3.2 Steady thermal lensing

The temperature field inside the substrate has, as a first effect, to change the refractive index by an amount
dn- δn(r,z) = dT T (r,z ), (3.39 )
where dn∕dT is the temperature index coefficient [23]. Therefore, the resulting index field, not being homogeneous, causes focusing effects on light called thermal lensing. The thermal lens is the integrated excess optical path (EOP) for a light ray crossing the substrate. Neglecting diffraction over the thickness of the mirror substrate, the EOP can be evaluated using
dn ∫ h∕2 Z (r) = --- T(r,z)dz. (3.40 ) dT − h∕2
It is easy to obtain the result in the two cases discussed above.

3.2.1 Thermal lensing from coating absorption

From Equation (3.18View Equation) we obtain the thermal lens caused by coating absorption:

∑ Z (r) = -dn 𝜖P-- pn,ssinh-γn,sJ (k r∕a) (3.41 ) coat dT πK ζn,s d1,n,s n n,s s

3.2.2 Thermal lens from bulk absorption

From Equation (3.25View Equation), one finds

[ ] dn-βhP--∑ -ps- (2-χa∕ζn,sh-)sinhγs- Zbulk(r) = dT πK ζ2 1 − d Jn (kn,sr∕a). (3.42 ) s n,s 1,n,s

3.2.3 Equivalent paraboloid

In order to study the consequences of the focusing properties of the thermal lens, we can compute the nearest paraboloid, defined by the apex equation :

2 Zˆ(r) = cr + d, (3.43 )
where c is the curvature parameter, related to the mean curvature radius Rc of the lens by c = 1∕2Rc. d is called a piston. We define the nearest paraboloid by requiring the new lens ˆ − Z to carry out the best correction to the wavefront distorted by Z. If ψ (r,φ ) is the normalized mode amplitude, the Hermitian scalar product S of the perfect incoming field with the distorted-corrected one is
S = ⟨ψ,C ψ ⟩, (3.44 )
where
C (r,ϕ) = exp [ik(Z (r,φ) − ˆZ(r)] (k ≡ 2π∕ λ) (3.45 )
(λ being the laser wavelength). For a small difference, this is, at second order,
∫ ∫ 2 k2 2 2 S = 1 + ik [Z (r,φ) − ˆZ (r)]|ψ(r,φ)| rdrdφ − 2-- [Z (r,φ) − Zˆ(r)]|ψ(r,φ )| rdrdφ. (3.46 ) ℝ2 ℝ2
The first integral represents a phase that can always be cancelled out by a suitable choice of d. The second integral represents the coupling loss due to imperfect correction of the wavefront. Parameters c and d are found by requiring a minimum loss. If we define the average < f > of any function f(r,φ ) by
∫ 2 < f >= 2 f(r,φ )|ψ (r,φ)|rdrd φ. (3.47 ) ℝ

In words, if all averages are performed with the weighting function |ψ (r,φ)|2 (i.e., the normalized beam intensity), then the determination of the best paraboloid amounts to the classical least-squares formulas:

< Zr2 > − < r2 > < Z > c = -------4---------2--2---- (3.48 ) < r > − < r >
d = < Z > − c < r2 > . (3.49 )

3.2.3.1 Averaging with LG modes

In the case of Laguerre–Gauss beams, the weighting function in the averaging process, is of the form (see Equation (3.27View Equation))

2 (n) (n) |ψ (r,φ)| = R m (r) + R m (r)cos(2n φ + 2φn ), (3.50 )
whereas the thermal lens is of the form (see Equation (3.29View Equation)
Z (r,φ) = Z0 (r) + Zn,m (r)cos(2nφ + 2 φn). (3.51 )
with
∑ Z0 (r) = Z0,sJ0(ζ0,sr∕a) (3.52 ) s>0
∑ Zn,m (r ) = Zn,m,sJ2n(ζn,sr∕a). (3.53 ) s>0
We have the following intermediate results:
∫ ∞ 2 < r2 >= 2π R (n)(r)r3dr = w-(2m + n + 1) (3.54 ) 0 m 2
∫ ∞ 4 < r4 >= 2π R (n)(r)r5dr = w--[6m (n + m + 1) + (n + 1)(n + 2 )] (3.55 ) 0 m 4
so that
4 2 2 w4- < r > − < r > = 4 [2m (m + n + 1) + n + 1]. (3.56 )
Now,
∫ ∞ ∫ ∞ < Z >= 2 π R(mn)(r)Z0(r)rdr + π R (nm)(r)Zn(r)rdr, (3.57 ) 0 0
making clear that we need the two integrals
∫ ∞ (n) ℐ0 = < J0(ζ0,sr∕a ) >= 2π 0 R m (r)J0(ζ0,sr∕a )rdr (3.58 )
∫ ∞ ℐn,m =< J2n(ζn,sr∕a) >= 2π R (mn)(r)J2n(ζn,sr∕a )rdr. (3.59 ) 0
These two integrals are analogous to those giving the FB coefficients of the intensity (see Equations (3.33View Equation) and (3.36View Equation)), so that we have immediately
ℐ0 = e−y0,sLm (y0,s)Ln+m (y0,s) (3.60 )
m! n − y (n) 2 ℐn,m = ---------(yn,s) e n,sLm (yn,s). (3.61 ) (n + m )!
To compute < Zr2 >, we need the integrals
∫ ∞ 𝒦0 = 2π R(n)(r)J0(ζ0,sr∕a)r3dr (3.62 ) 0 m
∫ ∞ (n) 3 𝒦n,m = 2π R m (r)J2n(ζn,sr∕a)r dr. (3.63 ) 0
These are easily obtained from the preceding definitions of ℐ0, and ℐn,m using the Bessel differential equation; namely, we have, for arbitrary κ,
( ) 2 2 1 < J0 (κr )r >= − ∂κ + --∂κ < J0(κr) > (3.64 ) κ
( 1 4n2) < J2n(κr)r2 >= − ∂2κ + -∂κ − -2-- < J2n(κr) > . (3.65 ) κ κ
Let us note that
2 2 c = <-J0-(ζ0,sr∕a)r->--−-<--r->-<-J0(ζ0,sr∕a-) > (3.66 ) 0,s < r4 > − < r2 >2
< J2n(ζn,sr∕a )r2 > − < r2 > < J2n(ζn,sr∕a ) > cn,m,s = -----------------4--------2---2-------------. (3.67 ) < r > − < r >
After some algebra, we get (here y ≡ y0,s)
2 c0,s = --------------------------2 e−y [(2m + n + 2 − y)Lm (y)Ln+m (y) [2m (n + m(+ 1) + n + 1 ]w ) + (m + 1) L (y)L (1) (y) − L (y)L (1) (y) m+(1 n+m−1 m n+m ) ] (1) (1) + (n + m + 1) Lm − 1(y)L n+m+1 (y) − L m (y)Ln+m (y) (3.68 )
and (here, y ≡ yn,s)
c = ------------2--------------yne− y n,m,s [2m (n + m + 1) + n + 1]w2 [ (n) 2 (n) (n+1) (2n − y)Lm (y ) + 4(n + m − y)L m (y)Lm −1 (y)− ( )] 2(n + m ) L (nm)(y)L (nm+−12)(y) + L(mn+−11)(y )L(mn−)1(y) (3.69 )
(with the convention (n) L m ≡ 0 [m < 0]). Now the curvature coefficient is simply
[ ] ∑ 1- c = c0,sZ0,s + 2cn,m,sZn,m,s . (3.70 ) s>0

3.2.3.2 Averaging with flat modes
In the case of an ideally flat mode (crude model) of radius b, we get

2 b4 V[r ] = --- (3.71 ) 12
2J1(ζ0,sb∕a ) < J0(ζ0,sr∕a) >= ------------ (3.72 ) ζ0,sb∕a
and
2 2 2 a-- < J0(ζ0,sr∕a )r > − < r > < J0(ζ0,sr∕a) > = ζ2s [4J0(ζsb∕a)+ (ζsb∕a − 8a∕bζs)J1(ζsb∕a)]. (3.73 )
The result for the curvature is
c = ∑ c(F)Z (3.74 ) s 0,s s>0
with
[ ( ) ( ) ( ) ] (F) 12a2 ζ0,sb ζ0,sb 8a ζ0,sb cs = -4-2- 4J0 ----- + -----− ----- J1 ----- . (3.75 ) b ζ0,s a a ζ0,sb a

3.2.4 Coupling losses

The differences in thermal lensing between axisymmetric and nonaxisymmetric high-order beams are very small, because diffusion of heat rapidly produces a quasihomogeneous temperature with respect to the polar angle. This is why we now restrict the discussions to axisymmetric modes. Existence of a thermal lens Z (r) causes a mismatching of the beam, which has passed through the lens, with the ideal one. The amplitude coupling coefficient is given by the Hermitian scalar product

⟨Ψ, eikZΨ ⟩ =< eikZ >, (3.76 )
where the average < ...> is weighted, as usual, by the normalized intensity of the readout beam. Mismatching results in turn in coupling losses due to the distortion of the wavefront of the transmitted beam. These distortions have been summarized by a spurious radius of curvature for which we have given formulas in the preceding paragraph (??). But it is clear from the shape of the lenses that the distortion is not parabolic. In general, there is some departure of the wavefront from a parabola. Thus, we have two contributions to coupling losses: a harmonic part (by reference to the harmonic oscillator potential) and a nonharmonic part. It is possible to compare the two contributions. For the parabolic or harmonic part, we have the following coupling coefficient γ for a spurious curvature radius R:
∫ a 2 γ = 2π eikr ∕2R |Ψ (r)|2rdr. (3.77 ) 0
In the case of LG modes, we can be more specific:
m! 4 ∫ a 2 2 2 γn,m = ------------ eikr ∕2R(2r2∕w2 )nL (mn)(2r2∕w2 )2e−2r∕w rdr. (3.78 ) (n + m )!w2 0
Assuming negligible clipping losses, we can replace the upper integration bound by + ∞ and take:
∫ ∞ γn,m = ---m!---- xnL (n)(x)2e−(1− iF)xdx, (3.79 ) (n + m )! 0 m
(with 2 F ≡ πw ∕2λR) so that we obtain
m ( ) ( ) -------1-------∑ m n + m 2 s γn,m = (1 − iF )2m+n+1 s s (− F ) . (3.80 ) s=0
In the case of a flat mode, the integral is trivial and we get:
|γflat|2 = sinc(πb2∕2 λR )2, (3.81 )
while the power coupling losses are simply
2 L = 1 − |γ| . (3.82 )

In Figure 20View Image, we have plotted the evolution of the coupling losses versus the dissipated power on the coating of a mirror. The solid and dashed curves correspond respectively to the total losses by a numerical integration of Equation (3.76View Equation) with the overall thermal lens and to harmonic losses. We see that all modes have almost only harmonic losses for weak dissipated losses (roughly below 100 mW). The anharmonicity appears soon for the LG00 mode of width 2 cm, and for a fraction of W, nonharmonic losses are significant. For Ex2 and Ex3, we see that the coupling losses are weaker and practically harmonic. The curve for LG55 is practically identical to the curve of the flat mode, which is why we did not plot it.

View Image

Figure 20: Coupling losses as functions of the dissipated power on the coating. Solid line: total losses, numerical integration of Equation (3.76View Equation). Dashed line: harmonic losses after Equation (3.80View Equation).

For weak dissipated power, in the case of LG modes we use the lowest-order approximation of Equation (3.80View Equation), so that we get

( πw2 )2 Ln,m = [2m (m + n + 1) + n + 1] ----- + 𝒪 (R −4). (3.83 ) 2λR
We note that, owing to Equation (3.56View Equation), this is nothing but a variance of (weighted by the intensity profile)
( )2 [ 2] Ln,m = -π-- V [r2] = V πr-- . (3.84 ) λR λR
Now for the flat mode, we get
1( πb2 )2 L flat = -- ----- + 𝒪 (R −4), (3.85 ) 3 2λR
which is again
[ ] πr2- Lflat = V λR . (3.86 )
NB: The curvature radii are expressed in m.W, the losses in W–2.

3.2.5 Numerical results

The thermal lensing is almost identical for 1 W coating or bulk absorption. In Figure 21View Image, we have plotted the thermal lens profile and the best fit paraboloid Zˆ(r) for the case of heating by internal absorption.

View Image

Figure 21: Thermal lens, heating by 1 W bulk absorption. The dashed line is the nearest paraboloid ˆ Z (r) (in the sense of least squares, weighted by the normalized beam intensity).

Some numerical results can be seen in Table 2. The losses are computed from the parabolic approximation [see Equations (3.83View Equation) and (3.85View Equation)] valid for weak dissipated power.


Table 2: Thermal lensing from coating and bulk absorption (abs.)
results (coating abs.) LG0,0 w = 2 cm Flat b = 9.1 cm LG5,5 w = 3.5 cm
curvature radius 328 mW 9,682 mW 27,396 mW
piston 3.23 μm/W 1.43 μm/W 1.08 μm/W
coupling losses 3.24 /W2 0.53 /W2 0.51 /W2
results (bulk abs.) LG 0,0 w = 2 cm Flat b = 9.1 cm LG 5,5 w = 3.5 cm
curvature radius 317 mW 9,164 mW 25,926 mW
piston 3.39 μm/W 1.52 μm/W 1.15 μm/W
coupling losses 3.47 /W2 0.59 /W2 0.56 /W2

Table 3 contains results for some LG modes having w parameters tuned for 1 ppm clipping losses on a 35 cm diameter mirror.


Table 3: Thermal lensing curvature radii (Rc) for LG modes having 1 ppm clipping losses, and associated coupling losses (L) in the weak power approximation [Equation (3.83View Equation)] (mirror diameter: 35 cm)
order (p,q) w [cm] Rc [mW] (coat. abs.) L [W2] Rc [mW] (bulk abs.) L [W2]
(0,0) 6.65 4,400 2.20 4,184 2.43
(0,1) 5.56 8,566 1.42 8,139 1.57
(1,0) 6.06 8,130 0.89 7,713 0.99
(0,2) 4.93 12,113 1.14 11,497 1.27
(1,1) 5.23 11,608 0.97 11,006 1.08
(2,0) 5.65 11,414 0.51 10,822 0.57
(0,3) 4.49 14,870 1.00 14,106 1.11
(1,2) 4.70 14,430 0.92 13,677 1.02
(2,1) 4.97 14,499 0.70 13,736 0.78
(3,0) 5.35 14,484 0.34 13,729 0.38
(0,4) 4.17 17,219 0.91 16,328 1.01
(1,3) 4.32 16,731 0.87 15,855 0.97
(2,2) 4.52 16,889 0.73 15,997 0.82
(3,1) 4.76 17,237 0.53 16,324 0.59
(4,0) 5.11 17,368 0.25 16,462 0.27
(0,5) 3.91 19,117 0.85 18,123 0.95
(1,4) 4.03 18,686 0.82 17,705 0.92
(2,3) 4.18 18,787 0.74 17,792 0.82
(3,2) 4.36 19,204 0.60 18,182 0.67
(4,1) 4.58 19,790 0.42 18,738 0.46
(5,0) 4.91 20,104 0.19 19,055 0.21

The difference between the flat beam and the mesa beam, regarding thermal lensing, is shown in Figure 22View Image. One sees that using the crude flat beam yields a small overestimation of the lensing.

View Image

Figure 22: Thermal lens for 1 W absorbed from the mesa mode (solid line) and the flat mode (dashed line)

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