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LIGO-‐G1401391-‐v1 Koji Arai – LIGO Laboratory / Caltech ¡ Longitudinal sensing and control § Plane wave calculation was sufficient ¡ Alignment, mode matching, mode selection § higher order modes need to be taken into account ¡ ¡ Solution of Maxwell’s equation for propagating electromagnetic wave under the paraxial approximation => Laser beams change their intensity distributions and wavefront shapes as they are propagated => Any laser beam can be decomposed and expressed as a unique linear combination of eigenmodes In this sense, a (given) set of eigenmodes are ortho-‐normal basis ¡ HG modes (TEM mods) : one example of the eigenmodes A. E. Siegman, Lasers, University Science Books, Mill Valley, CA (1986) H. Kogelnik and T. Li, Appl. Opt. 5 (1966) 1550-‐1567 Wikipedia http://en.wikipedia.org/wiki/Transverse_mode ¡ Astronaut ¡ World cup football Trivia ¡ There are infinite sets of HG modes § A TEM00 mode for an HG modes can be decomposed into infinite modes for other HG modes ¡ The complex coefficients of the mode decomposition is invariant along the propagation axis § Where ever the decomposition is calculated, the coefficients are unique. ¡ No matter how a beam is decomposed, the laser frequency stays unchanged! (sounds trivial but frequently misunderstood) omplex radius of curvature qN(z)is related tobeam the spot size w(z) and the important parameters ef this gaussian can thei} be relat,ed of at any z by by the the definition the plane ratio z/zR formulas ist curvature spot size R(z) 'wo and ¡ Beam size at z 11 A (rk)2, W(Z) = UJo 1 + qN(i) E R(i) -2 Tw2(z)' ¡ Wavefront curvature at z this parameter (2) .+ Ei?L. law obeysR(z) the == propagation v ,2 Gouy phase ◆ qN(z)=th(z) qo +z= tan-1 =z+ ✓1'zR, (3) z 1 ⌘(z) = tan zR ral ¡ words, along the entire gaussian beam is characterize Rayleigh rangepattern value the field by the single parameter wo (or q'"o,or zR) at the beam waist, plus ngth A in the medium . ¡ qo ==2 perture e (zi) value Transmission of A in these rw,2 formulas =IZR¥ is always (4) A cf. Huygens’ principle the wavelength of the radi- ¡ Gouy phase shift: Relative Phase shift between the transverse modes § Different optical phase of the modes for the same distance => Different resonant freq in a cavity § “Near field” and “Far field” (will see later) 5. Modal Analysis ' d 1 d2 § Ulm+ + Ulm+ ax + Ulm+ a2x Upq+ dxdy (5.30) 2 dx 2 dx 5.6. Modal Expansion of the Misaligned Beam ¡ On the assumption that 1 >> ax /w0 and the input beam is the fundamental Gauss- Lateral shift: ian beam in an arbitrary coordinate system, we can neglect the power translation to the modes higher than first off-axis mode (see Appendix A). After carrying out the above expansion, we obtain the expression for the laterally misaligned modes as " µ ∂ # 1 ofaxthe2 beam. ax Figure 5.6: Angular tilt ofUthe beam.U10+ Px (ax ) § UFigure y, z)Parallel ' 1displacement ° (5.31) 00+ (x,5.5: 00+ + 5. Modal Analysis 2 w0 w0 " # µ ∂ 3 ax 2 ax ¡ Rotational s hift: Px (ax ) § U10+ (x, y, z) ' 1 ° U10+ ° U00+ (5.32) 2 w w Inwhere the same way as thecoefficient parallel displacement, neglect 0 defined the expansion < pq + |lm+0 we > iscan by the0 modes higher than the first off-axis mode on condition the inequality 2 1 >> Æx /Æ0 >> Æ0 is satisfied. Zcan Z that From the above expressions, we see that (a /w ) is0 the order of the optical power x 0 0 § 0 0 < pq + beams |lm+ >are¥expanded Upq+by (x, the y, z)U lm+ (x , y , z )dxdy modes of the tilted The misaligned Hermite-Gaussian that is transferred from one mode to others by the parallel transport. # ZZ " d coordinates to the second order of the perturbation as 1 d2 2 § ' Ulm+ + Ulm+ ax + U a U dxdy (5.30) lm+ x pq+ 2 dx 2 dx " µ ∂ # 1 Æx 2 Æx 5.6.2 Angular Rx (Æx ) Tilt § U00+ (x, y, z) ' 1 ° U ° i U10+ (5.35) 00+ Figure 5.6: Angular2tiltÆof0 the beam. Æ0 " µ ∂2 # 3 Æinput Æx z axis (Fig. 5.6). x beam On that the assumption that >> beam GaussSuppose there angular tiltax' Æ/w x 0between Rx (Æxis) §an U10+ (x,1 y, z) 1and ° thethe U10+and °isitheUfundamental (5.36) 00+ 2 Æ0 Æ0 K. Kbeam awabe Ph.D thesis: http://t-‐munu.phys.s.u-‐tokyo.ac.jp/theses/kawabe_d.pdf in an arbitrary coordinate system, we can neglect the to In ian this case, the two coordinate systems are related to each other aspower In the same way as the parallel displacement, we can neglect the modestranslation higher than 'L: I 1 d..e - - L- - k -. k Nk ill A gaussian ii N•q t Ng trapped beam by two can mirrors be of the N proper ¡ curvature and spacing. TEM modes with matched wavefront RoC l'!-'7----' / L waist ))c':rl pt:f:I(i(' ( ︿ N w8 .- Ml, Rl Zl Z=O i U - M2 R2 , Z2 FIGURE 192 Notation and analytical model for analyzing a simple stable twomtrror cavlty. These two mirr()rs can thus trap the gaussian beam as a sta. nding wave between the two n'}irrors, with, if the inirrors are large enough in size, negligible A. E. Siegman, Lasers, niversity S cience Mill of Valley, CA (1986) diffraction or U `"spillover" losses pastBooks, the edges the mirrors. The two mirrors ¡ Due to different Gouy phase shifts between TEM modes, their resonant frequencies are different 19.3 GAUSSIAN fFSR = c/2L FAwax pt TRANSVERSE MODE FREQUENCIES fTMS = fFSR x ζ/(2 pi) N kAWtrans near pla n ar confocal ζ: cavity round trip Gouy phase shift A. E. Siegman, Lasers, University Science Books, Mill Valley, CA (1986) 1 - 9ig2) only 746 on y2(1 - yiy2) r the resonator CHAPTER 19i STABLE y para,meters RESONATORS defined TWO-MiRROR in t,he pre- Liantit,y av/Zi5('71il whichwe willdiscussin th() following. ¡ g-‐factors gl Ei 1-L/R1 g2 =- 1-L/R2 - Diagram Rl FIGURE The R2 19.3 resonator g parameters. obvious froin Equations 19.4 t() 19.7 that real an(i finit,e an beain paraineters and spot sizes (;an exist oi'ily if t}'ie nfined to a s(,al)ility ¡ Stability criteria rai}ge.defined by JN--ir Z L 748 provides. us with three equations, CHAPTER 19i STABLE TWO-MIRROR RESONATORS namely, O S gig2 S 1¥ R(zl) ==zl + zft./zl = R(Z2)==Z2+ Zft./Z2= - Rl , (i) (8) + R2, ability range because this is also exactly the condition with radii Ri and R2 and spacing L to form (2) a stable The minus sign in the first of these equations arises because of a difference in the for asLconventions analyzed in Chapter 15. A. E. rays, Siegman, asers, that we earlier sign use ii} in describing beam wavefronts or in describing resonator Bmirrors. University Science ooks, The gaussian wavefront curvature R(i) is usually taken as gaussian resonator thisfor astability was for a divergingtheory beam, or negative converging beam, criterion traveling to Mill Valley, Cpositive A ( 1986) the right; whereas the mirror curvatures Ri and R2shown are usually taken as positive nto the resoiiator stability diagram in Figure 19.4. numbers for mirrors that are concave inward, i.e., as seen looking out from within and L= z2 --- zl. ,dt,,,,i,,, FIGURE The 19.4 stability diagram for a two-mirror optical resonator. out A + DCq +in2+ D 0 1. (11) 4 where qin and qout are the q-parameters for the input and output beam respectively. A, A + Dof+the 2 ABCD matrix of the optical system. This formula B, C, and D are the elements As we will find in Section 3, for a Fabry-Perot cavity corresponds to the product means that di↵erent optical systems with the same ABCD matrix result in identical beam 4 oftransformations. the g-factors g1 g2 . Therefore Eq. (11) is equivalent to Eq. (3). ¡ General case InItthis note, the interpretation of Eqs. and arerelated extended for general cavities. It is is also known that the quantity (A +(10) D)/2 is (11) closely to cavity stability. The cavity derived that the accumulated round-trip Gouy phase shift can only eigenmodes are supposed to fulfill the beam-reproducing condition qoutbe = qcomputed in . By solving this from the round-trip matrix of the (i.e. cavity condition, Eq.(8) withABCD the unitarity condition ADas: BC = 1), we obtain two solutions ✓ ◆ for the forward and backward beams: A+D 1 ⇣ = sgnB · cos p , (12) A D ± (A +2D)2 4 q= (9) 2C where the value range of cos 1 x is defined to be ¡ AsThe avity is stable 3 when this quantity ζ exists q is accomplex number , the stability criteria is given by 0 cos 1 x < 2⇡ ( 1 < x 1). (13) A+D 2 1 1. Therefore, the product of the generalized g-factors is always positive. There is no sign ambiguity. (10) 2 3 cf. q = z + izR , where zR is the Rayleigh range. Equivalently, this can be expressed as page 2 A+D+2 0 1. 4 (11) A+D+2 As we will find in Section 3, for a Fabry-Perot corresponds to the product T1300189 “On the accumulated round-‐trip Gouy phase shift cavity for a general 4 optical cavity” Kgoji https://dcc.ligo.org/LIGO-‐ T1300189 of the g-factors Therefore Eq. (11) is equivalent to Eq. (3). 1 gA 2 .rai 746 ¡ CHAPTER 19i STABLE TWO-MiRROR RESONATORS gl Ei 1-L/R1 g2 =- 1-L/R2 To match the input beam axis and the cavity axis - Rl FIGURE The R2 19.3 resonator g parameters. Z L `,A ¡ Corresponds to the suppression of TEM01/10 mode in the R(zl) ==zl + zft./zl = beam with regard to the cavity mode (i) provides. us with three equations, namely, - Rl R(Z2)==Z2+ Zft./Z2= , + R2, and L= z2 x --- zl.(translation, rotation) ¡ 4 d.o.f.: (Horizontal, Vertical) (2) Note: it is most efine the trans/rot at the aist in the The intuitive minus sign into thedfirst of these equations arises because of w a difference sign conventions that we use ii} in describing beam wavefronts or in describing resonator mirrors. The gaussian wavefront curvature R(i) is usually taken as positive for a diverging beam, or negative for a converging beam, traveling to the right; whereas the mirror curvatures Ri and R2 are usually taken as positive ¡ To move numbers the m irrors r concave to minward, ove i.e., the beam? for mirrors that o are as seen looking out from within the resonator, and as negative numbers for mirrors that are convex as seen from 'L: b-l11 1 d..e - - -r-f I L- lll - k -. k Nk Xli lli ill FIGURE 19.1 A gaussian ii N•q t Ng trapped beam by two can be mirrors of the N proper ¡ curvature and spacing. To match the waist size and position of the input beam to these of the cavity / l'!-'7----' waist L ))c':rl pt:f:I(i(' ( ︿ N w8 .- Ml, Rl Zl Z=O i U - M2 R2 , Z2 FIGURE 192 Notation and analytical model for analyzing a simple stable twomtrror cavlty. ¡ Corresponds to the suppression of TEM02/11/20 mode in These two mirr()rs trap the m gaussian the beam with regard to tcan he thus cavity ode beam as a sta.nding wave between the two n'}irrors, with, if the inirrors are large enough in size, negligible diffraction or `"spillover" losses past the edges of the mirrors. The two mirrors thus form an optical resonator which can support both the loweft-order gaussian rnode, and also higher-order Hermite-gaussian or Laguerre-gaussian modeg., as resonant modes of the (avity. We will see in this section that this simple description is ¡ Wave Front Sensing § Misalignment between the incident beam and the cavity axis § The carrier is resonant in the cavity § The reflection port has ▪ Prompt reflection of the modulation sidebands ▪ Prompt reflection of the carrier ▪ Leakage field from the cavity internal mode Carrier Sidebands RF QPD (WFS) E Morrison et al Appl Optics 33 5041-‐5049 (1994) no signal spatially distributed amplitude modulation ¡ Wave Front Sensing § WFS becomes sensitive when there is an angle between the wave fronts of the CA and SB § Can detect rotation and translation Sensitive at the far field Sensitive at the near field of the beam separately, depending on the “location” of the sensor § Use lens systems to adjust the “location” of the sensors. i.e. Gouy phase telescope Frequent mistake: Beam diameter [mm] What we want to adjust is the accumulated Gouy phase shift! Not the one for the final mode! 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 LENS1 f=2000 WE_End (ideal) LENS2 f=200 0 1 2 3 4 5 6 7 8 9 10 8 9 10 Distance from the front mirror [m] Accumulated Gouy phase shift [deg] ¡ 180 120 LENS1 f=2000 WE_End (ideal) 60 0 -60 LENS2 f=200 target Gouy phase for end -120 -180 0 1 2 3 4 5 6 7 Distance from the front mirror [m] ¡ aLIGO implementation: Separate Gouy phase of a set of two WFSs with 90 deg WFS_A WFS_B zR zR Sensors (WFS). Each core optic is instrumented with an optical lever and viewed by a CCD camera. In addition, the transmitted beam of each optic will be monitored by a quadrant photo detector (QPD). The ASC sensor signals are processed and send to the optics through the Suspension (SUS) system. ¡ Combine WFS, DC QPD, digital CCD cameras QPD From Input Optics QPD QPD WFS QPD WFS WFS To LSC WFS WFS To LSC WFS To LSC ¡ PRC/SRC Degeneracy ¡ Sigg-‐Sidles instability & alignment modes § G0900594 ¡ Impact on the noise § G0900278 / P0900258 ¡ Parametric Instability § HOM in the arm cavity -‐>Rad Press. -‐>Mirror acoustic mode -‐>Scattering of TEM00-‐>HOM