5.1 Transient temperature field

We consider the time-dependent heat equation with an internal source of heat (bulk absorption of light) and a surface source on the coating. The Fourier equation is
[ρC ∂t − K Δ ]T (t,r,z) = S1(r), (5.1 )
where ρ is the density of the material and C its specific heat. S1 accounts for the dissipation of light passing through the substrate. As in the static case, it is independent on z because the attenuation of the beam is extremely weak; moreover, we assume it is independent of time. In a recycling interferometer, we would expect the transient lens to change the reflectance of the cavity, the recycling rate, in turn, and eventually the source S1, so that S1 is a function of t. But we consider here an isolated mirror. The case of a complex system with feedback can be treated sequentially (see [19]) using any propagation code, but this is beyond our present goal. For instance, we assume some servo loop acting to maintain a constant power flow. The boundary conditions are still
[ ] ∂T- 3 − K ∂z = − 4 σfT0T (r,− h ∕2) + S2 (r), (5.2 ) r,z= −h∕2
where S2 accounts for the dissipation of light on the reflective coating. The next two boundary conditions are
[∂T ] − K --- = 4σfT 30T(r,h ∕2) (5.3 ) ∂z r,z=h ∕2
and
[ ] ∂T 3 − K -∂r = 4σedgeT0T (a,z). (5.4 ) r=a,z
We already know the steady state solution T∞ for coating or bulk dissipation in the general form
∑ T ∞(r,z) = Ts(z)J0(ksr), (5.5 ) s
where ks ≡ ζs∕a and {ζs : s = 1,2,..} have the same definition as in Equation (3.11View Equation). T∞ (r,z) satisfies all boundary conditions plus the inhomogeneous heat equation. Thus, it is a special solution of Equation (5.1View Equation). We now look for a general time-dependent solution of the homogeneous heat equation
[ρC ∂t − K Δ ]Ttr(t,r,z) = 0 (5.6 )
and satisfying homogeneous boundary conditions. We search this transient temperature field under the form
∑ [ ′ ′ ′′ ′′ ] Ttr(t,r,z) = 𝜃s,j(t)cos(κ jz ) + 𝜃s,j(t)sin(κjz) J0 (knr ), (5.7 ) s,j
so that the general solution of Equation (5.1View Equation) will be
T (t,r,z ) = T ∞(r,z) + Ttr(t,r,z). (5.8 )

Now, the time functions ′ 𝜃s,j(t) and ′′ 𝜃s,j(t) must satisfy

′ ∂𝜃s,j-+ K--(k2 + κ′2)𝜃′ = 0 (5.9 ) ∂t ρC n j s,j
and
∂𝜃 ′′ K ( ) ---s,j+ --- k2n + κ ′′j2 𝜃′′s,j = 0, (5.10 ) ∂t ρC
whose solutions are
𝜃′ (t) = 𝜃′ exp (− t∕τ′ ) (5.11 ) s,j s,j s,j
𝜃′′ (t) = 𝜃′′ exp(− t∕τ′′ ), (5.12 ) s,j s,j s,j
where the time constants are, respectively,
′ ---τ---- τs,j = γ2 + u2 (5.13 ) s k
′′ --τ----- τs,j = γ2+ v2 (5.14 ) s j
with the main time constant
ρCh2 τ ≡ ------ (5.15 ) 4K
(about 0.8 h for a regular Virgo mirror substrate). The boundary conditions on the faces lead to the following equations:
( ) ( ) ′h ′h 4σfT03h ′h κj--sin κj-- − -------cos κ j-- = 0 (5.16 ) 2 2 2K 2
( ) 3 ( ) κ ′′j h-cos κ′j′ h- + 4σfT0h-sin κ′j′h- = 0. (5.17 ) 2 2 2K 2
In terms of the reduced radiation constant χ′ = χh∕2a, the first equation is of the form
usin u − χ′cosu = 0. (5.18 )
This equation admits an infinite discrete family of solutions ∗ {uj, j ∈ ℕ }, giving us ′ κ l = 2uj∕h. In the same way, we have κ ′′j = 2vj∕h, where the constants {vj,j ∈ ℕ ∗} are all solutions of
vcos v + χ′sinv = 0. (5.19 )
Owing again to the Sturm–Liouville theorem, the functions {cos(2ujz∕h),j ∈ ℕ ∗} form a complete orthogonal set, and a basis for symmetric functions of z defined in [− h∕2,h∕2 ]. Functions {sin(2v z∕h ),j ∈ ℕ∗} j form also a complete and orthogonal set for antisymmetric functions on [− h ∕2,h∕2 ]. The two sets are obviously mutually orthogonal. Moreover, we have
∫ h∕2 ′ ′ ′ cos(κjz) cos(κ j′z)dz = gjδjj′ (5.20 ) −h∕2
with the normalization constant
[ ] ′ h- sin(2uj) gj = 2 1 + 2uj (5.21 )
and in the same way
∫ h∕2 cos(κ′j′z) cos(κ ′′j′z)dz = gj′′δjj′ (5.22 ) −h∕2
with
[ ] g′′= h- 1 − sin(2vj) . (5.23 ) j 2 2vj
At this point, all constants have been determined, except ′ 𝜃s,j and ′′ 𝜃s,j. This is done depending on the initial conditions on the temperature. The total temperature field being
T (t,r,z) = T∞ (r[,z ) + Ttr(t,r,z ) ] ∑ ∑ ( ′ ′ ′′ ′′ ) = Ts(z) + 𝜃s,j(t) cos(κ jz) + 𝜃s,j(t)sin(κjz) J0(ksr), (5.24 ) s j
Thus, the initial temperature is
T(0,r,z) = T ∞(r,z) + Ttr(t,r,z) ∑ [ ∑ ] = T (z) + (𝜃′ cos(κ′z) + 𝜃′′ sin (κ′′z)) J (k r), (5.25 ) s s,j j s,j j 0 s s j
and if we require heating from room temperature, i.e., T(0,r,z) = 0, considering the orthogonality of the functions J0(ksr), we are led to the equation
∑ [ ] Ts(z) + 𝜃′s,j cos(κ′j) + 𝜃′s′,j sin (κ ′′j) = 0 (5.26 ) j
giving
∫ h∕2 𝜃′s,j = − -1 Ts(z) cos(κ′jz)dz (5.27 ) g′j −h∕2
∫ ′′ 1 h∕2 ′′ 𝜃s,j = − g′′- Ts(z)sin(κjz)dz, (5.28 ) j −h∕2
which completes the determination. Finally, the temperature field is
[ ( ) ( ) ] ∑ ′ − t∕τ′s,j ′ ′′ −t∕τs′′,j ′′ T (t,r,z) = − 𝜃s,j 1 − e cos(κ jz ) + 𝜃s,j 1 − e sin(κjz) J0 (ksr). (5.29 ) s,j
It is now possible to specialize the result to the two cases of coating and bulk absorption.

5.1.1 Transient temperature from coating absorption

In the case of coating absorption, we know from Equation (3.18View Equation) specialized to axisymmetry that

[ ] 𝜖P cosh(ζsz∕a) sinh(ζsz∕a) Ts(z) = 2πKa--ps ----d-------− ----d------ , (5.30 ) 1,s 2,s
(γs ≡ ζsh∕2a). After some elementary algebra, we have
′ 𝜖P-hps------cosuj---------1---- 𝜃s,j = − 2πKa2 1 + sin(2uj)∕2uj u2j + γ2s (5.31 )
𝜃′′ = 𝜖P-hps------sinvj---------1---- (5.32 ) s,j 2πKa2 1 − sin (2vj)∕2vjv2j + γ2s
so that the final result for the transient temperature field is
[ 𝜖P h ∑ cos(uj) ( −t∕τ′ ) ′ T (t,r,z ) = 2πKa2-- ps (1 +-sin(2u-)∕2u--)(u2-+-γ2)- 1 − e s,j cos(κjz) s,j j j j s
] sin(vj) ( − t∕τ′′ ) ′′ − --------------------2----2- 1 − e s,j sin(κjz) J0(ksr) (5.33 ) (1 − sin(2vj)∕2vj)(vj + γ s)
The transient thermal lens is
∫ dn- h∕2 Z(t,r) = dT T (t,r,z )dz −h∕2 ( ) dn--𝜖P-h2- ∑ -------sin-(2uj-)∕2uj-------- −t∕τs′,j = dT 2 πKa2 ps (1 + sin(2uj)∕2uj)(u2 + γ2) 1 − e J0(ksr). (5.34 ) s,j j s
Figures 36View Image, 37View Image, and 38View Image show the time evolution of the thermal lens. The time constant for temperature evolution is very long, several hours, but the time constant of the focal length of the thermal lens is much shorter because the final profile of the lens is reached within about 1/2 h, after which the temperature keeps growing uniformly without changing the gradients.
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Figure 36: Time evolution of the thermal lens from room temperature to the steady state limit. Heating from coating absorption, LG0,0 mode, w = 2 cm
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Figure 37: Time evolution of the thermal lens from room temperature to steady state limit. Heating from coating absorption, LG5,5 mode, w = 3.5 cm
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Figure 38: Time evolution of the thermal lens from room temperature to the steady state limit. Heating from coating absorption, Flat mode, b = 9.1 cm
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Figure 39: Coating absorption: time evolution of the curvature radius of the thermal lens LG 0,0 mode, w = 2 cm
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Figure 40: Coating absorption: time evolution of the curvature radius of the thermal lens, LG 5,5 mode, w = 3.5 cm
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Figure 41: Coating absorption: time evolution of the curvature radius of the thermal lens, flat mode, b = 9.1 cm
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Figure 42: Coating absorption: time evolution of the curvature radii of thermal lenses for three examples

5.1.2 Transient temperature from bulk absorption

In the case of internal absorption, we have

βP p [ χ cosh(k z) ] Ts(z) = -----s 1 − --------s-- . (5.35 ) πK ζ2s d1,s
We have, obviously, 𝜃′′ = 0 s,j and, after some algebra, we find
′ βP-psh2----------sinuj∕uj---------- 𝜃s,j = − 2πKa2 (1 + sin (2u )∕2u )(γ2+ u2), (5.36 ) j j s j
so that the temperature field is
βP h2 ∑ sinuj∕uj ( −t∕τ′ ) ′ T (t,r,z) = ------2 ps--------------------2----2- 1 − e s,j cos(κjz)J0(ksr) (5.37 ) 2πKa s,j (1 + sin (2uj )∕2uj )(γs + uj)
and the transient thermal lens is
dn βP h3 ∑ (sin uj∕uj)2 ( − t∕τ′ ) Z (t,r) = ---------2 ps--------------------2----2- 1 − e s,j J0 (ksr ). (5.38 ) dT 2πKa s,j (1 + sin(2uj )∕2uj )(γ s + u j)
The time evolution curves are practically identical to those shown in Figures 36View Image, 37View Image, and 38View Image. See, for example, Figure 43View Image; in the same way, the final lens profile is reached long before the temperature reaches the steady state.
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Figure 43: Bulk absorption: Time evolution of the thermal lens curvature radii for three examples

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