Unified theoretical model for calculating laser-induced wavefront distortion in optical materials

Luis C. Malacarne; Nelson G. C. Astrath; Mauro L. Baesso

doi:10.1364/JOSAB.29.001772

Journal of the Optical Society of America B
Vol. 29,
Issue 7,
pp. 1772-1777
(2012)
•https://doi.org/10.1364/JOSAB.29.001772

Unified theoretical model for calculating laser-induced wavefront distortion in optical materials

Luis C. Malacarne, Nelson G. C. Astrath, and Mauro L. Baesso

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Luis C. Malacarne, Nelson G. C. Astrath, and Mauro L. Baesso, "Unified theoretical model for calculating laser-induced wavefront distortion in optical materials," J. Opt. Soc. Am. B 29, 1772-1777 (2012)

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Related Topics
Table of Contents Category
- Optics at Surfaces
Optics & Photonics Topics
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The topics in this list come from the Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Surfaces (240.6700)
- Photothermal effects (350.5340)
- Thermal lensing (350.6830)

About this Article
History
- Original Manuscript: April 3, 2012
- Revised Manuscript: May 15, 2012
- Manuscript Accepted: May 16, 2012
- Published: June 25, 2012
Virtual Issues
July 20, 2012 Spotlight on Optics

Abstract

Laser-induced thermal lens in optical components causes wavefront distortion of the laser beam and may affect performance and stability of optical systems such as high-power lasers. The bulging of the heated area, the temperature dependence of the refractive index, and the photoelastic effects are responsible for phase shifts damaging beam quality. The theoretical background for laser-induced beam distortion is well understood and applies only for axially symmetric thermal loadings, with the assumptions that the stresses follow thin-disk or long-rod approximations. This, in fact, limits the overall applications of this model. In this work, we developed an unified theoretical model for the optical path change in optical materials regardless of its thickness. The modeling is based on the solution of the thermoelastic equation and has a real description of the surface deformation caused in the optical element. In the appropriated limits, as expected, the model retrieves the thin-disk and the long-rod type distributions. Furthermore, we provided time-dependent radial expressions for the temperature, surface displacement, and stresses. The theory presented in this paper provides simple analytical tools for designing laser systems, and complements previous work allowing one to access optical distortions of materials ranging from thin-disk to long-rod-like distributions.

Wavefront distortion arising from thermal lens in optical materials poses a major problem in high-power laser systems and has been subject to exhaustive investigation over the past few decades [1–9]. In such optical components, nonuniform heating caused by the laser beam absorption compromises the system’s performance affecting the optical quality of the beam [1,2]. This effect has been mostly investigated regarding phase shift induced in optical windows. Particularly for windows, the wavefront distortion originates from the refractive index gradient thermally induced in addition to the thermoelastic deformation of the windows surface.

The theoretical treatment for this effect has been investigated in great detail [3–6] and its bases relies on the fact that windows, or optical elements, are subject to axially symmetric thermal loading and thermoelastic properties are isotropic in the plane of the windows. In addition, the stresses are constrained to follow either the plane-stress or plane-strain approximations. The plane-stress approximation applies to thin-disk geometries [4], where the windows thickness is much smaller than the windows radius. The plane-strain model, on the other hand, uses the long-rod approximation that satisfies the condition of much larger thickness compared to radius. In the absence of mechanical loading, the optical windows are subjected to cylindrical symmetric radial and azimuthal stresses that are related in a simple manner to the temperature distribution within the material. The phase shifts responsible for the distortion of the beam arise from the change in path length as the heated area deforms, from the temperature dependence of the refractive index, and from thermoelastic effects associated with nonuniform heating pattern. We show here that those assumptions are, in fact, too restrictive and develop a unified analytical theory for the optical path change in optical materials regardless of its thickness. The modeling is based on solution of the thermoelastic equation and has a real description of the surface deformation by the optical element. In the appropriated limits, the model retrieves the thin-disk (plane-stress) and long-rod (plane-strain) type distributions. This generalized model complements previous work [3–6] in the sense that it allows us to access optical distortions of materials ranging from thin-disk to long-rod like distributions, and could have significant impact on designing laser systems and predicting windows degradation of optical materials. Moreover, this effect is the foundation of many photothermal methods for material characterization, and particularly important for the thermal lens spectrometry, where the optical path change is directly linked to the thermo-optical properties of solid and liquid materials [10–12].

For an axial symmetric beam propagating along the $z$ direction, the optical path is defined as

S (r, t) = \int_{path} n (r, z, t) d z .

We consider here a low optical absorption material of thickness

l_{0}

(path length) and refractive index

n^{s} (r, z, t)

centered in

z = 0

, as shown in Fig. 1. The material is surrounded by a nonabsorbing fluid medium of refractive index

n^{f} (r, z, t)

. The optical path length can be then written as

S (r, t) = 2 \int_{0}^{b (r, t)} n^{s} (r, z, t) d z + 2 \int_{b (r, t)}^{z} n^{f} (r, \bar{z}, t) d \bar{z},

where

b (r, t) = (l_{0} + Δ l) / 2

and

Δ l = 2 | u_{z} (r + u_{r}, l_{0} / 2, t) | \approx 2 | u_{z} (r, l_{0} / 2, t) | .

u_{i}

are the laser-induced thermoelastic displacement components [13,14]. We define

n (r, z, t) = n_{0} + Δ n

, and with regard to the change in index,

Δ n

, we must consider not only the temperature dependence (

Δ n_{th}

) but also the stress induced thermoelastic effect (

Δ n_{st}

) as [4]

Δ n = Δ n_{th} + Δ n_{st} .

Note that

u_{z} ≪ l_{0}

. Using the linear transformation

z \to (z - l_{0} / 2)

, the optical path length results in five contributions:

S (r, t) = S_{0} + S_{th}^{f} (r, t) + S_{th}^{s} (r, t) + S_{st} (r, t) + S_{\exp} (r, t),

where

S_{0}

is the optical path before heating;

S_{th}^{s, f} (r, t)

are the contributions from temperature index variations caused in the sample (

s

) and in the fluid (

f

);

S_{st} (r, t)

is the contribution from thermal stress; and

S_{\exp} (r, t)

is the contribution from thermoelastic expansion.

Fig. 1. Schematic diagram of the optical element geometry.

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In a first approximation, the thermal effects on the refractive index can be easily formulated as linearly dependent over the temperature range of interest as

Δ n_{th}^{i} = {(\frac{\partial n^{i}}{\partial T})}_{th} T^{i} (r, z, t) .

{(\partial n^{i} / \partial T)}_{th}

is the thermo-optic coefficient as measured at the reference temperature for the sample (

i = s

) or for the fluid (

i = f

T^{i} (r, z, t)

is the laser-induced temperature rise distribution. In this approximation, we may write

S_{th}^{f} (r, t) = 2 {(\frac{\partial n^{f}}{\partial T})}_{th} \int_{l_{0}}^{z} T^{f} (r, \bar{z}, t) d \bar{z},

and

S_{th}^{s} (r, t) = {(\frac{\partial n^{s}}{\partial T})}_{th} \int_{0}^{l_{0}} T^{s} (r, z, t) d z .

The thermal stress index variation involves considerations relating to the thermoelastic effect and applies only to the sample as

Δ n_{st}^{s} = - \frac{1}{2} {(n_{0}^{s})}^{3} [q_{∥} σ_{r r} + q_{⊥} (σ_{ϕ ϕ} + σ_{z z})],

for plane waves polarized along the radial direction, and

Δ n_{st}^{s} = - \frac{1}{2} {(n_{0}^{s})}^{3} [q_{∥} σ_{ϕ ϕ} + q_{⊥} (σ_{r r} + σ_{z z})]

for azimuthal polarizations.

q_{∥}

and

q_{⊥}

refer to the stress-optic coefficients for stresses applied parallel and perpendicular to the polarization axis, respectively.

σ_{i j}

are the stress tensor components. From Eqs. (9) and (10), we can introduce the symmetric and antisymmetric combinations of the radial and azimuthal aberrations [4], as

S_{st} (r, t) = S_{st}^{+} (r, t) + S_{st}^{-} (r, t)

with

S_{st}^{+} (r, t) = \frac{{(n_{0}^{s})}^{3}}{4} \int_{0}^{l_{0}} [(q_{∥} + q_{⊥}) (σ_{r r} + σ_{ϕ ϕ}) + 2 q_{⊥} σ_{z z}] d z,

S_{st}^{-} (r, t) = \frac{{(n_{0}^{s})}^{3}}{4} \int_{0}^{l_{0}} [(q_{∥} - q_{⊥}) (σ_{r r} - σ_{ϕ ϕ})] d z .

S_{st}^{+} (r, t)

represents the averaged medium’s sensitivity to thermal lens. It combines the laser-induced change in index, and the contribution from bulging and photoelastic effect for the two polarizations.

S_{st}^{-} (r, t)

contributes only if the medium is stress-birefringent.

Finally, the contribution originated from the surface deformation yields

S_{\exp} (r, t) = (n_{0}^{s} - n_{0}^{f}) {Δ l (r, z, t) |}_{z = 0} .

Terms of second order proportional to the product $Δ n Δ l$ are neglected here. The above expressions show us that the optical path change induced by the laser beam is a result of thermal, stress, and strain effects. This complex problem could be simplified under approximative assumptions. For instance, if the beam radius is smaller than the radial dimension of the sample, for a relative short exposure time $t$ , the temperature at the edges of the sample can be assumed constant. Moreover, if air is used as fluid, which is basically the case for optical windows in laser systems, $n_{0}^{f} \approx 1$ , and the approximation of no heat flow across the interface material/fluid is valid [15]. Thus, the fluid contribution $S_{th}^{f} (r, t)$ can be safely neglected—and will be omitted from this point.

We consider the case of a continuous wave incident beam with a Gaussian ${TEM}_{00}$ intensity profile. It is assumed that the attenuation of light intensity along the material thickness can be neglected—low optical absorption approximation. Thus, the temperature rise distribution with no axial heat flow to the surroundings is given by the solution of the following heat conduction differential equation [16]:

\frac{\partial}{\partial t} T (r, z, t) - \frac{k}{c ρ} \nabla^{2} T (r, z, t) = Q_{0} e^{- 2 \frac{r^{2}}{ω^{2}}},

with the initial condition

T (r, z, 0) = 0

and boundary conditions

T (\infty, z, t) = 0

and

\partial T (r, z, t) / {\partial z |}_{z = 0} = 0

c

ρ

, and

k

are specific heat, mass density, and thermal conductivity of the material, respectively.

Q_{0} = 2 P_{0} (1 - R) A / (ρ c π ω^{2})

A

is the optical absorption coefficient at the laser beam wavelength,

R

P_{0}

, and

ω

are the material’s surface reflectivity, the laser power, and its radius at the sample surface, respectively.

The thermoelastic equation for the stress and surface displacement caused by a laser-induced nonuniform temperature distribution, in the quasi-static approximation, can be expressed in cylindrical coordinates by introducing the scalar displacement potential $Ψ$ and the Love function $ψ$ following by the Poisson’s equation [17],

\nabla^{2} Ψ (r, z, t) = \frac{1 + ν}{1 - ν} α_{T} T (r, z, t),

and the Biharmonic equation,

\nabla^{2} \nabla^{2} ψ (r, z, t) = 0 .

α_{T}

is the linear thermal expansion coefficient, and

ν

the Poisson’s ratio.

The components of the displacement vector ( $u_{z}$ ) and the stress ( $σ_{i i}$ ) are obtained from $Ψ (r, z, t)$ and $ψ (r, z, t)$ by the relations

u_{z} = \frac{\partial Ψ}{\partial z} + \frac{1}{1 - 2 ν} [2 (1 - ν) \nabla^{2} - \frac{\partial^{2}}{\partial z^{2}}] ψ

and

σ_{z z} = - \frac{E}{1 + ν} (\frac{\partial^{2}}{\partial r^{2}} + \frac{1}{r} \frac{\partial}{\partial r}) Ψ + \frac{E}{1 - ν - 2 ν^{2}} [(2 - ν) \nabla^{2} - \frac{\partial^{2}}{\partial z^{2}}] \frac{\partial ψ}{\partial r},

σ_{r r} = - \frac{E}{1 + ν} (\frac{\partial^{2}}{\partial z^{2}} + \frac{1}{r} \frac{\partial}{\partial r}) Ψ + \frac{E}{1 - ν - 2 ν^{2}} [ν \nabla^{2} - \frac{\partial^{2}}{\partial r^{2}}] \frac{\partial ψ}{\partial z},

σ_{ϕ ϕ} = - \frac{E}{1 + ν} (\frac{\partial^{2}}{\partial r^{2}} + \frac{\partial^{2}}{\partial z^{2}}) Ψ + \frac{E}{1 - ν - 2 ν^{2}} [ν \nabla^{2} - \frac{1}{r} \frac{\partial}{\partial r}] \frac{\partial ψ}{\partial z},

where

E

is the Young’s modulus and the Laplacian

\nabla^{2} = \partial^{2} / \partial r^{2} + r^{- 1} \partial / \partial r + \partial^{2} / \partial z^{2}

. Eqs. (16) and (17) are solved using free stress boundary conditions,

σ_{r z} = σ_{z z} = 0

, at the surfaces.

Recently [13,15,18], we have obtained semianalytical expressions for the temperature and thermoelastic potentials. In those solutions, we were assuming that the radial dimension of the sample was large enough for the temperature to be considered constant at the edges of the sample, that is, $T (\infty, z, t) = 0$ . In fact, the solutions were validated by all numerical finite elemental analysis modeling with real boundary conditions [15,18]. Using this approximation, and assuming no azimuthal dependence for the temperature distribution, we can write the temperature rise as

T (r, t) = \int_{0}^{\infty} T (α, t) J_{0} (α r) α d α .

J_{n} (ξ)

is the Bessel function of the first kind, and

T (α, t) = Q_{0} ω^{2} e^{- \frac{1}{8} ω^{2} α^{2}} (\frac{1 - e^{- D t α^{2}}}{4 D α^{2}})

is the Hankel transform of the

r

-dependent temperature (

α

space).

D = k / (ρ c)

represents the thermal diffusivity of the sample. The scalar displacement potential and the Love function are

Ψ (r, z, t) = - \frac{1 + ν}{1 - ν} α_{T} \int_{0}^{\infty} T (α, t) α^{- 1} J_{0} (α r) d α,

and

ψ (r, z, t) = α_{T} \int_{0}^{\infty} f (α, z) α^{- 2} T (α, t) J_{0} (α r) d α,

with

f (α, z) = \frac{ν - 1 + 2 ν^{2}}{l_{0} α + \sinh (l_{0} α)} \frac{e^{- (l_{0} + z) α}}{2 (ν - 1)} [e^{l_{0} α} (l_{0} α - z α - 2 ν) + e^{(l_{0} + 2 z) α} (- l_{0} α + z α - 2 ν) + e^{2 z α} (- z α + 2 ν) + e^{2 l_{0} α} (z α + 2 ν)] .

The above expressions allow us to write the displacement and stresses in a simple manner as

Δ l = 4 (1 + ν) α_{T} \int_{0}^{\infty} h (l_{0}, α) T (α, t) J_{0} (r α) d α,

σ_{r} = \frac{E α_{T}}{(ν - 1)} \int_{0}^{\infty} (\frac{1}{r} - \frac{4 ν h (l_{0}, α)}{r l_{0} α}) J_{1} (r α) T (α, t) d α,

σ_{z} = \frac{E α_{T}}{(ν - 1)} \int_{0}^{\infty} (1 - \frac{4 h (l_{0}, α)}{l_{0} α}) J_{0} (r α) T (α, t) α d α,

and

σ_{ϕ} = \frac{E α_{T}}{(ν - 1)} \int_{0}^{\infty} (1 - \frac{4 ν h (l_{0}, α)}{l_{0} α}) J_{0} (r α) T (α, t) α d α - \frac{E α_{T}}{(ν - 1)} \int_{0}^{\infty} (\frac{1}{r} - \frac{4 ν h (l_{0}, α)}{r l_{0} α}) J_{1} (r α) T (α, t) d α,

where

h (l_{0}, α) = \frac{\cosh (l_{0} α) - 1}{\sinh (l_{0} α) + l_{0} α},

and

σ_{i} = \frac{1}{l_{0}} \int_{0}^{l_{0}} σ_{i i} (r, z, t) d z .

The symmetric and antisymmetric combinations of the radial and azimuthal aberrations to the optical path length lead to

S_{st}^{+} (r, t) = ϑ \int_{0}^{\infty} (q_{∥} + 3 q_{⊥}) J_{0} (r α) T (α, t) α d α - ϑ \int_{0}^{\infty} \frac{[q_{∥} ν + q_{⊥} (2 + ν)] h (l_{0}, α)}{l_{0} / 4} J_{0} (r α) T (α, t) d α,

and

S_{st}^{-} (r, t) = ϑ \int_{0}^{\infty} (q_{∥} - q_{⊥}) [α - \frac{ν h (l_{0}, α)}{l_{0} / 4}] J_{2} (r α) T (α, t) d α,

with

ϑ = {(n_{0}^{s})}^{3} E α_{T} l_{0} / (4 - 4 ν)

. The stress contributions to the refractive index variation can be neglected for some optical materials because it is substantially smaller than the thermal contribution [4]. However, for materials which

{(\partial n^{s} / \partial T)}_{th} < 0

, the thermal expansion contribution can be suppressed and the stress contribution becomes relevant. As for

S_{st}^{-} (r, t)

, it contributes only if the material is stress-birefringent.

Using Eqs. (14) and (27), the thermal expansion contribution can be written as

S_{\exp} (r, t) = 4 (n_{0}^{s} - 1) (1 + ν) α_{T} \int_{0}^{\infty} h (l_{0}, α) T (α, t) J_{0} (r α) d α .

Finally, with Eqs. (8), (22), (33), and (35), we can write $S (r, t)$ for free stress-birefringent material, that is $q_{∥} \approx q_{⊥}$ , as

S (r, t) = \int_{0}^{\infty} l_{0} {{(\frac{\partial n^{s}}{\partial T})}_{th} + \frac{{(n_{0}^{s})}^{3} E α_{T}}{4 (1 - ν)} [(q_{∥} + 3 q_{⊥}) - \frac{4 [q_{∥} ν + q_{⊥} (2 + ν)] h (l_{0}, α)}{l_{0} α}] + \frac{4 (n_{0}^{s} - 1) (1 + ν) α_{T} h (l_{0}, α)}{l_{0} α}} T (α, t) J_{0} (r α) α d α .

The general solutions presented here can be simplified if one take the following limits. For

l_{0} \to 0

, the expression for

S (r, t)

recovers the plane-stress approximation,

S^{0} (r, t) = l_{0} χ^{0} \int_{0}^{\infty} T (α, t) J_{0} (r α) α d α,

and, for

l_{0} \to \infty

, it recovers the plane-strain approximation,

S^{\infty} (r, t) = l_{0} χ^{\infty} \int_{0}^{\infty} T (α, t) J_{0} (r α) α d α,

where

χ^{0} = {(\frac{\partial n^{s}}{\partial T})}_{th} + \frac{{(n_{0}^{s})}^{3} E α_{T}}{4} (q_{∥} + q_{⊥}) + (n_{0}^{s} - 1) (1 + ν) α_{T}

and

χ^{\infty} = {(\frac{\partial n^{s}}{\partial T})}_{th} + \frac{{(n_{0}^{s})}^{3} E α_{T}}{4 (1 - ν)} (q_{∥} + 3 q_{⊥}) .

Eqs. (37) and (38) describe the well-known results for thin-disk and long-rod geometries [3,4].

To illustrate and support the analytical solutions, Eqs. (36) to (40), we perform simulations considering physical parameters of a calcium fluoride windows ( ${CaF}_{2}$ ) [6]. Figure 2 shows the normalized temperature, $T (r, t) / T (0, t)$ , and surface displacement, $u_{z} (r, t) / u_{z} (0, t)$ , profiles for windows with different thicknesses at an exposure time of $t = 0.2 s$ and $ω = 50 μm$ . The temperature profile has a sharp shape compared with the surface displacement for thick windows, which is caused by the mechanical inertia of the surface to expand upon temperature change. Both the temperature and the surface deformation profiles become similar only for very small thickness. This, in fact, validates the thin-disk, or the plane-stress, approximation. However, it is important to note that approximating the surface deformation by the temperature profile is reasonably acceptable only for very thin samples.

Fig. 2. Radial temperature (open circles) and surface displacement (continuous lines) profiles for the ${CaF}_{2}$ windows glass.

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The limit cases, plane-stress and plane-strain, are better explored by analyzing the optical path change. Figure 3 shows the optical path $S (r, t)$ scaled by a factor $l_{0} Q_{0}$ for $r = ω$ and different thicknesses. $S (r, t)$ was evaluated by Eq. (36) and the limit cases by Eqs. (37) and (38), respectively. The results illustrate well the total phase shift moving from a plane-stress approximation, for small $l_{0}$ , to the plane-strain as $l_{0}$ increases.

Fig. 3. Time dependence of the optical path $S (r, t) / (l_{0} Q_{0})$ for $ω = 50 μm$ using the physical parameters of ${CaF}_{2}$ glass windows [6]. Solid lines were calculated using Eq. (36) and Eqs. (37) and (38) for plane-stress (open squares) and plane-strain (open circles) approximations, respectively.

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Figure 4 shows the time and radial evolution of the optical path $S (r, t)$ for a windows of thickness $l_{0} = 2.0 mm$ . The radial dependence of the optical path length induces aberration in the wavefront propagation. The corresponding phase shift is given by

ϕ (r, t) = \frac{2 π}{λ} [S (r, t) - S (0, t)],

where

λ

is the laser wavelength. In fact, Eq. (41) provides an expression for analyzing the thermal lens effect caused by a laser-induced optical element. This effect is important in high-power laser systems and has numerous applications in photothermal effect based techniques [10–12,15] for material characterization, such as in the thermal lens spectrometry. As for high-power laser systems, a criterion normally used to characterize the laser beam distortion is the Strehl ratio—the ratio of the intensity at the point of maximum intensity in the observation plane with and without phase aberration. For instance, in an untruncated Gaussian illumination with beam radius

ω

, it is given by [19]

S_{g} = \frac{{| \int_{0}^{2 π} \int_{0}^{\infty} e^{- r^{2} / ω^{2}} e^{i ϕ (r, t)} r d r d ϕ |}^{2}}{{| \int_{0}^{2 π} \int_{0}^{\infty} e^{- r^{2} / ω^{2}} r d r d ϕ |}^{2}} .

In a similar way, in the thermal lens spectrometry, the intensity

I (t)

of the center beam at the far field detector plane is given by [10]

I (t) = \frac{{| \int_{0}^{\infty} \exp [(1 + i V) g - i ϕ (g, t)] d g |}^{2}}{{| \int_{0}^{\infty} \exp [(1 + i V) g] d g |}^{2}},

in which

V

is a geometric parameter from the experimental setup,

g = {(r / ω_{p})}^{2}

, and

ω_{p}

the probe beam radius in the mode mismatched configuration [10–12].

Fig. 4. Time and radial evolution to the optical path $S (r, t) / (l_{0} Q_{0}) (\times 10^{- 8})$ for the ${CaF}_{2}$ [6] glass windows for $l_{0} = 2 mm$ .

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In conclusion, we have developed a unified analytical theory for the wavefront distortion caused by laser-induced thermal lens. The theory applies not only to plane-stress and plane-strain type distributions, but also for stress and strain contributions within these limit cases. Radial and time-dependent expressions for the temperature, surface displacement, and stresses have been obtained. This generalized model could have significant impact on designing laser systems and predicting laser-induced windows degradation in optical materials. Furthermore, the optical path description presented in this work has direct application in thermal lens spectrometry, correlating optical path change to thermo-optical properties of solid materials.

ACKNOWLEDGMENTS

The authors are thankful to the Brazilian agencies CAPES, CNPq, and Fundação Araucária for the financial support of this work.

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12. N. G. C. Astrath, F. B. G. Astrath, J. Shen, J. Zhou, P. R. B. Pedreira, L. C. Malacarne, A. C. Bento, and M. L. Baesso, “Top-hat cw-laser-induced time-resolved mode-mismatched thermal lens spectroscopy for quantitative analysis of low-absorption materials,” Opt. Lett. 33, 1464–1466 (2008). [CrossRef]

13. N. G. C. Astrath, L. C. Malacarne, P. R. B. Pedreira, A. C. Bento, M. L. Baesso, and J. Shen, “Time-resolved thermal mirror for nanoscale surface displacement detection in low absorbing solids,” Appl. Phys. Lett. 91, 191908 (2007). [CrossRef]

14. F. Sato, L. C. Malacarne, P. R. B. Pedreira, M. P. Belancon, R. S. Mendes, M. L. Baesso, N. G. C. Astrath, and J. Shen, “Time-resolved thermal mirror method: A theoretical study,” J. Appl. Phys. 104, 053520 (2008). [CrossRef]

15. L. C. Malacarne, N. G. C. Astrath, G. V. B. Lukasievcz, E. K. Lenzi, M. L. Baesso, and S. E. Bialkowski, “Time-resolved thermal lens and thermal mirror spectroscopy with sample-fluid heat coupling: A complete model for material characterization,” Appl. Spectrosc. 65, 99–104 (2011). [CrossRef]

16. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids (Clarendon Press, 1959), Vol. 1, p. 78.

17. W. Nowacki, Thermoelasticity (Pergamon, 1982), Vol. 3, p. 11.

18. N. G. C. Astrath, L. C. Malacarne, V. S. Zanuto, M. P. Belancon, R. S. Mendes, M. L. Baesso, and C. Jacinto, “Finite-size effect on the surface deformation thermal mirror method,” J. Opt. Soc. Am. B 28, 1735–1739 (2011). [CrossRef]

19. R. Herloski, “Strehl ratio for untruncated aberrated Gaussian beams,” J. Opt. Soc. Am. A 2, 1027–1030 (1985). [CrossRef]

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References

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Year
Author
Publication

C. Zhao, J. Degallaix, L. Ju, Y. Fan, D. G. Blair, B. J. J. Slagmolen, M. B. Gray, C. M. Mow Lowry, D. E. McClelland, D. J. Hosken, D. Mudge, A. Brooks, J. Munch, P. J. Veitch, M. A. Barton, and G. Billingsley, “Compensation of strong thermal lensing in high-optical-power cavities,” Phys. Rev. Lett. 96, 231101(2006).
[Crossref]
W. Winkler, K. Danzmann, A. Rüdiger, and R. Schilling, “Heating bu optical absorption and the performance of interferometric gravitational-wave detectors,” Phys. Rev. A 44, 7022–7036 (1991).
[Crossref]
M. Sparks, “Optical distortion by heated windows in high-power laser systems,” J. Appl. Phys. 42, 5029–5046 (1971).
[Crossref]
C. A. Klein, “Optical distortion coefficients of high-power laser windows,” Opt. Eng. 29, 343–350 (1990).
[Crossref]
C. A. Klein, “Describing beam-aberration effects induced by laser-light transmitting components: a short account of Raytheon’s contribution,” Proc. SPIE 4376, 24–34 (2001).
[Crossref]
C. A. Klein, “Analytical stress modeling of high-energy laser windows: Application to fusion-cast calcium fluoride windows,” J. Appl. Phys. 98, 043103 (2005).
[Crossref]
L. B. Glebov, “Intrinsic laser-induced breakdown of silicate glasses,” Proc. SPIE 4679, 321–331 (2002).
[Crossref]
Y. Peng, Z. Sheng, H. Zhang, and X. Fan, “Influence of thermal deformations of the output windows of high-power laser systems on beam characteristics,” Appl. Opt. 43, 6465–6472 (2004).
[Crossref]
W. Koechner and M. Bass, Solid-State Lasers: A Graduate Text (Springer, 2003).
J. Shen, M. L. Baesso, and R. D. Snook, “Three-dimensional model for cw laser-induced mode-mismatched dual-beam thermal lens spectrometry and time-resolved measurements of thin-film samples,” J. Appl. Phys. 75, 3738–3748 (1994).
[Crossref]
N. G. C. Astrath, J. H. Rohling, A. N. Medina, A. C. Bento, M. L. Baesso, C. Jacinto, T. Catunda, S. M. Lima, F. G. Gandra, M. J. V. Bell, and V. Anjos, “Time-resolved thermal lens measurements of the thermo-optical properties of glasses at low temperature down to 20 K,” Phys. Rev. B 71, 214202 (2005).
[Crossref]
N. G. C. Astrath, F. B. G. Astrath, J. Shen, J. Zhou, P. R. B. Pedreira, L. C. Malacarne, A. C. Bento, and M. L. Baesso, “Top-hat cw-laser-induced time-resolved mode-mismatched thermal lens spectroscopy for quantitative analysis of low-absorption materials,” Opt. Lett. 33, 1464–1466 (2008).
[Crossref]
N. G. C. Astrath, L. C. Malacarne, P. R. B. Pedreira, A. C. Bento, M. L. Baesso, and J. Shen, “Time-resolved thermal mirror for nanoscale surface displacement detection in low absorbing solids,” Appl. Phys. Lett. 91, 191908 (2007).
[Crossref]
F. Sato, L. C. Malacarne, P. R. B. Pedreira, M. P. Belancon, R. S. Mendes, M. L. Baesso, N. G. C. Astrath, and J. Shen, “Time-resolved thermal mirror method: A theoretical study,” J. Appl. Phys. 104, 053520 (2008).
[Crossref]
L. C. Malacarne, N. G. C. Astrath, G. V. B. Lukasievcz, E. K. Lenzi, M. L. Baesso, and S. E. Bialkowski, “Time-resolved thermal lens and thermal mirror spectroscopy with sample-fluid heat coupling: A complete model for material characterization,” Appl. Spectrosc. 65, 99–104 (2011).
[Crossref]
H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids (Clarendon Press, 1959), Vol. 1, p. 78.
W. Nowacki, Thermoelasticity (Pergamon, 1982), Vol. 3, p. 11.
N. G. C. Astrath, L. C. Malacarne, V. S. Zanuto, M. P. Belancon, R. S. Mendes, M. L. Baesso, and C. Jacinto, “Finite-size effect on the surface deformation thermal mirror method,” J. Opt. Soc. Am. B 28, 1735–1739 (2011).
[Crossref]
R. Herloski, “Strehl ratio for untruncated aberrated Gaussian beams,” J. Opt. Soc. Am. A 2, 1027–1030 (1985).
[Crossref]

2011 (2)

L. C. Malacarne, N. G. C. Astrath, G. V. B. Lukasievcz, E. K. Lenzi, M. L. Baesso, and S. E. Bialkowski, “Time-resolved thermal lens and thermal mirror spectroscopy with sample-fluid heat coupling: A complete model for material characterization,” Appl. Spectrosc. 65, 99–104 (2011).
[Crossref]

N. G. C. Astrath, L. C. Malacarne, V. S. Zanuto, M. P. Belancon, R. S. Mendes, M. L. Baesso, and C. Jacinto, “Finite-size effect on the surface deformation thermal mirror method,” J. Opt. Soc. Am. B 28, 1735–1739 (2011).
[Crossref]

2008 (2)

N. G. C. Astrath, F. B. G. Astrath, J. Shen, J. Zhou, P. R. B. Pedreira, L. C. Malacarne, A. C. Bento, and M. L. Baesso, “Top-hat cw-laser-induced time-resolved mode-mismatched thermal lens spectroscopy for quantitative analysis of low-absorption materials,” Opt. Lett. 33, 1464–1466 (2008).
[Crossref]

F. Sato, L. C. Malacarne, P. R. B. Pedreira, M. P. Belancon, R. S. Mendes, M. L. Baesso, N. G. C. Astrath, and J. Shen, “Time-resolved thermal mirror method: A theoretical study,” J. Appl. Phys. 104, 053520 (2008).
[Crossref]

2007 (1)

N. G. C. Astrath, L. C. Malacarne, P. R. B. Pedreira, A. C. Bento, M. L. Baesso, and J. Shen, “Time-resolved thermal mirror for nanoscale surface displacement detection in low absorbing solids,” Appl. Phys. Lett. 91, 191908 (2007).
[Crossref]

2006 (1)

C. Zhao, J. Degallaix, L. Ju, Y. Fan, D. G. Blair, B. J. J. Slagmolen, M. B. Gray, C. M. Mow Lowry, D. E. McClelland, D. J. Hosken, D. Mudge, A. Brooks, J. Munch, P. J. Veitch, M. A. Barton, and G. Billingsley, “Compensation of strong thermal lensing in high-optical-power cavities,” Phys. Rev. Lett. 96, 231101(2006).
[Crossref]

2005 (2)

C. A. Klein, “Analytical stress modeling of high-energy laser windows: Application to fusion-cast calcium fluoride windows,” J. Appl. Phys. 98, 043103 (2005).
[Crossref]

N. G. C. Astrath, J. H. Rohling, A. N. Medina, A. C. Bento, M. L. Baesso, C. Jacinto, T. Catunda, S. M. Lima, F. G. Gandra, M. J. V. Bell, and V. Anjos, “Time-resolved thermal lens measurements of the thermo-optical properties of glasses at low temperature down to 20 K,” Phys. Rev. B 71, 214202 (2005).
[Crossref]

2004 (1)

Y. Peng, Z. Sheng, H. Zhang, and X. Fan, “Influence of thermal deformations of the output windows of high-power laser systems on beam characteristics,” Appl. Opt. 43, 6465–6472 (2004).
[Crossref]

2002 (1)

L. B. Glebov, “Intrinsic laser-induced breakdown of silicate glasses,” Proc. SPIE 4679, 321–331 (2002).
[Crossref]

2001 (1)

C. A. Klein, “Describing beam-aberration effects induced by laser-light transmitting components: a short account of Raytheon’s contribution,” Proc. SPIE 4376, 24–34 (2001).
[Crossref]

1994 (1)

J. Shen, M. L. Baesso, and R. D. Snook, “Three-dimensional model for cw laser-induced mode-mismatched dual-beam thermal lens spectrometry and time-resolved measurements of thin-film samples,” J. Appl. Phys. 75, 3738–3748 (1994).
[Crossref]

1991 (1)

W. Winkler, K. Danzmann, A. Rüdiger, and R. Schilling, “Heating bu optical absorption and the performance of interferometric gravitational-wave detectors,” Phys. Rev. A 44, 7022–7036 (1991).
[Crossref]

1990 (1)

C. A. Klein, “Optical distortion coefficients of high-power laser windows,” Opt. Eng. 29, 343–350 (1990).
[Crossref]

1985 (1)

R. Herloski, “Strehl ratio for untruncated aberrated Gaussian beams,” J. Opt. Soc. Am. A 2, 1027–1030 (1985).
[Crossref]

1971 (1)

M. Sparks, “Optical distortion by heated windows in high-power laser systems,” J. Appl. Phys. 42, 5029–5046 (1971).
[Crossref]

Anjos, V.

Astrath, F. B. G.

Astrath, N. G. C.

Baesso, M. L.

Barton, M. A.

Bass, M.

W. Koechner and M. Bass, Solid-State Lasers: A Graduate Text (Springer, 2003).

Belancon, M. P.

Bell, M. J. V.

Bento, A. C.

Bialkowski, S. E.

Billingsley, G.

Blair, D. G.

Brooks, A.

Carslaw, H. S.

H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids (Clarendon Press, 1959), Vol. 1, p. 78.

Catunda, T.

Danzmann, K.

Degallaix, J.

Fan, X.

Fan, Y.

Gandra, F. G.

Glebov, L. B.

L. B. Glebov, “Intrinsic laser-induced breakdown of silicate glasses,” Proc. SPIE 4679, 321–331 (2002).
[Crossref]

Gray, M. B.

Herloski, R.

R. Herloski, “Strehl ratio for untruncated aberrated Gaussian beams,” J. Opt. Soc. Am. A 2, 1027–1030 (1985).
[Crossref]

Hosken, D. J.

Jacinto, C.

Jaeger, J. C.

H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids (Clarendon Press, 1959), Vol. 1, p. 78.

Ju, L.

Klein, C. A.

C. A. Klein, “Analytical stress modeling of high-energy laser windows: Application to fusion-cast calcium fluoride windows,” J. Appl. Phys. 98, 043103 (2005).
[Crossref]

C. A. Klein, “Describing beam-aberration effects induced by laser-light transmitting components: a short account of Raytheon’s contribution,” Proc. SPIE 4376, 24–34 (2001).
[Crossref]

C. A. Klein, “Optical distortion coefficients of high-power laser windows,” Opt. Eng. 29, 343–350 (1990).
[Crossref]

Koechner, W.

W. Koechner and M. Bass, Solid-State Lasers: A Graduate Text (Springer, 2003).

Lenzi, E. K.

Lima, S. M.

Lukasievcz, G. V. B.

Malacarne, L. C.

McClelland, D. E.

Medina, A. N.

Mendes, R. S.

Mow Lowry, C. M.

Mudge, D.

Munch, J.

Nowacki, W.

W. Nowacki, Thermoelasticity (Pergamon, 1982), Vol. 3, p. 11.

Pedreira, P. R. B.

Peng, Y.

Rohling, J. H.

Rüdiger, A.

Sato, F.

Schilling, R.

Shen, J.

Sheng, Z.

Slagmolen, B. J. J.

Snook, R. D.

Sparks, M.

M. Sparks, “Optical distortion by heated windows in high-power laser systems,” J. Appl. Phys. 42, 5029–5046 (1971).
[Crossref]

Veitch, P. J.

Winkler, W.

Zanuto, V. S.

Zhang, H.

Zhao, C.

Zhou, J.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

Appl. Spectrosc. (1)

J. Appl. Phys. (4)

C. A. Klein, “Analytical stress modeling of high-energy laser windows: Application to fusion-cast calcium fluoride windows,” J. Appl. Phys. 98, 043103 (2005).
[Crossref]

M. Sparks, “Optical distortion by heated windows in high-power laser systems,” J. Appl. Phys. 42, 5029–5046 (1971).
[Crossref]

J. Opt. Soc. Am. A (1)

R. Herloski, “Strehl ratio for untruncated aberrated Gaussian beams,” J. Opt. Soc. Am. A 2, 1027–1030 (1985).
[Crossref]

J. Opt. Soc. Am. B (1)

Opt. Eng. (1)

C. A. Klein, “Optical distortion coefficients of high-power laser windows,” Opt. Eng. 29, 343–350 (1990).
[Crossref]

Opt. Lett. (1)

Phys. Rev. A (1)

Phys. Rev. B (1)

Phys. Rev. Lett. (1)

Proc. SPIE (2)

C. A. Klein, “Describing beam-aberration effects induced by laser-light transmitting components: a short account of Raytheon’s contribution,” Proc. SPIE 4376, 24–34 (2001).
[Crossref]

L. B. Glebov, “Intrinsic laser-induced breakdown of silicate glasses,” Proc. SPIE 4679, 321–331 (2002).
[Crossref]

Other (3)

W. Koechner and M. Bass, Solid-State Lasers: A Graduate Text (Springer, 2003).

H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids (Clarendon Press, 1959), Vol. 1, p. 78.

W. Nowacki, Thermoelasticity (Pergamon, 1982), Vol. 3, p. 11.

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Fig. 1. Schematic diagram of the optical element geometry.

View in Article | Download Full Size | PPT Slide | PDF

Fig. 2. Radial temperature (open circles) and surface displacement (continuous lines) profiles for the

{CaF}_{2}

windows glass.

View in Article | Download Full Size | PPT Slide | PDF

Fig. 3. Time dependence of the optical path

S (r, t) / (l_{0} Q_{0})

for

ω = 50 μm

using the physical parameters of

{CaF}_{2}

glass windows [6]. Solid lines were calculated using Eq. (36) and Eqs. (37) and (38) for plane-stress (open squares) and plane-strain (open circles) approximations, respectively.

View in Article | Download Full Size | PPT Slide | PDF

Fig. 4. Time and radial evolution to the optical path

S (r, t) / (l_{0} Q_{0}) (\times 10^{- 8})

for the

{CaF}_{2}

[6] glass windows for

l_{0} = 2 mm

View in Article | Download Full Size | PPT Slide | PDF

Equations (43)

Equations on this page are rendered with MathJax. Learn more.

S (r, t) = \int_{path} n (r, z, t) d z .

S (r, t) = 2 \int_{0}^{b (r, t)} n^{s} (r, z, t) d z + 2 \int_{b (r, t)}^{z} n^{f} (r, \bar{z}, t) d \bar{z},

Δ l = 2 | u_{z} (r + u_{r}, l_{0} / 2, t) | \approx 2 | u_{z} (r, l_{0} / 2, t) | .

Δ n = Δ n_{th} + Δ n_{st} .

S (r, t) = S_{0} + S_{th}^{f} (r, t) + S_{th}^{s} (r, t) + S_{st} (r, t) + S_{\exp} (r, t),

Δ n_{th}^{i} = {(\frac{\partial n^{i}}{\partial T})}_{th} T^{i} (r, z, t) .

S_{th}^{f} (r, t) = 2 {(\frac{\partial n^{f}}{\partial T})}_{th} \int_{l_{0}}^{z} T^{f} (r, \bar{z}, t) d \bar{z},

S_{th}^{s} (r, t) = {(\frac{\partial n^{s}}{\partial T})}_{th} \int_{0}^{l_{0}} T^{s} (r, z, t) d z .

Δ n_{st}^{s} = - \frac{1}{2} {(n_{0}^{s})}^{3} [q_{∥} σ_{r r} + q_{⊥} (σ_{ϕ ϕ} + σ_{z z})],

Δ n_{st}^{s} = - \frac{1}{2} {(n_{0}^{s})}^{3} [q_{∥} σ_{ϕ ϕ} + q_{⊥} (σ_{r r} + σ_{z z})]

S_{st} (r, t) = S_{st}^{+} (r, t) + S_{st}^{-} (r, t)

S_{st}^{+} (r, t) = \frac{{(n_{0}^{s})}^{3}}{4} \int_{0}^{l_{0}} [(q_{∥} + q_{⊥}) (σ_{r r} + σ_{ϕ ϕ}) + 2 q_{⊥} σ_{z z}] d z,

S_{st}^{-} (r, t) = \frac{{(n_{0}^{s})}^{3}}{4} \int_{0}^{l_{0}} [(q_{∥} - q_{⊥}) (σ_{r r} - σ_{ϕ ϕ})] d z .

S_{\exp} (r, t) = (n_{0}^{s} - n_{0}^{f}) {Δ l (r, z, t) |}_{z = 0} .

\frac{\partial}{\partial t} T (r, z, t) - \frac{k}{c ρ} \nabla^{2} T (r, z, t) = Q_{0} e^{- 2 \frac{r^{2}}{ω^{2}}},

\nabla^{2} Ψ (r, z, t) = \frac{1 + ν}{1 - ν} α_{T} T (r, z, t),

\nabla^{2} \nabla^{2} ψ (r, z, t) = 0 .

u_{z} = \frac{\partial Ψ}{\partial z} + \frac{1}{1 - 2 ν} [2 (1 - ν) \nabla^{2} - \frac{\partial^{2}}{\partial z^{2}}] ψ

σ_{z z} = - \frac{E}{1 + ν} (\frac{\partial^{2}}{\partial r^{2}} + \frac{1}{r} \frac{\partial}{\partial r}) Ψ + \frac{E}{1 - ν - 2 ν^{2}} [(2 - ν) \nabla^{2} - \frac{\partial^{2}}{\partial z^{2}}] \frac{\partial ψ}{\partial r},

σ_{r r} = - \frac{E}{1 + ν} (\frac{\partial^{2}}{\partial z^{2}} + \frac{1}{r} \frac{\partial}{\partial r}) Ψ + \frac{E}{1 - ν - 2 ν^{2}} [ν \nabla^{2} - \frac{\partial^{2}}{\partial r^{2}}] \frac{\partial ψ}{\partial z},

σ_{ϕ ϕ} = - \frac{E}{1 + ν} (\frac{\partial^{2}}{\partial r^{2}} + \frac{\partial^{2}}{\partial z^{2}}) Ψ + \frac{E}{1 - ν - 2 ν^{2}} [ν \nabla^{2} - \frac{1}{r} \frac{\partial}{\partial r}] \frac{\partial ψ}{\partial z},

T (r, t) = \int_{0}^{\infty} T (α, t) J_{0} (α r) α d α .

T (α, t) = Q_{0} ω^{2} e^{- \frac{1}{8} ω^{2} α^{2}} (\frac{1 - e^{- D t α^{2}}}{4 D α^{2}})

Ψ (r, z, t) = - \frac{1 + ν}{1 - ν} α_{T} \int_{0}^{\infty} T (α, t) α^{- 1} J_{0} (α r) d α,

ψ (r, z, t) = α_{T} \int_{0}^{\infty} f (α, z) α^{- 2} T (α, t) J_{0} (α r) d α,

f (α, z) = \frac{ν - 1 + 2 ν^{2}}{l_{0} α + \sinh (l_{0} α)} \frac{e^{- (l_{0} + z) α}}{2 (ν - 1)} [e^{l_{0} α} (l_{0} α - z α - 2 ν) + e^{(l_{0} + 2 z) α} (- l_{0} α + z α - 2 ν) + e^{2 z α} (- z α + 2 ν) + e^{2 l_{0} α} (z α + 2 ν)] .

Δ l = 4 (1 + ν) α_{T} \int_{0}^{\infty} h (l_{0}, α) T (α, t) J_{0} (r α) d α,

σ_{r} = \frac{E α_{T}}{(ν - 1)} \int_{0}^{\infty} (\frac{1}{r} - \frac{4 ν h (l_{0}, α)}{r l_{0} α}) J_{1} (r α) T (α, t) d α,

σ_{z} = \frac{E α_{T}}{(ν - 1)} \int_{0}^{\infty} (1 - \frac{4 h (l_{0}, α)}{l_{0} α}) J_{0} (r α) T (α, t) α d α,

σ_{ϕ} = \frac{E α_{T}}{(ν - 1)} \int_{0}^{\infty} (1 - \frac{4 ν h (l_{0}, α)}{l_{0} α}) J_{0} (r α) T (α, t) α d α - \frac{E α_{T}}{(ν - 1)} \int_{0}^{\infty} (\frac{1}{r} - \frac{4 ν h (l_{0}, α)}{r l_{0} α}) J_{1} (r α) T (α, t) d α,

h (l_{0}, α) = \frac{\cosh (l_{0} α) - 1}{\sinh (l_{0} α) + l_{0} α},

σ_{i} = \frac{1}{l_{0}} \int_{0}^{l_{0}} σ_{i i} (r, z, t) d z .

S_{st}^{+} (r, t) = ϑ \int_{0}^{\infty} (q_{∥} + 3 q_{⊥}) J_{0} (r α) T (α, t) α d α - ϑ \int_{0}^{\infty} \frac{[q_{∥} ν + q_{⊥} (2 + ν)] h (l_{0}, α)}{l_{0} / 4} J_{0} (r α) T (α, t) d α,

S_{st}^{-} (r, t) = ϑ \int_{0}^{\infty} (q_{∥} - q_{⊥}) [α - \frac{ν h (l_{0}, α)}{l_{0} / 4}] J_{2} (r α) T (α, t) d α,

S_{\exp} (r, t) = 4 (n_{0}^{s} - 1) (1 + ν) α_{T} \int_{0}^{\infty} h (l_{0}, α) T (α, t) J_{0} (r α) d α .

S (r, t) = \int_{0}^{\infty} l_{0} {{(\frac{\partial n^{s}}{\partial T})}_{th} + \frac{{(n_{0}^{s})}^{3} E α_{T}}{4 (1 - ν)} [(q_{∥} + 3 q_{⊥}) - \frac{4 [q_{∥} ν + q_{⊥} (2 + ν)] h (l_{0}, α)}{l_{0} α}] + \frac{4 (n_{0}^{s} - 1) (1 + ν) α_{T} h (l_{0}, α)}{l_{0} α}} T (α, t) J_{0} (r α) α d α .

S^{0} (r, t) = l_{0} χ^{0} \int_{0}^{\infty} T (α, t) J_{0} (r α) α d α,

S^{\infty} (r, t) = l_{0} χ^{\infty} \int_{0}^{\infty} T (α, t) J_{0} (r α) α d α,

χ^{0} = {(\frac{\partial n^{s}}{\partial T})}_{th} + \frac{{(n_{0}^{s})}^{3} E α_{T}}{4} (q_{∥} + q_{⊥}) + (n_{0}^{s} - 1) (1 + ν) α_{T}

χ^{\infty} = {(\frac{\partial n^{s}}{\partial T})}_{th} + \frac{{(n_{0}^{s})}^{3} E α_{T}}{4 (1 - ν)} (q_{∥} + 3 q_{⊥}) .

ϕ (r, t) = \frac{2 π}{λ} [S (r, t) - S (0, t)],

S_{g} = \frac{{| \int_{0}^{2 π} \int_{0}^{\infty} e^{- r^{2} / ω^{2}} e^{i ϕ (r, t)} r d r d ϕ |}^{2}}{{| \int_{0}^{2 π} \int_{0}^{\infty} e^{- r^{2} / ω^{2}} r d r d ϕ |}^{2}} .

I (t) = \frac{{| \int_{0}^{\infty} \exp [(1 + i V) g - i ϕ (g, t)] d g |}^{2}}{{| \int_{0}^{\infty} \exp [(1 + i V) g] d g |}^{2}},