## Abstract

A statistically repeatable, hot-air optical turbulence generator, based on the forced mixing of two air flows with different temperatures, is described.
Characterization results show that it is possible to generate any turbulence strength up to
${{C}_{N}}^{2}\text{\hspace{0.17em}}\Delta h\approx 6\times {10}^{-10}{\text{m}}^{\text{1}/3}$, allowing a ratio of beam diameter to Fried's parameter as large as
$D/{r}_{0}\approx 25$ for one crossing through the turbulator or
$D/{r}_{0}\approx 38$ for two crossings. The outer scale
$\left({L}_{0}\approx 133\pm 60\text{\hspace{0.17em} mm}\right)$ is found to be compatible with the turbulator mixing chamber size
$\left(170\text{\hspace{0.17em} mm}\right)$, and the inner scale
$\left({l}_{0}\approx 7.6\pm 3.8\text{\hspace{0.17em} mm}\right)$ is compatible with the values in the literature for the free atmosphere. The temporal power spectrum analysis of the centroid of the focused image shows good agreement with Kolmogorov's theory. Therefore the device can be used with confidence to emulate realistic turbulence in a controlled manner. A calibrated
${{C}_{N}}^{2}$
profile, both in layer altitude and strength, is necessary for the testing of off-axis adaptive optics correction (multiconjugate adaptive optics). Testing was done to calibrate the
${{C}_{N}}^{2}$ profile using the slope detection and ranging technique. The first results, with only one layer, show the validity of the approach and indicate that a multiple-pass scheme is viable with a few modifications of the current setup.

© 2006 Optical Society of America

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### Equations (18)

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(1)
$$\mathrm{Re}=\frac{\text{kinetic \hspace{0.17em} energy}}{\text{dissipated \hspace{0.17em} energy}}=\frac{UL}{\nu}\gg 1,$$
(2)
$${D}_{T}\left(\stackrel{\rightharpoonup}{\rho}\right)={\u3008{\mathrm{\left[}T\left(\stackrel{\rightharpoonup}{r}+\stackrel{\rightharpoonup}{\rho}\right)-T\left(\stackrel{\rightharpoonup}{r}\right)\mathrm{\right]}}^{2}\u3009}_{\mathit{r}},$$
(3)
$${D}_{T}\left(\stackrel{\rightharpoonup}{\rho}\right)={D}_{T}\left(\mathrm{\left|}\stackrel{\rightharpoonup}{\rho}\mathrm{\right|}\right)={{C}_{T}}^{2}{\rho}^{2/3},$$
(4)
$${D}_{N}\left(\rho \right)={{C}_{N}}^{2}{\rho}^{2/3},$$
(5)
$${{C}_{N}}^{2}={\left(\frac{{\alpha}_{n}P}{{T}^{2}}\right)}^{2}\text{\hspace{0.17em}}{{C}_{T}}^{2},$$
(6)
$${{C}_{N}}^{2}\sim {\left(\frac{{\alpha}_{n}P}{{T}^{2}}\right)}^{2}{\left(\delta T\right)}^{2},$$
(7)
$${r}_{0}^{\text{\hspace{0.17em} \hspace{0.17em}}-5/3}=0.4234{\left(2\pi /\lambda \right)}^{2}\text{\hspace{0.17em}}{\displaystyle {\int}_{0}^{\infty}{{C}_{N}}^{2}\left(h\right)\mathrm{d}h},$$
(8)
$$\varphi \left(f\right)=\frac{0.023}{{{r}_{0}}^{5/3}}\text{\hspace{0.17em}}{\left({f}^{\text{\hspace{0.17em}}2}+\frac{1}{{{L}_{0}}^{2}}\right)}^{-11/6},$$
(9)
$$\eta \left[{l}_{0}\right]=\mathrm{exp}\left(-{{l}_{0}}^{2}{f}^{\text{\hspace{0.17em}}2}\right).$$
(10)
$$\eta \left[{l}_{0}\right]\approx \mathrm{exp}\left[-0.137{\left(n+1\right)}^{2}\left({l}_{0}/{D}^{2}\right)\right],$$
(11)
$${f}_{c}=0.3\left(n+1\right)\frac{V}{D},$$
(12)
$${\theta}_{\text{iso}}=0.314\frac{{r}_{0}}{\u3008h\u3009},$$
(13)
$${\u3008h\u3009}^{5/3}=\frac{{\displaystyle {\int}_{0}^{\infty}{{C}_{N}}^{2}{\left(h\right)h}^{5/3}\text{\hspace{0.17em}d}h}}{{\displaystyle {\int}_{0}^{\infty}{{C}_{N}}^{2}\left(h\right)\mathrm{d}h}},$$
(14)
$$\text{FWHM}\cong \sqrt{{\left({\text{FWHM}}_{\text{telescope}}\right)}^{2}+{\left({\text{FWHM}}_{\text{atmosphere}}\right)}^{2}}\cong \lambda /D\sqrt{1+{\left(D/{r}_{0}\right)}^{2}}.$$
(15)
$${\alpha}_{p}=\frac{{X}_{C}}{{F}_{L}}=\frac{{\displaystyle \iint \iint P\left(x,y\right)\frac{\partial W}{\partial x}\left(x,y\right)\mathrm{d}x\mathrm{d}y}}{{\displaystyle \iint \iint P\left(x,y\right)\mathrm{d}x\mathrm{d}y}},$$
(16)
$${{\sigma}_{\text{AoA}}}^{\text{2}}\left[x,y\right]={\left(2\pi \right)}^{4/3}0.033{{C}_{N}}^{2}\delta h\times {\displaystyle {\iint}_{{R}^{2}}{{f}_{X\text{,}Y}}^{2}{\left({f}^{\text{\hspace{0.17em}}2}+{{L}_{0}}^{-2}\right)}^{-11/6}\mathrm{exp}\left(-{{l}_{0}}^{2}{f}^{\text{\hspace{0.17em}}2}\right)}{\times \text{\hspace{0.17em}}\left[\frac{2{J}_{1}\left(\pi Df\right)}{\pi Df}\right]}^{2}\mathrm{d}{f}_{X}\mathrm{d}{f}_{Y}.$$
(17)
$${{\sigma}_{A\mathrm{oA}}}^{2}\left[x,y\right]=2.8375{{C}_{N}}^{2}\delta h{D}^{-1/3}=0.1698{\left(\lambda /D\right)}^{2}{\left(D/{r}_{0}\right)}^{5/3}.$$
(18)
$${{C}_{N}}^{2}\sim {F}^{-1}\left\{\frac{F\left[C({\delta}_{i},{\delta}_{j})\right]}{F\left[A\left({\delta}_{i},{\delta}_{j}\right)\right]}\right\}\left({\delta}_{i},0\right),$$