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Space based Gravitational Wave Observations I – M Hewitson from Lake Como School of Advanced Studies

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Waves on the lake: the astrophysics behind gravitational waves
May 28 – June 1, 2018

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Space based Gravitational Wave Observations I – M Hewitson from Lake Como School of Advanced Studies

  1. 1. M Hewitson Institut für Gravitationsphysik der Leibniz Universität Hannover SIGRAV May 2018 Albert Einstein Institute Max Planck Institute for Gravitational Physics and Leibniz Universität Hannover Space based Gravitational Wave Observations LISA and its pathfinder
  2. 2. M Hewitson, LISA Detector, SIGRAV, May 2018 Martin Hewitson: Who he? • Undergrad and PhD in Glasgow • At AEI in Hannover since 2004 • Worked on: • GEO600 calibration and commissioning • LISA Pathfinder Data Analysis and operations • LISA mission formulation and instrumentation 2
  3. 3. M Hewitson, LISA Detector, SIGRAV, May 2018 Overview of lectures • Lecture 1 • why low(er) frequency? • low frequency, compare to ground based • measurement principle for LISA • discussion of noise sources • Lecture 2 • TDI: the basics • Building the LISA noise curve • Lecture 3 • LPF: what did we test, and why? 3
  4. 4. M Hewitson, LISA Detector, SIGRAV, May 2018 Why go to low frequency? • why go to low frequency? • Larger mass systems merge at lower frequency • Schwarzschild radius: • Orbital frequency for binary system: innermost stable circular orbit 4 r = 2Gm c2<latexit sha1_base64="Ow1aoIhoHRTb2ewj3GfTXNZdllQ=">AAAB/nicdVDLSsNAFJ3UV62vqLhyM1gEVyVJK20XQtGFLivYB7SxTKaTduhMEmYmQgkBf8WNC0Xc+h3u/BunbQQVPXDhcM693HuPFzEqlWV9GLml5ZXVtfx6YWNza3vH3N1ryzAWmLRwyELR9ZAkjAakpahipBsJgrjHSMebXMz8zh0RkobBjZpGxOVoFFCfYqS0NDAPBDyDfV8gnDjwEvI0wbdOOjCLVsmql6uVMtRkDk3q9ZpjnUI7U4ogQ3NgvveHIY45CRRmSMqebUXKTZBQFDOSFvqxJBHCEzQiPU0DxIl0k/n5KTzWyhD6odAVKDhXv08kiEs55Z7u5EiN5W9vJv7l9WLl19yEBlGsSIAXi/yYQRXCWRZwSAXBik01QVhQfSvEY6SzUDqxgg7h61P4P2k7Jdsq2deVYuM8iyMPDsEROAE2qIIGuAJN0AIYJOABPIFn4954NF6M10Vrzshm9sEPGG+fp5OUqA==</latexit><latexit sha1_base64="Ow1aoIhoHRTb2ewj3GfTXNZdllQ=">AAAB/nicdVDLSsNAFJ3UV62vqLhyM1gEVyVJK20XQtGFLivYB7SxTKaTduhMEmYmQgkBf8WNC0Xc+h3u/BunbQQVPXDhcM693HuPFzEqlWV9GLml5ZXVtfx6YWNza3vH3N1ryzAWmLRwyELR9ZAkjAakpahipBsJgrjHSMebXMz8zh0RkobBjZpGxOVoFFCfYqS0NDAPBDyDfV8gnDjwEvI0wbdOOjCLVsmql6uVMtRkDk3q9ZpjnUI7U4ogQ3NgvveHIY45CRRmSMqebUXKTZBQFDOSFvqxJBHCEzQiPU0DxIl0k/n5KTzWyhD6odAVKDhXv08kiEs55Z7u5EiN5W9vJv7l9WLl19yEBlGsSIAXi/yYQRXCWRZwSAXBik01QVhQfSvEY6SzUDqxgg7h61P4P2k7Jdsq2deVYuM8iyMPDsEROAE2qIIGuAJN0AIYJOABPIFn4954NF6M10Vrzshm9sEPGG+fp5OUqA==</latexit><latexit sha1_base64="Ow1aoIhoHRTb2ewj3GfTXNZdllQ=">AAAB/nicdVDLSsNAFJ3UV62vqLhyM1gEVyVJK20XQtGFLivYB7SxTKaTduhMEmYmQgkBf8WNC0Xc+h3u/BunbQQVPXDhcM693HuPFzEqlWV9GLml5ZXVtfx6YWNza3vH3N1ryzAWmLRwyELR9ZAkjAakpahipBsJgrjHSMebXMz8zh0RkobBjZpGxOVoFFCfYqS0NDAPBDyDfV8gnDjwEvI0wbdOOjCLVsmql6uVMtRkDk3q9ZpjnUI7U4ogQ3NgvveHIY45CRRmSMqebUXKTZBQFDOSFvqxJBHCEzQiPU0DxIl0k/n5KTzWyhD6odAVKDhXv08kiEs55Z7u5EiN5W9vJv7l9WLl19yEBlGsSIAXi/yYQRXCWRZwSAXBik01QVhQfSvEY6SzUDqxgg7h61P4P2k7Jdsq2deVYuM8iyMPDsEROAE2qIIGuAJN0AIYJOABPIFn4954NF6M10Vrzshm9sEPGG+fp5OUqA==</latexit><latexit sha1_base64="Ow1aoIhoHRTb2ewj3GfTXNZdllQ=">AAAB/nicdVDLSsNAFJ3UV62vqLhyM1gEVyVJK20XQtGFLivYB7SxTKaTduhMEmYmQgkBf8WNC0Xc+h3u/BunbQQVPXDhcM693HuPFzEqlWV9GLml5ZXVtfx6YWNza3vH3N1ryzAWmLRwyELR9ZAkjAakpahipBsJgrjHSMebXMz8zh0RkobBjZpGxOVoFFCfYqS0NDAPBDyDfV8gnDjwEvI0wbdOOjCLVsmql6uVMtRkDk3q9ZpjnUI7U4ogQ3NgvveHIY45CRRmSMqebUXKTZBQFDOSFvqxJBHCEzQiPU0DxIl0k/n5KTzWyhD6odAVKDhXv08kiEs55Z7u5EiN5W9vJv7l9WLl19yEBlGsSIAXi/yYQRXCWRZwSAXBik01QVhQfSvEY6SzUDqxgg7h61P4P2k7Jdsq2deVYuM8iyMPDsEROAE2qIIGuAJN0AIYJOABPIFn4954NF6M10Vrzshm9sEPGG+fp5OUqA==</latexit> fISCO / 1 Mtot<latexit sha1_base64="yfBdRAIH/qoe+plwB7Uxg87NqRg=">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</latexit><latexit sha1_base64="yfBdRAIH/qoe+plwB7Uxg87NqRg=">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</latexit><latexit sha1_base64="yfBdRAIH/qoe+plwB7Uxg87NqRg=">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</latexit><latexit sha1_base64="yfBdRAIH/qoe+plwB7Uxg87NqRg=">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</latexit>
  5. 5. M Hewitson, LISA Detector, SIGRAV, May 2018 GW Spectrum 5 http://rhcole.com/apps/GWplotter/ small bighuge = 3000 km = 300 million km = 30 ⇥ 1012 km 10 M☉ 106 M☉>109 M☉
  6. 6. M Hewitson, LISA Detector, SIGRAV, May 2018 Compare space and ground strain 6 LISA aLIGO
  7. 7. M Hewitson, LISA Detector, SIGRAV, May 2018 Challenges of observing at low frequency • what limits ground based detectors at low frequency? • Isolation: • suspensions on ground, free-fall in space • what limits at low frequencies? • on ground: external displacements, external forces • in space: small forces on test masses • smaller, lighter • Interferometry challenges • temperature, thermo-mechanical stability • readout noise 7
  8. 8. M Hewitson, LISA Detector, SIGRAV, May 2018 Remember the goal • want to measure the relative proper motion of free-falling test masses • infer from that the effect of gravitational waves 8
  9. 9. M Hewitson, LISA Detector, SIGRAV, May 2018 Isolation on ground • Complex pendulum style isolation for test masses • Get 1/f2 isolation from each stage 9 Brett Shapiro, G1400964-v3 ~free-fall along beam axis at frequencies above the pendulum resonance
  10. 10. M Hewitson, LISA Detector, SIGRAV, May 2018 Typical ground motion 10
  11. 11. M Hewitson, LISA Detector, SIGRAV, May 2018 Isolation on ground 11 Brett Shapiro, G1400964-v3 Brett Shapiro, G1400964-v3
  12. 12. M Hewitson, LISA Detector, SIGRAV, May 2018 aLIGO limiting noise sources (design) 12
  13. 13. M Hewitson, LISA Detector, SIGRAV, May 2018 ET low frequency Interferometer 13
  14. 14. M Hewitson, LISA Detector, SIGRAV, May 2018 Low frequency disturbances • Going to lower frequency (mHz) is not feasible on ground • Go to space! • What is the measurement principle in space? • What are the challenges of low-frequency observations in space? 14
  15. 15. M Hewitson, LISA Detector, SIGRAV, May 2018 Measurement principle: fundamentals • Aim is to measure the relative motion of free- falling test masses • On ground test masses are suspended, but are ‘free-falling’ along the measurement axis and above the suspension resonance 15 test mass in free-fall
  16. 16. M Hewitson, LISA Detector, SIGRAV, May 2018 Measurement principle: fundamentals • Compare different pairs of test masses arranged orthogonally to combat technical (frequency) noise 16
  17. 17. M Hewitson, LISA Detector, SIGRAV, May 2018 Measurement principle: fundamentals • Compare different pairs of test masses arranged orthogonally to combat technical (frequency) noise 16
  18. 18. M Hewitson, LISA Detector, SIGRAV, May 2018 Ground based interferometers 17 Compare length changes of each arm frequency noise is common to both Laser wavelength (frequency) is our ‘ruler’ frequency noise is indistinguishable from GW signal Actually use ~equal arm- length interferometer, insensitive to frequency noise
  19. 19. M Hewitson, LISA Detector, SIGRAV, May 2018 Length stability • Advanced suspensions produce very quiet mirrors
 • Advanced optics and readout result in displacement sensitivity < 10-19 m/√Hz at 100Hz! • Achievable because mirrors are held at operating point • Actuation noise is all at lower frequency 18
  20. 20. M Hewitson, LISA Detector, SIGRAV, May 2018 aLIGO displacement sensitivity 19
  21. 21. M Hewitson, LISA Detector, SIGRAV, May 2018 Homodyne versus heterodyne • why can’t we lock on a fringe in space? • orbital motion of free-falling bodies produces large relative motions • would need very strong (and noisy!) feedback forces to maintain ‘fixed' arm-lengths • this would need to be a mix of force on S/C (thrusters) and force on test masses • Use another concept instead: heterodyne interferometry with phase tracking • can measure over many fringes 20
  22. 22. M Hewitson, LISA Detector, SIGRAV, May 2018 The LISA Concept 21
  23. 23. M Hewitson, LISA Detector, SIGRAV, May 2018 The LISA Concept • Probe the change in proper time between pairs of free-falling test masses caused by GWs • Free-falling test masses are housed in drag- free spacecraft separated by a few million km • Proper time is inferred by the time of flight of photons exchanged between the satellites • We have multiple transponder links from which we can form Michelson-like signals 21
  24. 24. M Hewitson, LISA Detector, SIGRAV, May 2018 LISA Science 22
  25. 25. M Hewitson, LISA Detector, SIGRAV, May 2018 LISA is a mature concept • 1993 M3 proposal for 4 S/C ESA/NASA collaborative mission • 1995 LISA selected as ESA Cornerstone • 1997 3 S/C ESA/NASA LISA
 appears • 2005-2011 Joint ESA-NASA Mission Formulation study • 2012-13 Mission reformulated as ESA-led eLISA (evolving LISA) • 2013 Gravitational Universe selected as L3 science theme • 2017 LISA proposal selected as ESA’s L3 mission 23
  26. 26. M Hewitson, LISA Detector, SIGRAV, May 2018 LISA Proposal for ESA’s L3 mission slot 24 https://arxiv.org/abs/1702.00786 www.lisamission.org Submitted January 2017
  27. 27. M Hewitson, LISA Detector, SIGRAV, May 2018 20th June 2017 25 LISA selected as ESA's L3 mission! “The LISA trio of satellites to detect gravitational waves from space has been selected as the third large-class mission in ESA’s Science programme, while the Plato exoplanet hunter moves into development. These important milestones were decided upon during a meeting of ESA’s Science Programme Committee today, and ensure the continuation of ESA’s Cosmic Vision plan through the next two decades.” http://www.esa.int/Our_Activities/Space_Science/
  28. 28. M Hewitson, LISA Detector, SIGRAV, May 2018 LISA Measurement concept: Challenges • protect test masses from external forces which move test masses away from perfect geodesics • e.g. solar wind
 • can’t do interferometry as on ground (locked cavities) • high dynamics • beam divergence means most of the light is lost • need a transponder scheme 26
  29. 29. M Hewitson, LISA Detector, SIGRAV, May 2018 Test mass free-fall in space • Shield free-falling test mass (TM) in a spacecraft (S/C) • Need to ensure local forces between S/C and TM are ‘low enough’ • less relevant at higher frequencies on ground as they fall off with 1/f2 • S/C has no mechanical connection to the TM • drag-free S/C control • issues of charge accumulation on the TM (see later) 27
  30. 30. M Hewitson, LISA Detector, SIGRAV, May 2018 Measurement principle • make 3 measurements: • TM to S/C • S/C to S/C (contains the GW signal!) • S/C to TM • Combine to get TM to TM 28 ~million km m1 m2s1
  31. 31. M Hewitson, LISA Detector, SIGRAV, May 2018 What can possibly go wrong? 29 ~million km m1 m2s1 S/C are ‘shaking’ thrusters for drag-free control external forces (e.g. solar wind) local forces disturb TM magnetic force thermal residual gas force gradients
  32. 32. M Hewitson, LISA Detector, SIGRAV, May 2018 Lasers over millions of km 30 w(x) defined where amplitude falls to 1/e ~ 0.37
  33. 33. M Hewitson, LISA Detector, SIGRAV, May 2018 Lasers over millions of km • laser beam with gaussian profile diverges as 
 
 30 w(x) = w0 s 1 + ✓ z zR ◆2 <latexit sha1_base64="3qqp8VwLikfLm5AxUU9OnM3oWRE=">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</latexit><latexit sha1_base64="3qqp8VwLikfLm5AxUU9OnM3oWRE=">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</latexit><latexit sha1_base64="3qqp8VwLikfLm5AxUU9OnM3oWRE=">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</latexit><latexit sha1_base64="3qqp8VwLikfLm5AxUU9OnM3oWRE=">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</latexit> beam width Rayleigh range beam waist diameter w(x) defined where amplitude falls to 1/e ~ 0.37
  34. 34. M Hewitson, LISA Detector, SIGRAV, May 2018 Lasers over millions of km • laser beam with gaussian profile diverges as 
 
 • where the Rayleigh range is given by 
 
 30 w(x) = w0 s 1 + ✓ z zR ◆2 <latexit sha1_base64="3qqp8VwLikfLm5AxUU9OnM3oWRE=">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</latexit><latexit sha1_base64="3qqp8VwLikfLm5AxUU9OnM3oWRE=">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</latexit><latexit sha1_base64="3qqp8VwLikfLm5AxUU9OnM3oWRE=">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</latexit><latexit sha1_base64="3qqp8VwLikfLm5AxUU9OnM3oWRE=">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</latexit> beam width Rayleigh range beam waist diameter zR = ⇡w2 0 <latexit sha1_base64="EZfbS9OPrscEaVixF2hAm46kkkU=">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</latexit><latexit sha1_base64="EZfbS9OPrscEaVixF2hAm46kkkU=">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</latexit><latexit sha1_base64="EZfbS9OPrscEaVixF2hAm46kkkU=">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</latexit><latexit sha1_base64="EZfbS9OPrscEaVixF2hAm46kkkU=">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</latexit> wavelength w(x) defined where amplitude falls to 1/e ~ 0.37
  35. 35. M Hewitson, LISA Detector, SIGRAV, May 2018 Lasers over millions of km • laser beam with gaussian profile diverges as 
 
 • where the Rayleigh range is given by 
 
 • Far from the beam waist (large z), we have
 30 w(x) = w0 s 1 + ✓ z zR ◆2 <latexit sha1_base64="3qqp8VwLikfLm5AxUU9OnM3oWRE=">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</latexit><latexit 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sha1_base64="3qqp8VwLikfLm5AxUU9OnM3oWRE=">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</latexit><latexit sha1_base64="3qqp8VwLikfLm5AxUU9OnM3oWRE=">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</latexit> beam width Rayleigh range beam waist diameter zR = ⇡w2 0 <latexit 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sha1_base64="EZfbS9OPrscEaVixF2hAm46kkkU=">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</latexit><latexit sha1_base64="EZfbS9OPrscEaVixF2hAm46kkkU=">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</latexit><latexit sha1_base64="EZfbS9OPrscEaVixF2hAm46kkkU=">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</latexit> wavelength w(x) ⇡ z ⇡w0<latexit sha1_base64="jtU3cqgiP4D/qb3M3MeUQuj+iiM=">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</latexit><latexit sha1_base64="jtU3cqgiP4D/qb3M3MeUQuj+iiM=">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</latexit><latexit sha1_base64="jtU3cqgiP4D/qb3M3MeUQuj+iiM=">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</latexit><latexit sha1_base64="jtU3cqgiP4D/qb3M3MeUQuj+iiM=">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</latexit> w(x) defined where amplitude falls to 1/e ~ 0.37
  36. 36. M Hewitson, LISA Detector, SIGRAV, May 2018 Lasers over millions of km • laser beam with gaussian profile diverges as 
 
 • where the Rayleigh range is given by 
 
 • Far from the beam waist (large z), we have
 • For 2.5 million km, and laser wavelength of 1064 nm
 
 30 w(x) = w0 s 1 + ✓ z zR ◆2 <latexit sha1_base64="3qqp8VwLikfLm5AxUU9OnM3oWRE=">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</latexit><latexit sha1_base64="3qqp8VwLikfLm5AxUU9OnM3oWRE=">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</latexit><latexit sha1_base64="3qqp8VwLikfLm5AxUU9OnM3oWRE=">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</latexit><latexit sha1_base64="3qqp8VwLikfLm5AxUU9OnM3oWRE=">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</latexit> beam width Rayleigh range beam waist diameter zR = ⇡w2 0 <latexit sha1_base64="EZfbS9OPrscEaVixF2hAm46kkkU=">AAACDnicdVDLSsNAFJ34rPUVdelmsBRclaSttF0IRTcuq9gHNDFMJpN26OTBzESpIV/gxl9x40IRt67d+TdOH4KKHrhwOOfemXuPGzMqpGF8aAuLS8srq7m1/PrG5ta2vrPbEVHCMWnjiEW85yJBGA1JW1LJSC/mBAUuI113dDrxu9eECxqFl3IcEztAg5D6FCOpJEcv3jqpxQN4kcFjaPkc4dSKKbxxjKtyllpMveShzNELRsloVGrVClRkCkUajXrZOILmXCmAOVqO/m55EU4CEkrMkBB904ilnSIuKWYky1uJIDHCIzQgfUVDFBBhp9NzMlhUigf9iKsKJZyq3ydSFAgxDlzVGSA5FL+9ifiX10+kX7dTGsaJJCGefeQnDMoITrKBHuUESzZWBGFO1a4QD5HKRKoE8yqEr0vh/6RTLplGyTyvFpon8zhyYB8cgENgghpogjPQAm2AwR14AE/gWbvXHrUX7XXWuqDNZ/bAD2hvn6vom+M=</latexit><latexit sha1_base64="EZfbS9OPrscEaVixF2hAm46kkkU=">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</latexit><latexit sha1_base64="EZfbS9OPrscEaVixF2hAm46kkkU=">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</latexit><latexit sha1_base64="EZfbS9OPrscEaVixF2hAm46kkkU=">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</latexit> wavelength w(x) ⇡ z ⇡w0<latexit sha1_base64="jtU3cqgiP4D/qb3M3MeUQuj+iiM=">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</latexit><latexit sha1_base64="jtU3cqgiP4D/qb3M3MeUQuj+iiM=">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</latexit><latexit sha1_base64="jtU3cqgiP4D/qb3M3MeUQuj+iiM=">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</latexit><latexit sha1_base64="jtU3cqgiP4D/qb3M3MeUQuj+iiM=">AAACEHicdVC7TsMwFHV4lvIKMLJYVIiyVElb1HarYGEsEn1ITVQ5rtNadRLLdmhL1E9g4VdYGECIlZGNv8F9IAGCI13p6Jx77XuPxxmVyrI+jKXlldW19dRGenNre2fX3NtvyCgWmNRxxCLR8pAkjIakrqhipMUFQYHHSNMbXEz95g0RkkbhtRpz4gaoF1KfYqS01DFPhtnRKXQQ5yIaQccXCCe30GH6hS6aJA6ncNixJrBjZqycVSmUigWoyQyaVCrlvHUG7YWSAQvUOua7041wHJBQYYakbNsWV26ChKKYkUnaiSXhCA9Qj7Q1DVFApJvMDprAY610oR8JXaGCM/X7RIICKceBpzsDpPrytzcV//LasfLLbkJDHisS4vlHfsygiuA0HdilgmDFxpogLKjeFeI+0qEonWFah/B1KfyfNPI528rZV8VM9XwRRwocgiOQBTYogSq4BDVQBxjcgQfwBJ6Ne+PReDFe561LxmLmAPyA8fYJDlucnA==</latexit> w(x) defined where amplitude falls to 1/e ~ 0.37 w(x) = 2.5 ⇥ 109 [m] ⇥ 1064 ⇥ 10 9 [m] ⇡ 13.4[cm] ⇡ 6.4 km <latexit sha1_base64="DDr8x5c6sHCUEE5Tz7LVAFfwpCw=">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</latexit><latexit sha1_base64="DDr8x5c6sHCUEE5Tz7LVAFfwpCw=">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</latexit><latexit sha1_base64="DDr8x5c6sHCUEE5Tz7LVAFfwpCw=">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</latexit><latexit sha1_base64="DDr8x5c6sHCUEE5Tz7LVAFfwpCw=">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</latexit>
  37. 37. M Hewitson, LISA Detector, SIGRAV, May 2018 Lasers over millions of km 31 transmit aperture receive aperture
  38. 38. M Hewitson, LISA Detector, SIGRAV, May 2018 Lasers over millions of km • if we send out a 2W beam from S/C 1, how much power do we intercept at S/C 2? • depends on: • distance, receiving aperture (telescope), etc 31 transmit aperture receive aperture
  39. 39. M Hewitson, LISA Detector, SIGRAV, May 2018 Lasers over millions of km • if we send out a 2W beam from S/C 1, how much power do we intercept at S/C 2? • depends on: • distance, receiving aperture (telescope), etc • calculations on previous page assume we can get all the light out of the transmitting telescope • in practice, not practical, requires too large a telescope 31 transmit aperture receive aperture
  40. 40. M Hewitson, LISA Detector, SIGRAV, May 2018 Lasers over millions of km • if we send out a 2W beam from S/C 1, how much power do we intercept at S/C 2? • depends on: • distance, receiving aperture (telescope), etc • calculations on previous page assume we can get all the light out of the transmitting telescope • in practice, not practical, requires too large a telescope • Rather we should optimise transmit beam properties to maximise intensity over receiving telescope 31 transmit aperture receive aperture
  41. 41. M Hewitson, LISA Detector, SIGRAV, May 2018 Optimising beam waist and received power 32 [1] Holmes D, Avizonis P and Wrolstad K 1970 Appl. Opt. 9 2179–80 [2] S Barke et al 2015 Class. Quantum Grav. 32 095004
  42. 42. M Hewitson, LISA Detector, SIGRAV, May 2018 Optimising beam waist and received power • According to [1], the intensity of light received on-axis from a focused, apertured gaussian beam can be expressed as
 
 32 [1] Holmes D, Avizonis P and Wrolstad K 1970 Appl. Opt. 9 2179–80 [2] S Barke et al 2015 Class. Quantum Grav. 32 095004 I0 = 2P ⇡!2 " exp ⇥ (a/!)2 ⇤ exp ⇥ (b/!)2 ⇤ 2 + 4 exp ⇥ (a2 + b2 )/!2 ⇤ sin2 ⇥ ⇡(b2 a2 )(z R)/2 zR ⇤ ( z/⇡!2)2 + [(z R)/R] 2 # , <latexit sha1_base64="3Blp4YY1rX1rIE/c+jnh6tnCFg4=">AAADD3icdVJJb9NAFB6brYSlKRy5PBFVcgRJbDeozQGpggvcQkXaSrETjSfjZFRv8owRqeV/wIW/woUDCHHlyo1/w4uXqmV50mje+77vLbN4SSCkMs1fmn7t+o2bt7Zut+7cvXd/u73z4FjGWcr4hMVBnJ56VPJARHyihAr4aZJyGnoBP/HOXm74k3c8lSKO3qp1wt2QLiPhC0YVQvMdbRfg9dyE5+D4KWW5DeMidxIBThzyJZ3ZBTgB99W0oqvAcPj7pIJ7Bh1U0u7MdlKxXCkXenBZ4P0lqDaM4QkMr2hpiXkzuztoBmiKOlJEGJVCABzRQBm2QknXOO8ddQdI4sEXFM7hCOq0AnIwLvABpjV1q/Z1varARRZSAAU04dPWvN0x++Zob3+4B+iUhs5odGCbz8CqkQ6pbTxv/3QWMctCHikWUCmnlpkoN6epEizgRcvJJE8oO6NLPkU3oiGXbl6+ZwG7iCzAj1NckYISvZyR01DKdeihMqRqJf/kNuC/uGmm/AM3F1GSKR6xqpGfBaBi2HwOWIiUMxWs0aEsFTgrsBXFh1f4hTaX0JwU/u8c233L7Ftvhp3DF/V1bJFH5DExiEX2ySF5RcZkQpj2QfukfdG+6h/1z/o3/Xsl1bU65yG5YvqP32607oA=</latexit><latexit sha1_base64="3Blp4YY1rX1rIE/c+jnh6tnCFg4=">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</latexit><latexit sha1_base64="3Blp4YY1rX1rIE/c+jnh6tnCFg4=">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</latexit><latexit sha1_base64="3Blp4YY1rX1rIE/c+jnh6tnCFg4=">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</latexit>
  43. 43. M Hewitson, LISA Detector, SIGRAV, May 2018 Optimising beam waist and received power • According to [1], the intensity of light received on-axis from a focused, apertured gaussian beam can be expressed as
 
 • Ouch… apply some assumptions: • The arm-length is very long compared to any beam properties, so L→∞ • We set 
 
 where dtel is the diameter of the transmitting telescope, and α is the waist radius in units of the telescope radius • The outer radius of the annulus, b, is set to dtel/2 • The intensity is considered at the far SC, i.e., where z=L 32 [1] Holmes D, Avizonis P and Wrolstad K 1970 Appl. Opt. 9 2179–80 [2] S Barke et al 2015 Class. Quantum Grav. 32 095004 I0 = 2P ⇡!2 " exp ⇥ (a/!)2 ⇤ exp ⇥ (b/!)2 ⇤ 2 + 4 exp ⇥ (a2 + b2 )/!2 ⇤ sin2 ⇥ ⇡(b2 a2 )(z R)/2 zR ⇤ ( z/⇡!2)2 + [(z R)/R] 2 # , <latexit sha1_base64="3Blp4YY1rX1rIE/c+jnh6tnCFg4=">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</latexit><latexit 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  44. 44. M Hewitson, LISA Detector, SIGRAV, May 2018 Far-field intensity • Applying the assumptions gives an expression for the far-field intensity: 33 Irec = ⇡Pteld2 tel 2L2 2 ↵2 exp ⇥ 2/↵2 ⇤ exp ⇥ 1/↵2 ⇤ 1 2 , <latexit sha1_base64="Dw6nx/nowq+ks3vZpx72eJz+LOc=">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</latexit><latexit sha1_base64="Dw6nx/nowq+ks3vZpx72eJz+LOc=">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</latexit><latexit sha1_base64="Dw6nx/nowq+ks3vZpx72eJz+LOc=">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</latexit><latexit sha1_base64="Dw6nx/nowq+ks3vZpx72eJz+LOc=">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</latexit> maximise this by choosing beam waist
  45. 45. M Hewitson, LISA Detector, SIGRAV, May 2018 Received power • Received power depends on • far-field intensity • receiving aperture (telescope) 34 Prec = ⇡ ✓ Dtel 2 ◆2 Irec, <latexit sha1_base64="bXkJh4iT5kUagNrHM9u7sUWxzPU=">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</latexit><latexit sha1_base64="bXkJh4iT5kUagNrHM9u7sUWxzPU=">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</latexit><latexit sha1_base64="bXkJh4iT5kUagNrHM9u7sUWxzPU=">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</latexit><latexit sha1_base64="bXkJh4iT5kUagNrHM9u7sUWxzPU=">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</latexit> Some typical values for LISA: L = 2.5 Gm Pout = 1.25 W Dtel = 30 cm Irec = 10.25 [nW/m^2] Prec = 507.31 [pW]
  46. 46. M Hewitson, LISA Detector, SIGRAV, May 2018 So where are we? 35
  47. 47. M Hewitson, LISA Detector, SIGRAV, May 2018 So where are we? • We use lasers to measure relative motion of free-falling test masses 35
  48. 48. M Hewitson, LISA Detector, SIGRAV, May 2018 So where are we? • We use lasers to measure relative motion of free-falling test masses • But received power is too low to do a round- trip measurement 35
  49. 49. M Hewitson, LISA Detector, SIGRAV, May 2018 So where are we? • We use lasers to measure relative motion of free-falling test masses • But received power is too low to do a round- trip measurement • Homodyne detection is challenging due to large dynamic range 35
  50. 50. M Hewitson, LISA Detector, SIGRAV, May 2018 So where are we? • We use lasers to measure relative motion of free-falling test masses • But received power is too low to do a round- trip measurement • Homodyne detection is challenging due to large dynamic range • We need another scheme: • transponder-like scheme 35
  51. 51. M Hewitson, LISA Detector, SIGRAV, May 2018 Transponder-like scheme in LISA • send out one beam to far S/C • in far S/C compare received beam to local beam • control difference of the two laser frequencies • Laser 2 is now locked to Laser 1 • f2-f1 is kept constant • Far S/C returns this Laser 2 to S/C 1 • measure phase difference between two lasers • encodes GW signal + S/C motion + frequency noise of Laser 1 36 f2f1 Δs frequency lock polarisation optics omitted
  52. 52. M Hewitson, LISA Detector, SIGRAV, May 2018 Full constellation • One master laser locked to, e.g., a cavity to stabilise its frequency • All other lasers locked to this (in a chain) 37 Barke, Inter-Spacecraft Frequency Distribution, PhD Thesis 2015
  53. 53. M Hewitson, LISA Detector, SIGRAV, May 2018 Features of this approach • Transmitted lasers are subject to doppler shifts due to orbital motions • Line-of-sight relative velocities of S/C impose frequency shifts on received light 38 f = f0 v c<latexit sha1_base64="opGhkqcbb686VAjeoor+egw0Tg0=">AAACCnicdZDLSgMxFIYzXmu9jbp0Ey2Cq5JpK20XQlEXLivYC3SGIZNm2tDMhSRTKEPXbnwVNy4UcesTuPNtTNsRVPSHwM93zuHk/F7MmVQIfRhLyyura+u5jfzm1vbOrrm335ZRIghtkYhHouthSTkLaUsxxWk3FhQHHqcdb3Q5q3fGVEgWhbdqElMnwIOQ+YxgpZFrHtlXlCsMfXgOfRdB2xeYpBkcT1Mydc0CKqJ6uVopQ23m0qZer5XQGbQyUgCZmq75bvcjkgQ0VIRjKXsWipWTYqEY4XSatxNJY0xGeEB72oY4oNJJ56dM4YkmfehHQr9QwTn9PpHiQMpJ4OnOAKuh/F2bwb9qvUT5NSdlYZwoGpLFIj/hUEVwlgvsM0GJ4hNtMBFM/xWSIdZhKJ1eXofwdSn837RLRQsVrZtKoXGRxZEDh+AYnAILVEEDXIMmaAEC7sADeALPxr3xaLwYr4vWJSObOQA/ZLx9AsMQmbU=</latexit><latexit sha1_base64="opGhkqcbb686VAjeoor+egw0Tg0=">AAACCnicdZDLSgMxFIYzXmu9jbp0Ey2Cq5JpK20XQlEXLivYC3SGIZNm2tDMhSRTKEPXbnwVNy4UcesTuPNtTNsRVPSHwM93zuHk/F7MmVQIfRhLyyura+u5jfzm1vbOrrm335ZRIghtkYhHouthSTkLaUsxxWk3FhQHHqcdb3Q5q3fGVEgWhbdqElMnwIOQ+YxgpZFrHtlXlCsMfXgOfRdB2xeYpBkcT1Mydc0CKqJ6uVopQ23m0qZer5XQGbQyUgCZmq75bvcjkgQ0VIRjKXsWipWTYqEY4XSatxNJY0xGeEB72oY4oNJJ56dM4YkmfehHQr9QwTn9PpHiQMpJ4OnOAKuh/F2bwb9qvUT5NSdlYZwoGpLFIj/hUEVwlgvsM0GJ4hNtMBFM/xWSIdZhKJ1eXofwdSn837RLRQsVrZtKoXGRxZEDh+AYnAILVEEDXIMmaAEC7sADeALPxr3xaLwYr4vWJSObOQA/ZLx9AsMQmbU=</latexit><latexit sha1_base64="opGhkqcbb686VAjeoor+egw0Tg0=">AAACCnicdZDLSgMxFIYzXmu9jbp0Ey2Cq5JpK20XQlEXLivYC3SGIZNm2tDMhSRTKEPXbnwVNy4UcesTuPNtTNsRVPSHwM93zuHk/F7MmVQIfRhLyyura+u5jfzm1vbOrrm335ZRIghtkYhHouthSTkLaUsxxWk3FhQHHqcdb3Q5q3fGVEgWhbdqElMnwIOQ+YxgpZFrHtlXlCsMfXgOfRdB2xeYpBkcT1Mydc0CKqJ6uVopQ23m0qZer5XQGbQyUgCZmq75bvcjkgQ0VIRjKXsWipWTYqEY4XSatxNJY0xGeEB72oY4oNJJ56dM4YkmfehHQr9QwTn9PpHiQMpJ4OnOAKuh/F2bwb9qvUT5NSdlYZwoGpLFIj/hUEVwlgvsM0GJ4hNtMBFM/xWSIdZhKJ1eXofwdSn837RLRQsVrZtKoXGRxZEDh+AYnAILVEEDXIMmaAEC7sADeALPxr3xaLwYr4vWJSObOQA/ZLx9AsMQmbU=</latexit><latexit sha1_base64="opGhkqcbb686VAjeoor+egw0Tg0=">AAACCnicdZDLSgMxFIYzXmu9jbp0Ey2Cq5JpK20XQlEXLivYC3SGIZNm2tDMhSRTKEPXbnwVNy4UcesTuPNtTNsRVPSHwM93zuHk/F7MmVQIfRhLyyura+u5jfzm1vbOrrm335ZRIghtkYhHouthSTkLaUsxxWk3FhQHHqcdb3Q5q3fGVEgWhbdqElMnwIOQ+YxgpZFrHtlXlCsMfXgOfRdB2xeYpBkcT1Mydc0CKqJ6uVopQ23m0qZer5XQGbQyUgCZmq75bvcjkgQ0VIRjKXsWipWTYqEY4XSatxNJY0xGeEB72oY4oNJJ56dM4YkmfehHQr9QwTn9PpHiQMpJ4OnOAKuh/F2bwb9qvUT5NSdlYZwoGpLFIj/hUEVwlgvsM0GJ4hNtMBFM/xWSIdZhKJ1eXofwdSn837RLRQsVrZtKoXGRxZEDh+AYnAILVEEDXIMmaAEC7sADeALPxr3xaLwYr4vWJSObOQA/ZLx9AsMQmbU=</latexit> relative velocity along line-of-sight laser frequency (emitted) speed of light frequency shift
  54. 54. M Hewitson, LISA Detector, SIGRAV, May 2018 Orbital motions 39
  55. 55. M Hewitson, LISA Detector, SIGRAV, May 2018 Relative velocities 40
  56. 56. M Hewitson, LISA Detector, SIGRAV, May 2018 Doppler Shifts 41
  57. 57. M Hewitson, LISA Detector, SIGRAV, May 2018 Side bar: Heterodyne interferometry • how does it work? 42 f1 f2 f2-f1 Phase of beat signal depends on path lengths
  58. 58. M Hewitson, LISA Detector, SIGRAV, May 2018 Interference 43 f1 f2 f2-f1 50/50 beam splitter 50% reflective coating 2 light fields combine at a beam splitter
  59. 59. M Hewitson, LISA Detector, SIGRAV, May 2018 Detection 44 f1 f2 f2-f1 Note: carrier frequency is neglected throughout. It is THz, and can’t be detected. We only detect average light-fields. photodetector Sum amplitudes of light field! photodetector is a ‘square law’ detector ⌫ = c = 299792458 ⇥ m s 1 ⇤ 1064 [nm] =0.28 THz<latexit sha1_base64="MZ8LFLu90Nsn0FYWyrkumiGQd34=">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</latexit><latexit sha1_base64="MZ8LFLu90Nsn0FYWyrkumiGQd34=">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</latexit><latexit sha1_base64="MZ8LFLu90Nsn0FYWyrkumiGQd34=">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</latexit><latexit sha1_base64="MZ8LFLu90Nsn0FYWyrkumiGQd34=">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</latexit>
  60. 60. M Hewitson, LISA Detector, SIGRAV, May 2018 Do the math… 45 s1 = a1e(i!1t+ 1) <latexit sha1_base64="6QNrb7AVmXq+fThlcNM/36N1+Uo=">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</latexit><latexit 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sha1_base64="26wMQHKP7IFyEnwqVrhfwRIg8Zk=">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</latexit> s2 = a2e(i!2t+ 2) <latexit 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  61. 61. M Hewitson, LISA Detector, SIGRAV, May 2018 Do the math… 45 s1 = a1e(i!1t+ 1) <latexit sha1_base64="6QNrb7AVmXq+fThlcNM/36N1+Uo=">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</latexit><latexit 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sha1_base64="26wMQHKP7IFyEnwqVrhfwRIg8Zk=">AAACDXicdVDLSgMxFM3UV62vqks3wSpUhDIRte1CEN24VLBW6NQhk962oZkHSUYoQ3/Ajb/ixoUibt2782/MtCOo6IGEk3Pu5eYeLxJcadv+sHJT0zOzc/n5wsLi0vJKcXXtSoWxZNBgoQjltUcVCB5AQ3Mt4DqSQH1PQNMbnKZ+8xak4mFwqYcRtH3aC3iXM6qN5Ba3lEvwEabmhpukzJ3Qh1760ngXO1Gfu2Rn5BZLdqVeO6jv72G7Yo+RkkNiVw8wyZQSynDuFt+dTshiHwLNBFWqRexItxMqNWcCRgUnVhBRNqA9aBkaUB9UOxlvM8LbRungbijNCTQeq987EuorNfQ9U+lT3Ve/vVT8y2vFultrJzyIYg0BmwzqxgLrEKfR4A6XwLQYGkKZ5OavmPWppEybAAsmhK9N8f/kaq9C7Aq5IKXjkyyOPNpAm6iMCKqiY3SGzlEDMXSHHtATerburUfrxXqdlOasrGcd/YD19gmzVZlv</latexit> s2 = a2e(i!2t+ 2) <latexit 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sha1_base64="Gu1EdMiggggElKTntSga7Byy/vo=">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</latexit><latexit sha1_base64="9kY5xf0tTjSlns4PUe0j9scsxwI=">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</latexit>
  62. 62. M Hewitson, LISA Detector, SIGRAV, May 2018 Do the math… 45 s1 = a1e(i!1t+ 1) <latexit sha1_base64="6QNrb7AVmXq+fThlcNM/36N1+Uo=">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</latexit><latexit 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sha1_base64="SLm0eeC37HFropyuiQND87bPHXI=">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</latexit><latexit sha1_base64="26wMQHKP7IFyEnwqVrhfwRIg8Zk=">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</latexit> s2 = a2e(i!2t+ 2) <latexit sha1_base64="EsfXMQd+1iVZYBQZcTTxjkU+sTk=">AAACDXicdVDLahsxFL2Tvhz3NU0gm2xE3UJKYdB4ksZeFEy7ydKG2gl43KlGvrZFNA8kTcAM/oFu8ivdZJEQuu0+u2z6LdXYKSSlOSBxdM69XN0T51JoQ+m1s/bg4aPHT2rr9afPnr946b7aGOisUBz7PJOZOoqZRilS7BthJB7lClkSSzyMjz9X/uEJKi2y9IuZ5zhK2DQVE8GZsVLkvtFRk3wkzN74tdwRYZbgtHoZ8p6E+UxEzXeLyG1QjwZtfz8g1As+BK3dirTbrWCPEt+jSzQ6W73f3wCgG7lX4TjjRYKp4ZJpPfRpbkYlU0ZwiYt6WGjMGT9mUxxamrIE9ahcbrMgb60yJpNM2ZMaslRvd5Qs0XqexLYyYWam//Uq8X/esDCT1qgUaV4YTPlq0KSQxGSkioaMhUJu5NwSxpWwfyV8xhTjxgZYtyH83ZTcTwZNz6ee3/MbnU+wQg224TXsgA/70IED6EIfOHyHH3AOF86pc+ZcOj9XpWvOTc8m3IHz6w9v5pt0</latexit><latexit sha1_base64="k/5dNYhxTx5NlbcYMGDe2I8yfjk=">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</latexit><latexit sha1_base64="k/5dNYhxTx5NlbcYMGDe2I8yfjk=">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</latexit><latexit sha1_base64="xIw4UzqXq2Ak9bUNHlOCdlYGrSA=">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</latexit> Ipd / |s1 + s2|2 <latexit sha1_base64="mU0tU3vIIZWgAxH+2NRTo/0kjv4=">AAACCXicdVA9SwNBEJ2LXzF+RS1tFqMgCOEuKTRd0Ea7COYDknjsbTbJkr27ZXdPCJe0Nv4VGwuDWAn+Azt/iL2bRDGKPhh4vDfDzDxPcKa0bb9Zibn5hcWl5HJqZXVtfSO9uVVRYSQJLZOQh7LmYUU5C2hZM81pTUiKfY/Tqtc7HfvVayoVC4NL3Re06eNOwNqMYG0kN43O3bghfSRaQ9QQMhQ6RAPlOugQKTc3uMq56YyTtSdA9jcp5Ar5Qh59WZni3vvoGQBKbvq10QpJ5NNAE46Vqju20M0YS80Ip8NUI1JUYNLDHVo3NMA+Vc148skQ7RulhdqhNBVoNFFnJ2LsK9X3PdPpY91Vv72x+JdXj3T7uBmzQESaBmS6qB1xZN4dx4JaTFKied8QTCQztyLSxRITbcJLzYbwP6nkso6ddS6cTPEEpkjCDuzCAThwBEU4gxKUgcAN3MEDjKxb6956tJ6mrQnrc2YbfsB6+QCQ7Jvk</latexit><latexit sha1_base64="WVpchXEjEv9JslRxBGlU9txHqNI=">AAACCXicdVDLSsNAFJ34rK2PqEs3g1UQhJKkC+2u6EZ3FewDmhom02k7dJIMM5NCSbt146+4cWERt/0Dd36Irp22ilX0wIXDOfdy7z0+Z1Qqy3o1FhaXlldWU2vpzPrG5pa5vVORUSwwKeOIRaLmI0kYDUlZUcVIjQuCAp+Rqt89n/jVHhGSRuG16nPSCFA7pC2KkdKSZ8JLL3FFAHlzCF0uIq4iOJCeDY+h9JzBjeOZWTtnTQGtb1JwCvlCHn5Z2eLB22jcy7yXPPPFbUY4DkioMENS1m2Lq0aChKKYkWHajSXhCHdRm9Q1DVFAZCOZfjKEh1ppwlYkdIUKTtX5iQQFUvYDX3cGSHXkb28i/uXVY9U6bSQ05LEiIZ4tasUM6ncnscAmFQQr1tcEYUH1rRB3kEBY6fDS8yH8TypOzrZy9pWdLZ6BGVJgD+yDI2CDE1AEF6AEygCDW3APHsHIuDMejCfjeda6YHzO7IIfMMYfiYydXg==</latexit><latexit sha1_base64="WVpchXEjEv9JslRxBGlU9txHqNI=">AAACCXicdVDLSsNAFJ34rK2PqEs3g1UQhJKkC+2u6EZ3FewDmhom02k7dJIMM5NCSbt146+4cWERt/0Dd36Irp22ilX0wIXDOfdy7z0+Z1Qqy3o1FhaXlldWU2vpzPrG5pa5vVORUSwwKeOIRaLmI0kYDUlZUcVIjQuCAp+Rqt89n/jVHhGSRuG16nPSCFA7pC2KkdKSZ8JLL3FFAHlzCF0uIq4iOJCeDY+h9JzBjeOZWTtnTQGtb1JwCvlCHn5Z2eLB22jcy7yXPPPFbUY4DkioMENS1m2Lq0aChKKYkWHajSXhCHdRm9Q1DVFAZCOZfjKEh1ppwlYkdIUKTtX5iQQFUvYDX3cGSHXkb28i/uXVY9U6bSQ05LEiIZ4tasUM6ncnscAmFQQr1tcEYUH1rRB3kEBY6fDS8yH8TypOzrZy9pWdLZ6BGVJgD+yDI2CDE1AEF6AEygCDW3APHsHIuDMejCfjeda6YHzO7IIfMMYfiYydXg==</latexit><latexit sha1_base64="peDpQyYi9p89+s+4G5zmJJIP9BU=">AAACCXicdVDLSsNAFJ34rPUVdelmsAiCUJJ0od0V3eiugn1AE8NkMmmHTpJhZiKUtFs3/oobF4q49Q/c+TdOH2IVPXDhcM693HtPwBmVyrI+jIXFpeWV1cJacX1jc2vb3NltyjQTmDRwylLRDpAkjCakoahipM0FQXHASCvon4/91i0RkqbJtRpw4sWom9CIYqS05Jvw0s9dEUMejqDLRcpVCofSt+ExlL4zvHF8s2SXrQmg9U2qTrVSrcAvqwRmqPvmuxumOItJojBDUnZsiysvR0JRzMio6GaScIT7qEs6miYoJtLLJ5+M4KFWQhilQlei4ESdn8hRLOUgDnRnjFRP/vbG4l9eJ1PRqZfThGeKJHi6KMoY1O+OY4EhFQQrNtAEYUH1rRD3kEBY6fCK8yH8T5pO2bbK9pVdqp3N4iiAfXAAjoANTkANXIA6aAAM7sADeALPxr3xaLwYr9PWBWM2swd+wHj7BMZ8mRU=</latexit> Ipd / |a1|2 + |a2|2 + 2|a1||a2| cos ((!2 !1)t + )<latexit sha1_base64="S2Hhk4/+IIjDEooVIkJU4lTn0dY=">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</latexit><latexit sha1_base64="Gu1EdMiggggElKTntSga7Byy/vo=">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</latexit><latexit sha1_base64="Gu1EdMiggggElKTntSga7Byy/vo=">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</latexit><latexit sha1_base64="9kY5xf0tTjSlns4PUe0j9scsxwI=">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</latexit> d.c. power

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