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SENSITIVITY ENHANCEMENT IN FUTURE INTERFEROMETRIC GRAVITATIONAL WAVE DETECTORS By STACY M. WISE A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2006 Copyright 2006 by Stacy M. Wise I dedicate this work to my family. ACKNOWLEDGMENTS Foremost I would like to thank my family for their unconditional support. I would like to thank Dr. David Tanner and Dr. David Reitze for kindness and numerous helpful discussions. Thanks go to Dr. Guido Mueller, for imaginative ideas, helpful explanations, and for regular and sound figurative posteriorkicking, as well as egodeflating. Thanks also go to Dr. Volker Quetschke, for cheerfully answering probably thousands of questions, and to Dr. Bernard Whiting, for very thorough reviewing of my calculations and repeatedly showing me the error of my mathematical vi, (and some very excellent contra dancing). For friendship that was worth every bit of occasional distraction from the academic task at hand, I must thank Joe Gleason, Amruta Deshpande, and Malik Rakhmanov. Some distractions I've had even longer Rachel Haimowitz, Ahmed Rashed, and James "Tv.. Ii Pence helped me get here. I could .,li.v ,il count on finding Dea's HalfFast Jam folks or 50 Miles of Elbow Room ready to pl i, soulful tunes on fiddle and ,,0io on a we!.di. night, fulfilling a musical need. TABLE OF CONTENTS page ACKNOWLEDGMENTS ................... ...... iv LIST OF TABLES ...................... ......... vii LIST OF FIGURES ................... ......... viii A B ST R A CT . . . . . . . . . x CHAPTER 1 INTRODUCTION TO GRAVITATIONAL WAVE EXPERIMENTS 1 1.1 General Relativity ........................... 1 1.2 Direct Observation of Gravitational Waves ............. 3 1.2.1 Resonant Bar Detectors .......... .......... 4 1.2.2 Interferometric Detectors ................... 5 1.3 Interferometer Noise .......... ............... 10 1.3.1 Seismic Noise .......... ....... ....... 10 1.3.2 Thermal Noise ........... ............. 10 1.3.3 Quantum Noise ............... .. .. .. 11 1.4 ThirdGeneration Detectors and Beyond . . ..... 12 1.5 In This Work ............... .......... .. 14 2 PARALLEL PHASE MODULATION FOR ADVANCED LIGO ..... 15 2.1 Interferometer Control .................. .... .. 15 2.1.1 Phase Modulation .................. .. 16 2.1.2 PoundDreverHall Cavity Locking . . ..... 17 2.1.3 Advanced LIGO .................. ... .. 19 2.1.4 Electrooptic Modulators .. ................. .. 21 2.2 Parallel Phase Modulation Noise .................. .. 24 2.2.1 Mirror Motions .................. .. .. 25 2.2.2 Fluctuations at Nonzero Frequencies . . 33 2.3 MachZehnder Parallel Phase Modulation Prototype . ... 34 3 NEARFIELD EFFECT HEAT TRANSFER ENHANCEMENT ..... 42 3.1 Nearfield Theory .......... . . ... 42 3.2 Fluctuational Electrodynamics Theory . . ..... 44 3.2.1 Green's Function Method .... . . 45 3.2.2 Fluctuation Dissipation Theorem. . . ...... 47 3.3 Numerical Simulations of Evanescent Coupling .. ......... 3.3.1 Effect of Dielectric Function .. .............. 3.3.2 Effect of L ,i ,. I M edia .. ................. 3.4 Noise Coupling Due to Fluctational Electrodynamics .... 3.4.1 Van der Waals and Casimir Forces .. ............ 3.5 Implementation in an Interferometer .. ............. 3.5.1 Configurations .. .. .. ... .. .. .. ... .. .. 3.5.2 Stability Requirements for Cold Mass .. .......... 3.6 Experimental Proof .. ..................... 4 WHITELIGHT INTERFEROMETRY .. ................ 4.1 The Sensitivity Theorem Quandary .. .............. 4.2 The LinewidthEnhanced Cavity .. ................ 4.2.1 The Nature of the Gainbandwidth Dilemma........ 4.2.2 "Whitel;III Cavities 4.3 Gravitational Wave Response in the 4.4 Revised Theory .. ......... 4.5 Planewave Treatment of a Grating 4.6 Tests of Corrected Theory ..... 4.6.1 Experimental Setup ..... 4.6.2 Experimental Results . 4.6.3 AOM s .. ......... 4.7 Alternative Solutions .. ...... 4.8 If Alternatives Succeed ....... Time Domain. Compressor . 5 CONCLU SION . . . . . . . . APPENDIX A OPTICAL TRANSFER FUNCTIONS .. ............... A. 1 FabryPerot Cavity .. ..................... A.2 ThreeMirror Coupled Cavity .. ................ B HEAT TRANSFER THROUGH FIBERS .. .............. C CIRCUIT DIAGRAM .. ....................... REFEREN CES . . . . . . . . . BIOGRAPHICAL SKETCH .......... ............... III LIST OF TABLES Table page 21 Advanced LIGO degree of freedom in terms of length in Figure 23 .20 22 Control matrix for 40m prototype without secondorder sidebands [20] 21 23 Control matrix for 40m prototype with secondorder sidebands [20] 22 C1 Values of elements used in the MZ feedback circuit . .... 85 LIST OF FIGURES Figure page 11 The effect of a pluspolarization gravitational wave propagating into page on a string of test masses .................. 4 12 The effect of a crosspolarization gravitational wave propagating into page on a string of test masses .................. 4 13 The powerrecycled, cavityenhanced optical configuration . 7 14 The dualrecycled optical configuration ............... 8 15 Noise budget for Advanced LIGO ................ 13 21 Schematic for PoundDreverHall FabryPerot cavity locking ..... .17 22 FabryPerot intracavity intensity and PoundDreverHall error signal. 19 23 Lengths in Advanced LIGO that require feedback control ...... ..20 24 Serial phase modulation of IFO input light .... . 23 25 Parallel phase modulation of IFO input light ..... . 23 26 Possible noise motion in the MachZehnder optics and resultant noise on Advanced LIGO input light. C: carrier, SB: sideband. . 26 27 Phasor diagram of MachZehnder commonmode fluctuation's effect on recombined carrier and one pair of sidebands. . .... 26 28 Phasor diagram of MachZehnder differentialmode fluctuation's effect on recombined carrier and one pair of sidebands. . .... 28 29 Ratio of frequency noise to relative phase noise transfer functions to FabryPerot common mode error signal . . ....... 30 210 Permissible residual differential displacement of MachZehnder arm lengths. .................. .............. .. 31 211 Layout of the prototype MZ phase modulation scheme . ... 35 212 Spectrum of light transmitted by parallelphase modulating MachZehnder interferometer .................. ........... .. 36 213 Expanded view of MZ output spectrum. Mixed sidebands are absent the 43.5 MHz peaks are due to higherorder spatial modes. . 37 214 Comparison of MZ errorpoint noise with required stability for freerunning and closedloop operation .................. .. 40 215 Outofloop measurement of residual MZ differential displacement noise 40 216 Attenuation transfer function of MZ feedback electronics . 41 31 Comparison of quantummechanical sensitivity limit to test mass internal thermal noise limit for Advanced LIGO [22] . 43 32 Geometry of dielectic media for nearfield effect calculation two semiinfinite slabs separated by a vacuum gap with width d. ......... ..45 33 Power flux per unit area as a function of vacuum gap separating two semiinfinite chromium masses at 40 and 10 K . . ... 51 34 Power flux per unit area as a function of vacuum gap separating two semiinfinite doped silicon masses at 40 and 10 K for various dopant concentrations .................. .......... .. 52 35 Power flux per unit area as a function of doped silicon metalicity at a 1.4 micron separation. .................. ..... 53 36 Possible configurations for evanescent cooling of a mirror ...... ..54 41 Gratingenhanced cavity .................. ..... .. 60 42 Comparison of roundtrip phases for gratingenhanced and standard FabryPerot cavities .................. ....... .. 61 43 Linewidths of gratingenhanced and standard cavities . ... 62 44 Whitelight cavity gravitational wave response . ..... 68 45 A pair of identical, parallel, facetoface diffraction gratings and two mirrors ( ll and M2) form a resonant cavity. .......... ..69 46 Experimental design to test grating phase . . ....... 72 47 Measured and theoretical data for motion parallel to the grating face 75 48 Mislignment angle as determined from fits of measured data . 76 A1 Electric fields in a FabryPerot cavity ................. .. 82 A2 Electric fields in a 3mirror coupled FabryPerot cavity . ... 82 C1 Diagram of analog ciruit used for MZ feedback control . ... 86 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy SENSITIVITY ENHANCEMENT IN FUTURE INTERFEROMETRIC GRAVITATIONAL WAVE DETECTORS By Stacy M. Wise li ,v 2006 C'! In': David B. Tanner Major Department: Physics Mankind is poised to directly detect gravitational waves for the first time. To improve event rates of sources these detectors will be able to measure, we will need to increase the signaltonoise ratio of the current first generation instruments in upgraded versions. Three techniques for improving interferometer sensitivity to gravitationalwave strains are explored in this work. The first is a parallel phase modulation technique for adding two radiofrequency sidebands to the laser input of an interferometric detector like LIGO without sideband crossproducts. This allows for a diagonal control matrix in the interferometer topology planned for Advanced LIGO. The stability requirements for a MachZehnder interferometer with an electrooptic phase modulator in each arm is derived. We construct a prototype parallel phase modulator to test the viability of these requirements, with positive result. The second method is cryogenic cooling of optics within an interferometer by evanescent field coupling. We quantify the enhanced heattransfer with fluctuational electrodynamics theory for a variety of dielectric materials and evaluate the effect of stratified media on thermal coupling. Noise force couplings and the implications for an advanced interferometer are discussed. In the last chapter we investigate whether a diffractive element placed in a FabryPerot cavity can increase the cavity bandwidth without loss of peak signal buildup. We study the interaction of such an optical resonator with a gravitational wave. A diffraction grating technique is tested and ultimately proven unable to enhance the bandwidth of gravitational wave detectors. CHAPTER 1 INTRODUCTION TO GRAVITATIONAL WAVE EXPERIMENTS 1.1 General Relativity Just when the world was beginning to think nothing else remained for physicists but to measure the physical constants of the universe to the next decimal place of precision, the early 20th century brought the wonders of quantum mechanics and relativity. In 1916 Albert Einstein revolutionized our understanding of the universe with his general theory of relativity (GR) [1]. Initially seeking a frameindependent description of the laws of motion, he came to the startling conclusion that matter and energy warp space and time. NonEuclidean geometries arise in this fourdimensional spacetime "< ,i n II i, which, in turn, directs the motion of mass/energy. In this way, we see gravitation as a property of spacetime itself. When velocities are small compared to the speed of light, general relativity's consequences are not much different from Issac Newton's description of gravity. In regions of strong curvature or high velocity, however, general relativity is markedly and measurably different. The very first test of the new hypothesis was the bending of light by our Sun's gravitational field. Tod i, GR is part of everydi life, used for example in the timing of the Global Positioning System satellites. Although most predictions of Einstein's theory have now been satisfactorily verified by experiment, direct confirmation of one important prediction remains outstandingthat of gravitational waves. One of the primary motivations for pursuing theoretical alternatives to the pre20th century theory was that Newton's gravity allowed for instantaneous communication. If a mass moved to a new location in space, the effect on a test mass would be immediate, regardless of the distance separating these masses. Information encoded in the motion of an object on one side of the Milky Way Galaxy could theoretically be read (assuming a sensitive enough instrument exists, an issue we'll get to later) on the other side with no time delay. Einstein insists in the special theory of relativity that no information can travel faster than the speed of light [1]. Like waves spreading outward from a hand sweeping across the surface of a pond, changes in spacetime curvature from a moving mass propagate out from their source at the finite, although fleet, speed of light, about 3 x 108 meters per second. Just as accelerating charges produce electromagnetic waves, accelerating mass is needed for gravitational waves (GW). It is a much weaker force than electromagnetism. For example, the gravitational force's attraction between two electrons is 1042 times weaker than the electromagnetic repulsion. Moreover, whereas the dominant component of electromagnetic radiation is the dipole term, the absence of a negative mass corresponding to negative charge means the lowest order and dominant term in gravitational waves is the quadrupole. We will have waves whenever the second time derivative of a mass distribution's quadrupole moment Q is nonzero. An oftcited example [2] of gravitational wave generation in a laboratory demonstrates the difficulty in detecting (or creating) these small ripples. In some enormous diabolical research facility, scientists could create their own waves by rotating a M 500,000 kg iron rod about an axis perdendicular to its cylindrical symmetry axis at the maximum possible angular velocity. This max = 28 rad/s is determined by the tensile strength of iron. The bar is L = 20 m long. The gravitational wave power radiated is 2 M2L4G6C P 2.35 W 45 c" (where G is the gravitational constant and c the speed of light), which is a small fraction of the rotational energy ML22 /3 = 5.23 x 1010 J. Gravitational perturbation of spacetime geometry can be interpreted as a relative length change, or strain h, of space. In the nearNewtonian limit (i.e., gravitational waves are a small perturbation to otherwise flat spacetime), the strain is an estimated GQ h C4r where r the distance from source to observer. This strain is very small. For two solarmass neutron stars in the Virgo Cluster that are nearly in the merger phase of their coalescence, h is of order 1021. Einstein himself was very skeptical about whether humans could ever detect these waves directly. The brightest sources we expect to see in the universe are compact and massive objects such as coalescing black holes and coalescing neutron stars, and may be moving at relativistic velocities, as in the case of supernova explosions. Although the waves can carry enormous amounts of energy, their effect on matter is very weak because spacetime is stiff and not very responsive. We do have some indirect evidence for gravitational waves. The 1993 Nobel Prize for Physics went to Hulse and Taylor who carefully timed radio signals from a pulsar and found the star was losing energy that matched the predicted energy radiated away in gravitational waves [3]. 1.2 Direct Observation of Gravitational Waves The way to see gravitational waves directly is to measure the resultant tidal forces using test masses free to respond to perturbation of spacetime curvature. Far away from even the most ferocious source, the wave will produce only a small deviation in the geometry of spacetime. The geometry is described by the "metric" g,, such that the interval ds between two points in fourdimensional spacetime is d)2 g,,,dx dxV. SOCOO o Ir/4Q ir/2Q 3ir/4Q ir/Q time Figure 1 1. The effect of a pluspolarization gravitational wave propagating into page on a string of test masses 00000 I I I I I i 0 rc/4Q 7c/2l 371/4Q 7C/l time Figure 12. The effect of a crosspolarization gravitational wave propagating into page on a string of test masses Let us assume the wave is propagating through an otherwise flat Minkowski spacetime geometry. In the transverse and traceless gauge the perturbed metric due to a wave travelling in the zdirection is 1 0 0 0 0 1 + h+ hx 0 gp = 0 h 1 h+ 0 0 0 0 1 Gravitational waves have transverse force fields. The subscripts on the strain components h+ and hx denote the two possible polarizations of the wave. If it has the h+ pIIl" polarization, the wave's effect on a ring of test masses would be as in Figure 11. The linearly independent "( .. polarization hx has the effect shown in Figure 12, with the displacements rotated 450 from the other polarization state. 1.2.1 Resonant Bar Detectors Joseph Weber built the first gravitational wave detector in 1960 [4]. His instrument was a large (1 ton) aluminum cylinder with piezoelectric transducers to read out relative acceleration of the end surfaces. Gravitational waves excite oscillatory modes in this resonant bar detector when they contain the proper frequencies, much like a hammer striking a massive, puretoned bell. Gravitational signals make very small hammers, however, so resonant bar experiments todi use clever multistage transducers to register small bar movements and cryogenic cooling to reduce noise. Although Weber claimed a GW detection, subsequent experiments cast doubt on its authenticity, and we are still waiting tod i' for the first direct observation of gravitational waves. Modern resonant mass GW detectors include ALLEGRO at LSU, USA, EXPLORER at CERN, the Italian AURIGA in Legnaro and NAUTILUS in Frascati, and NIOBE in Perth, Australia [5]. There are also spheres, which work in the same way but can with limited precision locate the source in the sky. These include Brazil's Mario Schenberg [6] project and MiniGRAIL in the Netherlands [7]. 1.2.2 Interferometric Detectors Weber's instrument could have seen gravitationallyinduced strains as small as 1016/vHz; current bars can reach 1021/v/Hz, but the bandwidth is very limited for any such experiment. There is alv, a compromise between sensitivity and bandwidth because a high quality factor Q,,u bar resonance will be excited by a smaller signal, but will have less bandwidth, since bandwidth Af w fo/Qul. We want to be able to register a broader span of frequencies, both for sensitivity to a large range of sources, and high fidelity for a given signal. Interferometric detectors can have much greater bandwidth than bars. In them we use light to measure the relative acceleration of geodesictracking free masses. The quadrupole nature of gravitational waves makes a Michelson interferometer a natural choice for a detector. The mirrors at the end of the interferometer arms serve as test masses, and we measure their position fluctuations with a laser. Such a device automatically gains a factor of two in responsivity over the bars, as a gravitational wave compresses one arm while stretching the orthogonal arm, doubling the effect. An equalarmlength Michelson interferometer also has the same sensitivity to signals of all frequencies1 We gain greater advantage, however, if we sacrifice sensitivity in some frequency regions in favor of others. For this reason, the United States' interferometric effort, the Laser Interferometer Gravitationalwave Observatory or LIGO [8], as well the FrancoItalian VIRGO [9], Japanese TAMA [10], and BritishGerman GEO600 [11] have all altered their frequency response from that of the simple Michelson interferometer. The first three include FabryPerot optical cavities in the Michelson arms. The cavities have the effect of increasing the interaction time, and thus the strength, of gravitational signals within the cavity linewidth, while degrading performance outside the bandwidth. To counter the miniscule nature of GW strains in this part of the universe, we should make the length of the arms as great as possible. Earth's curvature limits how far we can take thisa mirror too far away will be below the optical horizon2 and the suspended mirrors pointing toward the center of the planet will not hang parallel enough to allow us to build an optical cavity. This and budgetary concerns for building platforms and such set a practical limit of several kilometers. LIGO has 4 kmlong arms, VIRGO near Pisa has 3 km, GEO near Hannover is 600 m long, and TAMA, in Tokyo, has 300 m cavities. Light is stored in the arm cavities to integrate the signal, but if we integrate for longer than half the GW period, the effect will begin to average out, so there is a limit to this approacha limit to the reflectivity of the cavitiy mirrors for a given length. To reduce the sensitivity to noise in the laser's frequency and amplitude 1 for strains that alter the arm lengths much less than the laser wavelength, which is true for all known sources of gravitational waves 2 even with light bending in the gravitational field of the earth SETM SI ITM BS PR output Figure 13. The powerrecycled, cavityenhanced optical configuration we tend to keep these interferometers' outputs "d1 il: in the rest position. That is, microscopic (laserwavelengthorder) length differences make the light exiting the antisymmetric port interfere destructively. Therefore most all the light will be reflected back toward the laser. To waste not, and to want not, LIGO, VIRGO, GEO, and TAMA add a partially reflective mirror between the laser and Michelson beamsplitter to the configuration to reflect this light back into the detector. It is positioned so that laser light reflected back toward the instrument will be in phase with fresh laser light passing through. We call this powerrecycling. GEO600 sits outside Hannover, Germany. It does not have FabryPerot cavities, but it improves signal integration time by adding a mirror to the interferometer output that preferentially reflects a certain band of signal frequencies back into the instrument for another round with the gravitational perturbation. This scheme is known as signal recycling. Basic powerrecycledcavityenhanced and signalrecycled Michelson interferometer configurations are shown in Figures 13 and 14. This work will focus on LIGO, present and future. TM Laser7^ LBS PR SSR output Figure 14. The dualre' vi 1, optical configuration The LIGO project currently consists of three interferometric detectors at two sites. A powerrecycled Michelson with 4 km FabryPerot arm cavities sits in the swampy pinewoods of Livingston, LA. About 2000 miles (or 10 millilightseconds) away there is another 4 km and a colocated 2 km device of the same design at the Hanford Reservation in eastern Washington. At the time of this writing, LIGO is in the middle of its fifth "Science Run," S5, a period of coincident datataking in hopes of detecting the first gravitational wave or, at least, setting an c1 ii i 11 y relevant upper limit on event rates for some sources. On the best d i it is reaching its design sensitivity but for an excess of 1.5 to 3 times the predicted noise in the 40 to 150 Hz band, a critical region. S5 will last about one and one half years, with a goal of a 711' triple coincidence duty cycle and the ability to see 1.5 solar mass neutron star inspirals out to 10 Mpc for the 4000 meter instruments and 5 Mpc for the 2k. LIGO is scheduled to acquire 1 year of data in triple coincidence to be searched for gravitational signals at the design sensitivity before we upgrade it to the next incarnation. Research for this Advanced LIGO has been concurrent with LIGO commissioning and science. The basic thrust of Advanced LIGO design is, unsurprisingly, greater sensitivity [12], so as to make a GW detection practically guaranteed. The event rate for signals that would be visible to initial LIGO is unfortunately quite low. For example, some reasonable and popular cosmological models predict binary neutron star coalescences with strains visible to initial LIGO to occur at a rate of up to 1 every 1.5 years [13]. Advanced LIGO will have ten times the sensitivity, and thus a view of a thousand times the volume of outer space. The upgrade will be able to peer into the rich Virgo cluster. This brings the estimated event rates for Advanced LIGO to 20 to 1000 per year [13]. According to Jun Mizuno's interferometer sensitivity theorem [14], the total integrated sensitivity of a detector over all signal frequencies is a function only of the stored light power. To improve chances of a detection, one can increase the laser power or cavity storage times, or tailor the frequency response to favor regions where either cosmological signal strengths are greatest or noise interference is least, while the total sensitivity remains unchanged. Advanced LIGO gets a little of both improvements. We first increase the input laser power 20 times, from 6 to 120 Watts. It is difficult to produce a clean singlespatial mode laser with greater than about 200 W at present, and beyond this power thermal lensing effects in our optics make increases more problematic than advantageous. The upgrade will also use a signalrecycling mirror to move the peak signal gain frequency of the arm cavities from 0 Hz (DC) to about 300 Hz. Although anticipated signals become greater as frequency decreases, a roomtemperature groundbased instrument is subject to thermal and seismic noise disturbances that destroy lowfrequency sensitivity anyway. Therefore we place our greatest hopes for detection as close as possible to this noise i. 1". 1.3 Interferometer Noise The physics of a gravitational waveinterferometer interaction is relatively simple. The most challenging task is to create a device quiet enough to measure these miniscule effects. Some have compared gravitational wave searches to trying to detect a boat launched from the coast of Africa by looking at the waves on Florida's shore.3 A daunting task, indeed. There is so much extraneous din on the earth. The sensitivitylimiting noise will be of three basic types: seismic, thermal, and quantum noise. 1.3.1 Seismic Noise Recall our estimated neutron star inspiral strain estimate of h = 1021. In initial LIGO this would cause a 10s8 m apparent test mass displacementless than the width of an atomic nucleus. Throughout most of the spectrum in which we are trying to make measurements, the earth itself is moving far more than a GW strain produces. To decrease sensitivity to ground motion noise, and because we must in some way keep the test masses from falling to earth, the mirrors hang as pendula from seismic isolation stacks. Although constrained in the vertical direction, they are free to respond to local spacetime curvature in the longitudinal degree of freedom. Above their resonance frequencies wr, the test masses' response to perturbations of the suspension points decreases as (w,/w)2. With n cascaded harmonic oscillators, we get (w/Uy )2" noise attenuation. Even so, our best efforts push the resonance frequency of the pendulum/isolation stack to about 1 Hz, so seismic noise will set the lowfrequency sensitivity limit. 1.3.2 Thermal Noise If the masses whose positions we measure so carefully are at a finite temperature, their constituent molecules will fidget in thermal excitation. This motion can excite 3 Thanks, Malik Rakhmanov, for the imagery. resonances in the mass or the suspensions that move the mirror surface along the optical axis, mimicking the motion due to a gravitational wave. In the pendulum suspension wires, we attempt to make these resonances as narrow in frequency as possible, to concentrate all the noise motion into a band that we will have to sacrifice. The masses are also made of material with highquality mechanical resonance. Additionally, we can tune the excluded frequencies out of the GW measurement band by changing the test mass aspect ratio. At present, none of the interferometric detectors is cryogenically cooled. Another thermal noise source occurs when low thermal conductivity and different laser intensity on different parts of the test mass cause localized refractive index and radius of curvature changes in the coating. The microscopically bubbling test mass surface is exhibiting thermorefractive noise. Thermoelastic noise [15] results from local temperature fluctuations causing localized thermal expansion. Sampling a large part of the surface with a wide laser beam makes the effect average out somewhat. One can also fight thermoelastic and thermorefractive noise with novel laser intensity profiles, perhaps using a beam with a flat intensity profile instead of Gaussian shape [16]. 1.3.3 Quantum Noise In interferometry there will be noise in a measurement simply due to the fact that light, despite its wavelike behavior, is also composed of discrete particles [17]. Light of a given intensity I is associated with an average number N of photons, with an uncer' li:v in that number due to the fact that light particles obey Poissonian statistics: AN  N Clearly, the greater the light intensity, the smaller the uncertainty in its measurement. To see gravitational waves on the interferometer output we must store sufficient power in the arms that photoncounting statistical errors are smaller than the change due to a wave. Because with arm cavities LIGO has frc qu',i, dependent light intensity, the shot noise also has frcE iiqu, dependent behavior. As Einstein noted, photons carry momentum. When N AN photons per second strike LIGO's suspended mirrors, they create a fluctuating force, and thus displacement, known as radiation pressure noise. As the radiation pressure noise in the arms is uncorrelated, it will appear in the interferometer gravitational wave channel output. Radiation pressure noise increases with laser power. We can fight the effect by using heavy test massesinitial LIGO has 10 kg fused silica arm cavity mirrors, and the upgrade will use 40 kg mirrors. Lowering the laser power is not the preferred solution because shot noise scales inversely with intensity. In a machine such as LIGO the quadrature sum of radiation pressure and shot noise defines a I ,ir, .ird quantum limit" (SQL) of sensitivity for a given laser power. With the addition of signalrecycling it has been noted that these quantum noise terms can become correlated, leading to better than SQL sensitivity at some frequencies [18]. All other noise sources must contribute less spurious GWsignal than these main three. A combination of interferometer imperfections (inevitable) and noisy input light can cause additional noise to appear on the detector output. This occurs when some common mode signal "leaks" to the antisymmetric port or when control signals used to stabilize the instrument are noisy. A typical ii. bl, I plot for a LIGOlike interferometer is shown in Figure 15 [22]. 1.4 ThirdGeneration Detectors and Beyond After Advanced LIGO, innovations and progress will continue to reach farther out into the universe. LISA is the Laser Interferometer Space Antenna, a gigameterlong interferometer that will fly in formation along the earth's solar orbit. It will feature three laserlinked test masses surrounded by spacecraft that shield the test mass from solar wind and other sources of drag. Without terrestrial 10 yquant Int thermal Susp thermal 10 Residual Gas Total nolse 1021 1022 ..............   102 10 10 10 10 10 f/Hz Figure 15. Noise budget for Advanced LIGO seismic constraints, LISA's most sensitive bandwidth will be in the milliHertz range. This project presents formidable engineering challenges, but research on the myriad subsystems that will make a space antenna is well underway. LISA is currently scheduled to launch in 2012. On earth, we expect future interferometric detectors will address seismic, thermal and shot noise in imaginative v iv. It will be difficult to make a test mass both cryogenically cooled and seismically isolated, but if we hope to see farther than Advanced LIGO we must find a way to lessen test mass internal thermal noise. Another idea for the future is to beat thermal lensing on the optics with an allreflective design. We would then be able to increase the input laser power (reducing shot noise) manyfold. Some have si .. I. 1 using diffraction gratings as beamsplitters and cavity inputcouplers. To alleviate the effect of seismic noise we could put detectors underground, and somewhat escape seismic harassment from human activity such as planes, trains, and logging operations, as well as natural phenomena like wind and earthquakes. Because most of a seismic wave travels along earth's surface, gravity gradient fluctuations are much less in subterranean spaces. TAMA's upgrades, as well as the panEuropean EURO project, will be underground. Looking even farther ahead (and to more distant GW sources), detector design becomes even more fantastic. Can we find some trick or innovation that defeats the rule that bandwidth must be sacrificed for sensitivity, and vice versa? 1.5 In This Work Each of these strategies improves a detector by amplifying the size of the signal relative to noise sources presentthe signaltonoise ratio, or SNR. This dissertation presents three techniques aimed at improving the SNR of interferometric detectors beyond initial LIGO and its siblings. The first is an improvement for Advanced LIGO's control system. A parallel phase modulation scheme is needed to enjoy a diagonal control matrix. The second technique also reduces noise, thermal noise, by cooling test masses with nearfield electromagnetic coupling of hot mirror to cold. The last proposal discussed in this work sirl 1  that signal strength can be improved without bandwidth loss by putting diffractive elements in FabryPerot cavities. Is there some way to defeat the sensitivity theorem [14]? The ultimate and thus far rather distant goal of all work on gravitational wave detectors is gravitational wave ,i1 'i11r, i. All signals we currently use to map out and understand our cosmic surroundings are electromagnetic variants, from the nakedeye astronomy of ancient (and current) man to microwave and gammaray antennae. As each new part of the electromagnetic spectrum opened for us, we discovered wonders in the universe we before had never even imagined. What awaits us in a new gravitational wave paradigm? CHAPTER 2 PARALLEL PHASE MODULATION FOR ADVANCED LIGO Any interferometric detector requires control mechanisms to hold the mirrors in the proper locations. The baseline control scheme for Advanced LIGO requires phase modulation of the interferometer input light at two frequencies Q1 and f2. Beat signals between these additional "sideband" frequencies provide error signals for some interferometer degrees of freedom. To diagonalize the control matrix one must avoid injecting light that is phase modulated at the beat frequencies ~1 Q2 into the interferometer. For this reason we must change the input optics' optical layout from serial modulation, as in initial LIGO, to a parallel configuration. In this chapter we analyze how noise generated by a MachZehnder (ilZ) interferometer with a modulator in each arm could appear in the gravitational wave channel of Advanced LIGO to determine the MZ IFO's stability requirements. We then construct a prototype that meets these requirements and demonstrates the desired phasemodulating characteristics. 2.1 Interferometer Control Interferometric detector optics must be stabilized against seismic noise, thermal expansion, and earthmoon tidal forces to keep these fluctuations from affecting the gravitational wave signal output and to keep the appropriate resonance conditions within the interferometer. Macroscopic distances determine what laser spatial mode couples to the space, but microscopic lengths decide whether light can build up in the cavity. One must measure with great precision changes in the relative separation of mirrors and feed a corrective signal back to positioning actuators.1 At present, the gravitational wave signal is read in the corrective signal. Because light makes such a superb ruler, we use an optical/electrical heterodyne technique for this lengthsensing and control. The light injected into the interferometer will contain several different frequencies, usually separated from the fundamental teraHertz laser frequency by tens of MegaHertz. In the desired optical configuration each of the radiofrequency "sidebands" (SB) or original unaltered laser light (the "( ii,, i frequency in radio nomenclature) resonates in different combinations of cavities that make up the interferometer. By measuring the beat note between two frequencies we obtain information about relative length changes of the cavities each component favors. 2.1.1 Phase Modulation Initial and Advanced LIGO use electrooptic modulators (EOMs) to convert some of the input laser light into sidebands. EOMs consist of a birefringent (i rI I1 across which an electrical potential is applied. The field reorients dipoles in the material and changes its index of refraction proportionally. If one applies a sinusoidal electrical signal, light passing through the crystal perpedicular to the field will have a sinusoidal phase modulation. For a small relative index change m, the laser gains frequency sidebands. Put a signal with time variation sin (Qt) on the EOMs crystal, and an originally monochromatic light field Eoei"t becomes Eoeiwt+imsin Qt EoeWt (Jo (m) + J, (in) eit + iJ (m) eit) iwt (J ()) 2it J2 (n) e2iQt) (2.1) where Jo, J1, and J2 are the zeroth and first and secondorder Bessel functions. The first term in parentheses is the carrier, and the second and third are the upper 1 Coilmagnet pairs attached to the test masses or other parts of the suspension 17 beamsplitter LZ I/ laser EOM RF osci lator Figure 21. Schematic for Ml Sphotodetector   PDH error signal mixer PoundDreverHall FabryPerot cavity locking and lower firstorder sidebands, separated from the carrier frequency by Q. The remaining two terms are secondorder sidebands at 2Q. 2.1.2 PoundDreverHall Cavity Locking The basic heterodyne technique is known as PoundDreverHall (PDH) locking, and our control scheme is a variation on the same. For a simple optical resonator, PDH works as follows. A FabryPerot cavity shown in Figure 21 is made from two partially transmissive mirrors, M1 and M2, separated by a length L. A laser introduces light into the cavity from the left. The first and second mirrors the light encounters have amplitude reflectivities and transmissivities rl, tl, and r2, t2, respectively. In the steadystate, once transient effect from turning on the laser have dissipated, the intracavity field Eca, and reflected field Er of the system in terms of the input Eo are (derived in Appendix A): Ecav itleikL Eo 1 r ir_. , r1 (72 + 2) t262ikL 1 i = T (k). Eo 1 r1r_.  (2.2) I I / The ratios Tc, and T, are the cavity transfer functions. Given input light with carrier and a pair of sidebands, with amplitude E, = Eo (et + i Me(r+()t + ie)t 2 2 which is an approximation of Eq. 2.1 valid for very small m. The reflected total amplitude from the cavity will be Eto = Eo (Tc T+ + T i( + i Tei(w )t where T = T, (ko), with ko the wavenumber of the carrier light. The quantities T are similarly defined for the sidebands. A photodetector upon which this signal falls will register an intensity Itot ID + m Elo 2 (iT (Tet + T_e t)) E 2 R (T Te2iQt) where R denotes the real part of an expression. The next step is to demodulate the photodetector output current with the frequency equal to the part of signal we wish to view. For the FabryPerot cavity the second term, with sinusoidal Qt variation, makes a good monitor of the cavity length. Demodulate the signal by multiplying it by sin (Qt + Ad) and integrating over a period. This is accomplished electronically with an ACcoupled mixer whose output we then subject to a lowpass filter. The result is an "error" signal for the cavity's length, which can be fed back to a mirror positioner to keep resonance conditions satisfied. If the length is nearly an integer number of carrier wavelengths and nonresonant for the sidebands, Tc alone will depend upon a small length change 6L from the resonant length. Under these circumstances Tc = Tc (6L), and the PoundDreverHall error signal is S(6L, d) = 2m Eo 2 [R (TA (6L) (T+ + T_)) cos(d) 04 0 Error sig Trans Int 03 0O n 01 0) 01  02] 0 23 3 2 1 0 1 2 3 degrees from resonance Figure 22. FabryPerot intracavity intensity (light line) and PoundDreverHall error signal (heavy black line) as functions of length detuning from resonance (T (L) (T+ + T))sin(d)] (2.3) and is shown for d = 0, along with the cavity transmitted light intensity, for various cavity length tunings in Figure 22. Here the sidebands serve as "local oscillators" that don't depend upon the quantity we are trying to measure, but allow us to bring the relevant signals from MHz to nearDC levels through the beatnote phenomenon. We demodulate the amplitude "b, ii signal at the difference frequency to remove the AC portion of the signal. What remains will be proportional to the relative phase of the two light frequencies. Different information is obtained by demodulating with a sinusoidal signal at the difference frequency that is in phase with the light phase modulation, resulting in an appropriately named "inlph i error signal, or with a sinusoid that is 90 out of phasethe "quadrature" signal [19]. 2.1.3 Advanced LIGO With the new signalrecycling (SR) mirror at the interferometer output, Advanced LIGO will have five lengths, longitudinal degrees of freedom (DOF), 20 C SB ITM ETM SD E BS PR F  SR Figure 23. Lengths in Advanced LIGO that require feedback control that must be controlled to keep the instrument at its operating point. These are illustrated in Figure 23. The five DOFs are the common and differential FabryPerot arm cavity lengths, L+ and L_, respectively, the differential length of the short Michelson arms, 1_, the length of the powerrecycling cavity composed of the common short Michelson arms and beamsplittertopowerrecycling mirror lengths, 1+, and the distance from the beamsplitter to the signalrecycling mirror, 1,. In terms of the lengths AF shown in the figure, the DOFs are defined as in Table 21: In order to monitor the various lengths, we use a heterodyne technique Table 21. Advanced LIGO degree of freedom in terms of length in Figure 23 DOF definition L+ (C + E)/2 L_ (C E)/2 1+ A+(B+D)/2 l_ (B D) /2 1, F+(B+D) /2 similar to PoundDreverHall to measure changes in the phase of light that has bounced off of different parts of the IFO. A dualrecycled interferometer requires two pair of frequency sidebands. The arm cavity common and differential error signals come from interference between a sideband (SB) and the carrier. The control scheme we expect to use in Advanced LIGO uses beats between sidebands for the three remaining degrees of freedom. The 40 meter Advanced LIGO prototype at Caltech tests a control scheme with this division of labor between light frequencies. Ideally, one would find an error signal for each DOF that was independent of all others, but, realistically, some coupling alvx exists between DOFs. This can be minimized making various frequencies resonant in different parts of the IFO, concentrating their power in these areas, and by appropriate choices of demodulation phases. Done well, this will produce a pleasingly diagonal control matrix. Table 22 is an example from the 40 m prototype of a normalized matrix showing the dependence of each error signal on each degree of freedom. Such a control scheme would be relatively easy to implement. Scientists at Caltech noticed, however, that when they injected light into their instrument that was already at the difference frequencies they wished to measure, their matrix included significant offdiagonal elements, as shown in Table 23 [20]. For this reason, Advanced LIGO will need an alternate method of adding phasemodulation sidebands to the input laser. Table 22. Control matrix for 40m prototype without secondorder sidebands [20] Dem. wj Dem. Q L+ L_ 1+ l_ s 21 100 1 4 x 109 1 x 103 1 x 106 2 x 106 22 271 5 x 109 1 1 x 10 1 x 103 2 x 10s Q1 x 22 189 2 x 103 3 x 104 1 3 x 102 1 x 101 Q1 x 22 50 6 x 104 2 x 103 8 x 103 1 7 x 102 Q1 x Q2 164 4 x 103 3 x 103 5 x 101 2 x 102 1 2.1.4 Electrooptic Modulators In initial LIGO, and, the initial 40 m prototype, the IFO input optics include a series of electrooptic phase modulators (EOMs) through which the laser passes. I f f f f f 22 Table 23. Control matrix for 40m prototype with secondorder sidebands [20] )em. wo Dem. Q L+ L_ 1+ l_ 1 t1 100 1 1 x 108 1 x 103 1 x 106 6 x 10 2 271 1 x 107 1 1 x 105 1 x 103 7 x 106 21 x Q2 1840 7 3 x 104 1 3 x 102 1 x 10 21 x 22 50 6 x 104 32 7 x 101 1 7 x 102 t1 x 22 1610 3 2 2 x 101 4 x 102 1 6 1 If the laser is fed through two sequential EOMs, driven with frequencies Q1 and Q2, the result is Eo [Jo (umi) Jo (n2) Ce + zJo (n2) J1 (n) e ) + Jo (n2) J1 (in) e )t +iJo (mi) Ji (m2) e(+"2)t + iJo () Ji (n2) Ce( "2)t eiwt (Jo (mn) J2 (in2) e2iQ2t + Jo (2in) J2 (nn2) e2il2t) eiwt (Jo (im2) J2 (2in) e2,01t + J (n2) 2 (1in) e2i01t) eit (Ji (mi) J1 (m2) 6e 1+"2)t + J1 (n1) (n2) eC(Q"Q2)t) eCt (+J1 (min) Ji (172) Cei(Q+Q2)t + J1 (min) J1 (m12) e i(Q2)t) . In the last two lines of the above equation, we find we've made sidebands of sidebands at the sum and difference of the EOM driving frequencies. As seen above, this mixes the error signals that feedback mirror positioning control in an undesireable way. There are two immediately obvious solutions. It is possible to drive a single EOM with a complex signal and produce multiple modulation frequencies without SBs of SBs2 Another, perhaps easier solution is to add sidebands in parallel rather than in series. We chose to investigate the latter in greater detail. Instead of sending the laser through a sequence of modulators, as shown in Figure 24, 2 QiTse Shu, unpublished. Some current research is also underway. (2.4) SUopLcs to IFO Figure 24. Serial phase modulation of IFO input light BS1 Q1I BS2 optics Figure 25. Parallel phase modulation of IFO input light to IFO we'll use an arrangement that makes mixed sidebands impossible. Split the laser with a beamsplitter, as shown in Figure 25. Send the transmitted light through one EOM and the reflected light through another. Recombine with another beamsplitter. This configuration is commonly called a MachZehnder ( \!Z) interferometer. What could be simpler? Naturally, however, there are a few caveats for the scientific emptor to consider. Because each EOM interacts with only half the input laser intensity, and because the resultant sidebands are halved again by the second beamsplitter, with the same radiofrequency power at the EOM, our parallelcreation sidebands will have only a quarter of the power in their series counterparts. With Advanced LIGO's large laser power and very highfinesse optical cavities this should present no problem to finding adequately sensitive error signals with the same SNR. Mirror motions or index of refraction changes in the two arms, particularly if differential, can create noise that interferes with gravitational wave detection, so we will likely need to actively stabilize the MZ in some way. Parallel phase modulation amounts to having one more degree of freedom to control than the series case. 2.2 Parallel Phase Modulation Noise The basic noise requirement for components of LIGO can be summarized as follows: no technical noise source should produce a detector output that is more than 10' of that made by the weakest gravitational wave the instrument is designed to register. The amplitude, frequency and pointing (transverse spatial mode) cleanliness of the MZ's output must be regulated. Additionally, because the combined carrier and sidebands no longer travel a common path, noise in their relative phases, amplitudes, and pointing can occur. A gravitational wave signal can be read from the detector output with either a heterodyne or homodyne technique, and the noise requirements will depend upon which is used. With heterodyne or "RF" readout, radiofrequency sideband light is ever present at the antisymmetric IFO port and acts as a local oscillator. The GW signal is in the beat between the sideband, which does not resonate in the FabryPerot arm cavities, and any carrier light that leaks out when a GW strain changes the arms differentially. In the "DC" readout strategy, we measure the effect of a GW strain as a change in intensity of carrier light itself at the output port. As one would expect, there are advantages and disadvantages associated with each method. Overall, the DC readout technique has more lenient carrier and sideband noise requirements, and has been selected for the baseline design [21]. 2.2.1 Mirror Motions Mirrors in the MZ interferometer may fluctuate in either their longitudinal or angular degrees of freedom. Both longitudinal and angular noise motions can be divided into those that are common to both arms, and those which are differential. In the case of angular noise a further division occurs between common and differential movement of optics in a single arm and that between optics in different arms. Figure 26 is a tree diagram that exhausts the possible mirror motions and consequent noise on the detector input light. The effects of the two main branches can be mixed if, for example, a beam is not centered on an optic or hits the mirror with nonnormal incident angle. Let us deal with each effect in turn. When we find that the parallel phase modulation configuration adds noise in a way similar to serial modulation, we will be satisfied that we need only recreate the stability of that original scheme. Longitudinal Common Motion Longitudinal commonmode fluctuations, when the mirrors move along the optical axis of the MZ, will cause the same phase (and thus frequency) noise to appear on the carrier and sidebands. This is illustrated in the phasor diagram in Figure 27. Let the field travelling a clockwise path around the MachZehnder IFO Mirror Motions Longitudinal Common Differential / C: overall phase C: amplitude SB: overall phase SB: phase rel. to C A Single arm common C. and SB pointing Common Both arms common C. an SB displaceme Angular Differential Single arm Both arms V V Id nt relative SB displacement relative SB pointing Figure 26. Possible noise motion in the MachZehnder optics and resultant noise on Advanced LIGO input light. C: carrier, SB: sideband. + Figure 27. Phasor diagram of MachZehnder commonmode fluctuation's effect on recombined carrier and one pair of sidebands. I: in terms of the original input field amplitude Eo be Ec 1/ /Zoe ikL1 6im si'n(Qt) E, 1/ /2EoeikL2 eim2 sin(Q2t), where Ec and Ec, are the clockwise and counterclockwise fields, respectively, Li and L2 the respective arm lengths, and m and m2 the modulation depths. Defining an average arm length L = (L1 + L2) /2 and differential length AL = (L1 + L2) /2, the recombined field at the second beamsplitter is Ett = E k [iCkAL (Jo(7rn) +iJ1 ( l) lt + iJi, (Mi)eilt) +eik (Jo(In2) + z2 *t + (Jl (M2) iQ2t)] (2.5) As the noise in L shows on both carrier and sidebands equally, this case is fundamentally no different than that found in serial modulation. Its effect on the GW signal is a function of mismatch between the Michelson arms, which allows light noise to leak to the dark port. Arm .mimmetry may be due to different losses or reflectivities of test mass mirrors, or to length disparity. The common carrier and sideband frequency noise, and the carrier amplitude noise resulting from MZ common arm length changes are both suppressed by independent feedback loops in Advanced LIGO. A lownoise highpower photodiode monitors the laser intensity behind the triangular modecleaner cavity in the input optics, and a correction signal fed back to the laser power. Another feedback loop locks the laser frequency to the common mode of the FabryPerot arm cavities. Longitudinal Differential Motions When optics move differentially along the optical axis, two noise effect occur. First, the recombined carrier light suffers amplitude fluctuations. Amplitude and intensity stability of the carrier is critical with DC readout [21], but, as mentioned, noise will be suppressed by feedback loops. + t Figure 28. Phasor diagram of MachZehnder differentialmode fluctuation's effect on recombined carrier and one pair of sidebands. The second noise effect is the relative phase between carrier and sidebands. With longitudinal differential motions the MachZehnder configuration allows for fluctuations in carrier and sideband relative phase, which may be interpreted as noise in the sideband frequency. This has the same effect as EOM phase noise in the serial modulation case (which will also occur in parallel modulation). We expect that relative phase noise between the carrier and control sidebands will present the largest problem of all noise terms discussed here as it appears as frequency noise on the laser via the frequency stabilization loop, as currently conceived, and no feedback loop supresses it. The mode cleaner that serves as a filter for laser frequency, pointing, and spatial mode noise is locked to the FabryPerot arm cavities' commonmode, and in turn serves as a frequency reference for the laser. The feedback loop is poorly equipped to deal with this kind of noise, and cannot distinguish sideband frequency changes from laser frequency changes. With relative phase noise, the control loop imposes a misguided correction to the laser frequency, and thereby relative phase noise becomes laser frequency noise, which, as mentioned above, can interfere with GW detection. Asymmetries in the arms of the Michelson interferometer, of both the intentional and the inadvertent but inevitable varieties, allow such commonmode noise as frequency fluctuations to appear in the dark port signal, where they are not distinguishable from a gravitational wave signal. To make MZassisted parallel phase modulation work for Advanced LIGO, one must calculate the allowable magnitude of carriersideband relative phase noise from Advanced LIGO's frequency noise requirement. Using DCreadout for the gravitational wave signal, Advanced LIGO has a 1 mismatch between its two resonator cavities. Given this operating point for the interferometer, there is a frequency noise requirement [22] for the input light made by comparing detector output due to carrier frequency noise to output resulting from the waves we seek. We wish to translate this into a requirement on relative phase noise amplitude. The procedure is to compare the effect of laser frequency noise on the FabryPerot commonmode signal to the effect of relative phase noise [23]. The L+ error signal is in the beat between carrier and f2 sideband light that is reflected from the IFO back to the laser. Enlisting the FINESSE modelling tool developed by Andreas Friese3 we calculated the ratio of laser frequency noise's effect on the commonmode error signal S (f) to the effect of relative phase noise. The ratio of these transfer functions is shown in Figure 29. It is relatively flat over the region of interest for our detectors. Given the allowable frequency noise Af (f) discussed for longitudinal common motion, we can find a corresponding relative phase noise limit A4Q,, (f): AKrif (f) f () Af (f) (2.6) df dSldAf,, where dS/df and dS/dK are the transfer functions of frequency and relative phase noise, respectively. A limit on relative phase noise translates into a limit on differential displacements ALMZ of the MZ arms by ALMZ A rel 27 3 available at www.rzg.mpg.de/ adf 30 Ratio of Transfer Functions to CM Signal 28 2 75  27o g 2 65  26  25 1 : 0I 10 I I 102 10 100 10 102 103 noise frequency (Hz) Figure 29. Ratio of frequency noise to relative phase noise transfer functions to FabryPerot common mode error signal Permissible ALMZ (f) is shown in Figure 210. It should be noted that this assumes the frequency stabilization loop's gain is high enough that it imposes practically all of the relative phase noise on the laser frequency. This will be the most stringent stability requirement for the MachZehnder interferometer. As mirrors on an optical table such as the input optics in Advanced LIGO will rest upon generally have much larger fluctuations, the MZ will need active control. Angular Motions When mirrors have angular fluctuations, a disparity arises between the optical axes of the MachZehnder IFO and Advanced LIGO's core optics. Gaussian optics describe the amplitude of a light beam in the plane perpendicular to its propagation in the paraxial approximation (that is, when its transverse variation is much less than the longitudinal variation.) The Gaussian modes constitute an orthonormal basis set into which any spatial light distribution may be decomposed. An interferometer will generally use light beams made almost entirely from the lowestorder term of the basis set, TEMoo. This is the term that has the no intensity zeroes and a Gaussian shape. When there are small displacements 6 Stability Requirement for MZ 10 10 i 1 10 11 10 12 10 102 10 noise frequency (Hz) Figure 210. Permissible residual differential displacement of MachZehnder arm lengths or tilts 0 in the beam relative to the first optical axis, the new spatial profile, when expanded in the original basis set, will contain small contributions from higherorder Gaussian modes [24]. For a laser to be resonant in a given optical cavity, it must have not only the proper wavelength (the cavity length divided by an integer), but also a particular spatial mode defined by the length and mirror curvatures. Spatial mode noise is a problem because it causes apparent laser amplitude noise since nonTEM ,,, modes couple less efficiently to the resonator cavities. Noise also results from the reconversion of noisy higherorder modes to the fundamental mode via misaligned core optics [25]. In the HermiteGaussian eigenbasis, in which a transverse spatial amplitude is the product of two onedimensional functions, the first three modes are: ik. Uo (x, z) \114 / e q ,the fundamental mode U (x, z) = ( / e2 ,the "tilt" mode U2 (x, z) (14 V T 2) e ,the bull'seye mode with wavenumber k, beamsize w(z), Guoy phasor '(z), and complex radius q(z). Let 0 and Q be beam tilts relative to the x and y axes, respectively. In addition, let there be small displacements 6 and T] of the beam from the core optics axis. The total field at the symmetric port of the MachZehnder IFO will be Etot oc 1/4 ((L1) 61)2 ik0x + ) w(L)J ( ( \1/4 (1)2 Cik ikL1 S((2 ) 1/4 i (L2 (_,2)26ikO Li S(() 1/4 ( L (Y72)2ik 2Y) ikL2 (2.7) Note the assumption of equal losses in the arms. This may also be written E oc 2eik(2+y2 )/2q ikL ei&ikSx/q ikOx ik y TW CikzALC 'T CikASx/q 6ik9AOx ikAny/qyikA y (1 + U) CikAL CiAl CikA,6xl/q ikAOxikAy/q ikAy + ( 1 w) Where a quantity labeled a is defined as the average 2 = a1 + a2, and corresponding differential quantity is 2Aa = a2 a2. We now expand the expression (1 + Aw/w) to linear order in small perturbations. The field is E o 2 1 ik(x2+y2)/2q ikL ei&6ik6x/q ikOx ' Ciky W, 2 cos x + Aw kAq x kAqy 2i Wsin kAL+ AT + + + kAOx + + (2.8) w q q Terms such as eikO refer to MZ commonmode tilts and displacements, and these are the same phases found in the electric fields of serial modulation schemes. When the new pointing of the MZ is expressed in terms of the IFO optical axis, there are small contributions from higherorder tilt modes. In the xdirection, E (x) oc 2k2A6x Aw kAsx Aw + AL 2i 2i kAOx q w qw which is 2Uo (x) + we2 ( + kO + A +kAOAT k2A6 Aw kA Aw +AL i i kAO UI (x) q q W w 2Uo (x) + atitUi (x) (2.9) Eq. 2.9 defines a tilt mode amplitude atilt. We see from the expression 2.8 that differential pointing effects are all of second order, and therefore most likely no problem at all. 2.2.2 Fluctuations at Nonzero Frequencies Now assume that the MZ mirror pointing noise occurs not at DC, but at a frequency f such that 6i sin(27ft), for example. Any Uo1 or Uoi light input into the main Advanced LIGO interferometer (IFO) can be transformed into Uoo light therein if core optics are misaligned. The 34 maximum permissible amplitude of a tilt4 mode injected into the IFO is given by atilt /2.5 x 103 21010rad 1 af2(f) = \ + (5 x 108) A T . Sf AOlTM VH Note that this assumes a 1010 rad differential tilt of the input test masses (IT\ i). It also includes a factor of 10 safety margin, keeping technical noise at least a decade below the interferometer's intrinsic quantum noise. As an exercise, we use 2.9 to calculate permissible commonmode tilts in MachZehnder optics such as mirrors and beamsplitters. The input mode cleaner supresses higherorder modes by an additional factor of 2000. We assume a beamsize on the optic in question of 5 x 104 m and that half the allowable fluctuations are allocated to tilts, and half to displacements. As an example, the maximum common tilt 0 is found from S< t 8.5 x 109rad 2kw H/z at 1 kHz. 2.3 MachZehnder Parallel Phase Modulation Prototype Having found the stability requirement for parallel phase modulation, a tabletop prototype will explore the attainability of these limits on differential optical path changes. For our purposes the MachZehnder interferometer should be compact, inexpensive, and easily controlled with a feedback loop. In this design the optics are all mounted to a single aluminium plate. These are small, offtheshelf mirrors and mounts. One mirror is glued to a stack of piezoelectric ceramics (PZT), which grants us length control access with only small driving voltages (2V can cover a full laser wavelength). To avoid adding pointing noise with our 4 "Tilt" here refers to either a U1o or U01 spatial mode, as distinguished from a tilt 0 in a single MZ beam rf signal 2 SBS polarizer rf signal 1 I Figure 2 11. Layout of the prototype MZ phase modulation scheme  EO12 D ph( todiode gain & frequency filter mixe * Figure 211. Layout of the prototype MZ phase modulation scheme actuator we use a combination of polarizing beamsplitters and A/4waveplates in the corners of the MZ, as shown in Figure 211. In this way, the PZT pushes and pulls the mirror along the light propagation axis. As a bonus, the waveplates also allow one to make the impedance matching of the arms nearly perfect, so that dark interference fringes at the output can be truly black. The EOMs emplovi l1 are offshelf LiNbOs resonant phase modulators. Their frequencies in the clockwise and counterclockwise arms (as viewed in the figure) are 31.5 and 12.0 MHz, respectively. It is with the 12 MHz sideband we lock the MZ differential mode. This requires a highspeed photodetector ours is a 1 GHz Si photodiode. We mix the AC portion of the photodiode ouput signal with the RFsignal that drives the 12 MHz EOM. The lowpassfiltered error signal is amplified and frcqu ii ,i,filtered by an analog circuit before it is fed back to the MZ length control PZT. Although one should easily be able to guess the spectral content of the MZ interferometer output, experimental science can be full of surprises (see ('!i Ilter 4), so we put the MachZehnder output light through a highfinesse optical spectrum Spectrum of MZ Output Light 018 016 014 012 01 0 08 0 06 004 002 0, 0602 3 15 I I  80 63 31 5 24 12 0 12 24 315 40 frequency (MHz) Figure 212. Spectrum of light transmitted by parallelphase modulating MachZehnder interferometer analyzer to see what frequencies were created. The interferometer's behavior was as expected, however, as seen in Figures 212 and 213. There are two pair of frequency sidebands. The j I., d peaks to the right of the carrier resonance are higherorder Gaussian modes. We look closely at the mixture frequencies. Small second harmonic sidebands are visible for both frequencies, so we know we would be able to discern whether SBs of SBs were present. No peak is present at 19.5 MHz, but a few small peaks (a mixed sideband should have approximately the same amplitude as a secondorder sideband) appear at 43.5 MHz. Upon closer inspection, looking at the light corresponding to these peaks on a CCD camera, we found that these were higherorder spatial modes resonating in the analyzer cavity, and not mixed sidebands. To get a sufficiently quiet MZ interferometer we will need a quiet error signal with strong dependance on the quantity it intends to regulate. Our technique is heterodyne length sensing in the beat signal between the f =12 MHz sideband and the recombined carrier. The field at the symmetric port of the MachZehnder IFO 37 Detail of MZ Output Spectrum 0 01 0 009 0 008 0 007 S0 006 2 0 005 0 004 0 003 0 002 0001 60 50 435 315 24 195 12 0 frequency (MHz) Figure 213. Expanded view of MZ output spectrum. Mixed sidebands are absent the 43.5 MHz peaks are due to higherorder spatial modes. (with the effect of the 31.5 MHz EOM omitted for clarity) is Esym = EleikLi C isin(Qt) + E26ikL2, (2.10) and the intensity is Isym, = IEi 2 + E2 2 + 2E1E2 Cos (kAL + msin (Qt)). (2.11) If m is small, and in our experiments it was a very small 0.001 for the 12 MHz EOM, the slowly timedependent part of Eq. 2.1 is Isym,AC 2E1E2 [cos (kAL) msin (kAL) sin (Qt)] (2.12) Demodulate the photodiode voltage with a sin (Qt) wave from the EOM's voltage source to get the MachZehnder error signal. The signal is zero when the MachZehnder's interference is at an extremum, and becomes either positive or negative if the differential length deviates in one direction or another from this value. Thus, if fed back to a bidirectional positioning device such as our PZT, the error signal "locks" the interferometer at a bright or dark fringe condition. The difference between bright and darkfringe locking is inversion of the signal. Unlike the FabryPerot cavity analyzed above the MachZehnder heterodyne error signal has an infinite locking range. That is, no matter how far removed the differential length becomes from the proper interference condition, the PZT will alvi push (or pull) the length closer to perfection, seeking that zero in the error signal. The disadvantage relative to FabryPerot PoundDreverHall locking is that we do not have sharp resonances that heighten our awareness of length changes through steep error signal slopes. The strength of a corrective signal is in the size of its derivative with respect to the parameter it controls. As we have a feeble sinusoidal variation in our error signal as a function of AL, we must add slope electronically. To determine how must amplification the error signal needs we measure the noise in the freerunning (no feedback) MachZehnder output and compare to the stability requirement. An analog proportionalintegrating amplifier circuit with adjustable gain makes up the difference. We would be content to set the gain to the highest level necessary and feed this back at all noise frequencies, but we must be more careful. Mechanical resonances in the system, particularly in the actuator, will be excited if the gain is greater than 1 at these frequencies. At 10 kHz, the PZT's internal resonance is the lowestfrequency and most problematic. Fortunately, measurements of the freerunning MZ's errorpoint noise indicate that it is already sufficiently quiet at 10 kHz, so the feedback loop needs no gain at this or higher frequencies. In Figure 214 one sees the performance of the stability feedback loop for various noise frequencies compared to the requirement. The openloop measurement indicates the interferometer needs active feedback to supress excess noise below about 500 Hz (where, unfortunately, there is a small resonance due to the PZT mount). With the feedback loop closed, the residual differential motion noise falls within an acceptable range of the calculated requirement. Noteworthy exceptions are the ..'.i ivatingly everpresent 60 Hz line noise spikes, as well as the 500 Hz resonance. We also made an independent measurement of the residual differential motion by demodulating the photodiode output with the radiofrequency signal that drove the second EOM. The measurement point is labelled Sot in Figure 211. Figure 215 depicts the outofloop noise measurement alone versus the target. Although a small amount of excess noise persists below about 100 Hz, we feel this result demonstrates the relative ease with which one can achieve the necessary stability for parallel phase modulation. If we implemented this scheme in Advanced LIGO, we could improve the noise performance by raising the frequency of the PZT resonance by gluing a lighter mirror to it, which would allow the feedback loop a higher unity gain frequency to suppress noise below 100 Hz. The noise attenuation transfer function for the electronics used in this experiment is found in Figure 216 (Appendix C contains the circuit diagram). Ground loops and line noise could be reduced with the various black magic techniques electrical engineers know. The entire MZ interferometer could also be enclosed in vacuum for better passive isolation. In the end, parallel phase modulation is vying with several other techniques for inclusion in Advanced LIGO's lengthcontrol scheme. Complex modulation of a single EOM has tantalizingly fewer degrees of freedom to control. Its strictest requirement is coordinating the phase modulation with amplitude modulation, that is, locking two oscillators. It may be that doubledemodulation for the inner Advanced LIGO DOFs is abandoned entirely for a different mix of beat signals. Along with the Caltech prototype group we have investigated the requirement for and performance of a possible solution, and await a coalition decision as the time to begin construction the the upgrade nears. 40 MZ Differential Mode Stability  requirement outloop means inloop meas open loop meas II , I i 200 400 600 800 1000 noise frequency (Hz) 1200 1400 1600 Figure 214. Comparison of MZ errorpoint noise with required stability for freerunning and closedloop operation MZ Differential Mode Stability 1013 200 400 600 800 1000 noise frequency (Hz) 1200 1400 1600 Figure 215. Outofloop measurement of residual MZ differential displacement noise 10 I 10 10 I, I f I I ' Feedback Attenuation Transfer Function noise frequency (Hz) Figure 216. Attenuation transfer function of MZ feedback electronics CHAPTER 3 NEARFIELD EFFECT HEAT TRANSFER ENHANCEMENT 3.1 Nearfield Theory Interferometric detectors will disclose the hitherfore imperceptible passage of a gravitational wave through careful timing of photons travelling between the mirrors that serve as test masses for the field. Unfortunately, if the mirrors themselves move with respect to another part of the instrument, the effect imitates a gravitational wave. In the current and advanced versions of LIGO, we have several strategies for quieting these noise motions. The test masses hang as pendula from seismic isolation stacks. In addition to active feedback outside the GW measurement band, initial LIGO has single stage suspensions, but Advanced LIGO will have four masses hanging, one from the other, ending with a very quiet final test mass at the bottom If all this works perfectly to hold the mirror, as a macroscopic object, at position steady to within 1018 to 1021m/v H (for LIGO and Advanced LIGO, respectively), one still has to contend with the fact that the mirror is composed of molecules with thermally excited motions all their own. These thermal fluctuations excite resonant modes of the test mass in a process described by the fluctuation dissipation theorem (FDT), discussed below. Levin et al. [26] calculated the noise in the GW channel of the IFO for a roomtemperature Advanced LIGO test mass. Test mass internal thermal noise is shown in green in Figure 31. According to the classical equipartition theorem, there is {kBT of energy in each mirror 1 VIRGO, currently under construction, has sevenstage pendula.[9] 43 1019 yquant Int thermal S. Susp thermal 10 20 Residual Gas STotal noise 1021 TN 1022 10 1023 1024 1025 i . 100 101 102 103 f/Hz Figure 31. Comparison of quantummechanical sensitivity limit to test mass internal thermal noise limit for Advanced LIGO [22] degree of freedom, where kB is Boltzman's constant and T the temperature in degrees Kelvin. Levin found internal thermal noise is proportional to the mass's temperature. In Advanced LIGO room temperature test masses prevent our reaching the quantum limit of detector sensitivity, a limit due to the corpuscular nature of light, in some frequency regions. The mirrors need to be cooled to 10K to have thermal noise comfortably2 below the target detector sensitivity, but we have yet to develop an economical active cooling method that meets our exacting engineering requirements. Chief among these is maintaining seismic isolation. One will commonly cool an object by attaching a cryogenic "cold f:i,; i that extends from a very cold reservoir. Naturally, we are not about to shortcircuit the isolation system by touching test masses so carefully separated from the environment with any kind of finger, cold or otherwise. The only routes through which heat can escape the masses suspended in vacuum are radiation and conduction through 2 a factor of ten the suspension fibers. The effect of the latter in suspensions like those in Adv. LIGO will be far insufficient.3 By the StefanBoltzmann law, the electromagnetic radiation from an object at finite temperature T in the farfield is Q =e(l5.67x m2K ) T4, where c is the emissivity of the material. The T4 factor in the StefanBoltzmann law means that radiative heat loss diminishes rapidly with temperature. A fused silica mirror (c = 0.9) with 1 m2 area, at 10 K, therefore, would radiate 510 pW. With Advanced LIGO's intracavity circulating powers of 10 kW and the anticipated mirror absorptivities, however, the input test mass will be collecting 2 W from the laser. Advanced LIGO would clearly require active cooling to beat the current thermal limit. In the LIGO upgrade the mirrors will remain at room temperature, but when in even more futuristic incarnations of interferometric detectors the standard quantum limit of sensitivity is much lower, we will certainly need to actively cool the test masses. How can we both enhance heat loss and maintain seismic isolation? Nearfield electromagnetic effects provide the answer. We cannot touch the test masses with a cold finger, but we can bring a cold object close enough that photons can tunnel through the vacuum gap from hot to cold mass [27]. Classically, the effect may be described as penetrating the region of significant evanescent fields so that these fields can transfer energy, a phenomenon also known as frustrated total internal reflection (FTIR). 3.2 Fluctuational Electrodynamics Theory To quantify the heat transfer enhancement, we use the fluctuational electrodynamics method developed by S. Rytov [28]. Thermal excitations within a material cause 3 See Appendix B for quanification of heat transfer through the fibers 1 2 3 Figure 32. Geometry of dielectic media for nearfield effect calculation two semiinfinite slabs separated by a vacuum gap with width d. oscillating currents J that serve as electromagnetic field sources, described by Maxwell's equations: V x H = 47opJ + i cE (3.1) V x E iwB (3.2) assuming the resultant fields have sinusoidal time dependence and the material is nonmagnetic. Our calculation will be assisted by two physical/mathematical theorems that of Green, and the fluctuation dissipation theorem (FDT). 3.2.1 Green's Function Method With Green's theorems one can find the electric and magnetic fields due to the thermallyinduced currents using Green's functions. E (r, w) ipow GEE (r, r', u) J (rl, ru) dVI (3.3) H (r, w)= x GHE (r, r/, a)) J (r/, w) dVI (3.4) where GXE (r, r/, w) is the dyadic Green's function relating the field at r to the source at r/. Because the Green's functions reflect the geometry of the problem, we must specify this to proceed. Let us consider two semiinfinite halfspaces filled with isotropic and nonmagnetic matter separated by a vacuum gap of width d. Orient the Cartesian axes such that the zaxis is perpendicular to the interfaces, and the plane z = 0 coincides with the surface of one semiinfinite chunk as seen in Figure 32. The three regions 1, 2, and 3 are defined as shown. Instead of a complete Fourier decomposition of Green's function into plane waves of the form Eei(krwt), which are not in general guaranteed to satisfy the freespace Maxwell equations, we will use a partial spatial transform Eei(K'r t)+ikzz, where K is a vector in the xy plane. The zcomponent kz is subject to the constraint K2 + k = k 2, with ko the photon wavenumber. Thus the Green's function encompasses both propagating waves, when K = KI < ko: k = k0 _ K2 k, y k as well as evanescent v ', when K > ko and kZ i K2 kO. Let us introduce further simplifying notation. We define the wave vector u K k'z. The unit vectors is defined as s K x 2, andp = kK + K+ z, so that u,s, and p constitute one orthonormal set, and K,s, and z another. Our R = x + y. The Green's function tying fields in region 3 to sources in region 1 is GE (r, r/, o) d2Kl (s ti+ 3t 1) eik 3z ik zi iK(RR'). (3.5) where the frame decoration G denotes the dyadic form [29], and the Fresnel coefficients are 2ki 13 kl, + k3 2nin3ki 1cki + 63kA3 3.2.2 Fluctuation Dissipation Theorem Our hero once again, Albert Einstein first identified the fundamental principle behind the fluctuation dissipation theorem in his work on Brownian motion [30]. A couple decades later, in 1928, Nyquist found a connection between voltage fluctuations in electronic circuits and electrical resistance that relies on the same underlying postulate [31]. The generalized theorem, of which Brownian motion and Johnson noise are examples, was published by Callen and Welton in 1951 [32]. The FDT states that in thermal equilibrium a system with dissipation D (w) generates a fluctuating force F 2 " (F2) J D (w) (w, T) dw, 7 Jo where O (a, T) = h with the Boltzman constant k = 1.38 x 1023 J/K, is the frequency distribution of an oscillator at temperature T, known as the Planck function. We need to find the correlation of forces which are due to fluctuating dipoles. The material has polarization density P. By the FDT (Pa,) j ( (c) (r, r', w)) (w, T) dw. The corresponding function for current density J = iwP is (J ) (, T) 2 (c) (r, w) 6 (w w') 6 (r r') 6ab du (3.6) Jo 27r Here, S (c) (r, w) is the imaginary part of the emitting body's dielectric function. Note that this form of the FDT presumes a null correlation length, an assumption which could prove problematic for materials in which the mean free path of charged particles is long relative to wavelengths given by S (w, T). The other assumption is that energy levels in the system are densely spaced relative to driving energies, even at these low temperatures. This enables use of the classical version of the FDT result. With the first of these two clues, Eq. 3.3 becomes E (r, Lw) P dV d2Kt t eik3ze i'iK(RR') f dV d2Kt t J( JK) eikz3zikzlz'iK(RR') (3.7) 872 J J k U 1 U1 and H is found from Eq. 3.2. Transforming the operator Vx to iu3x, the magnetic field in region 3 is H(r, w)= IdV Id2K u3 1x p3 1 ( kJK)) 8z v kl Ul U / .s1 ikz3z iklz' iK(RR') 31JC C dv/ d2 l tK' tl( 3) 1 +tP ( z K1J) K 87 V k,I U1 U\ ikz3Z ikzlz'6iK(RR') 3. Once we know E and H, we find the heat flux from one object to another by means of the ensembleaveraged Poynting vector S. (S)= R (E (r, w) x H* (r, w)) (3.9) The crossproduct required for the power flux calculation is E (r, w) x H* (r, w) p dV' f dV" f d2K' d2K" 3 ik'3 ik z6 :6iK(RR')6iK'(RR") (8 2)2 J J J (k' ') S+ z1 I 1, (3.10) 33 (l Ut1 1 1 I where J = Ja (r', w). When one takes the ensemble average of the Poynting vector, terms with the form (JJb*") arise repeatedly. This is the spatial correlation of the fluctuating currents. We evaluate this quantity with the help of the fluctuation dissipation theorem (Eq. 3.6). The heat flux perpendicular to the interfaces is (Sz2T) }= (u;,OT) dl f K fKf *dz' 12C (S (z, ) (16 73)2 () o K i(k'3k' ~z i(k'zlk )z' ( Wkl 2 2 < z' 49 1 tU 12 2 \ 2 a/ (3.1 1) Because the problem has rotational symmetry about the zaxis, we can substitute KdKdO, with 0 the angle K makes with the xaxis, for d2K. Another simplification is to evaluate the integral 0 ei(k, dzk  o 2' (kzi) A third simplifying equation is that =( 2) (kji) (k i). Remembering that the Fresnel coefficients are 2ki 13 ki + k3 2nin3ki ciki + C3k3 the expression in Eq. 3.11 may finally be written (S (z, T)) = 4 J KdK d 2 1 k2 2 (ki) (k S r1r3"2iz2d 2  (k + ki) (k+2 + k,3) 2 I1C2 2 1k2 12 2 (c(k*3) R (Ck3) ] l t rp, rp 2id2 z z3~j 12 (3.12) 21 23 113 i l 2 Ikz,2 + 2ki) (C3kz2 + 62kz3) 2 The integral is divided into two regimes. When IKI < Iko k ~ is real, and the Poynting vector describes propagating electromagnetic waves. The quantity k,3 is imaginary when IKI > Iko and a factor of e2i kz3 makes these terms die away exponentially with distance from the emitter. These are evanescent fields. Note that Eq. 3.12 is the same if the roles of emitter and absorber are reversed. To calculate the net heat flux P between two dielectrics we integrate the difference between rightmoving and leftmoving flux and integrate over all positive frequencies: P (d, T1, T2) [(Sz (d, w, T)) (S, (0, w, T2)) d (3.13) JOO At small vacuum gap widths, well into the evanescent field region, the heat flux scales with the inverse square of d. 3.3 Numerical Simulations of Evanescent Coupling We may evaluate Eq. 3.13 numerically once the dielectric functions and vacuum gap between the media are defined. Care must be taken at the juncture of propagating and evanescent waves as the equations are prone to infinities here if not handled well. Our first test of our newfound calculational ability was a pair of metallic masses with various microscopic separation. One mass was at 10 K and the other at 40 K. We use a Drude model for the dielectric function c (w) of chromium: Nqw fo C [LI) = Q (L)) + i mU (Y7 iu) where eb is the dipole contribution, N is the number of molecules per unit volume, qe the charge of an electron, fo the fraction of free electrons to total electrons per molecule, m the electronic mass, and 7o a phenomenological damping force. The calculated heat transfer in W/m2 is shown in Figure 33. A nearfield enhancement of several orders of magnitude is evident when the vacuum gap becomes much less than the dominant thermal wavelength Ath, given by Wien's law: 2.9 x 103m K Ath T For the hot (that is, 40 K) mass, this number is about 72 microns. For large vacuum gaps propagating modes dominate, and the heat transfer per unit area settles to a constant value coinciding with the StefanBoltzman law. At small distances the propagating waves contribute very little, as the density of resonant modes in the gap diminishes. Evanescent modes more than compensate, however, as these increase exponentially with proximity. 51 103 102 10 10, 10 10I 10 10 10' 10 10 10' vacuum gap size (m) Figure 33. Power flux per unit area as a function of vacuum gap separating two semiinfinite chromium masses at 40 and 10 K 3.3.1 Effect of Dielectric Function To test what kind of materials have the strongest evanescent coupling effect, we evaluated the heat flux between doped silicon masses. Silicon is of particular interest as it is a candidate material for allreflective interferometer designs. The dielectric function used is evaluated from ( 47r 2 i4z' ( w)) o Qco + w w2+w)) \2 + LL 2 (aL2 + ) where the plasma frequency up is given by 2 Ne2 2 2 (LLp n * with N the dopant concentration, e the electron charge, and m* the reduced electron mass. Heat transfer enhancement for various dopant concentrations is shown in Figure 34. The StefanBoltzman radiation for a perfect pair of emitters is depicted by the horizontal line at 0.15 W/m2. The more metallic samples are clearly poorer emitters in the far field, but in the near field the behavior is much more complex. Doped Si 1o17 10 10 1019 1020 1~* 10 10 1 102 100 103 10 108 10 10 10 10 10 vacuum gap (m) Figure 34. Power flux per unit area as a function of vacuum gap separating two semiinfinite doped silicon masses at 40 and 10 K for various dopant concentrations There is no monatonic progress with dopant concentration. In fact, if the plot is sliced along an axis extending into the page, plotting heat transfer versus dopant concentration for a single vacuum gap width (Figure 35), we see rather complex behavior as the material's dielectric function changes. There is an enhancement in thermal coupling when resonances in the absorber correspond to the thermal wavelengths of the emmitter. This makes clear that if we want to use evanescent cooling in LIGO, we will have to carefully select materials appropriate to the temperatures, geometries, and gap widths we wish to use. Overall, however, the effect of dielectric function specifics on heat transfer is much less than the effect of simply closing the gap width. 3.3.2 Effect of L vli. 1t Media We also investigated whether 1 ., 1 dielectric on the material surfaced could enhance heat transfer by acting as antireflection coatings. To do this one changes the Green's functions in the calculation slightly to include reflections at multiple interfaces. This changes the t',Ps. We found that this had almost no effect on 53 3. ..... i .... ... S d=1 39 microns 25 2 15 1 05 1016 10" 1018 101 1020 1021 1022 Dopant Concentration Figure 35. Power flux per unit area as a function of doped silicon metalicity at a 1.4 micron separation the thermal coupling. The reason is that such coatings affect the propagating modes primarily, which at these small vacuum gap widths contribute little to heat transfer. 3.4 Noise Coupling Due to Fluctational Electrodynamics Although we can extract heat from the interferometer test mass without touch using the nearfield effect, it is not true that the cold mass exerts no force on the hot mass. In fact, the very same dipole interaction that gives us heattransfer coupling also leads to the Van der Waals interaction [33]. 3.4.1 Van der Waals and Casimir Forces The similarity to proximityenhanced heat transfer is evident in the equation for the attractive force between two flat semiinfinite dielectrics separated by d: FVdW (d, T) 2 Z j dK duK2U20 (, T) x 27 r2c Jo Jo (kzj + K) (kz3 + K) .2iKwdc 1 lk, + K) (c3z3 + K) 2iKwd/c 1 [(k K) (kz3 K) (e)kzi K) (3kz3 K) (3.14) or hot cold hot cold Figure 36. Possible configurations for evanescent cooling of a mirror If d is large relative to Ath, the above equation becomes Aa Fvdw (d, T) 3 (3.15) where a is the area and A is a materialspecific number known as the Hamaker constant. Metallic masses will also experience an attraction due to vacuum energy fluctuations in the space between. The Casimir force at separations comparable to the conditions that make Eq. 3.15 valid is Cher2 Fcas (d, T) (3.16) 240d4 The dependance of heat transfer on 1/d2 for small distances, and the dependence of noise forces on 1/d3 and 1/d4 means that there will be a best distance from the hot test masses at which to place a cold mass. If too far, evanescent modes will not couple the two strongly. If too close, noise couplings will require us to suspend and isolate the cold mass as well as the hot, and the problem of heat transfer simultaneous with isolation is one step removed, but still present. 3.5 Implementation in an Interferometer 3.5.1 Configurations There are several v,v a future interferometric detector might take advantage of the nearfield heat transfer enhancement. The small separation distance could be along the laser optical axis, as shown in the lefthand side of Figure 36. It could be a ring with an aperture for the light beam if the mirror is a transmissive optic. In an allreflective design we could access more area, and thus transfer more heat. Alternatively, one may choose to place the cooling surfaces along the sides of the optics, as seen in the righthand side of Figure 36. This way, noise forces make the mirror jitter about in a direction transverse to the cavity lengths we are nii, Iii: greatly reducing how strict stability requirements will be. 3.5.2 Stability Requirements for Cold Mass Displacements of the IT\ i are related to the force exerted by the near cold object by d(f dF (f) 4rmf2' The Fourier transform of dF (t) is dF (f)=1 + dF dF (f) (ei(w+n)t + i(w )t) dt 2 dF [6 (w + ) + 6 ( Q)] , this from the fact that 6 (w + ) =( tdt. The Advanced LIGO displacement requirement at 10 Hz is 1019 m Therefore N  dF = (019 ) (47) (40 kg) (10 Hz)2 5.0 x 1015 N What requirement does this place on the distance between the two planes? dF = VdF 5.7 x 1015 2 2 H/z With dF specified, we can calculate the maximum allowable displacement of the cold mass at the noise frequency (10 Hz, in this example): N Aa x dz dF = 5.7 x 1015 VHZH 27z4 With A = 6.6 x 1020J (for SiO2), a = x (0.2m)2, and zo = 10pm, dz may be no greater than 4.3 x 1014 m/vHz. This number can be achieved without suspending the cold mass. This calculation assumes the worstcase scenario in which we must put the distance d along the optical axis. 3.6 Experimental Proof In the near future, the University of Florida's LIGO group will conduct experiments to measure the enhanced heat transfer effect between large, flat objects. They will evaluate the pertinence of these techniques to future versions of LIGO. CHAPTER 4 WHITELIGHT INTERFEROMETRY In this chapter we are peering into the misty distant future of gravitational wave interferometry. Every foreseeable innovation is either a way of reducing noise or of moving sensitivity .1liv from unavoidably noisy frequency regions. Without simply adding more stored light power to the detector, the sensitivity theorem [14] insists that these are the only options for catching more gravitational waves. When detector bandwidth is broad, with broader cavity resonances, the sensitivity hits a shot noise ceiling rather quickly. When the cavities have high finesse we stub our toes against the edges of a cramped bandwidth. We want spacious accommodations in both dimensions. What we would really like is to defeat the sensitivity theorem. Warning: such an ambitious chapter cannot end well. 4.1 The Sensitivity Theorem Quandary Advanced LIGO [12] will have a peak sensitivity at about 300 Hz, with a 1/f decline in responsiveness above the peak. The anticipated sources in the several 100 Hz range are pulsars, the fundamental frequency of the intermediate phase of a neutron starneutron star merger, and the final fundamental frequency of small black holeblack hole coalescences. High frequency pulsars, the harmonics of neutron star and black hole mergers, stellar core collapses, and neutron star oscillations, however, are all expected to emit gravitational waves within reach of Advanced LIGO's sensitivity, but above its bandwidth. An instrument with at least 20 kHz of bandwidth is needed for these sources. The bandwidth of LIGOlike terrestrial interferometric gravitational wave detectors is set by the pole of the FabryPerot cavities within the arms of the Michelson interferometer. This constraint arises because the gain of gravitational waveinduced signal 57 sidebands is limited to frequencies within the linewidth of the cavities. The nature of standard FabryPerot cavities is such that one cannot independently adjust for increased gain without suffering a loss of bandwidth. If these quantities could be decoupled, the resulting improvement in bandwidth may lead to viable high frequency detectors. We guessed that a diffractive element placed within an optical resonator could increase the cavity bandwidth without loss of peak intensity. As we discovered, this expectation was based on erroneous (but not uncommon, even among optical scientists) understanding of diffraction grating function. 4.2 The LinewidthEnhanced Cavity 4.2.1 The Nature of the Gainbandwidth Dilemma Consider a simple FabryPerot cavity with arm length L and input and output mirror amplitude reflectivities and transmissivities rl, tl and r2, 2, respectively. The normalized intensity of light of a particular frequency w = 27rc/A within the cavity is t2 I (o) 1 2 (4.1) 21 + 2 + rj2 Or2coS (I (O)) with ) (w) = 2wL/c, a frequ''i i dependent roundtrip phase shift. When, as in initial and Advanced LIGO, r2 1, and the mirror transmissivities and losses are very small, the maximum of this function can be approximated as i (Wo) 4 (4.2) The intensity is maximum for one frequency of light (and periodically for every free spectral range thereafter) for which < (A) = 27r, with n an integer. The intracavity intensity decreases for larger or smaller frequencies at a rate determined by ri and r2. The full width at half maximum of the Airy peaks in the intensity that correspond to cavity resonances is FWHM = FSR/F, with the free spectral range FSR = c/2L and the finesse F = x/rr2/ (1 rir2). Again consider the LIGO arm cavities, with highly reflective mirrors. One finds that 2wr F 27 FWHMcN t2 t2 (4.3) It is clear that linewidth and peak light intensity are inextricably intertwined in a standard FabryPerot cavity. As a result, gravitational wave scientists, who would like to maximize both quantities, must compromise in choosing the cavity's parameters. 4.2.2 "Whitel5!lII Cavities The logic of the diffractionenhanced cavity was as follows. The roundtrip phase shift's variation with frequency has been identified as the source of the gainbandwidth dilemma, immediately ii 1ii; it as the focus of design alterations. Perhaps K could be made invariant with frequency if the the optical path length inside the cavity were also freC qui' ,idependent: 2wL(w) S(U) 2 constant c (4.4) Making the constant an integer multiple of 27 would presumably ensure that light of any frequency resonates inside the altered cavity. We deemed this the S.litehl;h cavity. We posited that a pair of antiparallel diffraction gratings placed inside the cavity create a frec iiu' , dependent cavity length that cancels the freespace dispersion of the resonator. The arrangement shown in Figure 41 would work as follows: 1. Monochromatic light is injected into the cavity. 2. A gravitational wave modulates the lighttransit time between the input mirror and the distant first diffraction grating such that frequency sidebands are generated on the original laser light. Figure 41. Gratingenhanced cavity 3. The parallel gratings allocate a different path length to light of different frequencies. 4. As seen in the figure, the redder lower sideband (dashed line) travels a longer distance than the bluer carrier (solid line) or upper sideband (dotdashed line) light. This additional path can cancel the variation of the round trip phase shift relative to the carrier light. Figure 42 shows the roundtrip phase resulting from the fre uii' in'dependent optical path length and compares this to the freu'ii_',invariant + (w) of a standard cavity with the same nominal length. If changing length with frequency were the only effect gratings had on the light phase, the gratingenhanced cavity would have a superior linewidth wherever its slope is less than that of the standard cavity. Figure 43 shows the intensity buildup in a LIGOscale enhanced cavity as one varies the laser frequency, using the theoretical calulations above. This curve is the theoretical performance of a gratingenhanced cavity with 4134 m total oneway length, a spacing between the grating planes of 71 m, a grating constant of 1633 lines/mm, a grating incident angle of 54, and a laser wavelength of 1064 nm. The plot assumes the slightly idealized case of lossless gratings. Also shown is the resonance width of a standard cavity. The gratingenhanced linewidth has increased by about a million times. We note at this point, however, that this static response to changing frequency is not equivalent to gravitational wave response, as discussed below. Real gratings with losses will require greater laser power to reach the same maximum signal buildup inside the cavities as the standard cavity case. There is no theoretical limit to diffraction grating efficiency. Grating designs 2 6e+10 phase (rad) 5e+12 4e+12 3e+12 2e+12 le+12 0 le+12 2e+12 3e12 4e12 5b 12 frequency (Hz) Figure 42. Comparison of roundtrip phases for gratingenhanced (solid line) and standard (dashed) FabryPerot cavities. Frequencies on abscissa S f aser that use methods beyond the scalar approximation routinely produce efficiencies of essentially 10 r'. [34]. Actual grating fabrication is rapidly catching up to theory. Finally, then, if one puts mirrors at the places shown in the figure, and adjusts the lengths of the common paths and the intergrating spacing D correctly, it should be possible to arrange for each color that the ratio of freespace path to the wavelength is the same integer value. If this were so, and if the gratings had no other effect on the phase of the light waves, then the device shown would be a cavity resonant for all wavelengths, a v .itelh,!i cavity. Detailed calculations [35] show that the bandwidth of this cavity would in fact be finite (because of the nonlinear dispersion of the gratings) but would be many orders of magnitude larger than the bandwidth of the typical FabryPerot cavity, such as the ones in the arms of the LIGO detector. 4.3 Gravitational Wave Response in the Time Domain If antiparallel diffraction gratings indeed make light's roundtrip phase shift within a cavity frc uii' i,'invariant, it would then be helpful to compute the gravitational wave interaction of such a cavity in the time domain. All previous calculations concerned only an effect equivalent to varying the input laser's 10, 10 \ 10 N \ \ 10 \ 10 , 103 100 10' 102 103 104 105 106 107 10 frequency (Hz) Figure 43. Linewidths of gratingenhanced (solid line) and standard (dashed line) cavitites. The standard cavity curve is cut off at 75 kHz so as not to obscure the enhanced cavity plot. Frequencies = f flaser frequency, which is not the same. We begin by transmitting an input field Eoe(iwto) through the lefthand mirror, ITM, with amplitude reflectivity and transmissivity rl and tl. The parameters for the end mirror ETM are similarly defined. After transmitting through ITM, the field at time t is E(t) = itlEoe(it). Consider the traceless, transverse gauge interpretation of the gravitationalwave effect. The positions of the IT\ i and ET\ i remain fixed, but the spacetime metric does not, resulting in a change in the time in which light travels from one mirror to the other. Two assumptions are used in this calculation. First, assume that the GWstrain is sinusoidal: h (t) = ho sin (Qt). Second, if the strain h is small, we may take for granted that the difference between a null ray path and a geodesic will be of order h2, and thus negligible. A null ray is described by (ds)2 (dt)2 + (1 + h+)d2 + (1 h+)dy2 + dz2 = 0 Thus the time in which light travels from the ITA\ [to ET\ I a spatial coordinate distance L/2 away is described by p14) t i L / dt + / + h, cos (Qt)dx Jto C 0 Here, the zero of the xcoordinate is the coated surface of the IT\ i mirror. The ETA F resides at position x = A. The above equation may also be written: L t(1 ) dt h (t o cos(Ot))dt 2c to + ho cos (t) io 2 St(L) to h[ sin (t (L)) sin (Oto) (4.5) 2/ 20/ / 2 Keeping terms of linear order in ho, that is, substituting to = t () within the argument of the sine function, we approximate to: to (L L h, (L sin ( ((L\ L\\ (L\ L ho L L L =sinm cos (0 t (L 2 2c 02 4c 2 4c} We now substitute the new interpretation of to into the original input laser field. it Eoeiwto it1Eoeiw(t()_ ) l 2 sin e)(t()L) + eC(t() ) where we have expanded the quantity e sin( ) cos(2(t(4) )) to first order in ho and expanded the cosine term. One can now interpret the effect of the gravitational wave on the input laser light as phase modulation creating two frequency sidebands with frequencies w = W 2. The grating compressor within a whitelight cavity will also see phasemodulated light, and will donate an appropriate phase shift in reflection to each frequency component. Let the compressor length for the fundamental carrier frequency be ALo. The sidebands travel lengths ALL = ALo T 6L, where 6L assumes a linear dependence of the compressor length on the gravitational wave frequency 0, a valid assumption as the GW frequencies are much smaller than the optical frequencies. With this approximation we define: The quantity t (f) is to be interpreted as the time the sideband fields must reach the first grating of the compressor in order to emerge simultaneously (in the local frame) with the carrier light. We now replace t (L) ofor the three frequency components. The carrier: it1r2Eoe~w(t(f+ALo)L _o) The upper sideband: i ho QlL\ L+n)L(t(+ALo)L) i ia itlr2Eo sin 2 ce 2c e 4c 2Q 4c 9 The lower sideband: iwh0o QL i(> n)(t( +AL0o) At iL imL 2 s 4c itlr2E 2Q ( ( 2e g e Iterating the process above, the time at which the light returns to the ITM may be approximated: t + ALo) t (L + ALo) sin ) cos t (L + ALo) 2 2c Q 4c 4c In propagating the light back to the ITM, we assume that phase modulations of phase modulated light may be neglected, as these are of order h2, and the sidebands merely return with the appropriate phase. Modulations of returning carrier light, however, are significant. The carrier light becomes: iw(t(LAL) ) i + sin i(t(L+AL) .c. 2Q ( 4c)I and we note that further frequency sidebands are born. The upper sideband returns to the ITM: itrE iwho. QL i t(L+ALo) i itlEo sm e c e 2c e 4c 2Q 4c 4 Likewise the lower sideband: who L\ )i(wn)(t(L+ALo)_ itlr2Eo sm e t~r2 0 2Q 4c,2 Thus the total returning field is: it1r2Eo [e(t(L+Lo) L AL)] iT2lrEoi h sin (IL ei(w+Q)t(L+ALo) i2h0 ( L4c itr2E ho sin OL Q)t(L+ALo) C Ztl2Eo sin e e 20 4c s itr iEho \ i OLh C i(w+Q)t(L+nA C (L 20 sin (4c AL 'iL ML c / 2c e 4c iw(L+ALo) c e i(L+ALo) MfL c e 4c AL+) ( +aL+) c P c ']1 . itlr2Eo sin 202Q (OL ei(_n))t(L+ALo) e 4c i(L ) i(3L +AL+) e Thus the total field may be expressed in an illuminating manner as follows. First the carrier component: itr2Eoe w(t(L+ Lo) Ao) with an upper signal sideband: tlr2Eowho sin 20 i(L i(w+Q)t(L+zALo) (L+ALo) ML i+ L 3iQL iALo iQ6L S4ce ) C 4c c C 4c c 4c and the lower: tir2Ewho (L nei( )t(L+ALo)e L+Lo) e L i6L 3iL MiALo iQL ,nsm e e 4c + c C 4c C c C c 2 4c The design of the whitelight cavity, in its idealized form, ensures the cancellation of the variation in phase with wavelength for the intracavity field. The whitelight condition is: d d (27rLw (A) where Li (A) is the total halftrip cavity length for a particular wavelength. The above equation leads to: dLi (A) L.i (A) dLwl (wi) Lwi (Lwi) dA A dwji i UwI Converting this equation to the language of our previous calculations, in which dL (uw ) = ':L, duwi = +, L (uw ) = L + ALo T 6L, and uwi = a Q, we find, for the upper sideband: TL I[ L ( \1 i dL = dw dwu = Ltot In + JAL j Lou ) 0 S(L + ALo 6L) => u6L = QL + QALo 6L When this whitelight condition is applied to the above equation for the upper sideband, its amplitude and phase becomes: tl2Ewh, sin L i(w+Q)t(L+ALo) i(L+ALo) [iQL L] 2 2 \ 4c,) tEouho I i(w+n)tL (L+ALo).1 L (rr2,) C C c. sin 1) 2Q 2c for light that has made one roundtrip of the cavity. Carrier light that has made n roundtrips of the gratingenhanced cavity will have upper sidebands: t Eowho (L\ cL+ALo)* ei(w+)t tE sin I n (rir2) e ot 20 ( 2c The positions of the optical components are fixed while the gravitational wave is interpreted as a phase modulation of the cavity's internal light field. The diffraction gratings (assumed perfectly efficient) will "see" phase modulated light, and will diffract each of the three frequency components into a different path. The lower sideband, carrier, and upper sideband are now also separated in time, emerging sequentially from the compound mirror, though all with the same phase. The gravitational wave's manipulation of spacetime fashions further frequency sidebands from the carrier light as it returns to the cavity input mirror, where it interferes with the incoming laser light before it propagates again through the cavity. Summing over all sidebands created in the carrier from the infinite past yields an amplitude transfer function for the upper signal sideband of the form1 twLsin ) 2icL TWLC () Lsin 2 (4.6) 2c 1 rir2e cI Let us compare this equation to that of a standard cavity, whose upper sideband transfer function has the form: 2i(w+Q)L S) t1Lsinc ( rL) rir2e2i( TcP Li)=c ii(Lw+)t (4.7) 2c 1 rIr2e c 1 r1r2e Because the cavity's length is locked to the laser, the term e2iwL/c = 1, hence, the e2iL/c term in the denominator of 4.7 is the cause of the standard cavity's limited 1 B. F. Whiting, S. Wise, G. Mueller, 2003, unpublished. 10, 102 10 10 : 1\ 10' E\ Figure 44. Whitelight cavity's gravitational wave response. The solid line is the normalized intracavity intensity for a whitelight cavity; the dashed is a standard cavity. Both curves have minima at multiples of the FSR. bandwidth. Figure 4 4 compares the sidebands intensities within standard and perfect whitelight cavities with identical lossless mirror reflectivities (r = 0.995, r2 = 1) and nominal cavity lengths (4134 m). The gravitational wave strain has 10' 10' \\ 10'  10 10\ 10, 10, 104 10, maximum amplitude h0 = 1023. The zeo of the whitelight curve, due to the s in unction in the numerator, correspond to frequencies for which the cavityhe dashed is a time causes the signal to be integrated over a full cycle. This effect is not evident in LIGO's sensitivityandard cavity. Both curves, where the arm cave minima at multiples ofdiminish the sensitivity well before these fFigurequencies. We see then that a effect of the alleged whitelight perfect whitelight cavities with identical lossless mirror reflectivities (r2 0.995, 2 cavity non the LIGO bandwidth The gravitational wave stincrease. 4.4 Revised Theory maximumWe now describe measurementhe zeros of the phase shift of light curve, due to the sin grating set which show that the gratingo frequencies fsor whitelight cavity concept is time causes the signal to be instead, the pair of gratings provides a wavelengthdependent phase shift nearlnsitivy cancelling the phase from the additional freespach the path lengthivity shown in Figurre these 5 [36, [frequencies. We see then that a effsuspect the above analyssleged whitelight cavity on the LIGO bandwidth would be a three orderofmagnitude increase. 4.4 Revised Theory We now describe measurements of the phase shift of light by such a parallel grating set which show that the gratingcompressor whitelight cavity concept is almost ,, ,ml. /. h/ wrong. Instead, the pair of gratings provides a wavelengthdependent phase shift nearly cancelling the phase from the additional freespace path length shown in Figure 45 [36], [37]. We began to suspect the above analysis when SM2 0 Figure 45. A pair of identical, parallel, facetoface diffraction gratings and two mirrors (\ll and M2) form a resonant cavity. repeated measurements of gratingenhanced FabryPerot cavities, Michelson and MachZehnder interferometers all has virtually identical frequency behavior to their gratingless counterparts. For example, measurements of the resonance bandwidth of FabryPerot cavities containing highefficiency gratings and configured to be v liteh;ht cavities were unchanged if we removed the gratings and restored the same nominal optical path length. We found no enhancement of bandwidth. Here we show why the expected enhancement does not occur. Moreover, we also show that the phase depends not only on the intergrating I' ii. but also on the exact relationship between the grating features as seen by the light [38]. This dependence leads to the nonintuitive result that the phase is modulated strongly if one of the gratings is translated parallel to its face, even though the optical paths in Figure 45 are wholly unaffected. 4.5 Planewave Treatment of a Grating Compressor The error lies with an inappropriate mix of geometrical and physical optics. Consider the effect of the parallel grating pair on infinite plane waves. In Figure 45, parallel reflective gratings are located in the y = 0 and y = D planes. We will calculate the electric field at the two gratings and in a plane normal to the outgoing light (where the righthand mirror in Figure 45 is located). The light field impinging from the left on the first grating is Ein= Eoeik(xsin aycosa)(48) The grating at y = 0 bestows a spatial phase modulation on the incoming plane wave. The phase factor is e6kG x0o); the periodic function G (x x') represents the grating profile, with origin at x'. This phase factor may be expanded in a Fourier series: C eik0(r <) meimP(r<9 (4.9) where g = 27/d. Each term of the series is a diffraction order. In the following we consider only the m = 1 order, set C_1 = and chose x' = 0 for the first grating. The light field leaving that grating is EF,out = Eoei[(ksin ag)x+kycos/] (4.10) where we have used the grating equation to substitute k sin a + mg for k sin / < 0. When the light reaches the second grating, on the y = D line, it again receives a spatial phase modulation ei9g(oa). The quantity xo is the xoffset of the second grating's periodic modulation with respect to that of the first grating. Note that the second grating is reversed relative to the first, so that its local coordinate runs in the x direction. We again use m = 1, making light of all wavelengths leave the second grating parallel to the incident light. The outgoing electric field at the second grating is E2,out Eoi[k( in a+D cosp)gxo (4.11) When the light finally arrives at a point (x, y) on the righthand mirror of Figure 45, the electric field will be em = Eoei[k{xsin a(yD)cosca+Dcos/3}gxo] (4.12) We must now analyze how the phase of the field at M2 changes with light frequency. The phase K (w, x, y) is S= [x sin a (y D) cos a + Dcos ] gxo (4.13) L (w) g (D tan 0 + o), (4.14) c where L (w) is the total, wavelengthdependent, geometric path from the first grating (at the origin) to the end mirror [35]. We compute the dispersion 0/0uw, using the grating equation to eliminate 3/j0w, and find S xsina (y D)cosa + D + tan K sin a (4.15) OUj c cos )\ c Eq. 4.15 makes it clear that the variation of phase with frequency cannot be set to zero. The earlier calculations based on the grating equation alone included only the geometric pathlength contribution in Eq. 4.15, leading to the (incorrect) prediction that a0)/aw could become zero, thus allowing for the possibility of a white light cavity [35]. Missing from reference [35] was the second term in the righthand side of Eq. 4.14, present due to the positiondependent phase shift light receives upon reflection from (or transmission through) the grating. The result of Eq. 4.15 is familiar to shortpulse laser physicists as the group d. 1 v [39]. The exact form of the additional phase shift is rarely a concern, as the pulse compressor does not depend upon absolute phases. To our knowledge, direct experimental verification of the phase shift's form has never been published. The phase shift of Eq. 4.13 may be expressed in a particularly illuminating way as + (w) = [Lo + D (cos a + cos3)] gxo (4.16) c where Lo is the perpendicular distance from the origin to the plane of M2, defined by L = x sin ay cos a. Analyzing the grating compressor with plane waves reveals the origin and significance of the positiondependent phase shift on reflection from S/ to photodiode c ^ Y actuator stage 'Cact5 V laser put to leHe N r lterferometer Figure 46. The second of a pair of parallel gratings within one arm of a MachZehnder interferometer is placed on an xy translation stage. Michelson interferometers with HeNe lasers monitor the motion of the stage. the gratings. We may now resume calculations with the geometrical optical path, so long as we do not neglect the additional phase associated with the gratings. This theory makes a very specific prediction, which may be experimentally confirmed, about how the oneway phase shift depends on the distance D between gratings and the spatial offset xo between grating profiles. 4.6 Tests of Corrected Theory 4.6.1 Experimental Setup To test that the phase shift does have the specific form of Eq. 4.16, we incorporated a pair of gratings into one arm of a MachZehnder interferometer, as shown in Figure 46. We used reflective gratings with 1500 grooves/mm and a design input angle a = 420 (so that 3 = 680) We also used an input angle of a = 500 ( = 57.30) in some of our trials. The gratings had a high efficiency, with !I! 'I.'. of the incident light diffracted into the firstorder by each of the two. We placed the second grating of our grating pair on a twoaxis translation stage. This stage allowed us to vary the parameters xo and D of the grating pair. We aligned the grating face with one of the two orthogonal axes of the stage, which we will call x' and y', as well as was possible with a naked human c A small misalignment angle 0 inevitably remains between the grating axes (x, y) and the stage axes (x', y'). The axes are related by x' = cos 0 + y sin 0. Assuming the construction of the translation stage is better than our ability to place the grating, we also have y' = sin 0 + y cos 0. As the stage moves along the x' direction, it will produce a combination of the effects on phase due to the phase's xo and D dependence; however, because 0 is small, the influence of xo, with a period equal to the grating period, will dominate. The converse is true when one moves the stage along y'. To calibrate the displacement of the translation stage, we attached to it the end mirror of a simple Michelson interferometer illuminated by a heliumneon laser. In fact, there are two mirrors (and two interferometers) set perpendicular to the two motions of the stage. To ensure that perpendicularity, we move the stage in the orthogonal direction, so that the mirror slews crabwise across the HeNe beam, and adjust its angle relative to the stage until we reduce the number of output intensity fringes to a minimum. Again, this technique relies upon good inherent perpendicularity in the stage's crossed axes. Whenever the stage moves, we monitor the output intensity of both interferometers. We quantify the stage motion by counting the HeNe fringes. The input to the MachZehnder interferometer is a 1064 nmwavelength gratingstabilized diode laser. For good contrast, the physical lengths of the two arms are nearly equal. We move the second grating along the x' and y' axes and observe the intensity fringes of the infrared interferometer and compare with theory. 4.6.2 Experimental Results While the light input angle a and the grating period d are known, 0 remains as a fitting parameter. The output intensity of the MachZehnder is fit to I (x') = A + B cos [K (x') + C], (4.17) where 2xr 2w KP (x') = (cosa + cos ) x' sin0 + x' cos 0, (4.18) A d for the x' motion, or to I (y') A + B cos [ (y') + C] (4.19) with 2w 2w S(y') (cos a + cos /) y' cos 0 y' sin 0, (4.20) A d for the orthogonal direction. The quantities A, B, and C are rather unimportant fitting parameters; the period of the output fringes determined by + is key. We make a leastsquare fit of the theory to our data by adjusting A, B, C, and 0. Figure 47 shows examples of typical results for a trial with a = 50. Figure 47 (top) shows the interference seen for movement along the x'direction, i.e., when the grating moves parallel to its face. This motion gives strong fringes; the measured fringe contrast is in the 92 !' range. Now, motion parallel to the grating face has no effect on the geometric path lengths inside the interferometer. Thus, our initial expectation (based on the geometric path length) was that the light phase would be unaffected by this motion. In contrast to this expectation, the phase of the light goes through a full cycle as the grating is translated by an amount d. In Figure 47 (bottom), we show the interference signal observed for motion along y'. We also plot, in addition to the predicted output from the theory above, the output intensity that would be observed if only the geometric path length were 0.6  S0.4  0.2 012345678910 0 1 2 3 4 5 6 7 8 9 10 Displacement along x'(pm) 0.8 S0.6  0.4  0 .2 ' 0 1 2 3 4 5 6 7 8 9 10 Displacement along y'(pm) Figure 47. (Top) Measured (crosses) and theoretical (solid line) data for motion parallel to the grating face. The light is incident at 500. (Bottom) The crosses show the data for motion perpendicular to the grating face. Of the two calculations, the results favor the planewave treatment, with the additional phase shift on reflection (solid line) over one based on geometric path length alone (dotted line). 05 04 03 0 2 02 0 3 0 4 0 5 0 5 10 15 20 25 trial Figure 48. Misalignment angle 0 as determined from fits of measured interference data to theory. The meaning of the symbols is as follows. Crosses: motion along x' with a = 500; Circles: motion along y' with a = 500; Asterisks: motion along x' with a = 420; x's: motion along y' with a = 420. The data in Figure 47 is from trials 2 and 8. changed by the grating motion. Use of geometric path alone predicts a period for the interference pattern that is different from the measured one, whereas the theory that incorporates the positiondependent phase shift predicts the period that we measured. The outcome makes clear that the setup cannot be understood with only the diffraction angle and the geometric path length. The additional positiondependent phase shift is real [40]. Figure 48 shows an indication of the agreement between experiment and the theory presented here. In it, we plot the misalignment angle 0 of the grating for a number of trials. Eleven of the twentythree measurements are derived from grating motion parallel to its face, and twelve from perpendicular motion. In every case, we obtained highcontrast fringes with agreement with theory comparable to what is shown in Figure 47. The error bars on each datum reflect uncertainty in a, d, 0, A, and the motion of the stage. Clusters of specific values for 0 indicate the systematic error in the alignment of the grating on the stage, but the overall errors are very small. The quality of the fit to the measured interference pattern is evident in Figure 47 and in the small values for the misalignment angles in Figure 48, averaging 0 = 0.030 0.120, a reasonable value for alignment by human The phase of light reflected by or transmitted through a diffraction grating cannot be deduced from the grating equation alone. That equation omits the curious result, derived above, that the absolute phase is proportional to the distance along the grating face at which the light strikes. Indeed, the flat gratings behave as mirrors tilted at angles Otilt = sin'(mA/2d) relative to the xaxis shown in Figure 45. We confirmed this theory by testing the dependence of light phase on the position of the grating. For a gratingcompressor setup, we found good agreement between this theory and the change of light phase as the mirror moved both parallel and perpendicular to its face. Our result shows that whitelight cavities cannot be built from grating pairs. In fact, one might have conjectured that causality should prevent whitelight cavities from being built in a much wider class of nondissipative systemsnot just grating pairs. Indeed, we have found that a pair of prisms has a similar effect to the gratings on the phase of light passing through them. Finally, we note that the phase effect discussed here is not unique to the grating pair and would arise in an experiment utilizing a single grating. In our arrangement, the first grating is fixed and serves to preserve the beam width and to keep the angle of the light leaving the second grating constant as wavelength is adjusted. Otherwise, it is equivalent to a mirror. Except for a loss of contrast, we expect that the data of Figure 47 would be identical if the first grating were replaced by an appropriately oriented mirror. 4.6.3 AOMs Interestingly enough, it seems a clue to the missing phase ili. i, was under our noses, indeed, in our optical cabinets, all along. If one wishes to make further measurements of the effect of a moving grating on diffracted light phase, an acoustooptic modulator (AOM) is a good candidate. In an AOM one applies an acoustic waveform to a crystal, sending pressure waves moving through the material at speeds on the order of thousands of meters per second. Light diffracted off this moving grating inside the crystal should exhibit the same phase shift behavior found in our slowmoving grating experiments. How would one see the phase shift effect? It would fre lpii modulate the diffracted light. In fact this Dopplershifted diffraction frequency shift is a wellknown phenomenon, but is rarely if ever connected in literature or discussion to grating compressors! 4.7 Alternative Solutions The extra phase shift due to transverse position along the grating at first appears to cancel the linewidthbroadening perfectly by coincidence. However, similar experiments with prism pairs, where no such hidden phases exist, yielded the same result. We believe this is a manifestation of nature's jealous guardianship of causality and speed limits. Our initial analysis confused group delay with phase d 1 i We speculate that the only way a white or even broadened cavity is possible is when there is a way to delay the light significantly with resonant absorption or a gain medium. There are several v in which this delay modification may be accomplished. The required anomalous dispersion is found in the center of atomic absorption lines, but the absorption would generally suppress any gravitational waveinduced signals. Wicht et al. [41] studied optically pumped atomic resonance systems and proved that one could have an appropriate anomalous dispersion for a whitelight cavity at a point of vanishing absorption and optical gain. 4.8 If Alternatives Succeed The final design for a full whitelight interferometer will include two whitelight cavities in its arms. Because we no longer need worry about limiting our bandwidth by increasing the arm cavity mirror reflectivities, these may be matched to the losses in the bandwidthenhancing black box for optimal coupling. Thus we may forego both power and signalrecycling mirrors at the symmetric 79 and antisymmetric ports, respectively. This has the added advantage of removing power from the thick substrates of the beamsplitter and input mirrors, reducing thermal loading problems. CHAPTER 5 CONCLUSION To move from first detection of a gravitational wave to gravitational astronomy with future detectors will require innovative noise suppression and signal enhancement. The parallel phase modulation technique for lengthcontrol is straightforward in theory and execution. We derived a stability requirement for the extra degree of freedom in the parallel versus serial modulation scheme that was achievable and reasonable. This technique could be applied to the next generation of interferometric detectors with little problem, we believe. The nearfield cooling effect is robust in principle, but we still lack definitive experimental proof that Rytov's heattransfer calculation method is correct for planar geometries. Forthcoming proofofprinciple experiments at this institution will be difficult. We will be challenged to keep two objects separated by a tiny distance, fighting Van der Waals and Casimir forces. We will compare the outcome of such experiments with alternative cooling methods such as those used in the cryogenic Japanese LCGT [42]. As for whitelight cavities, it seems there will be no such thing as a free lunch. Despite a (too) promising theoretical calculation, linewidth broadening remained elusive, and we realized we had repeated a surprisingly common mistake in optics. The sensitivity theorem is not to be taken lightly. If we could yet achieve the necessary dispersion with atomic resonances, we still have a far more complicated system than the failed diffractiongrating setup. Nonetheless, the rewards of a broadband FabryPerot cavity would be a tremendous boon to gravitational wave detection. We retain an iota of hope for this technique. APPENDIX A OPTICAL TRANSFER FUNCTIONS A.1 FabryPerot Cavity Using the convention that fields passing through a partially transmissive optic receive a 90 phase shift relative to reflected fields, the amplitudes immediately to the right of the first optic is Ea = itlEiE + rlEb Eb r22ikLEa, where k is the wavenumber of the light. Combining these equations, one finds the total steadystate intracavity field: itlEj, L itl (A.1) Ea 1 rir_  from which the cavity transmitted field can be found by transmission through the lefthand optic: tlt2eikfLE, Et = it2eikLEa 2 Tt (k, L) Ej,. (A.2) 1 r~1ri2AL The reflected amplitude is the sum of the light promptly reflected from and the amplified light leaking through the input mirror: r + (r2 + t2 2ikL Er = rEi, + itlEb 1 l i I T, (k, L) E (A.3) A.2 ThreeMirror Coupled Cavity Understanding of the threemirror coupled cavity aids in a simplifying perception of any more complicated optical system, as these usually behave in a similar manner. A threemirror nested cavity is decipted in Figure A.2, with Ea Eb r1, tl Figure A1. Electric fields in a FabryPerot cavity En Er Et r2 t2 r3 t3 Figure A2. Electric fields in a 3mirror coupled FabryPerot cavity Ein Er Et 83 mirror parameters as indicated. Having found the transfer function of fields reflected from a single cavity above, we can treat the middle and rightmost optic as a single mirror with a complex reflectivity given by Eq. A.3. For example, the intracavity field between the left and middle mirror is then 3 it E (A.4) 1 rircave2ikL1 where r 2 (r + t2) 2ikL2 cav 2r i'2 1 r2r 2 APPENDIX B HEAT TRANSFER THROUGH FIBERS A suspension fiber of diameter D and length Lfiber, connecting two heat reservoirs at temperatures T1 and T2 conducts a power P p kBD2 T1 T2 4 Lfiber which for silica fibers with a conductivity K = 1.4K with D 0.1 mm mK Lfiber = 0.2 m, TI = 40 K, and T2 = 10 K, gives [43] P 1.65pW. APPENDIX C CIRCUIT DIAGRAM All operational amplifiers are OP27. The circuit elements are as follows: Table C 1. Values of elements used in the MZ feedback circuit element value R1 1 kQ R2 1 kQ R3 1 kQ R4 1 kQ R5 100 kQ R6 1 kQ R7 1 kQ R8 1 kQ R9 1 kQ R10 2 kQ Rll 1 kQ R12 1 kQ R13 220 kQ C1 0.1/pF Vout +15Vo R7 R1 R12 R10 15V  Figure C 1. Diagram of analog ciruit used for MZ feedback control REFERENCES [1] A. Einstein, "Relativity: The Special and the General Theory," R. Lawson, trans., (Crown Trade Paperbacks, New York), 1961. 1.1 [2] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (W. H. Freeman and Co., San Francisco), 1979. 1.1 [3] J. H. Taylor, L. A. Fowler, and J. M. Weisberg, \!. irements of General Relativistic Effects in Binary Pulsar PSR1913+16," Nature, vol. 277, pp. 437, 1979. 1.1 [4] J. Weber, "Detection and generation of gravitational waves," Phys. Rev., vol. 117, p. 306, 1960. 1.2.1 [5] L. Ju, D. G. Blair, and C. Zhau, "Detection of Gravitational Waves," Rep. Prog. Phys., vol. 63, pp. 13171427, 2000. 1.2.1 [6] 0. D. Aguiar, L. A. Andrade, L. Camargo Filho, C. A. 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