Analytical models of thermal aberrations in massive mirrors heated by high power laser beams

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Analytical models of thermal aberrations in massive

mirrors heated by high power laser beams

Patrice Hello, Jean-Yves Vinet

To cite this version:

Analytical

models of thermal aberrations

in

massive mirrors

heated

_{by high power}

laser beams

Patrice Hello and Jean-Yves Vinet

Laboratoire

_{d’Optique Appliquée,}

Ecole

_{Polytechnique,}

ENSTA, 91120 Palaiseau, France

Groupe

de Recherche sur les Ondes de Gravitation, Bât. 104, Université Paris-Sud, 91405

_Orsay,

France

(Reçu

le 15 décembre 1989,

accepté

le 28

_{février 1990)}

Résumé. 2014 Nous traitons le

problème

de l’échauffement des

supermiroirs

par des sources laser intenses. L’échauffement vient de

_{l’absorption}

dans le substrat de silice ou de la

_dissipation

dans les couches réfléchissantes. Dans chacun de ces _cas, nous construisons des modèles

analytiques

originaux

pour la

température

transitoire et les aberrations

_{thermiques qui}

en résultent. Nous

présentons

des formules

analytiques,

basées sur des séries

rapidement

convergentes, et des résultats

numériques

concernant l’évolution de la

_{température,}

et le

développement

des aberrations en séries de

_polynômes

de Zernike.

Abstract. 2014 We consider the

problem

of massive mirrors heated

_by

intense laser sources.

_Heating

occurs both

_{by absorption}

in the silica substrate and

_{by dissipation}

in the

_coating.

In each case we

derive

_{original analytical}

models of the transient _temperature and of the

_resulting

thermal aberrations.

Analytical

formulas, based upon

rapidly converging

series, and numerical results are

presented

for the time

_dependent

_temperature and for the time

_{dependent expansion}

of the aberrations in series of Zernike

polynomials.

Classification

Physics

Abstracts 04.80 - 42.78

1. Introduction.

The

_heating

of mirrors or other

_cylindrical

solids like YAG rods has been

_yet

studied in

various

_experimental

conditions and

_{approximation}

schemes. But the

resulting

type

of model

depends strongly

on the

_boundary

conditions as well as on the

_{adopted simplifications.}

Some authors consider two or three-dimensional models but restrict themselves to

_steady

state

_temperature

distributions

_[1-8].

Others take time

_dependence

into account but consider

only

one or two

_dimensions,

_building

_up

_simple

models of

_plates,

laser windows or slabs...

[9-10].

Some rare authors

_finally

consider three-dimensional mirrors and include the time

dependences by using

Green’s

_{integrals [11-13].}

In _{this paper,} we

_present

for the first time an

_{explicit analytical}

model

_including

finite

dimensions of the heated

solid,

the time

_dependence

and radiation or convective losses.

2.

_Physical

conditions.

The

_{starting point}

of our work is the

_study

of the thermoelastic behavior of the mirrors involved in interferometric

_{gravitational}

antennas.

The details of these antennas

_planned

for the

_foregoing

_years are now

_widely

known. In the Italian and French VIRGO

_project

as well as in the American LIGO

_project,

a

_{key subsystem}

is a

Fabry-Pérot cavity

with a

long

_storage

time,

i.e. with a

_high

surtension coefficient. In the

required

conditions for

_operation

at an

acceptable signal-to-noise

_ratio,

hundreds of watts

light

power will cross the

_input

mirror,

and tens of kilowatts will be

_continuously

reflected on its internal coated face.

Obviously,

materials with very low intrinsic

absorption,

and very low losses

coatings

are to

be used.

Nevertheless,

it is

_likely

that at least several watts will be

_dissipated

in the substrate and the

_{coating, resulting}

in

temperature

gradients.

These

temperature

gradients

will in turn

result in index

_gradients

and

_geometrical

alteration of the

_reflecting

face,

and

_finally,

the

optical properties

of the mirror will be affected.

The

_requirements

on

optical

surfaces and bulk index

homogeneity being

severe, an evaluation of the thermal behavior of the mirrors seems therefore necessary.

Moreover,

time-dependent

solutions are

_especially

useful in a

_dynamical

model of a

_Fabry-Pérot

_cavity

with a

high

surtension coefficient for

_example.

The mirrors that we consider are thick

_cylindrical

blocks of _pure

silica,

with a

_high

reflectance

_coating

on the inner

face,

and

eventually

an

_{antireflecting coating}

on the outer

face.

_Light

power is thus

expected

to be

_{dissipated by}

bulk

_absorption

in the silica

_substrate,

and

_by

the

_coating

losses. These mirrors are

_supposed

to be

_{suspended by}

thin wires in a vacuum vessel so that the heat losses are

_only

due to thermal radiation

_(see

_{Fig. 1).}

Fig.

1. - Sketch of the mirror. The

incoming light

can be absorbed either

by

the substrate

(bulk

absorption)

or in the

_coating

in the left face

_{(coating absorption) ;}

the mirror losses heat

only by

thermal

radiation.

temperature

T max

in the mirror is not much

higher

than the

_temperature.

_Text

₍

= 300

_K)

of the

walls of the vacuum vessel :

It is well known that the radiative heat losses of a surface element at

_temperature

T are

given by :

where a’ is the Stefan-Boltzmann constant corrected for

_emissivity.

With the above

assumption,

this becomes a linear

_{expression :}

The whole

_problem

becomes thus

linear,

and we can treat

_separately

the different contributions to

_{heating : dissipation}

in the

_coating

and bulk

_absorption.

Note that convective heat losses can also be treated

_by

the

_model,

_according

to the above

_expression

for

F ;

the coefficient of 8 T has

_simply

to be modified. 3.

_{Heating by absorption}

of _{power in the}

_coatings.

3.1 STEADY STATE SOLUTION. - We consider _a disk of radius a and thickness h. The radial

coordinate is 0 , r , a and the axial coordinate -

_{h /2 _}

z

h /2

(Fig. 1).

The

incoming light

power is

supposed

to be

_dissipated

in a thin

layer

on the z = - h

/2

face,

and _we shall

represent

it

_by

an

incoming

heat flux at z = - h

/2.

With no internal

_generation

of

_heat,

the

Fourier

_equation

is -

’

where

T ( t, r, z )

is the

_temperature

distribution

_(K),

we define in the

following

table the different coefficients and their numerical values in the case of silica :

p :

density

(2

202

kg

m-

3),

C :

_specific

heat

₍₇₄₅

J

_kg-1

1

_K-1),

K : thermal

conductivity (1.38

W

m-1

1

K-

_1).

We first consider the

_steady

state solution of the form

for which the

boundary

conditions are :

where

_I(r)

is the

_intensity

distribution in the

_{incoming light}

beam

_(W

_{m- 2),}

and e the

A

_sufficiently

_general

harmonic function solution of

_{equation (5)}

_may be

_expressed

as :

where the constant

_km

will be determined from condition

₍₂₎

and constants

_Am

and

Bm

from conditions

(3)

and

(4).

Equation (2)

is satisfied if:

Let’s call

_(m

the m-th solution of:

where we have introduced the reduced radiation constant :

The (m

may be obtained

numerically by

standard

techniques,

then

It is well known from the Sturm-Liouville

_problem

that the functions

form an

_orthogonal

basis for functions defined within the interval

_{[0, a}

_]

with normalization

constants r141 :

--Then we can _express

_{1 (r)}

as a Dini series :

so that

_{equations (3)}

and

₍₄₎

become :

with

from which the solution is

_completely

determined :

In the case of a Gaussian beam of waist w and

_{power P :}

in our numerical calculations w = 2 cm

_according

to the

_{Virgo specifications.}

It is easy to

compute

the coefficients _pm

_by

the formula

If a is assumed much

_larger

than the waist w, one finds :

In

_figure

2 we

_represent

the

_stationary

_temperature

distribution for eP = 1 W absorbed

power.

Fig.

2. - A

map of the

steady

state _temperature field for a

_{coating absorption}

of 1 W. Note : for all the

3.2 TRANSIENT SOLUTION.

3.2.1 General

_principles.

- A time

dependent

solution of

_{equation (1)}

_may be written as :

where

_{T 00 (r,}

_{z )}

is the

_steady

state solution found

above,

satisfying

the conditions

_{(2), (3)}

and

(4).

Ttr ( t, r, z )

is solution of

_{equation (1),}

and satisfies the

_{homogeneous boundary}

conditions :

But we add the

Cauchy

data :

where

_{To (r,}

z )

is an

_arbitrary

function. The heat

equation

is now satisfied

_by

a series of the

form : ’

with the time constants

in the

_following

we shall use the characteristic

_{time tc :}

The (m having

the same definition as

_above,

the condition

(13)

is fulfilled. Now the conditions

_{(14) give}

the

_{dispersion equations}

for the even and odd modes

_{respectively :}

The constant

_up

_(resp.

_vp)

is the

_p-th

solution of

_{equation (18) (resp. (19)),}

the

{u p, v p }

may be

numerically computed

with standard methods. The functions

We need the normalization

_{constants sp}

and

_cp,

defined

by

The initial condition

(15)

is satisfied if

Assuming

that the

_right

hand side of the

_preceding

equation

admits a Dini

expansion :

we _may determine the constants

_{A pm,}

_Bpm

_by

3.2.2

Heating

from

zero. _- When the initial state is thermal

_equilibrium

with uniform

temperature

equal

to the

_surrounding

vessel’s

(before

switching

on the

laser),

we have the

special

case

_{To (r,}

_{z )}

=

Text,

and we obtain thus from

equation (8) :

A direct calculation

using

the

_{dispersion equations gives}

for

and we find

In

_figure

_3,

we _represent the

_temperature

evolution of the warmest

_point

_{(z = - h /2,}

r =

0)

for 1 W absorbed _power. The characteristic time is about 30 h. In

figures

4 and 5 _we

show transversal and axial cuts at different times.

Fig.

3. - Evolution of the

temperature of the « warm

_{point »}

at the centre of the

_coating.

The reduced time is

_tltc

and the absorbed power is 1 W in the

coating.

Fig.

4. -

Successive transversal cuts of the _temperature field in the

_{coating plane}

at the times

Fig.

5. - Successive axial cuts of the

temperature field at the times,

tc / 1

000,

tc/100, t,110

and tc for 1 W absorbed in the

coating.

4.

_{Heating by}

bulk

_absorption.

4.1 STEADY STATE SOLUTION. - We consider now the case where the

_light

beam crosses the

transparent

disk with a weak distributed attenuation characterized

_by

the constant a

(m-1)

and no losses on the

faces,

so that there is

_only

internal heat

_generation

and zero

incoming

heat fluxes at the boundaries. The Fôurier

_equation

is now :

where the z

_dependence

of the

light intensity

due to attenuation was

_dropped

out as

giving

second order terms in a.

1er)

is

supposed

to have the Dini

_{expansion given by equations (9)}

and

(10),

so that we can write the

_steady

state solution under the form :

expecting

it to be even with

_respect

to z. The

_boundary

conditions are

_{homogeneous :}

Condition

₍₂₄₎

is fulfilled if

_km

has the définition

_(6),

and condition

_{(25) gives}

the value of

We

_get

the final

_{expression :}

This

_stationary

_temperature

distribution is shown in

_figure

6.

Fig.

6. - An

exemple

of map of the

steady

state _temperature field in the case of a bulk

absorption ;

the

absorbed power is

2 mW.m-’.

,

4.2 TRANSIENT SOLUTION. - As in the

previous

section,

the time

_dependent

solution of

equation (22)

will be taken of the form

where

_Ttr

is a solution of the

_homogeneous

heat

equation,

and

owing

to the

_required

even

1dependence

on z, may be

expressed

as :

where

Within the interval -

_/x/2

z z

h /2,

we can

expand

the z

_dependence

of

_Ta)

in a Fourier

series ;

Ta) having

the Dini

_expansion

with

(Jm(

( we find

where the coefficients

_{C pm}

are

_{given by :}

by

choosing

_{C pm - -}

C;m

we obtain the time

dependent

solution

satisfying

the initial condition :

Figure

7 shows the evolution of the warmest

_{point ( r}

= z =

0).

Figures

8 and 9 show

transversal and axial cuts of the

_temperature

field for different times. We took a

10- 5/m

absorption

rate which

corresponds

to

_optimistic

estimates for very pure silica and 200 W incident

_light

_power.

Fig.

7. - Evolution of the _temperature of the « warm

_{point » :}

the

_point

at the centre of the silica

Fig.

8. - Successive transversal cuts of the

temperature field in the central

_{plane (z}

= 0) at the times

tc/1 000, tc/100, tc / 10

and tc

when the absorbed power is 2

mW.m-1

1 in the bulk.

Fig.

9. -

Successive axial cuts of the _temperature field at the times

_tell

000,

tell 00,

tell 0

and te for on 2

mW.m-1

1

5. Thermal aberrations due to index

_gradients.

Transparent

dielectric materials such as silica have in

_general

_temperature

_dependent

refraction indices. The local index

_{n (x, y, z )}

_may be related to the local

_temperature

T(x, y, z ) by

where _no is the reference index

_{corresponding}

to the

_temperature

_To.

For

_silica,

we have

dn/dT = - 0.87 x

10- 5 K-1.

The

_length

scale of the

_spatial

variations of T

_being

much

_larger

than the

_wavelength

of the

light,

we _may use the

_geometric

_{optics approximation}

for

evaluating

the

phase

distortion of a wave

_{passing through}

the thermal lens. The

optical path obeys

the eikonal

equation

At first

order,

we _may

consider 0

as the sum of the reference

_{optical path}

plus

a

_perturbation

so

_that,

_{by neglecting}

_{the squares of the derivatives of}

_1/1,

_{equation (28)}

becomes :

The total

_{optical path}

distortion due to

_{thermal lensing}

in the substrate is thus after a

_passing

through

the silica :

’As

_{dn/dT is}

_negative

the thermal lens is a

_diverging

lens : the

_{optical path}

is

_bigger

at the borders than at the center of the mirror.

1

The

_axisymmetric

aberration may be

expanded

as

_usually

on a basis of Zernike

_polynomials

R2q(0)(r/b )

within the disk of radius b a

corresponding

to the

_optical

zone of the silica block.

(The

block is much wider than the

_light

_beam,

and

_possible

aberrations outside the beam are

meaningless :

in the case of a Gaussian

_{beam, b}

could be 3 times the waist

_w).

We shall thus

write :

where the coefficients

_{C2 q}

are obtained from

5.1 CASE OF HEATING BY COATING ABSORPTION. - When the

_temperature

distribution is

given by equation (21),

the

_preceding

method

_gives

the time

_dependent

thermal aberration :

An

_expansion

in a series of Zernike

_polynomials

such as

₍₃₀₎

_gives

in the case of a Gaussian beam of radius w, and EP =

P abs (absorbed power),

transient coefficients

We have

_plotted

in

_figure

10 the time

_dependence

of the first c coefficients. The coefficient co was omitted because it

_represents

_only

an uniform additional

_phase.

Fig.

10. - Evolution of the five first coefficients of the Zernike

polynomials

except co. The bulk

absorption

datas are

_plotted

in

_plain

lines and the

coating absorption

ones in dashed lines. In both cases

the total absorbed power is 1 W.

and the

_expansion

in Zernike

_polynomials,

with

_{P abs}

= haP,

gives

the coefficients :

see

_figure

10.

Finally figure

11 shows an

_example

of the thermal lens

profile

in the

steady

state for the two

cases

_(coating

and bulk

_absorption)

and for 1 W absorbed _power. The

profile

is limited to an

optical

zone of 5 cm radius

_according

to the VIRGO

_{specifications.}

Fig.

11. - The

shape

of the thermal lens when the

_steady

state is established. We

_give

the absolute value of the thermal aberration,

dn /dT

being negative.

The

_plain

line refers to the bulk

_absorption,

and the dashed one to the

coating absorption.

In both cases the total absorbed power is 1 W.

6. Conclusion.

A time

_dependent

modelization of the

_temperature

field in

_transparent

mirrors caused

_by

absorption

of

_light

either in the

_coatings

or in the bulk material has been derived and

numerically exploited.

It shows that the

_steady

state

_temperature

is about 13

K/W

in the first case

_{(coating dissipation)}

and about 2

_K/W

in the second one

_{(bulk absorption) ;}

nevertheless the two mechanisms of

_light

power

dissipation give roughly equal

thermal

_{leasing (about}

1.5

_tJbm/W

at the

center)

for

_{equal dissipated}

powers. When the substrate is made of pure

silica,

the

_dissipation

rate

_by

bulk

_absorption

is however

_expected

to be much smaller than the losses in the

coatings,

and we can therefore conclude that the

_{thermal lensing}

is

_mainly

due to

thermoelastic deformations of the mirror in the cases of

_absorption

either in the bulk or in the

coating.

Clearly, dissipation

of 1 W on the

coatings

would have a

_{huge perturbing}

effect on the

Virgo

cavities and we have either to find _very low

_dissipation

_materials,

or to correct for the induced aberration. But when such

_dissipative

elements are included in a resonant

_cavity,

a non-linear

_coupling

occurs between the thermal _system which determines the

_{optical tuning,}

and the

_optical

_system

which determines the rate of

_heating.

We are therefore

_developing

a numerical simulator of a

_{dissipative cavity, including}

the

_preceding

results.

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