University of Amsterdam
MSc Physics and Astronomy
Science for Energy & Sustainability
Master Thesis
Optical System Design of the Einstein
Telescope Pathfinder
by
Annabel Wolf
12316970 (UvA)
2651888 (VU)
60 ECTS
January 2020 - December 2020
Supervisors:
prof. dr. S. Hild
dr. S. Steinlechner
Examiners:
prof. dr. F. L. Linde
prof. dr. A. P. Colijn
ET Pathfinder
Abstract
ETpathfinder is an R&D environment to develop new technologies for third generation gravitational wave interferometers. The aim of this thesis is to design several aspects of the optical system of ETpathfinder. The FINESSE software is used for the simula-tions. Arm cavity parameters such as the complex beam parameter, stability, and radii of curvature of the test masses are investigated. Moreover, the consequences of radiation pressure coupling and the effect of an etalon in the input test mass are brought forward. Finally, several input mode cleaner designs are discussed, as well as mode matching be-tween these input mode cleaners and the input test mass of the arm cavity, and the influence of astigmatism on this. Overall, the results provide a basis for ETpathfinder’s optical design.
Acknowledgement
I would like to thank my supervisors in Maastricht, Sebastian Steinlechner and Stefan Hild, for their guidance and useful discussions. I also want to thank my examiners in Amsterdam, Frank Linde and Auke Pieter Colijn, for their time.
I very much appreciate the help of Ronald Westra and Gideon Koekoek, who helped me find this thesis research position. Also, thank you Lex Greeven for proofreading my thesis and providing me with many useful comments. I’m very grateful for the support of my family and friends during the past year. Finally, I’d like to thank the whole GWFP research group.
List of Abbreviations
AR Anti-Reflective CE Cosmic Explorer ET Einstein Telescope ETM End Test Mass
ETpathfinder Einstein Telescope Pathfinder
FINESSE Frequency domain INterferomEter Simulation SoftwarE FPMI Fabry-Perot Michelson Interferometer
FSR Free Spectral Range FWHM Full Width Half Maximum HG Hermite-Gaussian
HOM Higher Order Mode HR Highly Reflective IMC Input Mode Cleaner ITM Input Test Mass
KAGRA Kamioka Gravitational Wave Detector LG Laguerre-Gaussian
LIGO Laser Interferometer Gravitational-Wave Observatory RoC Radius of Curvature
Contents
1 Introduction 2
1.1 Gravitational Wave Detectors . . . 2 1.2 Einstein Telescope Pathfinder . . . 4 1.3 Research Objectives . . . 6
2 Theory 7
2.1 Gaussian Beams . . . 7 2.2 Optical Cavities . . . 10 3 Optical Simulations for ETpathfinder 14 3.1 Arm Cavity Parameters . . . 14 3.2 Input Mode Cleaner . . . 22 3.3 Beam Matching between IMC and ITM . . . 28
4 Conclusion 32
References 33
A Appendix 35
A.1 Optical Parameters . . . 35 A.2 Code . . . 35 A.3 IMC . . . 40
1
Introduction
Gravitational waves were first detected in 2015 by the Advanced LIGO gravitational wave observa-tory (Abbott et al., 2016). This accomplishment took approximately 100 years to be realized, since the prediction of gravitational waves by general relativity in the early 20th century. Detectable gravitational waves are emitted from sources with rapidly accelerating (vast) masses such as coa-lescing binary systems, either black hole – black hole, neutron star – neutron star, or black hole – neutron star; pulsars; supernovae; or the stochastic background (Bonnand, 2012). Even though currently operating gravitational wave detectors have observed many coalescences of compact bi-nary systems already (Nitz et al., 2020), increasing the detection rate will open up a whole new range of astrophysical discoveries and theories. A new generation of gravitational wave observa-tories, such as the Einstein Telescope, will make this possible. The Einstein Telescope Pathfinder (ETpathfinder) will play an important role to research and develop new technologies to improve the instrumentation for such facilities even further.
An introduction to gravitational wave detectors is given in this chapter, followed by a brief explanation of the interferometric detection principle. Subsequently, a description of the ET-pathfinder is given and the research objectives of this thesis are outlined. In Chapter 2 Gaussian beams and optical cavities are introduced. This is followed by a discussion of the optical component simulations and calculations in chapter 3, and the results are summarized in chapter 4.
1.1
Gravitational Wave Detectors
The first attempts to measure gravitational waves were done with resonant bar detectors at room temperature. Although the sensitivity of these measuring devices was improved over several decades, the interferometric method showed greater potential at the beginning of the 21st cen-tury (Losurdo, 2014).
Figure 1: Amplitude spectral density as function of frequency for several existing gravitational wave detectors and the future Einstein Telescope, produced via http://gwplotter.com/ (Moore et al., 2014). The yellow area outlines the amplitude of the GW150914 (first measured gravitational wave) signal and the pink marks the pulsar amplitude range.
The first generation of interferometric gravitational wave detectors, consisting of Virgo, LIGO, TAMA300 and GEO600, were not sensitive enough to measure any gravitational waves. However, many new technologies were developed during this phase. Implementing all these novel technologies led to second generation (ground-based) observatories: Advanced Virgo, Advanced LIGO, KAGRA and GEO HF (Cerdonio & Losurdo, 2012). Third generation detectors, such as the Einstein Telescope (ET) or Cosmic Explorer (CE), are expected to have the sensitivity enhanced by another order of magnitude (Losurdo, 2014) corresponding to a factor thousand in detection volume.
Figure 1 shows the measuring range of several first and second generation detectors, and the aimed sensitivity of ET. The square root of the power spectral density is the most commonly used quantity for sensitivity curves and provides us with the amplitude spectral density (Moore et al., 2014). This is in units of strain/√Hz, where strain is a measure of relative length change in the arm cavities of the observatories. In other words, it is a measure of the fraction of stretching or compression in the arms. The strain amplitude reach of current detectors is in the order of 10−23/√Hz. Not only will the overall sensitivity be enhanced in third generation detectors, but there will be a significant increase in the lower frequency detection domain (Punturo, L¨uck, & Beker, 2014) as can be seen for ET in figure 1.
Gravitational waves can be measured by extremely sensitive Fabry-Perot Michelson laser in-terferometers. The laser beam that enters the Michelson interferometer is split into two by a beamsplitter. The two beams then both travel to an arm cavity with two suspended test masses each, the input (ITM) and end test mass (ETM). The beams are reflected by the ETMs and, after a large number of round trips, travel back to the beamsplitter, where the recombined output beam is directed onto a photodiode. A schematic of such a laser interferometer is provided in figure 2.
Figure 2: Schematic of a laser interferometer, the laser source, beam splitter, test masses (sus-pended mirrors) and photodetector are indicated (Hough et al., 2005; Barish & Weiss, 1999).
In the event of a gravitational wave, the optical path lengths traveled by the two beams are not equal because the space is stretched in one direction and compressed in the perpendicular direction. The output photodetector then measures a phase difference between the two, proportional to the amplitude of the gravitational wave. A laser interferometer is commonly operating at (or close to) the dark fringe position, meaning the signal output is zero when no gravitational waves are interacting with the detector. This is caused by the destructive interference of the two laser beams. In the case of a gravitational wave event, two signal sidebands form around the recombined fundamental beam which are detected by the output photodiode (Vajente, 2014a). The offset field would be generated by a difference in optical path length between the arm cavities of the Michelson interferometer (DARM offset) or between the beamsplitter and the ITMs (MICH offset) (Vajente, 2014b). To improve sensitivity, several methods can be used to gain a larger signal to noise ratio. Different types of noise that are necessary to suppress are described in the next section.
Noise Sources
Laser interferometers have to deal with different types of noise sources, both fundamental and technical in origin. One of the challenges in the ETpathfinder research environment is to find ways to suppress the overall noise level for third generation interferometers. The sources that limit the sensitivity of gravitational wave detectors most significantly are mentioned in this section.
The fundamental noise limit is caused by the quantum nature of light. Quantum noise includes both shot noise and quantum radiation pressure noise, where the first is most prominent at high frequencies and the latter at low frequencies (Goda et al., 2008). Light behavior is not completely deterministic, but subject to random fluctuations putting a limit on the sensitivity of the measure-ment of the detected power. This is known as shot noise. Quantum radiation pressure noise arises from the force that photons exert on the mirrors, resulting in slight displacements. An increased laser power will decrease the level of the shot noise (Bonnand, 2012), however, this will increase the contribution from the radiation pressure noise. The use of (frequency dependent) squeezed light could help reduce these quantum noise effects (Abbott et al., 2017).
Another source that limits the sensitivity of laser interferometers is thermal noise. This origi-nates from the test masses, their coating, and the suspensions, and includes Brownian thermal noise and thermoelastic dissipation (relaxation). Brownian noise is a consequence of internal friction of the test masses’ materials, whereas thermoelastic noise results from thermodynamic fluctuations of the system’s temperature (Hough et al., 2005). The thermal noise level of the test masses and coatings can be lowered by reducing the temperature of the mirrors and increasing the laser spot size (Gr¨af et al., 2012). Suspension thermal noise can be decreased by using longer suspensions and a lower operating temperature. ETpathfinder will operate under cryogenic conditions to decrease this noise level.
Seismic noise is caused by the coupling of ground motions to the test masses. Earthquakes, weather conditions or human activity can influence this noise level. It is most prominent in the lower frequency regime and is almost negligible above 30 Hz (Abbott et al., 2017). ETpathfinder is built on an independent foundation and the test masses are suspended to suppress seismic noise. Noise caused by a variation of the local mass distribution in the ground or atmosphere, causing fluctuations in the local gravitational field, is known as Newtonian noise (Beccaria et al., 1998). The interferometer cannot be shielded from Newtonian noise as it is a consequence of direct coupling from the surrounding geology to the test masses (Beker, Van Den Brand, Hennes, & Rabeling, 2012). The only way to reduce Newtonian noise is to build the observatory in a quiet seismic environment.
Beam scattering noise and damping noise are both types of residual gas noise that influence the sensitivity of laser interferometers. Beam scattering noise is caused by optical path-length changes and rescattering of light, and damping noise results from residual gas impacting on the test masses (The ETpathfinder Team, 2020). Both are caused by non-perfect vacuum conditions, therefore, improving the vacuum level will decrease this noise level. There are several other tech-nical noise sources that limit the sensitivity of interferometric detectors. Examples include noise from electronics, control loops and charging noise (Sengupta, 2016).
1.2
Einstein Telescope Pathfinder
ETpathfinder is built to test and develop new technologies for third generation laser-interferometric gravitational wave detectors with the goal to highly increase their detection sensitivity compared to existing observatories. It is currently being constructed in Maastricht and is an international collaboration with more than 15 partners. The main differences in design with currently operat-ing detectors are the operatoperat-ing temperature, the mirror material, and the laser wavelength (The ETpathfinder Team, 2020).
Figure 3: Schematic of the ETpathfinder, showing the two Fabry-Perot Michelson Interferometers (FPMI) operating at different temperatures and wavelengths (The ETpathfinder Team, 2020). The arms are 20 m each and contain parallel arm cavities of 9.22 , illustration is not to scale. interferometers, arranged in two arms of 20 m each as shown in figure 3. One arm will be cooled to 15 K and tests a laser wavelength of 2µm, whereas the second will operate at 120 K with laser wavelength 1550 nm. The operating temperatures are this low to reduce the thermal noise level. The test masses in ETpathfinder are suspended and composed of very pure silicon, this material performs well under cryogenic conditions due to its high coefficient of thermal conductivity in contrast to fused silica (often used as test mass material). The specific temperatures mentioned above were selected because the coefficient of thermal expansion of silicon is approximately zero for 120 K (White, 1973) and close to zero for 15 K (a lower temperature would give a lower coeffi-cient, but that is technically challenging). As consequence, no or very little thermoelastic noise is produced by the test masses. The laser wavelengths in ETpathfinder are significantly larger than the laser wavelength of 1064 nm used in currently operating gravitational wave detectors, such as Advanced Virgo and Advanced LIGO (P˜old, 2014; Bonnand, 2012). The increased laser wave-length is necessary for the light to be transmitted through the silicon, as it is not transparent at 1064 nm (Punturo et al., 2014). The optical specifications of ETpathfinder are listed in appendix A.1.
ETpathfinder has a target sensitivity of 10−18m/√Hz at 10 Hz in the first phase (The ET-pathfinder Team, 2020), corresponding to a strain integrated over the cavity arm length (≈ 10 m) of the order of 10−19/√Hz. The strongest gravitational wave sources produce signals with strain am-plitudes of typically 10−21/√Hz (Hough et al., 2005), so even if the arm cavities in ETpathfinder were configured perpendicular to each other instead of parallel, they would not be expected to measure any gravitational waves.
General Layout of the Optical Configuration
The optical components of ETpathfinder are divided into four main sections: the pre-stabilized laser, the input optics, the core optics and the detection. The prestabilized laser enters the vacuum system with a power of approximately 1 W. It continues through the input mode cleaner and modulator before it reaches the core optics: the beam splitter and arm cavities. The laser is split at the beamsplitter, travels towards the parallel arm cavities, and after recombination the light travels through the output mode cleaner and is detected by a photodiode.
–1.5 –1.0 –0.5 0.0 0.5 1.0 1.5 0.0 –0.5 –1.0 –1.5 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 ITMs @ 6.406m, ETMs @ 15.706m two-inch mirror, suspension footprint 15cm x 10cm four-inch mirror, suspension footprint 20cm x 10cm 150cm
Figure 4: Preliminary optical layout plan of ETpathfinder (The ETpathfinder Team, 2020). The final design will be based on this, but is not complete at the time of writing.
Figure 4 shows a more detailed overview of the preliminary layout of the optical components. The arm cavities are located on the upper left and left side of this schematic. The triangular cavity on the bottom is the input mode cleaner (IMC). After the IMC, the beam goes through a system that mode matches the beam to the ITM mirrors of the arm cavities. The test masses have a diameter of 15 cm and are 8 cm thick. One side of each mirror is highly-reflective, the HR surface side; the other has an anti-reflective coating layer, the AR coating side. The schematic shown here is not the final design.
1.3
Research Objectives
The aim of this thesis research is to contribute to the design of several optical components of the ETpathfinder. The focus of research is on the arm cavity parameters, the input mode cleaner, and beam matching between the IMC and ITM. The main tool to make the simulations and models is FINESSE (Brown & Freise, 2014). All research in this thesis is focused on the input and core optics of the 1550 nm laser interferometer arm of ETpathfinder.
The theory to support the simulations and calculations is first explained, after which the following optical parts are studied:
• General arm cavity parameters such as the beam spot size, beam waist and its location, the cavity stability, and free spectral range.
• Optimal radius of curvature of the anti-reflective coating side of the input test mass. • The effect of radiation pressure coupling in the arm cavities.
• The effect of an etalon in the input test mass. • Several input mode cleaner designs.
2
Theory
The behavior of laser beams in gravitational wave interferometers is commonly described by Gaus-sian beam optics. An introduction to GausGaus-sian beams and optical cavities is given in this chapter.
2.1
Gaussian Beams
Generally, the propagation of light through optical systems can be described by Maxwell’s equa-tions. If it is assumed that the monochromatic light of a laser propagates along the optical axis z with small perpendicular, or transverse, components the field can be described by the paraxial diffraction approximation (Vajente, 2014a). One solution of this paraxial Helmholtz approxima-tion, assuming cylindrical symmetry, is the Gaussian beam solution. A schematic of a Gaussian beam is shown in figure 5.
Figure 5: Gaussian beam, showing the curved wavefronts of the beam with zR as the position of
the maximum curvature: the Rayleigh range. w0 indicates the beam waist and w(z) the beam
radius depending on the location on its optical axis z.
The beam radius w(z) and the wavefront radius of curvature R(z), given by:
w(z) = w0 r 1 + z zR 2 (1) R = z 1 +zR z 2 (2) are two useful relationships between quantities of Gaussian beams (Saleh & Teich, 2007). w0
is the beam waist, z is the distance to this waist, and zR the Rayleigh range. The beam radius,
also called spot size, is the radius at which the intensity is 1/e2times the maximum intensity, and the waist is the minimum spot size. The radius of curvature of the wave front is infinite at the beam waist, because the phase surface is flat here. The curvature of the wavefront increases when moving away from the waist until the Rayleigh range is reached, the point where the curvature is maximum, given by:
zR=
πw2 0
λ (3)
At this distance along the beam waist propagation axis, the area of the beam cross section is doubled.
The properties of a Gaussian beam at a specific point along the optical axis can be described by the complex beam parameter, defined as (Hodgson & Weber, 2005):
Higher Order Modes & Gouy Phase
As mentioned before, the fundamental Gaussian beam is one solution of the paraxial Helmholtz equation. In addition to this, there are other solutions, leading to the Hermite-Gaussian (HG, in cartesian coordinates) and Laguerre-Gaussian (LG, in cyclic coordinates) modes (P˜old, 2014). These describe the transverse higher order modes (HOMs) of a laser beam. HOMs have the same basic properties as the fundamental laser beam (such as the wavefront curvature), but differ in intensity pattern. A beam can be described by the sum of either its HG or LG modes, in this thesis it is more convenient to use HG modes. The modes are indicated with TEMmn, where n
gives the number of nodes in the horizontal direction and m the number of nodes in the vertical direction. Following this, TEM00 is the pure fundamental mode. The fundamental mode as well
as several higher order Hermite-Gaussian modes are shown in figure 6.
Figure 6: The intensity patterns of the fundamental mode TEM00, and several higher order
Hermite-Gaussian modes (Kogelnik & Li, 1966).
Gaussian beams have slightly slower phase velocities compared to plane waves. This additional longitudinal phase lag is called the Gouy phase shift and is more prominent for higher order modes. This phase shift is described by:
φ = (m + n + 1) arctan λz πw2
0
!
(5)
where m and n are the mode order and the second term the Gouy phase (Bond, Brown, Freise, & Strain, 2016). The Gouy phase difference between individual HOMs plays an important role in optical resonators, since it changes the resonance condition. This is the principle of mode-cleaning cavities, which selectively transmit only one particular mode.
ABCD matrices
In ray optics, ABCD matrices are used to relate the distance from the optical axis and the slope of a ray as it propagates through an optical system (a series of optical elements). The vertical position and slope at the final position, x2and x02respectively, are linearly dependent on the input
parameters x1 and x01such that:
x2 x02 = A B C D x1 x01 (6) In this paraxial approximation it is assumed that all optical elements are very thin and all angles of propagation are sufficiently small so that sin(θ) ≈ θ (Kogelnik & Li, 1966). For example, if a beam with certain vertical distance from the optical axis z starts at point z1 and travels to
point z2 under an angle of x01 in the same medium, the distance to the optical axis changes by
(z2− z1)x01 and the angle stays constant. This situation is illustrated in figure 7.
Figure 7: Propagation of a light ray along the optical axis z under an angle of x01= x02. x1and x2
are the vertical distances to the optical axis.
Thus, the matrix that transforms the beam from z1 to z2, distance d, is:
x2 x02 = 1 d 0 1 x1 x01 (7) The same approach can be used for light moving through lenses or mirrors. The ABCD matrix for propagation through a (thin) lens is described by the change in the slope of the beam. For parallel beams this is related to the location where it crosses the optical axis, the focal length f . A beam that encounters a mirror experiences a change in its slope with respect to the optical axis depending on the radius of curvature (RoC) of the mirror, Rc. The ABCD matrix of a spherical
mirror is closely related to that of a thin lens, as f = Rc/2. The change upon encountering a lens
or mirror is respectively described by: 1 0 −1 f 1 and 1 0 −2 Rc 1 (8)
Rc is negative for convex surfaces and positive for concave surfaces.
If a beam travels through different media, the refractive index of the materials also need to be taken into account. The propagation of a beam in a medium with refractive index n, or through a mirror substrate from a medium with refractive index n1 to n2are given by:
1 nd 0 1 and 1 0 n1−n2 Rc−n2 n1 n2 (9)
The same ABCD matrices can be used to describe the change in beam parameters for Gaussian beams. In this case, the ABCD matrix relates the complex beam parameters of different points in a system. The output complex beam parameter q2is related to the input complex beam parameter
q1 as:
q2=
Aq1+ B
Cq1+ D
(10) where A, B, C, and D are the elements of the ABCD matrix of the optical system. The overall ABCD matrix of a system is the product of the matrices of the individual components.
2.2
Optical Cavities
Optical cavities, also known as optical resonators, are systems in which a light beam resonates between two (or more) mirrors. An example of an optical cavity is the Fabry-Perot cavity, often used in laser interferometers to enhance the average photon storage time. Each cavity has a distinct eigenfunction, the cavity mode, for which the laser beam inside is resonant and the light power builds up. In such a situation the wavefront curvature of the light matches the radius of curvature of the mirrors of the resonator. A simple optical cavity with two mirrors is shown in figure 8.
Figure 8: An optical cavity with two mirrors, M1 and M2 (Brooker, 2003). The vertical dashed
line shows the location of the beam waist, w indicates the beam spot size and a is the radius of the mirror.
Light is resonant when a multiple of half-wavelengths fit inside the cavity. The spacing between the resonant frequencies in a cavity is called the free spectral range (FSR), and equals the speed of light divided by twice the arm cavity length. The detuning from the peak resonance frequency that gives half of the maximum power is known as the full width at half maximum (FWHM) (Vajente, 2014a), often referred to as the linewidth of a cavity. The ratio between these two quan-tities provides information about how narrow the resonance peaks are compared to the frequency spectrum:
F = F SR
F W HM (11)
where F is the finesse of a cavity. Cavity Stability
An optical cavity is stable if the laser beam will replicate itself after a round trip in the cavity, known as the eigenmode of the system. In other words, the complex beam parameter after one round trip is equal to the initial complex beam parameter, so it follows that:
q1= q2= q =
Aq + B Cq + D Rearranging leads to:
q(Cq + D) = Aq + B → Cq2+ (D − A)q − B = 0 This can be solved with the quadratic formula, resulting in:
q = (A − D) ±p(D − A)
2+ 4BC
2C
Knowing that the determinant, AD − BC, is equal to 1 (Kogelnik & Li, 1966), it follows that: 2Cq = A − D ±p(D + A)2− 4
Since q is describing a Gaussian beam and is therefore a complex number, the radicand has to have a negative value. This means that (D + A)2< 4.
The ABCD matrix of a linear cavity with length d is the product of the ABCD matrices for the first mirror with RoC R1, a propagation over the distance d, a reflection at the second mirror
with RoC R2, and another propagation of d. One roundtrip in the cavity then corresponds to:
A B C D = 1 0 − 2 R1 1 1 d 0 1 1 0 − 2 R2 1 1 d 0 1 = 1 −R2d 2 2d − 2d2 R2 4d R1R2 − 2 R1 − 2 R2 1 + 4d2 R1R2 − 4d R1 − 2d R2 Combining the above result with the requirement of (D + A)2< 4 we obtain that:
1 − 2d R2 + 1 + 4d 2 R1R2 − 4d R1 − 2d R2 !2 < 4 Taking the square root and dividing by 2 results in:
−1 < 1 + 2d 2 R1R2 − 2d R2 − 2d R1 < 1 Now adding 1 to both sides and dividing by 2 gives:
0 < 1 + d 2 R1R2 − d R2 − d R1 < 1 ↓ 0 < 1 − d R1 1 − d R2 < 1 (12)
The above terms,1−Rd i
, are defined as g1and g2respectively. These are called the g−factors.
The cavity stability criterion is thus:
0 < g1g2< 1 (13)
Only if the criterion in equation 13 holds, real and finite solutions are obtained for the beam waist and spot sizes (Diaz, 2018).
Sidles Sigg Instabilities
The two suspended test masses of the Fabry-Perot arm cavities in a gravitational wave detector experience independent torsional oscillations if no light is present. However, in a locked suspended cavity the intra-cavity laser power couples the two test masses (Hirose, Kawabe, Sigg, Adhikari, & Saulson, 2010). The resonating light now acts as an optical spring, where its strength is propor-tional to the laser power in the arm cavity (Liu et al., 2018). If there is any misalignment between the two mirrors, two angular modes can arise: the angular hard and soft mode. A hard mode has a tilted optical axis, whereas a soft mode has a shifted optical axis. Both angular modes are presented in figure 9.
Figure 9: Illustration of the hard and soft mode (Liu et al., 2018).
The hard mode is always stable, because the radiation pressure induced torque is working in the same direction as the mechanical restoring torque of the suspensions. The soft mode however, can cause the system to become unstable as it acts as optical anti-spring on the test masses. Thus, if the radiation pressure torque exceeds the intrinsic angular restoring torque, the cavity will be unstable. This optomechanical effect is also known as Sidles-Sigg instability (Sidles & Sigg, 2006).
The eigenvalues of the hard and soft mode are defined as Ωh and Ωsrespectively, given by:
Ωh= s −P L Ic g1+ g2−p4 + (g1− g2)2 1 − g1g2 (14) Ωs= s −P L Ic g1+ g2+p4 + (g1− g2)2 1 − g1g2 (15) where P is the circulating power in the cavity, L the cavity length, I the moment of inertia of the test mass, c the speed of light, and g1 and g2 the g−factors (Liu et al., 2018).
Kp= Iω02 (16)
ω0 is the resonant angular frequency of the suspended mirrors, so 2π times the common mode
frequency. The optical torsional spring constant due to radiation pressure is defined as:
Krp,h/s= IΩ2h/s (17)
with Ωh/s the eigenvalue of the hard or the eigenmode of the soft mode as given in equation
14 and 15 respectively. If the pendulum spring constant is larger than the radiation pressure induced spring constant, the restoring torque will bring the system to equilibrium. If not, the anti-torsional spring will cause the system to become unstable. Since the radiation pressure induced spring constant is dependent on the intra-cavity power, the system will become unstable once the threshold power Pthreshold is reached:
Pthreshold=
Iω2
0c(1 − g1g2)
Lg1+ g2+p(g1− g2)2+ 4
3
Optical Simulations for ETpathfinder
This chapter contains simulations and calculations for ETpathfinder’s arm cavity parameters, its input mode cleaner, and mode matching between the two. The discussions are focused on the interferometer with laser wavelength 1550 nm and temperature of 120 K. The software FINESSE, Frequency domain INterferomEter Simulation SoftwarE, was used to produce all simulations in this thesis except the ‘Etalon Effect’ plots (these were computed in Mathematica). FINESSE was specifically designed for the optical system simulation of interferometric gravitational wave detec-tors. It translates the input code into a set of linear equations, which are then solved numerically (Freise et al., 2004). Relevant parts of the FINESSE simulations are included in appendix A.2.
3.1
Arm Cavity Parameters
ETpathfinder consists of two Michelson interferometers with parallel Fabry-Perot resonators, formed by an ITM and ETM each. The radii of curvature of the ITM and ETM mirror’s HR surface are chosen to be 14.5 m. This choice is justified in the ETpathfinder design report chap-ter 6 (The ETpathfinder Team, 2020). The distance between the two test masses is 9.22 m, and the beam waist location is in the middle of the two mirrors since the RoC of both are the same. Symmetric cavities have the advantage of having large spot sizes on the test masses, lowering the thermal noise level (Hild et al., 2008). Using equation 1, 2, and 3 the beam radius at the mirrors and the beam waist of the resonator can be calculated as:
w = 2.210 mm and w0= 1.825 mm
The stability of the cavity is calculated by multiplying the g-factor of both mirrors, which are both 0.3641 according to equation 12. The cavity g-factor product is then:
g1g2= 0.1326
The cavity is stable, because the product of g−factors is between zero and one and thus the stability criterion is met.
1.0 1.5 2.0 2.5 3.0 3.5 4.0 f [Hz] ×107 0 100 200 300 400 500 Power [W]
Figure 10: Circulating power in the arm cavity as function of frequency. Two resonant frequencies are shown, the distance between them is the FSR, which equals c/2L ≈16.258 MHz.
The FSR of the resonant modes in the cavity is shown in figure 10 and is the distance between the peaks equaling 16.258 MHz.
Anti-Reflective Coating Side of ITM
The HR surface side of the arm cavity mirrors are curved to obtain a stable cavity. It is also important to consider the RoC of the AR surface side of the test masses, because if these were to be flat the laser beam would significantly diverge after leaving the cavity. The AR coating side should ideally cause the beam to be collimated when it reaches the steering mirrors, located 6.406 m in front of the ITM mirror. Moreover, the steering mirrors of ETpathfinder are only three inches (= 76.2 mm) in diameter, so it is important that the beam diameter after 6.406 meter is significantly smaller than the mirror’s diameter, to avoid diffraction losses. If this is ignored, some of the energy in the beam will leak away around the edges of the mirror (Saleh & Teich, 2007).
Assuming the angle of incidence on the steering mirror is 45◦, the radius of the beam is:
w = 1 2
√
2wincident (19)
where the incident beam radius is the radius of the beam on the mirror plane. Moreover, it is assumed only 80% of the three-inch mirror can be used due to the coating application process limitations. This number is a generally used estimate to ensure only the fully coated surface area of the mirror is considered. This leads to:
wincident, max= 0.8 × 1 2 × 76.2 mm = 30.48 mm wmax= 1 2 √ 2wincident,max= 21.55 mm
So the maximum beam radius for a three-inch mirror is 21.55 mm. However, it is desirable that the actual beam radius is a factor of three smaller than the maximum beam radius, so that the power losses are only about 1 ppm. This results in a beam size limit at the mirror of 7.18 mm.
Figure 11 shows the beam radius as function of RoC of the AR coating side of the ITM mirror. The blue and orange line show the beam radius after 6.406 m for a fused silica and silicon mirror respectively. The green and pink line, which are overlapping, represent the beam radius just after the ITM mirror. The dotted black line indicates the beam size limit at the steering mirror of 7.18 mm as was calculated above. The plot shows that any RoC of more than a few meters (≈ 2.5 m) results in a beam radius on the steering mirror that is well below the beam size limit. The interception points of the fused silica mirror are (1.196 m, 2.222 mm) and (3.304 m, 2.222 mm). For silicon test masses this is (4.841 m, 2.222 mm) and (9.039 m, 2.222 mm). So for the latter case the beam radius at 6.406 m is equal to the beam radius just after the mirror if the radius of curvature is 4.841 or 9.039 m. Deviation in the beam radius for small changes in the RoC is smaller for the second intersection point, because the slope of the beam radius curve is less steep compared to the first intersection point. Therefore a RoC of roughly 9 m is preferred.
Figure 12 shows the beam waist between the arm cavity and steering mirror for different radii of curvature of the AR coating side of the ITM for both fused silica (blue curve) and silicon (orange curve) mirrors. Again, the overlapping green and pink line represent the beam radius just after the ITM mirror. It shows that the beam waist for an RoC of 9 m is in the order of millimeters. It is important that the beam waist is not very small (in the order of micrometers or tens of micrometers), because then the system might be unstable. More details about the FINESSE code for both figures is included in appendix A.2.
Figure 11: The radii of the beams after 6.406 m vs. the RoC of the AR coating side of the ITM. The plot shows the curve for fused silica (n = 1.45) and silicon (n = 3.5) mirrors. The intersection points indicate the RoC values for which the beam is collimated at the steering mirror.
100 101 102 103 Radius of Curvature [m] 0 2 4 6 8 Radius [mm] w0, n = 1.45 w at ITM, n = 1.45 w0, n = 3.5 w at ITM, n = 3.5 beam size limit
Figure 12: The beam waists and beam radii vs. the RoC of the AR coating side of the ITM. The plot shows the curve for fused silica (n = 1.45) and silicon (n = 3.5) mirrors.
Radiation Pressure Coupling
When the length of the arm cavities of ETpathfinder is set to resonance for its laser wavelength, power is build up. Therefore, it is important to quantify if the radiation pressure coupling between the test masses leads to Sidles-Sigg instability, possibly causing an unstable system. Equation 18 can be used to calculate the threshold power inside the cavity for which the optomechanical effect needs to be counteracted. The threshold power is calculated for both the yaw (rotation around the vertical axis) and pitch (rotation around the horizontal axis) modes of the test masses.
For both modes, L is 9.22 m and g1 = g2 = 0.3641. The resonant angular frequency (ω0) and
torsional spring constant (Kp) of the suspended mirrors were provided by Eric Hennes (personal
communication, 17th of July 2020), who calculated these numbers with a finite element model. These values, and the resulting effective moment of inertia from equation 16 for the yaw and pitch mode are respectively:
ω0,yaw= 10.7 × 10−3Hz, Kp,yaw= 6.7 × 10−4N m rad−1 → Iyaw= 0.148 kgm2
ω0,pitch= 450 × 10−3Hz, Kp,pitch= 0.5 N m rad−1 → Ipitch = 0.0625 kgm2
Using equation 18, the following values are found for the threshold intra-cavity power for the yaw and pitch mode:
Pthreshold,yaw= 7.01 kW
Pthreshold,pitch= 5166 kW
The power in the cavity is thus limited by the yaw mode Sidles-Sigg instability that arises for a power exceeding 7.01 kW. This needs to be taken into account in the design of the control loops, where this instability can be countered by active control. Below the threshold power the system’s stiffness is large enough to compensate the optomechanical torque and stay balanced. Hence, for the initial ETpathfinder interferometer with a target cavity power of approximately 500 W (see figure 10), radiation pressure couplings will not play a significant role.
Etalon Effect
A resonant cavity can emerge between the front and back surface of a mirror if these are parallel, and match the beam circulating inside the cavity. However, if the temperature in the mirror varies the optical path length of the laser beam can change due to a change in the refractive index and/or thermal expansion of the mirror substrate (Brooks et al., 2020). This is known as the etalon effect, illustrated in figure 13. The etalon effect can be used to balance the arm cavities of an interferometer by tuning the cavity finesse.
Figure 13: Schematic representation of the etalon effect in a Fabry-Perot cavity (Brooks et al., 2020). AR is the anti-reflective coating and HR the highly reflective side. A resonant cavity is formed inside the mirror substrate, indicated by Lm.
The finesse deviation in ETpathfinder caused by this effect is investigated in this section. Equations from Hild et al. (2008) were used to produce the plots. The amplitude reflectivities ρi
and the reflectance of a lossless etalon ρE are given by:
ρi= p 1 − Ti− Li (20) ρE= ρ2− ρ3(1 − ρ22) e−2iφ− ρ 2ρ3 (21)
where ρ2is amplitude reflectance of the etalon front (ITM HR) and ρ3the amplitude reflectance
of the etalon back (ITM AR). The current design values for the ITM HR side are 7796 ppm for transmission and 50 ppm for the (average) losses. For the ETM HR side these values are 5.0 ppm and 50 ppm respectively. Transmission values are obtained by subtracting the absolute values of reflection (ρE) from 1, for different values of the ITM AR transmittance. Transmission as function
of etalon tuning is presented in figure 14. The reflection values of the ITM AR surface are plotted from 0.1% to 0.01%, assuming no losses.
0 50 100 150 7400 7600 7800 8000 8200
Etalon Tuning [deg]
Transmission of ITM [ppm ] No Etalon Reflection = 0.1% Reflection = 0.05% Reflection = 0.02% Reflection = 0.01%
Figure 14: Transmission of the ITM mirror depending on the etalon tuning for different reflection values of the ITM AR side assuming parallel surfaces.
The etalon modifies the reflectivity of the ITM, and as a result the finesse is changed. The finesse of the arm cavity is given by:
F = π 2arcsin 1 − ρ1ρE 2√ρ1ρE !−1 (22)
where ρ1 is the amplitude reflectance of the ETM HR side. The cavity finesse as function
of etalon tuning is shown in figure 15. The plot shows that a lower reflection results in a less prominent etalon effect in the ITM, the maximum finesse deviation is smaller.
The etalon effect can be very useful to balance the arm cavities in interferometers as is shown in the paper of Hild et al. (2008). However, for ETpathfinder it is desired to actively avoid this effect. Due to manufacturing imperfections, the reflectivities of the ITM mirrors won’t be exactly equal. Therefore, changing the temperature to control the finesse in the arms, will lead to slightly different finesse values for each Fabry-Perot cavity as the mirror temperatures cannot be controlled individually (the mirrors are located in the same arm). The etalon effect can be avoided by having different RoC values for the HR and AR surface sides, or by adding an additional wedge to the AR side of the ITM. First, the unequal RoC option is explored.
The laser beam is transformed when it is transmitted through the HR side of the ITM from the arm cavity. The new complex beam parameter can be calculated using the ABCD matrix for transmission through a mirror with a different refractive index, given by equation 9. In this matrix, n1 is the refractive index in the arm cavity, n = 1 (vacuum). n2 is the refractive index
of the mirror material, n = 3.5 (silicon). The complex beam parameter of the transmitted beam related to the cavity beam parameter qc is then:
0 20 40 60 80 740 760 780 800 820 840
Etalon Tuning [deg]
Cavity Finesse No Etalon Reflection = 0.1% Reflection = 0.05% Reflection = 0.02% Reflection = 0.01%
Figure 15: Arm cavity finesse vs. etalon tuning for different reflection values of the ITM AR side. The HR and AR surface are assumed to be parallel.
qt= Aqc+ B Cqc+ D = qc+ 0 qcR1−nHRn +1n = nqc qcR1−nHR+ 1
When the beam is reflected from the AR side of the ITM mirror, the complex beam parameter is altered by traveling through distance d between the two surfaces and the encounter of the second mirror. The ABCD matrix for this change is given by:
1 0 − 2 RAR 1 1 d n 0 1 = 1 d n − 2 RAR − 2d RARn+ 1
This results in the complex beam parameter of the reflected beam depending on qtas:
qr= qt+dn − 2 RARqt− 2d RARn+ 1
The coupling coefficient that relates the two beams inside the mirror substrate is calculated following the approach in Hild et al. (2008) and Bayer-Helms (1984). This number is used to find the influence of the mode mismatch on the finesse vs. etalon tuning curve. The strongest etalon effect will be present when the RoC of the front and back surface are equal. In ETpathfinder the HR side of the ITM will most likely have a RoC of 14.5 m whereas the AR side will be 9 m.
The coupling coefficient is defined as:
k0000(qr, qt) = √ 1 + K0 1 + K∗ (23) where: K = 1 2(K0+ iK2) = i(qr− qt)∗ 2Im(qt) (24)
K0= zR,r− zR,t zR,t (25) K2= zr− zt zR,t (26) The coupling coefficient is multiplied with the reflectance of the ITM AR surface in the finesse equation. The computed finesse of these non-parallel surfaces as a function of tuning is shown in figure 16. 0 20 40 60 80 770 780 790 800 810 820
Etalon Tuning [deg]
Cavity Finesse No Etalon Reflection = 0.1% Reflection = 0.05% Reflection = 0.02% Reflection = 0.01%
Figure 16: Arm cavity finesse vs. etalon tuning for different reflection values of the ITM AR side taking into account that the front surface has an RoC of 14.5 m and the back surface 9 m.
Figure 16 shows that the range of cavity finesse is smaller compared to the cavity finesse range in figure 15, where both sides of the mirror have the same RoC value. The maximum finesse deviations are written in table 1.
Maximum Finesse Deviation
Reflection AR (%) RoC HR and AR 14.5 m RoC HR 14.5 m and AR 9 m 0.1 99.6, 6.48% 56.1, 3.61%
0.05 70.4, 4.54% 39.7, 2.54% 0.02 44.5, 2.85% 25.1, 1.60% 0.01 31.5, 2.01% 17.8, 1.13%
Table 1: Maximum finesse deviation numbers for the plots shown in figures 15 and 16. The relative deviation from the nominal finesse value is also provided.
To evaluate the effect of other AR side RoC values, figure 17 was produced. The plot has 0 degree etalon tuning and the ITM HR side is assumed to be 14.5 m. The red shades show AR side RoCs smaller than 14.5 m, and the blue lines RoCs larger than 14.5 m. It is clear that the finesse deviation drops significantly if the RoCs of the HR and AR surfaces are different, and that the deviation drops quicker if the AR RoC is smaller than the HR RoC.
Another method to break the etalon effect is to tilt the back surface of the ITM mirror slightly. The coupling coefficient that takes this misalignment angle, γ, into account (Hild et al., 2008) is given by:
0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 750 760 770 780 790 ITM AR Reflectivity Finesse 24 m 22 m 20 m 18 m 14.5 m 11 m 9 m 7 m 5 m
Figure 17: Finesse as function of ITM AR reflectivity for different values of the AR RoC. The black line shows the finesse deviation for an AR coating side RoC of 14.5 m, smaller RoCs are indicated in shades of red and larger RoCs in blue. The HR side has a RoC of 14.5 m.
k0000(qr, γ) = exp −
π|qr|2sin2(2γ)
2λIm(qr)
!
(27)
The effect of introducing this additional wedge to the system is presented in figure 18. Wedges between 1 and 10 millidegree are added to the AR surface with RoC of 9 m, the HR surface has a RoC of 14.5 m. 0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 750 760 770 780 790 ITM AR Reflectivity Finesse 1 mdeg 2 mdeg 3 mdeg 4 mdeg 5 mdeg 6 mdeg 7 mdeg 8 mdeg 9 mdeg 10 mdeg
3.2
Input Mode Cleaner
The pre-stabilized laser source is mode cleaned by the IMC before it enters the arm cavities. The purpose of the IMC is to produce a very pure fundamental Gaussian beam. The IMC cavity does this by filtering out the higher order modes present in the laser beam, therefore, the IMC should be designed in such a way that the modes are non-degenerate resulting in a resonant fundamental mode and suppressed HOMs (R¨udiger et al., 1981). Additionally, the IMC stabilizes the frequency of the laser and can therefore be used as a reference cavity (Diaz, 2018). Different triangularly shaped mode cleaner designs for the IMC of ETpathfinder are considered in the following subsections. A bow-tie cavity, a configuration of four mirrors, can also be used as mode cleaning cavity but is not favored for suspended IMCs due to its complexity.
Large Suspended IMC
The first design considered for the IMC is a triangular cavity with sides of 2.8 m and a base of 0.2 m. This cavity would be distributed over two suspended optical benches located in two different vacuum towers, as shown in the optical layout in figure 4. A schematic of the large suspended IMC design for the ETpathfinder is shown in figure 19.
Figure 19: Configuration of the large suspended IMC, a triangular cavity with sides of 2.8 m and base of 0.2 m. The nodes that were used in the FINESSE simulation, see appendix A.2, are indicated in the figure. The schematic is not to scale.
10
14 × 10
06 × 10
02 × 10
1mHR RoC (m)
10
410
310
210
110
0relative mode power
m+n=0 m+n=1 m+n=2 m+n=3 m+n=4 m+n=5 m+n=6 m+n=7 m+n=8 m+n=9 m+n=10 m+n=11 m+n=12
Figure 20: The relative power of the fundamental mode and first 12 higher order modes as function of radius of curvature of the HR mirror of the large suspended IMC. RoC 6.7, 7.8, 9.4, 14, and 17 m are indicated.
A simulation of the relative location of the eigenmodes of this IMC cavity was computed to find the most optimal RoC of the HR side of the mirror. Figure 20 shows the relative power of the fundamental mode and the first 12 HOMs as function of RoC of the HR mirror of the IMC. Regions where only relatively high modes are resonant with the fundamental mode (or none at all) are suitable values for the HR RoC, this includes 6.7 m, 7.8 m, 9.4 m, 14.0 m, and 17.2 m. For these radii of curvature there is little overlap between the modes, even if the RoC is a few centimeters smaller or larger, see figure 20. This means the HOMs are not (fully) resonant in the cavity and will not contribute significantly to the beam that travels towards the arm cavity.
Table 2 provides the relative contributions of the HOMs for different RoCs, weighted by their mode numbers. As can be concluded from these values, the contribution of HOMs is the smallest when a mirror with a RoC of 6.7 m is used. The beam waist lies between the IC- and OC-mirror (indicated in figure 19) and has a value of 1.34 mm. The mode power of this IMC as function of tuning is shown in figure 21. The other mode scans are shown in appendix A.3. All RoCs in the range of the relative power plot, 4 to 20 m, produce cavities that are well within the stable region.
m+n 6.7 m 7.8 m 9.4 m 14 m 17.2 m 0 1.00 1.00 1.00 1.00 1.00 1 1.03 × 10−4 1.08 × 10−4 1.18 × 10−4 1.54 × 10−4 1.80 × 10−4 2 2.57 × 10−5 2.71 × 10−5 2.96 × 10−5 3.85 × 10−5 4.51 × 10−5 3 3.99 × 10−5 6.64 × 10−5 2.31 × 10−4 3.70 × 10−4 1.05 × 10−4 4 1.78 × 10−4 5.14 × 10−5 2.52 × 10−5 1.40 × 10−5 1.28 × 10−5 5 3.31 × 10−5 2.75 × 10−4 1.37 × 10−4 2.02 × 10−5 2.57 × 10−5 6 9.97 × 10−6 1.66 × 10−5 5.78 × 10−5 9.25 × 10−5 2.62 × 10−5 7 4.19 × 10−5 2.46 × 10−4 1.69 × 10−5 1.36 × 10−4 1.17 × 10−4 8 2.40 × 10−5 8.52 × 10−6 6.32 × 10−6 1.78 × 10−5 1.09 × 10−4 9 8.84 × 10−5 2.03 × 10−5 4.05 × 10−6 3.50 × 10−6 7.35 × 10−6 10 4.14 × 10−6 3.44 × 10−5 1.71 × 10−5 2.53 × 10−6 3.22 × 10−6 11 2.30 × 10−6 3.58 × 10−6 6.14 × 10−5 2.94 × 10−6 2.31 × 10−6 12 4.03 × 10−6 2.11 × 10−6 4.23 × 10−6 6.36 × 10−6 2.41 × 10−6 Total 5.55 × 10−4 8.60 × 10−4 7.09 × 10−4 8.58 × 10−4 6.36 × 10−4
Table 2: Relative HOM power weighted by 1/(m + n) for the IMC with base 20 cm and sides 2.8 m.
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Mode Scan RoC=6.7m
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Figure 21: The power of the fundamental mode and first 12 higher order modes in the large suspended IMC as function of tuning of the HR mirror with radius of curvature of 6.7 m.
Small IMC
In the previous section, the IMC was distributed over suspended optical benches in both towers. As the space in the towers is limited, it would be beneficial if the IMC was designed to be smaller and placed into one vacuum tower. Moreover, a smaller IMC is easier to install and align. To investigate if a smaller IMC would be able to filter out the HOMs as efficiently as the large IMC, the relative power of the fundamental mode and first 12 HOMs as function of HR RoC is shown for two smaller IMC designs. To minimize the incident beam angle on the mirror, the base length should be as small as possible. The minimum base length of the small IMC is dependent on the test mass suspension design and was calculated to be approximately 18 cm for the design at the time of writing, shown in figure 22. A base of 19 cm was used in the simulations to avoid risk.
Figure 22: Schematic of the suspended IMC base length. The mirrors inside the suspensions, indicated as blue lines, have a thickness of 3 cm. Therefore distance a is 5 cm (13.0 cm/2 - 3 cm/2). The angle between ac and bc are both 45◦, so b is also 5 cm and c has a value of √2 × 5 cm = 7.071 cm. d is half of 15.5 cm minus b, therefore d is 2.75 cm. The angle between df and ef are both 45◦, so e is 2.75 cm and f is√2 × 2.75 cm = 3.89 cm. The total minimum base length equals 2 × c + f = 18.03 cm. The schematic is not to scale.
The maximum side length that would fit inside one of the towers is about one meter. This configuration, as well as an IMC with a base of 19 cm and sides of 45 cm were investigated. The two relative power vs. RoC plots are shown in figure 23 and 24, for an IMC with sides of 45 cm and 100 cm respectively.
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03 × 10
04 × 10
06 × 10
02 × 10
1mHR RoC (m)
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0relative mode power
m+n=0 m+n=1 m+n=2 m+n=3 m+n=4 m+n=5 m+n=6 m+n=7 m+n=8 m+n=9 m+n=10 m+n=11 m+n=12
Figure 23: The relative power of the fundamental mode and first 12 higher order modes for an IMC with a 19 cm base and sides of 45 cm. Interesting RoCs to look at would be 3.2, 4.2, 7.5 and 9 m. Peak splitting is visible due to astigmatism.
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04 × 10
06 × 10
02 × 10
1mHR RoC (m)
10
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0relative mode power
m+n=0 m+n=1 m+n=2 m+n=3 m+n=4 m+n=5 m+n=6 m+n=7 m+n=8 m+n=9 m+n=10 m+n=11 m+n=12
Figure 24: The relative power of the fundamental mode and first 12 higher order modes for an IMC with a 19 cm base and sides of 100 cm. Interesting RoC to look at would be 6.4, 8.5 and 15 m. Interesting RoCs for the 19x45 cm IMC to investigate are 3.2, 4.2, 7.5 and 9 m, and for the 19x100 cm IMC 6.4, 8.5 and 15 m. For these values there is a relatively non-degenerate distri-bution of HOMs, especially the lower mode orders are relatively far away from the fundamental mode. Moreover, the overlap does not change significantly for small RoC changes (in the order of centimeters).
Table 3 and 4 provide the values of the relative weighted HOMs power of the small IMC with base 45 cm and 100 cm respectively. From these tables can be concluded that for the small IMC with sides of 45 cm, the RoC of 3.2 m for the HR mirror gives the lowest HOMs contribution. For the small IMC with sides of 100 cm this is 6.4 m. The mode scans for these RoCs are shown in figure 25 and 26 respectively. The other mode scans can be found in appendix A.3.
m+n 3.2 m 4.2 m 7.5 m 9 m 0 1.00 1.00 1.00 1.00 1 1.82 × 10−4 2.28 × 10−4 3.83 × 10−4 4.54 × 10−4 2 4.47 × 10−5 5.60 × 10−5 9.38 × 10−5 1.11 × 10−4 3 1.20 × 10−4 5.49 × 10−5 3.45 × 10−5 3.45 × 10−5 4 1.29 × 10−5 1.28 × 10−5 1.61 × 10−5 1.80 × 10−5 5 2.35 × 10−5 6.56 × 10−5 1.47 × 10−4 5.90 × 10−5 6 2.75 × 10−5 1.31 × 10−5 8.47 × 10−6 8.48 × 10−6 7 9.54 × 10−5 1.64 × 10−5 3.27 × 10−5 1.05 × 10−4 8 1.01 × 10−4 1.89 × 10−4 9.46 × 10−6 7.58 × 10−6 9 7.86 × 10−6 1.53 × 10−4 6.55 × 10−6 4.37 × 10−6 10 3.32 × 10−6 1.10 × 10−5 1.37 × 10−5 6.33 × 10−6 11 2.32 × 10−6 3.97 × 10−6 7.65 × 10−5 1.28 × 10−5 12 2.36 × 10−6 2.37 × 10−6 2.30 × 10−4 5.75 × 10−5 Total 6.24 × 10−4 8.07 × 10−4 1.05 × 10−3 8.78 × 10−4
Table 3: Relative HOM power weighted by 1/(m + n) for the small IMC with sides 45 cm and base 19 cm.
The stability of a cavity is determined by its g-factor and is stable if the value computed by FINESSE is between −1 and 1 (Freise et al., 2004). For RoCs values between 2 and 20 meters g is between −0.905 and 0.095 for the cavity with 100 cm sides and 19 cm base; and g is between −0.9455 and −0.455 for the cavity with 45 cm sides and 19 cm base. Thus, both small IMCs are stable in this RoC range.
m+n 6.4 m 8.5 m 15 m 0 1.00 1.00 1.00 1 1.79 × 10−4 2.26 × 10−4 3.75 × 10−4 2 4.46 × 10−5 5.63 × 10−5 9.34 × 10−5 3 1.13 × 10−4 5.22 × 10−5 3.40 × 10−5 4 1.29 × 10−5 1.28 × 10−5 1.60 × 10−5 5 2.46 × 10−5 7.69 × 10−5 1.34 × 10−4 6 2.78 × 10−5 1.29 × 10−5 8.47 × 10−6 7 1.40 × 10−4 1.72 × 10−5 3.50 × 10−5 8 8.91 × 10−5 1.53 × 10−4 9.51 × 10−6 9 6.98 × 10−6 1.18 × 10−4 7.02 × 10−6 10 3.14 × 10−6 1.02 × 10−5 1.58 × 10−5 11 2.30 × 10−6 3.82 × 10−6 1.19 × 10−4 12 2.47 × 10−6 2.33 × 10−6 1.18 × 10−4 Total 6.46 × 10−4 7.43 × 10−4 9.65 × 10−4
Table 4: Relative HOM power weighted by 1/(m + n) for the small IMC with sides 100 cm and base 19 cm.
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Mode Scan RoC=3.2m
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Figure 25: The mode power of the fundamental mode and first 12 higher order modes for an IMC with base 19 cm and sides of 45 cm, HR RoC is 3.2 m, as function of mirror tuning.
the beam has different beam waist parameters in the sagittal (y − z) and tangential (x − z) plane. These elliptical beams are called astigmatic (Bond et al., 2016). For an IMC with sides of 45 cm and a base of 19 cm, the angle of incidence is approximately 12.19◦. For an IMC with sides of 100 cm and a base of 19 cm, this is approximately 5.45◦. The beam waist (in x- and y-direction) and beam size for the IMC with base 45 cm and 100 cm vs. the RoC of the HR side are shown in figure 27 and 28 respectively. It is clear that the smallest IMC shows the strongest astigmatism. However, for smaller RoCs (such as the preferred value of 3.2 m) the difference in beam waist between the x- and y-direction is less prominent compared larger RoC values. Frequency splitting visible in several plots in this section is caused by these astigmatic beams.
After one roundtrip in a triangular cavity, the laser beam is mirrored around its vertical axis compared to the initial beam. This means that astigmatic beams, which are not symmetric in
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Mode Scan RoC=6.4m
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Figure 26: The mode power of the fundamental mode and first 12 higher order modes for an IMC with base 19 cm and sides of 100 cm, HR RoC is 6.4 m, as function of mirror tuning.
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0.6
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0.8
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1.1
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1.3
radius (mm)
waist mHRx
waist mHRy
beam radius mHR
Figure 27: Beam waist in the x- and y-direction as function of HR mirror RoC for an IMC with a 19 cm base and sides of 45 cm.
this direction, cannot fully constructively interfere and therefore won’t be completely resonant (Fulda, Kokeyama, Chelkowski, & Freise, 2010). Astigmatism in the IMC can be problematic for gravitational wave detection, because it impairs the mode matching to the interferometer (Barriga, Zhao, & Blair, 2005). The influence of the IMC design on mode matching is discussed in the next section.
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radius (mm)
waist mHRx
waist mHRy
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Figure 28: Beam waist in the x- and y-direction as function of HR mirror RoC for an IMC with base 19 cm and sides of 100 cm.
It is shown that all three triangular IMC designs have good HOMs filtering abilities. However, there is one other difference between the designs to consider besides the astigmatism. Using a smaller IMC requires mirrors with higher reflectivity to gain the same linewidth as a larger design. The reflectivity has to increase with the same factor as the length of the cavity decreases, to get the same small linewidth. This is important because a small linewidth, or high finesse, increases the filtering effect (Tacca, 2014). Hence, it can be a disadvantage to use a small IMC, as higher reflectivity cavities are more difficult and thus more expensive to manufacture.
3.3
Beam Matching between IMC and ITM
The beam that leaves the IMC has to be matched to the beam that exits from the ITM. Two mirrors are placed between these components for this purpose, the situation is illustrated in figure 29.
Figure 29: Schematic overview of the beam matching between the IMC and the ITM mirror of the cavity. Distances d1, d2and d3are known from the optical layout design, whereas the radii of
curvature of the two steering mirrors are undetermined.
the individual components, which are two spaces with certain distances and two mirrors. Moving through space d1and reflecting off mirror 1 gives:
1 0 − 2 R1 1 1 d1 0 1 = 1 d1 − 2 R1 1 − 2d1 R1 Adding space d2 and mirror 2 leads to:
1 d2 − 2 R2 1 − 2d2 R2 1 d1 − 2 R1 1 − 2d1 R1 = 1 −2d2 R1 d1+ d2− 2d1d2 R1 − 2 R1 − 2 R2 + 4d2 R1R2 1 − 2d1 R1 − 2d1+2d2 R2 + 4d1d2 R1R2 The complex beam parameter at the input test mass mirror of the cavity can then be described as the product of this final matrix and the complex beam parameter of the input mode cleaner mirror: qIT M = (1 − 2d2 R1)qIM C+ d1+ d2− 2d1d2 R1 (−R2 1 − 2 R2 + 4d2 R1R2)qIM C+ 1 − 2d1 R1 − 2d1+2d2 R2 + 4d1d2 R1R2
First, mode matching between the large suspended IMC design and the ITM is investigated, subsequently, the same method is used for the two smaller IMC designs.
Large IMC mode matching
Both the beam waist and the waist location of the IMC and ITM mirror can be obtained by simulations in FINESSE. With this information the complex beam parameters at the large IMC and ITM, qIM C,l and qIT M respectively, were calculated with equation 3 and 4. At the large IMC
the beam waist lies 10 cm before the output mirror and has a value of 1.34 mm. The waist of the ITM is 3.29 m before the ITM mirror (or 7.907 m after the second mode match mirror) and has a value of 2.080 mm. It is assumed that the ITM AR coating side has a RoC of 9 m. Using these numbers, the complex beam parameters at the large IMC and ITM respectively are:
qIM C,l= 0.1 + 3.63938i
qIT M = −7.9 + 8.76889i
Now, if the distances are fixed the only unknowns are the radii of curvature of the two mirrors. The distances between the components were estimated using the schematic layout of the optical components in the CDR document (The ETpathfinder Team, 2020), depicted in figure 4. Using d1 = 6.22 m and d2 = 3.25 m, the numerical solutions are:
R1= 17.70 m, R2= 26.25 m
R1= 2.70 m, R2= 3.06 m
A second method to mode match the beam from the IMC to the large ITM was used as well. The contours of all mirror RoCs with beam waist 2.080 mm and waist location 7.907 m after the second mirror were plotted, see figure 30, and the intersection points then give the values of the desired RoCs of mirror 1 and 2: (2.70, 3.06) or (17.70, 26.25).
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RoC mirror 1
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RoC mirror 2
0.002 0.002 -7.907 -7.907Figure 30: Contour plot of the beam waist (red) and waist location (blue). The intersection point are (2.70, 3.06) and (17.70, 26.25), these indicate the values for which the beam from the IMC and ITM are mode-matched.
Small IMC mode matching
The same analysis is used to find the radii of curvature to mode match the beams from the small design IMCs. However, the beams from the smaller IMCs showed to be astigmatic, thus the beam has to be matched either in the tangential or sagittal plane. At the small IMC the beam waists are located 9.95 cm before the output mirror. The complex beam parameters for the x-direction, indicated by s1x, and y-direction, s1y, of the 19x45 cm IMC are:
qIM C,s1x = 0.095 + 1.18645i qIM C,s1y = 0.095 + 1.21950i
And for the 19x100 cm IMC, the solutions for the x-direction, indicated by s2x, and y-direction,
s2y, are:
qIM C,s2x = 0.095 + 2.40358i qIM C,s2y = 0.095 + 2.41679i
The complex beam parameter at the ITM has not changed. Again, using d1 = 6.22 m and d2
x-direction → R1= 3.00 m, R2= 2.45 m
y-direction → R1= 2.99 m, R2= 2.47 m
and for the 19x100 cm IMC:
x-direction → R1= 11.80 m, R2= 58.79 m
y-direction → R1= 11.85 m, R2= 57.61 m
These results show the radii of curvature for the mode match mirrors have different optimal values in the x- and y-plane.
For all IMC designs applies that if the position or value of the beam waist of the ITM or IMC is shifted slightly, the beam parameters will be mismatched. It is important to take this into account, as the mode matching might not be perfect due to e.g. fabrication imperfections. Table 5 gives the mismatch values for displacements of the waist from the IMC side up to 50 cm. The code for finding the RoC solutions as well as the mismatch values is presented in appendix A.2.
Change in z (cm) Mismatch 5 8.128 × 10−6 10 3.251 × 10−5 15 7.315 × 10−5 20 1.300 × 10−4 25 2.032 × 10−4 50 8.122 × 10−4
Table 5: Mismatch values between the beam from the IMC and the beam from the ITM AR side. The mismatch values are the same for all calculated solutions.
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Conclusion
This research aimed to study and design several optical components of the ETpathfinder. Based on calculations and simulations, the most important findings are now summarized.
Firstly, various arm cavity parameters were discussed and calculated. The beam radius, beam waist, cavity stability and free spectral range were provided. Moreover, the optimal radius of curvature of the anti-reflective coating side of the input test mass was investigated. Simulations indicated that a radius of curvature of 9 m is preferred, as this provides a nearly collimated beam towards the steering mirrors. The beam spot size lies well below the beam size limit of the mirror. Then, the radiation pressure coupling between the input and end test mass of the locked arm cavity was researched. The soft and hard modes of the coupled system, and intrinsic torsional torque of the suspended mirrors were taken into account to calculate the cavity power threshold. The system can potentially become unstable if the power exceeds about 7 kW for the yaw mode, and 5166 kW for the pitch mode. Thus, if the intra-cavity power exceeds 7 kW, active control is needed to counteract the instability. The initial target cavity power of ETpathfinder is approximately 500 W, so the radiation pressure coupling does not play an important role in the first phase.
Furthermore, the etalon effect in the input test mass was simulated and two methods to avoid this effect were described. Changing the radius of curvature of the two surfaces (AR and HR) of the mirrors relative to each other, or adding a small wedge on the AR side of the mirror changes the coupling coefficient and can therefore break the etalon effect.
The resonant modes in one large and two small input mode cleaner designs with a range of different radii of curvature of the HR mirror were simulated and compared. For large IMC design with sides of 2.8 m and a base of 0.2 m, the preferred RoC of the HR mirror is 6.7 m. For a smaller IMC with sides of 1.0 m, and base of 0.19 m this is 6.4 m and for an IMC with sides of 45 cm and base of 19 cm 3.2 m is most optimal. However, the two small IMC design showed to produce astigmatic beams. This can impair the mode matching to the arm cavities and was therefore investigated further.
ABCD matrices and the complex beam parameter theory were used to mode match the beam from the input mode cleaner to the arm cavity of the interferometer. The radii of curvature of two mirrors in between these systems were calculated. This method was used for the three different IMC designs. The large IMC gave the solutions: (2.70 m, 3.06 m) and (17.70 m, 26.25 m). For the two small designs, mode matching in the x- and y-direction was compared. As expected the solutions for the mode matching mirror radii of curvature were different. Therefore, there is no perfect mode matching possible if the beam is astigmatic. The mismatch can cause power loss and additional noise coupling and should be avoided as much as possible.
Within this thesis several parts of ETpathfinder’s optical system were analysed, and the results and conclusions provide a foundation for the optical design for ETpathfinder.
