According to the shear approximation, the theoretical loss is the integral of the power outside the mirror. The solid line is an embedding of the same surface made with an axisymmetric algorithm.
Brief Description of a LIGO Interferometer
The LIGO-related original work reported in this thesis concerns mirror designs for these arm cavities.
The Arm Cavities of Initial LIGO
Arm Cavity Design in Advanced LIGO
The thermal noise of the suspension is also significantly improved, so that the internal thermal noise of the mirrors is dominant in the most sensitive area. Internal thermal noise is caused by imperfect averaging of time-dependent thermal fluctuations on the mirror surface.
Mesa Beams in LIGO
The power distribution at the mirror surface is the same for Mesa Beams supported by both near-flat and near-confocal mirrors. The electric field amplitude at the mirror surface is the same for Mesa Beams supported by both near-flat and near-confocal mirrors.
Hyperboloidal Beams in LIGO
The difference between a spherical mirror with radius equal to half the cavity length and the height of this mirror is exactly the height of the nearly flat Mexican mirror shown in Fig. The plot shows the theoretical power distribution of the Mesa Beam outside the LIGO mirror.
Conical Mirrors in LIGO
If Mesa beams prove too difficult to implement or use, and something better than the baseline Gaussian is still desired, one of the hyperboloidal beams in between can be used. When the scale of the perturbations approaches or exceeds the mirror radius, conical cavities produce much larger diffraction losses than Mesa cavities subjected to the same perturbation.
Isometric Embeddings of Black Hole Surfaces in Flat 3D Space
Why is this in my thesis?
Oddly enough, it was Kip Thorne, the only person I've ever obeyed, who advised me to do this. The Albert Einstein Institute, although, in my opinion, is more qualified to award Ph.D.s than most universities, it is not authorized by German authorities to do so.
Relationship With The Rest of The Thesis
In the vicinity of the mirror the phase of each minimum Gaussian varies as kδζ =kδz [Eq. 2.5). We first consider the inclusion of the visible horizon for the parameter dataset.
![Figure 2.1: Optical axes of the families of minimal Gaussians beams used to construct: (a) an FM Mesa beam [8], denoted in this paper α = 0; (c) our new CM Mesa beam, denoted α = π; (b) our new family of hyperboloidal beams, which deform, as α varies from](https://thumb-ap.123doks.com/thumbv2/123dok/11361701.0/34.918.177.537.117.614/figure-optical-families-minimal-gaussians-construct-denoted-hyperboloidal.webp)
Mesa Beams Supported by Nearly-Flat Mirrors (FM Beams; α = 0)
Then the minimal Gaussian in the immediate vicinity of the mirror plane, at ζ = `+δζ (z |δζ| b), has a simple form. Here and throughout the paper, we ignore partial corrections of the order λ/b∼ b/`~10−5.) Mesa-beam (FM-beam) field constructed by superposition of minimal Gaussian angles in the image. The index 0 on U0 is the value α = 0 of the angle of rotation of the Gaussian optical axes when looking at this FM beam from the perspective.
Mesa beams supported by nearly concentric mirrors (CM Beams; α = π)
Accordingly, these minimal Gaussians superimpose on the reference sphere to produce a CM field given by. Note that this Uπ(r, Sπ, D) is the complex conjugate of the FM field,U0(r, `, D), evaluated on the transverse planeez=`(the reference surface for the case of nearly planar mirrors); see cf.
Hyperboloidal Beams Supported by Nearly Spheroidal Mirrors
Apart from the phase, Uπ is the same on the mirror surface as on the reference sphere, Sπ, the intensity distribution of the light is the same on the mirror as on Sπ:. 2.12) Moreover, because Uπ(r, Sπ, D) is the complex conjugate of U0(r, `, D), both have the same moduli and intensity distributions; that is, the CM beam has the same Mesa-shaped intensity distribution as the FM beam (solid curve in Figure 2.3 below). In this figure, P is the point on the spheroid, Sα, at which we want to calculate the field. The field Uα(r, Sα, D) on the spheroid Sα is obtained by adding the minimum Gaussians (2.2) with $andδζ given by Eqs.
Replacing α with π−α and changing the sign of α is equivalent to the complex conjugation of Uα; therefore: distance between the mirrors), and both the radial and angular integrals can be performed analytically, giving the field on this sphere.
Conclusions
As for the FM and CM beams, so also for the hyperboloidal beam (2.16) and for the same reasons: The mirror's surface must be displaced longitudinally from the fiducial spheroid,z=Sα(r), byδz=Hα(r) , where . For α = 0.1π, to keep the diffraction losses at 2.68ppm, D is increased to 10.5 cm and the physical beam diameter is, accordingly, a little larger than in Fig. Forα= 0.2π,D was increased to D= 13.0 cm; for α= 0.252π, it is increased to D =∞, producing a beam shape that is Gaussian to the accuracy of our numerical calculations, but is significantly larger than the minimal Gaussian of Fig. 2.3 and is roughly the same as the baseline design for advanced LIGO. Our numerical calculations suggest that for D=∞and allα6= 0 orπ, the hyperboloidal bundle is Gaussian, with width varying from minimal, σ=b =p.
These considerations suggest that the optimal configuration for advanced LIGO will be close to α=π, but whether the optimum is exactly atα=π (the CM configuration) or at some modestly smaller α will depend on practicality and thermal noise -considerations not explored in this paper.
Acknowledgments
The mirror height is measured in units of λ, where λ= 1.06 µm is the wavelength of the light used in the interferometer. The solid line is an embedding of the same surface calculated with an axisymmetric algorithm. 4.6, where we show the embedding of the horizon of a black hole plus the Brill wave data set corresponding to the parameters.
Clearly, the non-axisymmetry of the metric components is largely a coordinate effect.
The Minimization Problem
- Radiation in a Cavity - Generalities
- Normal Modes
- Minimizing The Coating Noise – Statement of The Problem
- Gradient Flow with Constraint
- Minimization Strategy
Here the functions f(ω) and(ω) are connected in a complicated way, based on the positions of the source and the optical instrument. The scalar field that we expand into Ψ-modes is now one of the counterpropagating components. Due to time reversal symmetry, the phase front of this component passing through z = 0 on the optical axis must be perpendicular to the optical axis, and therefore coincides with the z = 0 plane.
Therefore, we choose an initial step size such that there is no temporary increase in the value of the perceived sound at any time during the descent.
Results
- Internal Thermal Noise
- Field Characterization at the Mirror
- Mirror
- Convergence and Conical Cavities with Fewer Coefficients
- Optical Modes Supported by Finite Nearly-Conical Mirrors and their Diffrac-
3.5, the power and electric field amplitude distribution for the beam we derived spans a much larger fraction of the mirror than the previously considered Mesa beam. As in previous work [1], we determined that the mirror surface is the phase front of the theoretical beam obtained by the minimization process. When fewer coefficients are used, the beam is spread over a smaller area of the mirror.
When using fewer Gauss-Laguerre modes in beam expansion, the beam does not extend to the end of the mirror.
Tolerance to Imperfections and Compatibility with LIGO
- Sensitivity to Mirror Tilt
- Mirror Figure Error
- Understanding the Impact of Various Frequency Components of the Mirror
- Sensitivity to Mirror Translation
- Driving the Conical Cavity with a Gaussian Beam
The diffraction loss is calculated as 1− |λi|, where |λi| is the absolute value of the axisymmetric propagator. We used this tool to study the effects of mirror tilt, mirror displacement, and mirror image errors, as well as to check the diffraction losses of the conical cavity. Adding about one-tenth the perturbation to conical mirrors produces about the same increase in diffraction loss as full perturbation in a Mesa cavity.
In Table 3.6 we show some more numerical values for diffraction losses of the conical cavity when one mirror is shifted from the optical axis.
Conclusions
The diffraction loss reaches 3 ppm after 500 reflections, while the theoretical prediction is 1 ppm according to the clipping approach. This results in the light power distribution being different at each bounce, as we can see in the case of Mesa. This only indicates that the cavity can hold lights in multiple resonating modes, but still one must be selected by fine-tuning the cavity length so that only the desired mode resonates.
Acknowledgements
Computing the components of the metric in the embedded surface in terms of the inherited coordinate system{θ, φ}, given the embedding relations (4.14), is not difficult. It is important to note that the definition of the embedding function F above is by no means unique. In the top panel of the figure we show the original surface and in the bottom panel the resulting embedding.
As a second example, we now consider the embedding of the apparent horizon corresponding to the black hole plus Brill wave data set with parameters.
Our Method
- A Direct Method for Horizon Embeddings in Axisymmetry
- Our General Method for Embeddings in Full 3D
We then have two natural sets of coordinates in the embedded surface: those inherited directly from the original surface via the embed mapping, and the standard angular coordinates in flat space. It is important to note that the metric components in the fitted surface will be the same as the metric components in the original surface only if we are using the legacy coordinate system, but not if we are using standard angular coordinates in flat space. The built-in area metric will now be completely determined by a set of alm, bnm, and cnm coefficients.
It is not difficult to find a real-valued function F defined on V that has a global minimum at the point P ∈ V for which the metric of the embedded surface is the same as the metric of the original surface.
Tests
- Recovering a Known Surface
- An Axisymmetric Example: Rotating Black Holes
- Black Hole Plus Brill Wave
- Application to Full 3D Spacetimes
The dashed line is the result of our minimization algorithm, and the solid line is an embedding of the same surface made with an axisymmetric algorithm in the embeddable region. This is easily seen by examining the residual function F for the embeddings of the Kerr horizons. In the figure, the dashed line shows the embedding obtained with our minimization algorithm, and the solid line shows the embedding of the same surface obtained in [24].
The dashed line is the embedding obtained with our minimization algorithm and the solid line the embedding of the same surface obtained by Anninos et al.
Conclusions
However, the mass energy carried by the non-axisymmetric modes is much smaller than the energy of the axisymmetric modes [28, 22]. As before, the value of F converges exponentially to zero, but the convergence is slower than in the previous example due to the higher degree of complexity of the surface. We observed that the non-asymmetry of the embedded surface is somehow smaller than you would expect from just looking at the metric.
Brill, “On positive definite mass of time-symmetric Bondi-Weber-Wheeler gravitational waves,” Ann.
