Design and Construction of the Suspended Fabry-Perot for Gaussian and Non-Gaussian Beam Testing. Preliminary test with Gaussian Beams

(1)

Università di Pisa

Facoltà di Scienze Matematiche Fisiche e Naturali

Corso di Laurea Specialistica in Scienze Fisiche

Anno Accademico 2003-2004

Tesi di Laurea Specialistica

“Design and Construction of a Suspended

Fabry-Perot Cavity for Gaussian and

Non-Gaussian Beam Testing. Preliminary Test with

Gaussian Beam”

Candidata

Relatori

Barbara Simoni

Dott.R.De Salvo

Prof. F.Fidecaro

Dott. E.D’Ambrosio

LIGO-P040037-00-R

(2)

(3)

1. Abstract

A non Gaussian, Flat-Topped, laser beam profile, also called Mesa Beam Profile supported by non spherical mirrors known as Mexican Hat (MH) mirrors, has recently been proposed [1] as a way to depress the mirror thermal noise and thus improve the sensitivity of future interferometric Gravitation Wave detectors including Advanced LIGO [2]. Non-Gaussian beam configurations have never been tested. The main motivation of this project is to demonstrate the feasibility of this new concept. The topic of this thesis is to design and build a Fabry-Perot cavity which can support a Mesa beam. The construction of the MH mirrors for this cavity requires more time than available to me. This thesis represents the initial work necessary to design this type of resonator. The construction of a mechanical structure capable to support both a Gaussian or a mesa beam and the testing of the structure with standard spherical mirrors and Gaussian beams is also part of this thesis. The test of the Mesa beams properties will be covered in a different thesis work when the new mirrors will become available in the fall. The first part of this thesis presents the motivation to develop this project. The second part presents requirements, design and construction of the experimental apparatus necessary to test the MH beams. The third part will presents the initial configuration and the preliminary results of the tests with spherical optics in the Fabry-Perot cavity

(4)

2. Introduction

Equation Chapter 2 Section 1

The Mesa beam has been proposed to mitigate the effects of thermal noise in the mirrors: thermoelastic noise in sapphire mirrors substrate and coating Thermal Noise in fused silica mirrors [3]. The experiment of which this thesis is part, aims to verify the feasibility and controllability of Mesa Beam Fabry Perot cavities. The verification of the actual improvement in the sensitivity with Mesa Beams is obviously not the aim of this thesis; it would require comparison between two twin and much more complex cavities. Nevertheless the following chapters will briefly present the expected thermal noise improvements from which interferometers could benefit from the introduction of the Mesa beams. Braginsky, Gorodetsky and Vyatchanin [4] showed that the spectral density

of the thermoelastic gravitational-wave noise scales as the inverse cube of the beam-spot radius . Also substrate thermal noise and thermal noise deriving from coating mechanical losses, benefit from a flat topped beam profile according to the following equations. ( ) h S f 0 r 3 0 2 0 0 ( ) 1 Thermoelastic Noise

( ) 1 Coating Thermal Noise

( ) 1 Brownian Thermal Noise

h h h s f r s f r s f r ∝ ∝ ∝ (2.1.1)

The sensitivity of a G.W. Interferometer would be proportional to the square root of the Eq. (2.1.1). The baseline design of Advance-LIGO [5] assumes the use of sapphire mirrors and suggests that if the thermoelastic noise can be reduced significantly, the advanced LIGO interferometers range for astrophysical sources will increase significantly. Equipped with mesa beams, these interferometers will even have a chance to confront with the Standard Quantum Limit (SQL). The same would be true for interferometers built with fused silica mirrors, in this case coating thermal noise would be the limiting factor depressed by the mesa beams.

(7)

The transition from the LIGO’s baseline design to non spherical mirrors does not significantly impact on the mechanical and topological design of the interferometer. The implementation of the Mesa beams simply requires the replacement of the mirrors, possibly only changing the profile of the end mirror of the main F.P cavities, to modify the mirrors of the injection system, and moderately increasing the alignment precision of the mirror controls [6]. Comparing this approach with the cryogenic approach shows a close analogy between the width of the laser beam and the temperature of the mirror in the sensitivity of the interferometer to mirror thermal noise. Thus Mesa beams offer similar advantages of cryogenics with much less complications for the interferometer design [7].

This project was developed to verify the predicted behaviour of Mesa beams. The FP cavity prototype is being designed to experiment how a MH mirror cavity is capable to transform an incoming Gaussian beam into a flat top beam profile, and how difficult is to lock and keep aligned such an optical cavity. This cavity represents a scaled down version of Advanced LIGO.

The prototype is a 7.3 m long, rigid, folded FP cavity, with Finesse =100, suspended under vacuum. The mechanics is built using INVAR, a material with low thermal expansion coefficient.

(8)

The experiment is a group activity. I had the honour to be asked to be the first leading researcher and integrator of the effort of all other colleagues.

The group includes:

Dr. E.D’Ambrosio, of LIGO, Caltech, one of the people that conceived the Mesa Beam idea to suppress thermal noise, expert in beam simulations and theoretical aspects.

Dr. Phil Willems, of LIGO Caltech, expert of laser interferometry and, with Dr. D’Ambrosio, one of the initial promoters of the test interferometers.

Dr. Riccardo De Salvo, of LIGO Caltech, expert in mechanics, suspension system and vacuum.

Dr Jean Marie Mackowsky and Dr.Alban Remillieux, expert in dielectric coating deposition and the inventor and the designer of the corrective coating technique that will make the M.H technically possible.

Dr.Juri Agresti, student of Dottorato di Ricerca of University of Pisa and LIGO Caltech, who is studying the theoretical aspects and simulations of the mesa beam optical properties, dynamic, and thermal noise suppression properties.

Dr. Maddalena Mantovani, of University of Pisa, that made a Tesi Specialistica on the development of seismic attenuation suspensions and contributed to the design of the suspension system.

Miss Yanyi Chen, undergraduated student of Electronic Engineering, of Louisiana State University, who, as a summer student project at LIGO, made the suspension control board.

Miss Nicole Virdone, Mayfield high school student, volunteer for the summer that very effectively assisted me in the assembly phase.

Invaluable contributions came from

Mr. Paul Russel, LIGO electronics technically Dr Flavio Nocera, LIGO Engineer

Mr. Ricardo Paniagua, supervisor of the Machine Shop at Caltech, and his team.

Mr. Gianni Gennaro of PROMEC, Bientina, Italy, the draft man of the mechanical design.

Mr. Carlo Galli of Galli &Morelli, Lucca, Italy that directed the production of the mechanics and vacuum components.

Dr.Chira Vanni Agresti assisted.

(9)

3. Thermal Noises

It is possible to generate a significant reduction in several source of mirror thermal noise using modified optics that reshape the beam from a conventional Gaussian profile into a mesa-beam shape. The interferometers’ output phase shift is proportional to the difference of the test masses’ average position, with the average being performed over the position of a mirrored test-mass face, weighted by the light’s energy flux. If the intensity distribution is flat in most regions of the mirror, then the adjacent valleys and bump created by all thermal noise will average out. If, instead, the laser beam energy flux used to probe the test mass surface is rapidly changing, like in a Gaussian beam, then the valleys and bumps will not average out well and the thermal noise will be high. Wider light beams, average better and induce lower measured noise. These considerations suggest that large-radius flat-topped beams with steep edges will lead to much smaller thermal noise than small-radius, peaked Gaussian beams with gradually sloping sides. The studies on possible use of mesa beam were stimulated by the thermoelastic problem. Thermoelastic noise is only relevant for materials, like sapphire, with large thermal expansion coefficient and it is irrelevant in all present fused silica interferometers and even for advanced LIGO if the sapphire mirror options will be rejected. It has to be noted that the dominating thermal noise source in fused silica mirrors is the Brownian thermal noise of the coating and, after that, the mirror’s substrate bulk thermal noise. These two important noise sources, not discussed here, are depressed by wider, flatter probe beams, and the interferometer sensitivity would profit from the implementation of mesa beams as illustrated in Eq (2.1.1).

The remaining of this chapter briefly presents the Thermoelastic noise for the bulk and for the coatings.

(10)

3.1. Thermoelastic Noise

Equation Chapter (Next) Section 1

The thermoelastic noise is associated with thermoelastic dissipation, which is caused by heat flow along the gradients of temperature. This noise is created by the stochastic flow of heat within each test mass (mirror) which produces stochastically fluctuating hot spots and cold spots inside the test mass The test-mass material expands in the hot spots and contract in the cold spots, creating fluctuating bumps and valleys on the mirrors faces. For Gaussian beams, the influence of beam radius on the thermoelasic noise has been pointed out by Braginsky, Gorodetsky and Vyatchanin [

0 r

8]

• “Bulk” Thermodynamical Fluctuations

Thermodynamical fluctuations (TD) of temperature in mirror are transformed due to

thermal expansion coefficient αBulkinto a noise which can be a limiting factor in the

sensitivity of laser interferometric gravitational antennae. The spectral density of displacement X can be presented for half infinite medium as follows:

(

)

2 2 2 2 3 2 0 1 8 ( ) 2 Bulk B TD Bulk v k T k S C r α ν ω ω π + = (3.1.1)

Here k_Bis the Boltzman constant, T is the temperature, ν is Poisson ratio, k is the thermal

conductivity and is the specific heat capacity per unit volume. This result has been

refined for the case of finite sized mirror by Liu and Thorne [

v

C

9]. However this physical result can be illustrated using the following qualitative consideration. We consider the surface fluctuations averaged over the spot with radius which is larger than the characteristic diffusive heat transfer length

0 r T λ . 0 T T v r k C λ λ ω = (3.1.2)

(11)

Where ω is the characteristic frequency (for LIGO it is about ). We can evaluate the characteristic diffusive heat transfer length for the Fused Silica and for the sapphire from the following table.

1 2 100s

ω_≈ π_i −

Table 3-1 Material’s parameters for fused silica and sapphire

Fused Silica Sapphire

7 0 1 5.5 10 K α ₌ _i − − _{5.0 10} 6 0 1 K α ₌ _i − − 5 0 1.4 10 erg k cms K = i 6 0 4 10 erg k cms K = i 3 2.2 gr cm ρ= 4.0 gr₃ cm ρ= 6 0 6.7 10 erg C g K = i 6 0 7.9 10 erg C g K = i

Fused Silica Sapphire

3 3.88 10 T cm λ ₌ _i − _{1.4 10} 2 T cm λ ₌ _i − 40 2 1.196 10 sec Bulk S = i − cm S_Bulk =0.763 10i −37cm2sec

Table 3-2 Thermoelastic noise characteristic length and amplitude for fused silica and sapphire substrates

For fused silica and for sapphire , in both cases the

condition 3 3.88 10 T cm λ ₌ _i − _{1.4 10} 2 T cm λ ₌ _i − 0 6 T r cm

λ ∼ is verified. Since in a volume 3

T λ

∝ the length variation due to

the temperature TD fluctuations T∆ is about 2 2 3 B T Bulk T Bulk T v T k T X T C α λ α λ λ ∆ = ∆ (3.1.3)

The number of such volumes that contribute to surface fluctuations is about 03

3

T

r

N _λ

and hence the displacement X averaged over the spot with radius consists of these independently fluctuating volumes displacements sum, and is approximately equal to

0 r

(12)

2 3 0 T Bulk T v B X k T X C r N α λ ∆ (3.1.4) Comparing Eq (3.1.4) with the spectral density of displacement, we can see that

TD Bulk

X S ∆ω (3.1.5)

Consequently the thermoelastic noise power of the bulk scales as 3

0 1 TD Bulk S r ∝

O-Shaughnessy, Strigin and Vyatchanin [10] have shown that the thermoelastic noise for mesa beam can be three times weaker in noise power than for the Advanced LIGO baseline Gaussian beams (with beam sizes that produces the same diffraction losses)

• TD Fluctuations in Thin Layer

TD fluctuations of temperature in mirrors may produce surface fluctuations not only trough thermal expansion in the mirror body, but also through thermal expansion in the mirror coating. This noise may be larger than the analogous one in mirror body due to larger coating materials thermal expansion coefficient despite the smaller effective volume. The coating of the mirrors is the deposition of many thin dielectrical layers on the mirror surface. The layer thickness d is much smaller than the diffusive heat transfer

characteristic length λT (for optical coatingd≤10µm), and therefore the following

condition is verified: 0 T T r d k C λ λ ρω = (3.1.6)

Where C is the specific heat and ρ is the density. This unequality means that the

oscillating temperature distribution can be approximated as adiabatic in the transversal direction, i.e. tangent to the surface [11]. Therefore we may consider the temperature fluctuations in the layer thickness d, to be the same as in the layer with thicknessλT. This

means that TD temperature fluctuations in the layer do not depend on the thickness d. In

this case, it is possible to estimate as follows the spectral density TD ( )

layer

S ω of surface

(13)

The fluctuations of the averaged displacement of mirror surface isXd ≈ ∆α Tdwhere α is

the effective thermal expansion coefficient given as difference between the thermal expansion coefficient of the bulk and the thermal expansion coefficient of the layer material

(

α α= _Layer −α_Bulk

)

2 2 2 3 2 2 0 0 2 2 0 d B T B v T v T B d v T X Td k T k T T C r C r k T X d C r α λ λ λ α λ ∆ ∆ = (3.1.7)

These formulae are in good agreement with rigorous formula for spectral density

( )

T

S_∆ ω of averaged temperature T∆ in layer obtained in [12]

2 2 0 2 ( ) B T v T k T S C r ω π λ ω ∆ (3.1.8)

And it is possible to estimate the spectral density 2 2 2 2 2 2 0 2 ( ) ( ) TD B layer T T d k T S d S C r α ω α ω π ρλ ω ∆ =

The gravitational wave antenna spectral sensitivity is proportional to the square root of the Spectral density.

(14)

4. Gaussian Beams

The cavity modes of a Fabry-Perot resonator with spherical mirrors and most optical beams from lasers have a Gaussian amplitude beam profiles. As we test our cavity using spherical optics, we review the Gaussian Beam formalism, the fundamentals of wave propagation and the practical properties of Gaussian Beams. This formalism has been used to calculate the mirrors and the lenses used in the injection optics of the F.P cavity built. This chapter can be dropped by a reader familiar with Gaussian beam properties and paraxial beam description.

4.1. Derivation of the Paraxial Wave Equation

Electromagnetic fields in any uniform and isotropic medium are governed in general by the scalar wave equation

2 2

[∇ +k E x y z] ( , , )= (4.1.1) 0

where is the phasor amplitude of a field distribution sinusoidal in time. We will

be concerned with optical beams propagating primarily along the z direction, so the

primary spatial dependence of will be an exponential variation . For any

beam of practical interest the amplitude and phase of the beam will generally have some transverse variation in x and y which specifies the beam’s transverse profile. This profile usually changes very slowly with distance z due to the diffraction and propagation effects. Similarly, for a reasonably well-collimated beam, given the short wavelength of light the transverse variations across any plane z will also be slow, compared to the

plane-wave variation in the z direction Macroscopically the light beam power is

( , , ) E x y z ( , , ) E x y z e−ikz ikz e−

(15)

contained in a small cross section that changes very slowly with propagation. As a consequence we can approximate it by monochromatic light wave, propagating in one direction and with small diffraction. Therefore:

( , , ) ( , , ) ikz

E x y z ≡u x y z e− (4.1.2)

where u is a complex scalar wave amplitude which describes the transverse profile of the beam. The scalar wave equation (4.1.2) can be written as:

2 2 2 2 2 2 2 u u u u ik x y z x ∂ ₊∂ ₊∂ ₋ ∂ ₌ ∂ ∂ ∂ ∂ 0 (4.1.3)

This slowly varying dependence of on z can be expressed mathematically by

the paraxial approximation:

( , , ) u x y z 2 2 2 u k z z u ∂ _<< ∂ ∂ ∂ (4.1.4)

By dropping the second partial derivative in z, we find the paraxial wave equation:

2 2 2 2 2 u u u ik x y x ∂ ∂ ∂ 0 + − ∂ ∂ ∂ = (4.1.5)

4.2. Paraxial Wave Propagation

Equation Section (Next)

The paraxial wave equation is fully adequate for describing nearly all optics resonator and beam propagation problems that arise with real lasers.

Consider wave diverging from a source point located at x’,y’,z’ and observed at an observation point x,y,z. If the axial distance z-z’ between the source and observation points is sufficiently large compared to the transverse coordinates x’,y’ and x,y, then the field distribution produced by this wave at point x,y on the plane located at distance z can be written using the paraxial approximation, in the form:

(4.2.1)

( , , ) ( , , ; ', ', ') ( ', ', ') ' '

u x y z =

∫∫

K x y z x y x u x y z dx dy

(16)

2 2 [( ') ( ') ] ' ( , , ; ', ', ') ' i x x y y z z i K x y z x y x e z z π λ λ − − + − − = − (4.2.2)

to the electromagnetic field u x y z( ', ', ')

2 2 2 2 2 1 [ ( ' ' ) ( ' ' 2 ( ') ( ') 2 2 ( ', ', ') ( ') ik x y x y R z w z u x y z e w z π − + + + = ) (4.2.3)

where w (z’) and R (z’) are respectively a measure of the decrease of the field amplitude with the distance from the axis and the radius of curvature of the wave front that intersects the axis at z.

(17)

4.3. Physical Properties of Gaussian Beams

A lowest-order Gaussian beam is characterized by a spot size and a planar wave front

in the transverse dimension, at a reference plane. The plane with the smallest size is known as the beam waist. For simplicity we define z=0 at the waist plane. The normalized field pattern of this Gaussian beam at any other plane z will then be given by:

0 w 0 R = ∞ 0 w [ ] 2 2 2 2 2 _{2 ( )} ( ) ( ) 2 2 ( , , ) ( ) x y x y ik R z ikz i z w z u x y z e e w z ψ π ⎡₋ + ₋ + ⎤ ⎢ ⎥ − + ⎢⎣ ⎥⎦ = (4.3.1)

Introducing the “Rayleigh range”

2 0 R w z π λ

= all the important parameters of this Gaussian

beam can then be related to the waist spot size w0and the ratio R z z by the formulas 2 2 2 0 ( ) 1 R z w z w z ⎡ _⎛ _⎞ ⎤ ⎢ ⎥ = _{+ ⎜ ⎟} ⎢ ⎝ ⎠ ⎥ ⎣ ⎦ (4.3.2) 2 ( ) zR R z z z = + (4.3.3) 1 ( ) tan R z z z ψ ₌ − ⎛ ⎞ ⎜ ⎝ ⎠⎟ (4.3.4)

The field pattern along the entire Gaussian beam is characterized entirely by the single

parameter at the beam waist, plus the wavelength in the medium. The net effect of the

phase shift 0 w

( )z

ψ for the lowest-order Gaussian mode, is to give an additional cumulative

phase shift of 90± on either side of the waist, or a total added phase shift of 1800 in

passing trough the waist. This added phase shift means in physical terms that the effective axial propagation constant in the waist region is slightly smaller, or that the phase velocity and the spacing between phase fronts are slightly larger than for an ideal plane wave.

The intensity of a Gaussian beam falls off rapidly with radius beyond the spot size w0

(18)

2

P=

∫∫

u dA (4.3.5)

Where dA integrates over the cross-sectional area. The radial intensity variation of a Gaussian beam with spot size w is given by:

2 2 2 2 2 ( ) P r w I r e w π − = (4.3.6)

To allow the propagation of a Gaussian beam with losses less than an arbitrary fraction ε , a practical aperture must be larger than the effective diameter of a cylindrical beam

with the same power distribution of the Gaussian and truncation effects of less than ε on

the Gaussian beam tails. In other words the fractional power transfer for a Gaussian beams of spot size w passing trough a centred circular aperture of diameter 2a will be given by: 2 2 ₂ 2 2 2 0 2 power-transmission = 2 1 w a r w a re dr e π π − = −

∫

− _w2 (4.3.7) An aperture with a=w transmits ~86% of the total power in the Gaussian beams, while an aperture with

2

a=π w will pass just over 99% of the Gaussian beam power.

The power cut-off effects are not the whole story, sharp-edged apertures, even though they may cut off only a very small fraction of the total power in an optical beam, will also produce aperture diffraction effects which are significant when a laser beam is used for high precision length measurement.

Another important question is how rapidly an ideal Gaussian beam will expand due to the diffraction spreading as it propagates away from the waist region or, in practical terms, over how long a distance can we propagate a collimated Gaussian beam before it begins to spread significantly beyond the radius of our available optics components. The

variation of the beam spot size with the distance is given by Eq (4.3.2). If the input

spot size at the waist is smaller, the beam expands more rapidly due to the diffraction. The distance which the beam travels from the waist before the beam diameter increase by

( )

w z

2is given by the “Rayleigh range”. If we consider a Gaussian optical beam transmitted

from a source aperture of diameter D, the relation between the collimated beam distance and the transmitting aperture size, using the 99% criterion [13] is

2 2 0 2 collimated range 2z_R πw D λ πλ = = ≈ (4.3.8)

(19)

4.4. High order Gaussian Modes

We consider only the standard set of higher-order modes in simple optical resonators.

The free-space Hermite-Gaussian TEMmn solutions can be written, in either the x or y

transverse dimension, emphaysing the spot size w (z) and the Guoy phase shift ( )ψ z in

the form: [ ] 1 2 2 1₄ 2 2 (2 1) ( ) 2 ( ) ( ) 2 2 ( , ) 2 ! ( ) ( ) kx x i n z ikz i R z w z n n n e x u x z H e n w z w z ψ π ⎡ ⎤ + ⎢− − − ⎥ ⎢ ⎥ ⎣ ⎦ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ =_{⎜ ⎟} _⎜_⎜ _⎟_⎟ × _⎜_⎜ _⎟_⎟ ⎝ ⎠ _⎝ _⎠ _⎝ _⎠ (4.4.1)

Note, the important point is that the higher-order modes, because of their more rapid transverse variation, have a net Guoy phase shift of (n+1/2) ( )ψ z in travelling from the

waist to any other plane z, as compared to only ( )ψ z for the lowest-order mode. This

differential phase shift between Hermite-Gaussian modes of different orders is of fundamental importance in explaining why higher-order transverse modes in a stable laser cavity will have different oscillation frequencies

4.5. Transverse Modes in Optical Resonators

This section describes the transverse mode properties of laser resonators and how these modes should be analyzed.

Optical resonators have open sides, so they always have diffraction losses of energy leaking out the sides of the resonator. We consider the portion of the optical energy

travelling in the +z direction and contained within some short axial segment of length z∆

within the cavity. It is possible to think of the radiation in this segment as a thin slab of

radiation whose axial thickness z∆ is small compared to the length L of a typical laser

cavity and large compared to the optical wavelengthλ.

The variation of the field can be written as:

( ) ( , ,

( , , ) ( , , ) i t kz i x y z

x y z e E x y z e ω φ

(20)

The complex phasor amplitude describes the transverse amplitude and phase variation of the beam. The transverse intensity profile is given by:

( , , )

E x y z

2

( , , ) ( , , )

I x y z = E x y z

And the transverse phase profile is given by the transverse phase variation ( , , )φ x y z .

The propagation of each such slab is unaffected by the radiation in the previous or subsequent axial segments because the optical radiation in each axial segments is more or less independent from the other slabs.

This section describe how to calculate the propagation effects for an optical pulse through one round trip in a resonator and how to find the transverse mode

patternsEnm( , )x y that self reproduce themselves in one round trip. Of course without

external power feeding these patterns would have reduced amplitude after each turn due to the diffraction and any other kind of losses during each round trip.

The total propagation through one round trip in an optical resonator can be described mathematically by a propagation integral which will have the general form:

(4.5.2)

(1) (0)

0 0 0 0 0 0

( , ) ikp ( , ; , ) ( , )

E x y =e−

∫∫

K x y x y E x y dx dy

The propagation integral in Equation (4.5.2) is a linear operator equation: the linear

propagator K acts on the optical field at a reference plane on one round trip to

produce a new optical field one round trip or one period later. For a given

resonator or kernel exist a set of mathematical eigenmodes 0_{( , )} E x y 1_{( , )} E x y ( , ) nm E x y and a corresponding

set of eigenvalues γnm such that each one of these eigenmodes after one round trip

satisfies the round-trip propagation expression

(4.5.3)

0 0 0 0 0 0

( , ) ( , ; , ) ( , )

nmEnm x y K x y x y Enm x y dx dy

γ ≡

∫∫

The eigensolutions that satisfy Equation (4.5.3) always exist; these eigensolutions are the self-reproducing transverse eigenmodes we seek for the optical resonator.

This self-reproducing eigenmodes are the mathematical definition of “transverse modes”. A transverse wave pattern that is bound within a finite width will always spread out due to diffraction as it propagates. In an open-sided resonator with finite-diameter mirrors some of the radiation will spread out past the mirror edges each round trip, and the magnitudes of the transverse eigenvalues will therefore always be less than unityγnm <1.

(21)

Hence, even with perfectly lossless but finite size mirrors, the nm-th eigenmode of an optical resonator will always have a power loss per round trip given by:

2

fractional power loss per round trip 1= − γnm (4.5.4)

These losses results from diffraction losses at the mirror edges and will continue to occur on all subsequent round trips. If no optical gain or external power feeding is present, the amplitude of each individual transverse mode will decay with successive round trips in the form: 0 ( , ) ( , ) k k nm nm nm E x y E x y =γ (4.5.5)

Because of the kernel for open-sided optical resonators is not a hermitian operator, this means that is not automatically guaranteed the existence of a complete and orthogonal set

of eigensolutions to Equation (4.5.2). The set of modesE_nm( , )x y are generally

biorthogonal (without complex conjugation) to an adjoin set of modes which

represent the transverse modes travelling in the opposite direction in the same cavity

† _{( , )} nm E x y † ( , ) ( , ) nm pq nm pq E x y E x y dx dy =δ δ

∫∫

(4.5.6)

It is also not guaranteed rigorously that the transverse eigenmodes of an optical resonator form a complete set. However, the Hermite-Gaussian or Laguerre-Gaussian functions that approximate the eigenmodes in ideal stable resonator do form a complete basis set, and in most practical situations people simply proceed as if the resonator eigenmodes always form a complete set.

The propagation kernel in a typical optical resonator depends only very

slightly on the exact frequency or wavelength of the radiation, so the diffraction effects

experienced by a transverse mode function in a round trip will be essentially the

same for any carrier frequency. Hence the transverse mode properties and the axial frequency properties of a given cavity can be treated almost completely separately from each other. 0 0 ( , ; , ) K x y x y ( , ) nm E x y

If we consider also a laser medium with transversely uniform round-trip voltage gain inside the optical cavity, the total round-trip amplitude gain and phase shift become:

mpm

eα

(1)_{( , )} mpm ikp 0 _{( , )}

nm nm nm

E x y =γ eα − E x y (4.5.7)

The transverse eigenmodes for any given laser can be calculated based only on the mean laser wavelength, and all of the axial modes will then have the same set of transverse

(22)

eigenmodes and eigenvalues of the laser cavity. Similarly, in the case of a cavity with external power feeding, the modal structure is only determined by the cavity itself.

4.6. Resonant Optical Cavities

In this section, we will discuss a generic Fabry Perot resonator. In its original form the FP cavity consisted of two closely spaced and highly reflecting mirrors, with mirror surfaces adjusted to be as flat and parallel to each other as possible. The standard formulas describe the resonant frequencies and the transmission properties of a FP as a function of the mirror spacing, the optical wavelength and the angle of incidence. To excite any such cavity in just one of the transverse modes, it is necessary to shape and focus the input beam using lenses and other so-called mode matching optics, in order to couple properly into the desired transverse mode of the cavity. In this chapter we analyze the resonant properties of the beams inside and outside such cavity

g_rt trans inc refl circ r1 r2

(23)

The Circulating Intensity, the Cavity Resonance, the Axial-Mode-Spacing, the Finesse and the Optical Cavity Tuning

The circulating signal just inside the input mirror consists of the vector sum of the portion of the incident signal which is transmitted through the input mirror plus a contribution representing the circulating signal which left this same point one round time earlier, travelled once around the cavity, and returned to the same point after passing trough all the elements and bouncing off mirror M1 as well as the other mirrors in the cavity. The total circulating signal just inside mirror M1 can thus be written in the form:

1 ( )

circ inc rt circ

E = jt E +g ω E (4.6.1)

Where ( )g_rt ω is the net complex round trip gain for a wave making one complete transit

around the interior of the resonant cavity. If the round trip optical path is passive and

contains material with voltage absorption coefficientα₀, or other internal losses, the

round trip power reduction is . The circulating signal after one complete round trip

will then return to the reference plane with a net round trip transmission factor which is given for a passive lossy cavity by:

0 4 L e−α 0 2 2 1 2 ( ) L c L j rt g ω =r r e⎡⎣−α − ω ⎤⎦ (4.6.2)

It is possible to relate the circulating signal inside the cavity to the incident signal outside the cavity 0 1 1 2 2 1 2 1 ( ) 1 L c circ L j inc rt E jt jt E = −g ω = ₋_{r r e}⎣⎡−α − ω ⎤⎦ (4.6.3)

The signal inside the optical resonator exhibits resonance behaviour each time the round trip phase shift 2 L

c

ω _{happen to be equal to an integer multiple of}₂_{π . These resonant}

frequencies are known as cavity axial modes, and the frequency interval between them is

known as the free spectral range of the cavity. The resonant frequencies ω_qand the free

spectral range ∆ω_axof the optical cavity are thus given by:

1 q ax q q c q L c L π ω π ω ω ₊ ω = ∆ = − = (4.6.4) The power transmission through a resonator is often written in the form:

(24)

( )

(

)

2 max 2 2 2 ( ) ( ) ₁ _sin ax trans inc E T E πω π ω ω ω ℑ ∆ = + (4.6.5)

Where T_maxis the peak transmission through the cavity and ℑ is the Finesse of the cavity.

The Finesse represents the resolving power of the cavity used as a transmission filter. ℑ

is the ratio of the free spectral range to the cavity bandwidth

1 rt rt g g π ℑ ≡ − (4.6.6)

Very small changes in the length of an optical cavity can be used to detune the resonant frequencies of the cavity by sizable amounts. From the resonant frequency expression Eq

(4.6.4) we can see that changing the length of the cavity by small amount δLat fixed q

tunes each of the axial mode resonant frequencies by an amount

2 q q q a L L L λ x δω δ ω δ δω ω ≈ − ≈ − ∆ (4.6.7)

In other words, changing the length of the cavity by one half wavelength, shifts each of the axial mode by an amount equal to the spacing between axial modes. It is very easy for a FP to range through many axial modes. Interferometer cavities are commonly stabilized to lock a mode to the laser frequency by a combination of a temperature tuning, and the use of piezoelectric actuators. This combination allows to move the mirrors back and forth in a controlled way by a few optical wavelengths.

4.7. Stable Gaussian Resonator Modes

When a Gaussian beam with a certain waist size and location has a wavefront radius that matches exactly a mirror’s radius of curvature, if the transverse size of the mirror is sufficiently larger than the Gaussian spot size, it reflects the Gaussian beams exactly back on itself, with exactly reversed wavefront curvature and direction. It is then said that the beam profile matches the mirror. The same is true for a beam hitting a FP and matching the radius of curvature of the F.P modes. In practice, instead of starting from a Gaussian beams and looking for the mirror to fit the beam, it is to common to consider the two

(25)

curved mirrors of a cavity and their spacing to find the Gaussian beam that properly fit the modes confined between the two mirrors. Assuming that the Gaussian beam has an initial spot size w0 and that the mirrors are located at distance z1 and z2 from the location

of the beam waist, the essential conditions are that the wave front curvature R[z] of the Gaussian beam must match the mirror curvature (R1 and R2) at each mirrors. This provides 3 equations: 2 1 1 1 2 2 2 2 2 1 ( ) ( ) R R z 1 2 R z z R z z R z z R z L z z = + = − = + = = − (4.7.1)

The minus sign in the first of this equation is a convention. The Gaussian wave front curvature is usually defined as positive for a diverging beam travelling to the right.

Before inverting the Eq (4.7.1) we introduce the g Parameters describing the transversal beam confinement in the cavity. These parameters are given by

1 1 2 2 1 1 L g R L g R ≡ − ≡ − (4.7.2) In terms of these parameters, the trapped Gaussian beam has a unique Rayleigh range

given by [14]:

(

)

(

1 2 1 2

)

2 2 2 1 1 2 1 2 R g g g g z g g g g − = + − 2 L (4.7.3)

(26)

(

)

(

2 1

)

1 2 1 1 2 1 2 g g z g g g g − = + − L (4.7.4)

(

)

(

1 2

)

2 2 1 1 2 1 2 g g z g g g g − = + − L (4.7.5)

It is also useful to write the waist spot size and the spot size at the end of the

resonator . 0 w 1 w w₂

(

)

(

)

(

)

(

)

1 2 1 2 2 0 2 2 1 1 2 2 2 1 1 1 2 2 1 2 2 1 2 1 2 1 1 g g g g L w g g g g g L w g g g g L w g g g λ π λ π λ π − = + − = − = − (4.7.6)

These quantities depend only on the resonator g parameters and on the quantity Lλ

π . A stable cavity must confine its resonant modes, i.e. do not allow them to defocus to infinity. It is obvious that real and finite solutions for the Gaussian parameters and spot sizes can exist only if the g parameters are confined to a stability range defined by:

1 2

0 g g≤ ≤ (4.7.7) 1

Each pair of mirror forming an optical resonator can be represented by a point in the plane. If this point falls in the shaded region of Figure 4-1, the mirrors correspond to a stable periodic focussing system that confines the beam.

1 2 g g -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 g1 g1g2=1 g1g2=1

(27)

In order to minimize Thermal Noise it is important to have the widest beam profile compatible with the available optics size and the accepted diffraction losses. For this reason in an optimized G.W Interferometer, a symmetric cavity is usually chosen and, to

maximize the spot size a condition is chosen. These two conditions

correspond to an almost flat flat configuration and to an almost concentric one, both non confining, degenerate conditions. Being forced by Thermal Noise consideration to design barely confining cavities, close to either g = 0 or g = 1, makes the cavity very sensitive to either alignment issues a thermal focussing instability. In order to maintain the beam just barely focussed (large spot) a minimal focussing curvature (large additional radius of curvature) is added to the mirrors in either the g =1 or g =0 configuration. To maintain the resonant mode within the mirrors confines, the two mirrors must be aligned to each other within a small fraction of the mirror size to radius of curvature ratio. Similarly, thermal focussing (which heating and expanding the material at the beam spot generally contributes a defocusing component) must be small compared with the minimal mirror focussing curvature. These considerations will be important when considering the case of the flat beam profile.

1 or 0

g ∼ g∼

The most commonly resonant condition used is g< 1

In preparation of testing the Mexican Hat Mirror, we have assembled the cavity as a half-symmetric resonator with spherical optics. An elementary half half-symmetric resonator consists in two mirrors of which one is flat, and the other is curved. In this type of configuration, g₁ =1and ₂

2

1 L

g g

R

= = − . This resonator is equivalent to half of a

symmetric system (2 mirrors with the same curvature at twice the separation). In the symmetric F.P the waist would be located at the location of the flat mirror of the “half” cavity, with spot size given by:

2 2 0 1 ₁ L g w w g λ π = = − (4.7.8) On the curved mirror the spot size would be:

(

)

2 2 1 1 L w g g λ π = − (4.7.9)

An alternative configuration that has been studied for this project is the Quasi Concentric Resonator which is characterized by large spot sizes at the end mirrors. For a

(28)

near-concentric resonator, in which the cavity length L is less than the sum of the two radii by the amount Error! Objects cannot be created from editing field codes., the resonator parameters are given by _R₁ _≈_R₂ ₌L₂+ L , _∆ _g₁_≈_g₂ _{= − +}1 ∆L_R_.

The spot size at the central waist is then given by : 2 0 L for L L 4L L w λ π ∆ ≈ ∆ (4.7.10)

And the end mirror spot sizes by: 2 2 1 2 4 for L L L L L w w λ π ≈ ≈ ∆ ∆ (4.7.11)

We have also studied this configuration because it is more insensitive to misalignment of either mirror. Tilting of either mirror still leaves the center of curvature located on the other mirror surface, and simply displaces the optics axis of the resonator by a small amount[15]

(29)

5. Non Gaussian Beams

These results are the work of E. D’Ambrosio and other, and the object of the doctoral Thesis of J.Agresti. I was involved in the specific calculations to adapt these results to our 7.3m cavity.

To reduce the impact of the mirror’s surface thermal noise on the laser beam it is useful to optimize the shape of the beam using a flat light profile. The ideal power distribution to minimize thermal noise would be a flat distribution over the entire mirror surface. This distribution, of course would produce very large, totally unacceptable, diffraction losses. Therefore we have to find a compromise to minimize the diffraction losses and still produce a flat power distribution over the largest possible fraction of the mirror surface. It was chosen to approximate a rectangular shape, described as an overlap of an infinite number of delta functions, as the superposition of a limited number of narrow Gaussian functions.

Figure 5-1 The form of the flat-topped-beam at the waist can be viewed as the superposition of narrower Gaussian Functions uniformely distributed over a circle.

(30)

To design the beam, first we have to define the intensity of the flat topped beam at its waist position (where the wave front is exactly flat). We started from the step function, the flattest intensity we may imagine:

2 2 2 2 2 2 1 for ( , ) 0 for s 2 x y p p u x y x y p π ⎧ ₊ _≤ ⎪ = ⎨ ⎪ ₊ _> ⎩ (5.1.1)

Where p is the waist radius.In principle we can obtained this with an uniform distribution

of delta functions. We quickly found more useful to replace the delta functions with a distribution of overlapping narrow Gaussian functions to generate a flat profile:

( ) (2 )2 0 0 2 0 2 2 2 0 0 1 0 0 2 0 0 2 ( , ) w x x y y when 0 x y p w u x y e dx dy wπ p ⎡ ⎤ − _⎢_⎣ − + − _⎥_⎦ + ≤ =

_∫∫

₀ → (5.1.2)

For each of the narrow Gaussian beams, the spot size becomes quite large in the propagation, making the field spread too much. The optimal choice turned out to choose a larger value for w₀ while keeping the ratiow0

psmall. The minimum spot size on the

Fabry-Perot cavity mirror is obtained for each individual Gaussian beam whose waist is 0 2 L L w k λ π = = (5.1.3)

withw₀= waist of each Gaussian function and L=length of the optical cavity.

Then the beam is propagated toward the ends of the cavity, we obtained:

( ) (2 )2 0 0 2 0 2 2 2 0 0 0 1 2 0 0 2 0 ( , ) ' ' ( , : ', ') ( ', ') 1 ( , ) 2 i x x y y w x y p u x y dx dy K x y x y u x y i u x y e dx dy wπ + ⎡ ⎤ − _⎢_⎣ − + − _⎥_⎦ + ≤ = + =

∫∫

(5.1.4)

using the symmetric kernel ( ) ( )

2 2 2 _' _' 2 ( , : ', ') i x x y y L i K x y x y e L π λ π λ ⎡ ⎤ − _⎢_⎣ − + − _⎥_⎦ =

The wave front of the field generated by the distribution of Gaussians, Eq (5.1.2), is defined by:

[

] [

]

( , ) ( , ) (0,0)

wf x y u x y u

Φ = Φ − Φ .

If we are able to build a cavity with mirrors whose surfaces match this wavefront, the beam profile described by the Eq (5.1.2) would automatically be the fundamental eigen mode of that cavity.

In order to match the wave front of the beam, the height of the mirror must be:

{

( , ) [ (0, 0)] [ ( , )] 2 h x y λ u u x y

}

π = Φ − Φ (5.1.5)

Of course, even if the power distribution of the narrower Gaussian of Eq (5.1.2) falls faster than a single Gaussian beam profile, a sizeable fraction of the mirror surface must still be shaped according to Eq (5.1.5) to allow for sufficiently small diffraction losses.

(31)

Although different solutions are possible, typically 36% of the mirror surface can be covered with the flat beam profile, to be compared with the 9 % of the F.W.M.H of a Gaussian beams with similar diffraction losses [16].

The corresponding profile of the beam intensity and of the mirror profile are shown in Figure

5-2

Figure 5-2 Intensity profile of the beam and mirror profile at the end mirror calculated for our 7.3 m test interferometer. The red mark indicates the minimum mirror radius necessary to have the diffraction losses

below 1 ppm

(32)

6. Conceptual Design of the Experiment

Equation Chapter 6 Section 1

The conceptual design of the test interferometer was a full group, iterative activity, involving the theorist that brought the idea of MH, the interferometer specialists, mirror coating manufacturer, the mechanics, seismic isolation and vacuum specialists. I was involved in this process and was required to do, or re do for cross check, all calculation and estimations involved. I also took care with all contacts with the manufactures and contractors and to execute the approved purchases.

The resonator we designed and built for this project is a scaled version of Advanced LIGO.

Advanced LIGO will be a symmetric confocal cavity. To save space we designed a “half” symmetric cavity with a single spherical or MH end mirror at one end and a flat input mirror at the point that would be the waist. To further save space we decided to fold the cavity inside a rigid structure

The nominal parameters of Advanced LIGO are:

• L=4000 m Length of the Fabry-Perot interferometer • d=32 cm Diameter of the mirror

• w=6 cm Spot size of the beam

• _λ ₌₁_.₀₆₄_⋅₁₀−6m Wavelength of the laser

Since we are keeping the same wavelength and the MH mirror construction constraints dictate a mirror of at least 2 cm diameter, the size of the mirror scaled as:

2 '

' L

L

d ≈d

This would correspond to a single folded “half” cavity L’= 8 m long. The length of available INVAR bars then fixed the cavity length at 7.32 m

The 3.65 m long cavity structure is built using material with low thermal expansion coefficient (INVAR ),and suspended under vacuum. The Appendix B shows the technical draft of this project.

(33)

.

6.1. General Description of the Experimental Device

I. Mechanical Constraints

The requirement to design this resonator was to build a scaled down version of Advanced LIGO. At the beginning we thought to build an easy to assemble table top experiment, but building a cavity 1m long would have meant to work with mirror with 0.7 cm of

diameter and it proved technically impossible to build MH mirrors with this size1. Also

optical bench vibrations would have disturbed the interferometer’s operations. We then enquired on the smallest feasible MH mirror and we found it to be about 2 cm. This consideration and the scaling law then decided the scale of the necessary cavity, 16 m for a simmetric cavity. We then chose to build a rigid suspended cavity to eliminate the vibration problems. Given the constraints from the size of the laboratory, and from mechanical stability requirements we decided for a folded “half” cavity. We designed a rigid cavity structure using low thermal expansion coefficient materials, for longitudinal stability and to prevent possible alignment instabilities due to differential thermal expansions of the material in the interferometer. At the beginning we investigated the possibility to use [17] Zerodur expansion class2, a glass ceramic with _α ₌_{0.1 10 / K}_i −6 −1

or Ule [18] an ultra-low expansion titanium silicate material , but because

of the fragility, the long delivery time and the high price, it was decided to use INVAR

( ), which is more economic and much easier to procure in a reasonable

time. Also INVAR is available in suitable lengths, while ceramic bars of the desired length would have to be composite. An additional thermal shield was added to the design to compensate for the relatively higher thermal expansion coefficient of INVAR After choosing the material, the length of the cavity was adjusted to the maximum length available for the Invar rods. This fixed the length of the rigid structure to 12 feet (3657mm) instead of the desired 4m.

9 3.10 / K α ₌ − K / 10 18 . 1 _⋅ −6 = α 1_{See chapter 6.2}

(34)

Once decided the constructing material we had to develop the design of the structure. The technical development took a long time work to reach the final configuration. The technical drafts were made by PROMEC in Italy. A very simple schematic drawing of this device is shown in Figure 6-1

Figure 6-1 Schematic drawing of the cavity.

The cavity is a suspended structure made by 3 INVAR rods of 31.75 mm diameter and

3657mm long spaced at on a 336.5 mm diameter inside the vacuum pipe. A thermal

shield is used to minimize variations of temperature. Sufficient rigidity is obtained by means of 5 spacer plates (3 are shown in the Figure 6.1) solidly clamping the rods at equal intervals. The second and the fourth spacers are also used to suspend the cavity for seismic isolation. The flat folding mirror is mounted on the end plate at one end of the structure. The two end mirrors of the F.P cavity are side by side at the end plate at the other side of the structure. The mirrors support scheme is shown in the Figure 6-2 .

0 120

(35)

The control of the mirrors 2 is made by means of a triplet of micrometric screws and

piezoelectrics mounted at the periphery of each mirror at 1200. The reflective surfaces of

the mirrors are close to the ends of the INVAR rods. The composite thermal expansion of the mirror support structure, including end plates, micrometrers, piezos, and the mirror substrates were calculated to be sufficiently small, to satisfy our requirements discussed in paragraph II and Section 7.3

Because ground vibrations can excite resonance in our interferometer structure and disturb the operations, this entire structure is suspended by a system of four maraging

wires for horizontal isolation and MGAS blades for vertical isolation.3 The position and

the structure of the suspending device are shown in Figure 6-3. It is formed by a system of

4 couples of maraging springs designed to suspend 22 kg each4. The suspension point

will be finely adjusted by parasitic springs and stepping motors and will include the low frequency control system development by Maddalena Mantovani [19] to decrease the resonant frequency of this suspending system. For optical stability and thermal isolation the entire system is enclosed in a custom vacuum tank

Figure 6-2 Drawing of the support for the input and end mirrors

MH mirror Flat mirror

2_{See Section6.4} 3_{See Section6.3} 4_{See Section 7.2} piezotranslator Mirror spring 2 Mirror holder

(36)

(37)

II. Optical Constraints

After the development of the mechanics of this project we had to develop the optical counter part. First of all we had to choose the finesse of the cavity. Initially, for lock ease,

we intended to build a cavity withℑ =30. We finally chose because of the

following argument. The final aim of the experiment is to feed a Gaussian beam in a Mexican-Hat mirror profile F.P cavity and study how efficiently this cavity transforms the Gaussian beam profile into a Mesa flat beam profile. Comparing the calculated

graphics of the power profile prediction for different values of finesse, and

, we found that at the Gaussian input beam still contaminates by several

percents in power the central part of the mesa beam profile. Instead we found that with the beam is almost purely flat topped and the Gaussian mode is filtered out.

100 ℑ = 30 ℑ = 100 ℑ = ℑ =30 100 ℑ =

Figure 6-4 Variation from the beam profile from the pure mesa-beam profile at the MH end mirror with Finesse =30 and Finesse =100

(38)

Figure 6-5 Corresponding beam profile at the input mirror at low finesse. Since the beam power is injected or extracted from this port, this would be the beam profile contamination measured by our experiment.

Once chosen the Finesse of the cavity we have to choose the reflectivity of the MH and of the other two mirrors to produce the desired Finesse. In a G.W interferometer the F.P. mirror reflectivity is asymmetric. The end mirror is always fully reflective (R~0.999999) while the mirror close to the beam splitter is partially transparent, thus allowing to feed power in the cavity and determining the cavity Finesse. In a real interferometer we would feed the beam through one of the two spherical or MH mirror, and we would have to mode match the feed beam to the cavity with a suitable converging beam, prepared by a mode matching telescope. In our case we can more simply feed the cavity through the flat mirror. In this case the matching injection beam would have a waist at the injection point. To find the reflectivity of our input mirror we first assume no losses in the other mirror, we solve the equation for the Finesse for r1 and find it to be r1=0.969.

2 1 2 1 1 2 100 1 0.969 1 r r r r r r π ℑ = = ℑ = ⇒ = − (6.1.1)

Finally we cross check that the reflectivity of the other two mirrors is large enough not to spoil the desired finesse.

(39)

The specifications for the 3 mirrors are: Reflectivity MH Reflectivity folding mirror Reflectivity input mirror > 0.999 > 0.999 0.969

Table 6-1Specifications of the mirrors reflectivity

The mirrors for the Gaussian beam test were ordered with the same reflectivity of Table 6.1 to have the same Finesse of the MH test. The radius of curvature of the spherical end mirror was chosen to be 7.8m in order to be equivalent to the Advanced LIGO design and generate a scaled beam spot. Mirror of 8 m of curvature was readily commercially available it was deemed to be a suitable approximation and was used. This choice of mirror coupled to our cavity length corresponds to a Gaussian beam spot size of 3mm at the end mirror and of 0.8mm at the input mirror. The injection beam will have to be matched to this size and divergence. A different injection optics configuration will be necessary when the MH mirror will be implemented.

After deciding the length and the finesse of the cavity and after the calculation of the

beam profile5 we finally agreed on the MH mirror specification with our LMA

collaborators. The MH mirror is presently in production.

(40)

6.2. Technical Problems to manufacture the MH mirror

This section is reported for completeness even if I had no part in this process. The production of the MH mirror is completely under the responsibility of our colleagues at

LMA Lyon. The MH profile is obtained by differentially depositing SiO2 glass with the

desired radial profile over a flat profile. .

The construction of a MH mirror basically consists in 2 main steps [20]: • General shape coating:

This process deposits a coating with the general shape of the MH with a precision of 60 nm. This is made interposing a static mask between the sputtering source and a rotating mirror substrate. The mask profile is calculated according to the desired mirror profile, folding into its shape a 1/R weight to keep into account the substrate rotation. Figure 6-6 shows the principle of the General Shape Coating. The maximum thickness that can be deposited with this technique is the order of a few thousand nm.

Thin mirrors warp under the surface stress of the deposited coating materials more than the allowed tolerances. In order to maintain sufficiently tight surface tolerances, the thickness of the substrate must not be less than 0.4 times the diameter. We decided on a 50 mm diameter times 30 mm thick substrate even if the MH profile only extends on the central 20 mm.

(41)

Figure 6-6 General shape coating

• Corrective coating:

The second step is a more precise correction of the ``general shape'' which is used to achieve the desired profile with the desired precision. This time a small orifice mask produces a small pencil of sputtered atoms and the substrate is moved around to paint atoms where the profile is too low. In this process the first step is to measure the surface achieved with the first coating step, to compare it with the theoretical shape of the MH and to compute a correction map. The measurement of the achieved mirror shape is performed with a wave front interferometer measurement of the mirror after the general shape coating. The comparison between the achieved and the desired mirror shape generates a corrective file that is used to move the robot arm (the precision of the robot

varies between 0.2 and 1mm) that positions the mirror in front of the corrective SiO2

molecular beam correcting the defects of the mirror profile. The desired local correction thickness is controlled by adjusting the dwell time of the robot in each location. This method corrects the coating thickness with a precision of less than 10 nm PV (Peak-to-Valley). Because of the much lower deposition speed, with this technique it is not practically possible to correct more than 100 nm of thickness. The main limitation of the

(42)

corrective technique comes from the measurement of the wave front, from the robot arm

movement precision and from the size and sharpness of the SiO2 corrective beam.

Figure 6-7 Corrective coating procedure

The smaller is the mirror, the more difficult it is to get the necessary precision. It is also difficult to generate the steep slopes required by small size mirrors.

Given the small size mirrors required by our short cavity, it was decided to use a small coater with a small robot with 0.2 mm of precision. The limiting factors in building our small size MH are the transversal spatial resolution (0.35mm) of the wave front sensing interferometer used to measure the mirror shape achieved by the general coating step and the maximum measurable slope (500 nm/mm). The available resolution is just sufficient to generate the correction file for the corrective technique for the 20 mm diameter MH mirror supporting a 10 mm FWHM spot and 1 ppm diffraction losses.

The most difficult problem in building a small MH is that the slope of the corrective coating at the edge of the substrate (470nm/mm) is very close to the limit of what can be measured with the interferometer (500nm/mm). Incidentally, this is the reason that limited the minimal feasible size of the MH mirror and consequently the cavity length.

(43)

Additionally it should be noted that the measurement is less precise at the edge than in the central part and some level of defects will be expected in the rim of small diameter mirrors.

Building larger diameter MH mirrors is much easier. In Figure 6-8 it is shown an example

for a flat mirror 156 mm of substrate diameter, 100 mm thick, before and after the corrective technique. In this case the corrective technique was used on one side to generate the flattest possible mirror. After this treatment the Peak to Valley value of the

surface is 10.8nm. Figure 6-9 shows the first MH prototype attempted on the other face of

the same flat mirror. The data shows the mirror before and after the complete MH production process.

It should be noted that the small dip in both the flat and the MH mirrors, is due to a mechanical defect (later repaired) in the robot arm movement that prevented the molecular beam to reach the central spot. Building our smaller mirror is requiring substantially additional research and development.

(44)

Figure 6-9 MH mirrors before and after the corrective technique

• Other consideration:

I was assigned the calculations for a possible, easier, deposition scheme starting from a spherical, rather than plane, mirror substrate

To reduce the difficulty to deposit 1 micron of coating above a flat mirror to build the MH rim at the outer diameter, we considered to deposit the MH profile over a commercial spherical mirror profile. This would minimize the maximum required thickness of coating. We calculated that a spherical mirror with radius of curvature of 100 m would be optimal. Figure 6-10 shows the comparison between the MH mirror profile and the spherical profile. Subtracting the MH profile from the spherical mirror profile we obtained a minimized coating thickness. Figure 6-11 shows the MH mirror profile and the deposition profile over the spherical mirror. In this way the maximum thickness be deposited is 0.5 micron instead 1 micron. This technique reduces the maximum amount of coating thickness to be applied, but does not help with the problem of measuring the obtained surface profile (limited to a maximum slope of 500nm/mm) between the two steps of corrective coatings. Also this idea would require deposition of a thicker layer at the center of the mirror, where the beam is maximally sensitive to possible build–up defects. The idea of starting the MH mirror from a spherical mirror was then abandoned.

To minimize the slope measurement problem, considering that the profile slope peaks at the outer rim, we calculated the minimum acceptable mirror diameter as a function of a

(45)

specified diffraction loss and truncated the deposition at that diameter. To do that, E. D’Ambrosio simulated the diffraction losses as a function of the size of the mirror to find the best compromise between the diffraction losses and the diameter of the mirror which, in its turn, fixes the maximum thickness and slope of coating. Figure 6-12 shows these diffraction losses. According to this simulation, to limit diffraction losses to about 1ppm the radius of the MH mirror have to be between 10 and 11 mm.

(46)

Figure 6-11 Difference from deposition over MH profile and over spherical mirror

Figure 6-12 Diffraction losses as a function of a mirror radius

(47)

6.3. Suspending System

The suspending system of this apparatus consists in 4 couples of Maraging Blade MGAS (Monolithic Geometric Anti-Spring). Maraging steel (acronym of “martensitic ageing”) is a high-nickel, low-carbon content steel [21]. Its microstructure is strengthened by suitable ageing treatment which causes precipitation of thin inter-metallic particles reducing the motion of the dislocations [22]. To design the blade for this suspending device we used a Mathematica program developed by Youiky Aso [23]. The suspension system is shown in the Figure 6.3. It is composed by the MGAS Maraging blades, Maraging wires, the LVDTs, the force actuators, the parasitic springs and the stepping motors.

¾ MGAS

The MGAS filter is constituted by two opposing Maraging blades. The geometric configuration permits us to mechanically tune the resonant frequency of the system because the compression of the blades against each other changes the geometric

anti-spring component. This mechanism can be easily understood from the Figure 6-13.

Increasing the compression of the blade we generate a positive vertical force gradient, null at the working point.

Figure 6-13 Schematic drawing of the geometric anti-spring mechanism

¾ LVDT

The position reading system of the apparatus is made by LVDTs (Linear Variable Differential Transformer) which are high precision position sensors, free from every kind of contact and force and can be used under vacuum [24.] An LVDT is constituted by three coils, two large ones in series, coiled in opposite direction, which act as receiver and one smaller coil which is the emitting coil. Typically the two receiver coils are mounted on a reference structure while the emitter coil is mounted on the moving mechanical component.