Optical Cavity Designs for Interferoetric Gravitational Wave Detectors Pablo Barriga 7 August 9
Assignents.- Assuing a cavity of 4k with an ITM of 934 radius of curvature and an ETM of 45 radius of curvature. Calculate the following paraeters: - Waist sie and position - Spot sie on both ETM and ITM - Stability g factor for both irrors and the whole cavity - Free spectral range - High order ode spacing.- To the previous cavity we add two irrors with the following radius of curvature: 34 and -4.4. Distance between ITM and the first irror is 9.7, fro this irror to the second irror is 5.. Assuing a vacuu environent and fused silica substrate deterine the radius of curvature of the wavefront at a distance of 5.8 fro the second irror. PR -4.4 L 5.8 PR3 34 L 5. L 9.7 ITM L 4 ETM
Gaussian Bea Coplex bea paraeter: Spot sie: ω ( ) R( ) ω ( ) q ( ) j λ λ ω o ωo Rayleigh range: ω o R λ Bea divergence: λ θ ω o ω o waist ( ) Gouy phase: θ R ω o waist radius λ wavelength Radius of curvature of the wavefront: ψ R ( ) ω o ωo λ ( ) arctan R R() ω()
Herite Gaussian: Laguerre Gaussian: ( ) ( ) ( ) { } ( ) ( ) ( ) ( ) ( ), exp!! exp,, y x R y x jk k j y H x H n n j y x U n n n ω ω ω ω ψ ( ) ( ) ( ) ( ) ( ) { } ( ) ( ) ( ) ( ) ( ) ( ),,, exp cos exp!! 4,, r R r jk jk r L r l j l l r U l l ω ω ω φ ω ψ δ φ Transverse Modes Higher Order Modes
Herite Gaussian: Laguerre Gaussian: ( ) ( ) ( ) { } ( ) ( ) ( ) ( ) ( ), exp!! exp,, y x R y x jk k j y H x H n n j y x U n n n ω ω ω ω ψ ( ) ( ) ( ) ( ) ( ) { } ( ) ( ) ( ) ( ) ( ) ( ),,, exp cos exp!! 4,, r R r jk jk r L r l j l l r U l l ω ω ω φ ω ψ δ φ Transverse Modes Higher Order Modes
Phase shift: Frequency of Transverse Modes total phase shift fro one end of the cavity to the other, including the Gouy phase φ ( ) kl ( n ) [ ψ ( ) ψ ( )] ψ ( ) ψ ( ) ( ± g ) arccos g Bea radius and Gouy phase shift along the propagation direction for a bea in air with 64 n wavelength and µ radius at the waist. The positions at plus and inus the Rayleigh length are arked.
Frequency of Transverse Modes Frequency of higher order odes for a given axial ode: ( ) ( ) arccos g g n q qn ± free spectral range This frequency and the finesse of the cavity defines the suppression factor sin 4 n S n Therefore we can define a transission factor ( ) sin 4 n n r r t t T
Higher Order Modes Tuning Assuing syetric cavity R L 4 g irror Ψ rad 3 4 5 f ax f TEM kh 37.47 74.95 f [kh] Near Planar!
Higher Order Modes Tuning Assuing syetric cavity R 5446 L 4 g irror.96 Ψ.39 rad f ax f TEM 4.6 kh kh 3 4 5 37.47 74.95 f [kh]
Higher Order Modes Tuning Assuing syetric cavity R 4 L 4 g irror Ψ / rad f ax 3 f TEM 4.6kH 8.7 kh 4 Half FSR 5 37.47 Con-focal! 74.95 f [kh]
Higher Order Modes Tuning Assuing syetric cavity R 76.5 L 4 g irror -.96 Ψ.76 rad f TEM 3.9 kh f ax 3 37.47 74.95 f [kh] f 4.6 kh
Higher Order Modes Tuning Assuing syetric cavity R g irror - L 4 f ax Ψ rad FSR f TEM 37.5 kh 3 37.47 74.95 f [kh] Concentric!
Higher Order Modes Tuning High Order Modes Transission 9 8 7 6 5 9 4 8 7 3 6 9 8 5 7 9 3 4 5 6 7 Transission [a.u.] - 8 9 -.5..5..5.3.35.4.45.5 Ψ: Gouy phase shift (rad) g cos (Ψ)
Ray Transfer Matrices ' ' x x D C B A x x x corresponds to the position and x corresponds to the slope (or angle) of the bea. Matrix representation of ray trace: It is necessary to create one atrix per optical eleent Ray transfer atrices for Gaussian beas ω λ j R q q D C B A k q q B A q D C q A. E. Siegan, University Science Books, Sausalito, California (986) H. Kogelnik and T.Li, Laser beas and resonators, Appl. Opt. 5, 55 (966) ï
Ray Transfer Matrices For siple optical coponents Propagation in free space or in a ediu of constant refractive index Refraction at a flat interface Refraction at a curved interface Reflection fro a flat irror Reflection fro a curved irror Thin lens Eleent Matrix d n n n n Rn n n R f d distance n initial refractive index n final refractive index. R radius of curvature, R > for convex (centre of curvature after interface) n initial refractive index n final refractive index. Identity atrix Rearks R radius of curvature, R > for concave f focal length of lens where f > for convex/positive (converging) lens. Valid if and only if the focal length is uch greater than the thickness of the lens.
Stable Resonator Fabry-Perot Cavity ITM ETM In 897 Fabry and Perot constructed an optical resonator for use as an interferoeter.
Fabry-Perot Cavity Original (898) odel of the Fabry-Perot interferoeter.
Stable Resonator Fabry-Perot Cavity ITM Free Spectral Range: 3 ( 4 ) ETM 8 FSR c 3 37. L 5 kh Finesse: ω ω ax cav r r ( r r ) Refer to notes fro lecture 6 for coplete analysis
Fabry-Perot Cavity Stability factor: L g R L g R Cavity stability: g L L g R R Stable cavity: gg Gouy phase: g cos ( ψ )
Fabry-Perot Cavity
Advanced Interferoeters ETM Y M3 ITM Y PRM BS ITM X ETM X Laser M M SRM To Detector Bench
Advanced Interferoeters ETM Y Input Mode-cleaner Fabry Perot Cavities M3 ITM Y PRM BS ITM X ETM X Laser M M Power Recycling Cavity SRM Signal Recycling Cavity Output Mode-cleaner To Detector Bench
Length Degrees of Freedo ETM Y Degrees of freedo L Y Fabry-Perot Cavity L (L x L y )/ L-(L x -L y )/ Input Mode Cleaner M3 ITM Y l l pr (l x l y )/ l-(l x -l y )/ l src l sr (l x l y )/ L IMC l y PRM l pr BS ITM X L X ETM X Laser M M l x Fabry-Perot Cavity l sr SRM L OMC Output Mode-cleaner To Detector Bench
Length Sensing and Control ETM Carrier: Resonant in PRC and Ars RF Sideband f: Resonant in PRC RF Sideband f: Resonant in PRC and SRC L l East Fabry-Perot Cavity ITM PRM BS ITM L ETM l pr l sr l South Fabry-Perot Cavity SRM To Detector Bench
Rule of Thub Carrier should be resonant in the ars and the PRC. Carrier resonant in the SRC for resonant sideband extraction (RSE), and anti-resonant for signal recycling. SB should be nearly anti-resonant in the ars, and resonant in the PRC. SB also nearly anti-resonant in the ars, and resonant in the PRC. One of the SB should be resonant in the SRC and the other nearly anti-resonant.
Sidebands Selection For transission of odulation sidebands by the ode-cleaner, L IMC and f ust satisfy: f n c L IMC For sideband coupling into the recycling cavity, L PRC and f ust satisfy: f n Sideband ust not resonate inside the ain ars, but also not exactly anti-resonant: n f L c c L PRC Ar 3 n 3 not and integer
Sidebands Selection Signal recycling cavity is different and depends on the operation schee selected for the interferoeter. Only the higher frequency f is used to deterine the length of the SRC. L SRC c LSRC 4 f ( n φ ) s φ s ï Resonant Sideband Extraction In order to do this we also need to introduce a difference in the ars lengths in the recycling cavities, known as Schnupp Asyetry. l l l l x y c 4 f
Mode-cleaner Design A ring cavity will act as an optical filter transitting only the fundaental ode TEM A field distribution syetric with respect to the vertical axis closes in itself whenever the total cavity length (L) is an integer ultiple (q) of the optical wavelength (λ) Syetry with respect to the y-axis iplies that the spatial dependences of the field with x is an even function E(x,y) E(x,y) If the field distribution is anti-syetric with respect to the vertical y-axis, the resonance condition is achieved when L (q/)λ for a cavity fored by an odd nuber of irrors and when L qλ for a cavity with an even nuber of irrors. E(x,y) E( x,y)
Syste of Coordinate M x l º y M3 M l º Syste of coordinate: is along the direction of propagation x is parallel to the plane of incidence on the cavity irrors y is perpendicular to the sae plane of incidence
Frequency of Higher Order Modes We can foralise this effect by writing two equations: kl kl ( n ) arccos( g ) q ( n ) arccos( g ) q ñ is even ñ is odd Or a ore general expression: ( ) ( ) ( ( ) ) n arccos g q kl ( ( ) ) ( ( ) ) for,, 4, 6 for, 3, 5
Fro here we can deduce the frequency of any higher order ode ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) arccos arccos qn g n q L c FSR c k q g n kl ω Frequency of Higher Order Modes
The delta frequency between the fundaental ode and any higher order ode is then given by: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) arccos arccos arccos arccos arccos n n n n g n g n g q g n q g q n Frequency of Higher Order Modes
How effective the cavity will be for filtering the higher order odes is then given by the suppression factor of the cavity ( ) ( ) ( ) ( ) arccos sin 4 sin 4 sin 4 n n n n n g n S S L c S Suppression of Higher Order Modes
The suppression factor also deterines the transission factor for the cavity The transission of the fundaental ode by this factor will deterine the transission of any given higher order ode ( ) 3 r r r t t T S T T n n ( ) ( ) ( ) ( ) 3 arccos sin 4 ± n R L n r r r t t T Transission of Higher Order Modes
Frequency of Higher Order Modes High Order Modes Frequency..9 TEM () TEM ().8.7 Transission factor.6.5.4.3. TEM TEM TEM TEM TEM3 TEM TEM4 TEM TEM4 TEM TEM TEM3 TEM3 TEM3.. 7.8.6 3.4 6.3 8.5 9.4 3.96 3.79 33.76 36.537 frequency [H]
Mode-cleaner Geoetry M3 Concave end irror l º99 Incident angle.579 o ω o M M l º Incident angle 44.7 o Flat Mirrors used as input and output couplers
Coating and Substrate Absorptions High reflectivity coating absorption produces astigatic theral lensing. The spot ellipticity produce different distribution between X and Y axis. M used as output coupler the diagonally transitted bea produces strong astigatic theral lensing.
Eccentricity Variation with Power Eccentricity.6.4. Eccentricity variation with Power M3 Spot Waist Sie M/M Flat M3.5 4 6 8 4 6 8 Eccentricity.6.4. M3 Spot Waist Sie M/M M3 3.3 Eccentricity.6.4. 4 6 8 4 6 8 Input Power [W] M3 Spot Waist Sie M/M 47 M3 3.3 4 6 8 4 6 8 Input Power [W]
Output Mode-cleaner Soe light will leak to the dark port reducing fringe contrast and increasing the noise. An OMC should reject all the coponents of the contrast defect. The OMC design will depend on the readout configuration Heterodyne (RF) or Hoodyne (DC). RF uses the sidebands as local oscillator ï long OMC with suspended irrors. DC uses a sall aount of carrier light as local oscillator ï no need of sidebands ï sall OMC in a bow-tie configuration.
Output Mode-cleaner
Exaple: GEO 6 design 4 irrors cavity Finesse of 5 Cavity g factor of.735 Round trip length of 66 c Output Mode-cleaner Results fro Finesse: cavity is stable! Eigenvalues: qx(-.353.5868i), wx443.8558u x-35.9 qy(-.353.586335i), wy445.64u y-35.37 finesse : 55.4, round-trip power loss:.396964 opt. length: 658.89, FSR: 455.4864MH FWHM:.9364545MH (pole:.46873mh) Preliinary results of transission: Sidebands:.96 TEM :.39 TEM :.8 TEM3 :. TEM4 :.6 TEM5 :.6
Ar Cavity Paraeters Cavity Length ITM (ETM) radius of curvature ITM (ETM) diaeter Cavity g-factor Waist sie radius Spot sie radius Free spectral range High order odes separation 4 76 3 c.859.49 6.57 37.47 kh 3.88 kh
Case : Marginally stable Power Recycling Cavity Gouy phase (deg) PRM ITM ETM Total Gouy Phase 8 6 4 Distance () Original Advanced LIGO design: R R 76 with ar length of 4 Waist radius.49 Mirror spot sie radius 6.57 Waist position FSR 37.5 kh HOM frequency gap 3.9 kh Rayleigh range 389.87 Bea radius 76 λ 4 5 j 4.87 j9.39 q q ( ) R( ) ω ( ) ( ) j j389. 87 R