant factors. Thisalso shows that this signal is nulled at maximum SNR.In gravitational-wave detectors, the OMC transmis-sion is often held constant by a control system. If fd iswithin the control bandwith, but fb is not, then the car-rier alignment sidebands will be suppressed. Sufficientsuppression makes the beacon scheme approximatelyequivalent to the optimal scheme. This was the config-uration used by both the L1 and H1 interferometers inthe sixth LIGO science run [9].The technique of using a beacon modulation to sensemisalignment of the signal field can be generalized toother types of alignment sensors, for example: split quad-rant photodetectors detecting light picked off the beamentering the OMC or split quadrant detectors in re-flection of the OMC coupled with frequency or lengthdithered sidebands, also known as wavefront sensors.We define the standard carrier alignment signal asGstandard. We also define another signal, demodulatedfb away from the standard signal, as Gbeacon. PDC andP (fb) are defined as in (4). The generalized signal is:Goptimal = Gbeacon − P (fb)2PDC Gstandard. (7)We emphasize that the G signals are derived from thealignment sensor, while the P signals are derived fromthe OMC transmission.To summarize, we have introduced the concept of us-ing a beacon as a strong modulation close to signal fre-quencies to generate alignment signals that lead to in-creased SNR compared to the standard dither scheme.The beacon scheme demonstrates a factor of 3 improve-ment in SNR for the LIGO H1 detector. Finally, we pro-pose a detection scheme to give optimum SNR when usedto align an OMC or filter cavity.We gratefully acknowledge illuminating discussionswith Tobin Fricke, Jeff Kissel and Daniel Sigg. LIGO wasconstructed by the California Institute of Technologyand Massachusetts Institute of Technology with fund-ing from the National Science Foundation and operatesunder cooperative agreement PHY-0757058. This paperhas LIGO Document Number ligo-p1100025.References1. The LIGO Scientific Collaboration, Rep. Prog. Phys. 72,076901 (2009).2. The Virgo Collaboration, CQG 25, 184001 (2008).3. H. Grote and The LIGO Scientific Collaboration, CQG27, 084003 (2010).4. T. J. Kippenberg and K. J. Vahala, Science 321, 1172(2008).5. G. Breitenbach, S. Schiller, and J. Mlynek, Nature 387,471 (1997).6. D. Z. Anderson, Appl. Opt. 23, 2944 (1984).7. D. Sigg and N. Mavalvala, J. Opt. Soc. Am. A 17, 1642(2000).8. K. Kawabe, N. Mio, and K. Tsubono, Appl. Opt. 33,5498 (1994).9. T. Fricke, N. Smith-Lefebvre, R. Abbott, R. Adhikari,K. Dooley, M. Evans, P. Fritschel, V. Frolov, K. Kawabe,and S. Waldman, “DC Readout Experiment in En-hanced LIGO”, Preprint (2011).3

Optimal Alignment Sensing of a Readout Mode Cleaner Cavity

Optimal Alignment Sensing of a Readout Mode Cleaner Cavity

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