Full-field unsymmetrical beam shaping for decreasing and homogenizing the thermal deformation of optical element in a beam control system

Haotong Ma; Qiong Zhou; Xiaojun Xu; Shaojun Du; Zejin Liu

doi:10.1364/OE.19.0A1037

Optics Express
Vol. 19,
Issue S5,
pp. A1037-A1050
(2011)
•https://doi.org/10.1364/OE.19.0A1037

Full-field unsymmetrical beam shaping for decreasing and homogenizing the thermal deformation of optical element in a beam control system

Haotong Ma, Qiong Zhou, Xiaojun Xu, Shaojun Du, and Zejin Liu

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Haotong Ma, Qiong Zhou, Xiaojun Xu, Shaojun Du, and Zejin Liu, "Full-field unsymmetrical beam shaping for decreasing and homogenizing the thermal deformation of optical element in a beam control system," Opt. Express 19, A1037-A1050 (2011)

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Abstract

We propose and demonstrate the full-field unsymmetrical beam shaping for decreasing and homogenizing the thermal deformation of optical element in a beam control system. The transformation of square dark hollow beam with unsymmetrical and inhomogeneous intensity distribution into square dark hollow beam with homogeneous intensity distribution is chosen to prove the validity of the technique. Dual deformable mirrors (DMs) based on the stochastic parallel gradient descent (SPGD) controller are used to redistribute the intensity of input beam and generate homogeneous square dark hollow beam with near-diffraction-limited performance. The SPGD algorithm adaptively optimizes the coefficients of Lukosz-Zernike polynomials to form the phase distributions for dual DMs. Based on the finite element method, the thermal deformations of CaF₂ half transparent and half reflecting mirror irradiated by high power laser beam before and after beam shaping are numerically simulated and compared. The thermal deformations of the mirror irradiated by the laser beam with different powers and the influences of thermal deformation on beam quality are also numerically studied. Results show that full-field beam shaping can greatly decrease and homogenize the thermal deformation of the mirror in the beam control system. The strehl ratios of the high power laser beams passing through the beam control system can be greatly improved by the full-field beam shaping. The technique presented in this paper can provide effective guidance for optimum design of high power laser cavity and beam shaping system.

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Fig. 1 Configuration of the adaptive full-field beam shaping system based on SPGD algorithm.

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Fig. 2 Intensity distributions of the input and target beam, (a) input beam, (b) target beam.

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Fig. 3 Shaping results for square dark hollow flattop beam, (a) intensity distribution of the output beam, (b) corresponding evolutions of the relative J_fiterror as the algorithm proceeds.

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Fig. 4 Phase distributions generated for intensity redistribution and wave front compensation (the unit is radian). (a) for intensity redistribution, (b) for wave front compensation.

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Fig. 5 Far field intensity distribution of the output beam before and after phase compensation, (a) before phase compensation, (b) after phase compensation.

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Fig. 6 Phase distribution of the output beam after being compensated and the PIB curve, (a) phase distribution (the unit is radian), (b) PIB curve.

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Fig. 7 Illustration of a half transparent and half reflecting mirror with high power laser beam irradiating.

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Fig. 8 Power density distribution of the 150kW high power laser beam (the unit is W/m ²), (a) before beam shaping, (b) after beam shaping.

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Fig. 9 Thermal deformation of the half transparent and half reflecting mirror irradiated by the high power laser beam without and with beam shaping, when the power is 50kW, 100kW and 150k (the unit is m), the irradiation time is 5s. (a) before beam shaping, (b) after beam shaping.

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Fig. 10 Change of the thermal deformation of position a and position b of half transparent and half reflecting mirror along with the irradiation time. (a) Irradiated by 50kW high power laser beam, (b) Irradiated by 100kW high power laser beam, (c) Irradiated by 150kW high power laser beam.

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Fig. 11 Change of PV and rms value of thermal deformation of half transparent and half reflective mirror along with irradiation time. (a) with 50kW power, (b) with 100kW power, (c) with 150kW power.

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Fig. 12 Far field intensity distribution of the 50kW power laser beam after being reflected by the half transparent and half reflecting mirror (the unit is W/m ²), the irradiation time is 6s, (a) with beam shaping, (b) without beam shaping.

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Tables (1)

Table1 Properties of the CaF₂ Mirror (at 293K)

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Equations (18)

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ϕ_{j} (r, θ) = \sum_{i = 2}^{N} a_{i} L_{i} (r, θ)

L_{n}^{m} (r, θ) = B_{n}^{m} (r) \times {\begin{matrix} \cos (m θ) m \geq 0 \\ \sin (m θ) m < 0 \end{matrix},

B_{n}^{m} (r) = {\begin{matrix} \sqrt{\frac{1}{n}} [R_{n}^{0} (r) - R_{n - 2}^{0} (r)] n \neq m = 0 \\ \sqrt{\frac{2}{n}} [R_{n}^{m} (r) - R_{n - 2}^{m} (r)] n \neq m \neq 0 \\ \frac{2}{\sqrt{n}} R_{n}^{n} (r) m = n \neq 0 \\ 1 m = n = 0 \end{matrix} .

R_{n}^{m} (r) = \sum_{k = 0}^{\frac{n - m}{2}} \frac{{(- 1)}^{k} (n - k)! r^{n - 2 k}}{k! (\frac{n + m}{2} - k)! (\frac{n - m}{2} - k)!} .

J_{f i t e r r o r} = \sum_{x} \sum_{y} {[I_{a c t u a l} (x, y) - I_{t \arg e t} (x, y)]}^{2} .

J_{c o m p e n s a t i o n} = \iint I_{f a r f i e l d} {(x, y)}^{2} d x d y,

\begin{array}{l} I_{t \arg e t} = \\ \exp {- [a_{1} {(x - x_{o})}^{2 p_{1}}] - [b_{1} {(y - y_{o})}^{2 q_{1}}]} - \exp {- [a_{2} {(x - x_{o})}^{2 p_{2}}] - [b_{2} {(y - y_{o})}^{2 q_{2}}]}, \end{array}

\nabla^{2} T (r, φ, z; t) + \frac{\overset{•}{q}}{κ} = \frac{1}{α} \frac{\partial T (r, φ, z; t)}{\partial t} .

\frac{\partial T (r, φ, z; t)}{\partial r} |_{r = r_{0}} = \frac{h}{κ} (T - T_{\infty}),

\frac{\partial T (r, φ, z; t)}{\partial z} |_{z = 0} = \frac{h}{κ} (T - T_{\infty}),

\frac{\partial T (r, φ, z; t)}{\partial z} |_{z = d} = q (r, φ; t),

T (r, φ, z; t) |_{t = 0} = T_{\infty},

\nabla^{2} u_{r} - \frac{u_{r}}{r^{2}} + \frac{1}{1 - 2 ν} \frac{\partial ε}{\partial r} - \frac{2 (1 + ν)}{1 - 2 ν} α_{l} \frac{\partial T}{\partial r} = 0,

\nabla^{2} u_{z} + \frac{1}{1 - 2 ν} \frac{\partial ε}{\partial z} - \frac{2 (1 + ν)}{1 - 2 ν} α_{l} \frac{\partial T}{\partial z} = 0.

u_{r} |_{r = r_{0}} = u_{z} |_{r = r_{0}} = 0,

D_{P V} = D_{\max} - D_{\min},

D_{r m s} = \sqrt{\frac{\sum_{n = 1}^{M} {[D (n) - \sum_{n = 1}^{M} D (n) / M]}^{2}}{M}},

Δ ϕ (r, φ) = k \cdot 2 u_{z} (r, φ) \cdot \cos θ,

Property	Value	Unit
Density(ρ )	3180	kg/m ³
Thermal conductivity (κ )	10	W/mK
Specific heat (C)	870	Ws/kgK
Thermal expansion coefficient ( $α_{l}$ )	8.87×10⁻⁶	K ⁻¹
Poisson’s ratio	0.356
Young’s modulus	100.5	GPa
Radius of the mirror (r ₀)	0.1	m
Thickness of the mirror	0.03	m
Absorption coefficient of the coating (η ₁)	0.002	m ⁻¹
Absorption coefficient of the volume (η ₂)	0.12	m ⁻¹

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Equations (18)

Optics Express