E2E s Physics tools. Biplab Bhawal. Optics Electronics Mechanical Mathematical functions Data generation and output. Ligo doc. no.

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1 E2E s Physics tools Ligo doc. no. G E Date: Mar 18, 2002 E2E school, LLO Biplab Bhawal LIGO, Caltech Tools: Optics Electronics Mechanical Mathematical functions Data generation and output 1

2 Optics Laser & Field Field operations modulator, sideband-gen, sideband-filter, freq-shifter Beam operations propagator,shifter,wiggler Mirror, lens, telescope Summation cavities Fabry-Perot, triangular, recycling cavity, Measurements: power-meter, photodetector, 2

3 Laser (The field-gen module) Laser (a la field-gen ) generates field at a fixed central frequency It could be plane-wave or a Gaussian beam If it s a Gaussian beam, its basis (distance to waist and waist-size) needs to be specified amplitude of different spatial modes (in Hermite-Gaussian basis) can be specified Distance to waist, Z waist-size,w 0 3

4 Sideband generator SIDEBAND- FILTER SIDEBAND-GEN amplitude frequency LASER CARRIER Sideband-gen : E out = E in Exp( iγ ϕ sin( Ωt) ) Exp( Γ amp sin( Ωt) ) N N E out E in ( i) n i Ji( Γ ϕ ) I n i ( Γ amp ) n = N i = N exp( inωt) 4

5 Phase and Frequency noise (as input to Laser) P o white noise generator Digital Filter spectrum same as of dp(t) power LASER white noise generator Digital Filter spectrum same as of df(t) Integrator (dig. filter) phase Note: All frequencies of carrier and sidebands are constants during the simulation implementation of frquency noise: t φ() t = ωt ()t d = 0 ω 0 t + t δω() t 0 At any other point use Field-modulator module out = in * (1+del_amp) * exp(i*phi) 5

6 Time-domain modal model (w 0,-z ) substrate (w 0,z) z w 0 w(z) (w 0,z ) coating waist position (w 0,-z) coating Field carries two modal info: waist-size, dist-to-waist Tilt, shift, curvature mismatch are treated using mode decomposition matrix Modal basis changes after passing thru lens/curved mirror on reflection at an angle from a curved mirror 6

7 contents of Field In E2E modes upto 4th order (15 modes) can be considered spatial modes frequency carrier a power-meter module can measure power for any mode of any frequency For one freq: total power of all modes or modes of a certain order 7

8 Photodetector Pd-demod: photodetector+demodulation+low_pass_filter shape [0] y x [2] [1] [3] [4] Figure 1: shape number of detectors [6] [5] [7] [8] detmap y axis phi_end r_max r_min gap (dx0,dy0) phi_begin x axis Figure 2: Specification of a detector ready-made boxes y x circular_det.box xhalf_det.box yhalf_det.box Figure 3: detector boxes quad_det.box 8

9 misalignment effects (in Hermite-Gaussian basis) Initial beam : k - mode no. ; w - waist AU 0 Beam-wiggler : deviation(p)ofbeambyanangle A U 0 + jp kw U 2π 1 X θ y Z Propagation Laser Beam wiggler 9

10 misalignment effects Beam-shifter : lateral displacement (d) 2 A U -- 0 π 1 2 d + ---U w 1 X x Z 10

11 misalignment effects in Mirror mirror tilts : rotation (r) by an angle (pitch or yaw) A U 0 + jr kw U 2π 1 mirror s lateral shift : eqv. to rotation d 11

12 Mode Mismatch Mode mismatch: change in waist size shift in beam position Waist-position mismatch (b) : b A U 0 + j kw 2 { U 0 + U 2 } Waist-size mismatch (s) : s A U U 2w 2 X new waist position Z z 12

13 Mirror Reflection and Transmission Operations Reflection Operation (of mirror) on field : multiplication by amplitude-reflectivity with appropriate sign changes in transversal modal-basis of the beam if reflected at an angle from a curved mirror appropriate phase factor for small longitudinal displacement a composite matrix representing small perturbations in mirror, like rotation, shift, mismatch between mirror surface and phase-front, etc. parity operation which flips the beam about vertical axis Transmission operation (of mirror) on field : multiplication by amplitude-transmissivity changes in transversal modal-basis of the beam (lensing) 13

14 Propagation (prop and telescope) Effects (in vacuum) longitudinal phase gouy phase change of basis (distance to waist changes) time-delay telescope l 2 length d 2 output field calculated here 0 (l 1,f 1 ) (l 2,f 2 ) (l n,f n ) Prop module effects propagation thru vacuum and may or may not include time-delay (user s choice) Telescope module can propagate light thru a series of lens placed in vacuum with all changes in guoy phases properly included The telescope effect can also be specified by setting the basis and gouy phase information at the output instead of specifying details of lens & length 14

15 EM Composite Systems (What are summation modules?) t=0 t=2 t=2τ τ r c D[1] D[2] D[3] D[4] D[4+1] D[1] IM B[1] B[2] B[3] B[4] B[1] RM coupled cavity 15

16 Composite Systems (summation modules) propagator propagator Using primitives: time-step = Length/speed of light summation cavity Using summation cavity: time-step = N X Length/speed of light Three modules so far Fabry-Perot cav triangular cavity power-recycled Michelson cavity or the LIGO recycling cavity 16

17 Summationcavity in LIGO 17

18 Triangular cavity 18

19 Digitizer Analog data: discrete time-series of e2e with small timesteps Digitized data: sampled at larger time-steps (e.g. 16K circuit) Digital filter characterised by poles, zeros, polepairs and zeropairs of equivalent analog filter proper handling of data in discrete time-steps is done using z-transform internally can be used as analog filter to use in a digital circuit, it needs to be used in conjunction with ADC and DAC 19

20 ADC & DAC ADC simulation points out(n-1) S(n-1) analog input out(n)=s(n-1)/ (n 1)τ nτ Figure 4: digitization in time (n+1)τ out( nτ) = ( n 1)τ input()t t d ( n 1)τ ADC( nτ) = floor( gain out( nτ) + 0.5) output can be an integer in between limits specified by setting maximum no of bits DAC 20

21 Mechanics susp3dmass d_yaw susppt d_pendulum Thickness y d_attach center of Mass d_pitch center of cylinder y force d_cm mirror position Diameter x End view z coated surface Side view Figure 5: Single Suspended Mass 21

22 Coordinate system: AxisRotation Y arm ETMr (0, 1/2 π, 0) MC3 (1/4 π, 1/2 π, 0) MC2 (-1/2 π, 1/2 π, 0) ITMr (π, 1/2 π, 0) ITMt ETMt (1/2 π, 1/2 π, 0) (-1/2 π, 1/2 π, 0) MC1 (3/4 π, 1/2 π, 0) RM (1/2 π, 1/2 π, 0) BS (-3/4 π, 1/2 π, 0) X arm Figure 6: Detector to e2e coordinate transformation z θ Z y ϕ ψ x Y X Figure 7: Euler angles 22

23 Math functions algebraic: madder (ax+by), sine, square_root, inverse... logic : and,or,xor, a>b, not, flipflop, switch limiter, delay, switch (a) connection view (b) connection when switch is non-zero Figure 8: hardswitch in action (c) connection when switch is zero In many cases, better to use FUNC() 23

24 Data input and output generating inputs clock (time) : may also use get_time() function in Func() random number (normal) random number (flat) data_in : can get a data into simulation data_reader : can read input data from an ascii file Output data_out : can get a data out of the simulation ( probe ) data_viewer: view data at a particular instant psd_out: accumulate input data, calculate psd and write to a file 24

Mode mismatch and sideband imbalance in LIGO I PRM

Mode mismatch and sideband imbalance in LIGO I PRM LASER INTERFEROMETER GRAVITATIONAL WAVE OBSERVATORY -LIGO- CALIFORNIA INSTITUTE OF TECHNOLOGY MASSACHUSETTS INSTITUTE OF TECHNOLOGY Technical Note LIGO-T04077-00- E Sep/0/04 Mode mismatch and sideband