## Abstract

The Hartmann test is a well-known technique for testing large
telescope mirrors. The Hartmann technique samples the wave front
under analysis by use of a screen of uniformly spaced array of holes
located at the pupil plane. The traditional technique used to
gather quantitative data requires the measurement of the centroid of
these holes as imaged near the paraxial focus. The deviation from
its unaberrated uniform position is proportional to the slope of the
wave-front asphericity. The centroid estimation is normally done
manually with the aid of a microscope or a densitometer; however, newer
automatic fringe-processing techniques that use the synchronous
detection technique or the Fourier phase-estimation method may also be
used. Here we propose a new technique based on a regularized
phase-tracking (RPT) system to detect the transverse aberration in
Hartmanngrams in a direct way. That is, it takes the dotted pattern
of the Hartmanngram as input, and as output the RPT system gives the
unwrapped transverse ray aberration in just one step. Our RPT is
compared with the synchronous and the Fourier methods, which may be
regarded as its closest competitors.

© 1999 Optical Society of America

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### Equations (16)

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(1)
$$\mathit{Hs}\left({x}_{1},{y}_{1}\right)=\sum _{n=-N/2}^{N/2}\sum _{m=-N/2}^{N/2}h\left({x}_{1}-\mathit{dn},{y}_{1}-\mathit{dm}\right),$$
(2)
$$x=\frac{r-L}{r}{x}_{1}+L\frac{\partial W\left({x}_{1},{y}_{1}\right)}{\partial {x}_{1}},y=\frac{r-L}{r}{y}_{1}+L\frac{\partial W\left({x}_{1},{y}_{1}\right)}{\partial {y}_{1}},$$
(3)
$$H\left(x,y\right)=\left\{\sum _{n=-N/2}^{N/2}\sum _{m=-N/2}^{N/2}\mathit{hi}\left[\frac{r-L}{r}{x}_{1}-L\frac{\partial W\left({x}_{1},{y}_{1}\right)}{\partial {x}_{1}}-\mathit{dn},\frac{r-L}{r}{y}_{1}-L\frac{\partial W\left({x}_{1},{y}_{1}\right)}{\partial {y}_{1}}-\mathit{dm}\right]\right\}P\left(x,y\right),$$
(4)
$$W\left({x}_{1},{y}_{1}\right)=B{\left(x_{1}{}^{2}+y_{1}{}^{2}\right)}^{2},$$
(5)
$$x=\frac{r-L}{r}{x}_{1}+4\mathit{LB}{\left(x_{1}{}^{2}+y_{1}{}^{2}\right)}^{2}{x}_{1},y=\frac{r-L}{r}{y}_{1}+4\mathit{LB}{\left(x_{1}{}^{2}+y_{1}{}^{2}\right)}^{2}{y}_{1}.$$
(6)
$${F}_{x}\left(x,y\right)=cos\left[{\mathrm{\omega}}_{0}x+K\left({x}^{2}+{y}^{2}\right)x+{\mathrm{\varphi}}_{x}\left(x,y\right)\right]P\left(x,y\right),$$
(7)
$$U\left(x,y\right)=\sum _{\left(\mathrm{\u220a},\mathrm{\eta}\right)\in \left[{N}_{x,y}\cap P\left(x,y\right)\right]}\left\{{\left[H\left(\mathrm{\u220a},\mathrm{\eta}\right)-cos{p}_{x}\left(x,y\right)\right]}^{2}+\mathrm{\lambda}{\left[{\mathrm{\varphi}}_{x}\left(\mathrm{\u220a},\mathrm{\eta}\right)-{\mathrm{\varphi}}_{x}\left(x,y\right)\right]}^{2}m\left(\mathrm{\u220a},\mathrm{\eta}\right)\right\},$$
(8)
$${p}_{x}\left(x,y\right)={\mathrm{\omega}}_{0}x+K\left({x}^{2}+{y}^{2}\right)x+{\mathrm{\varphi}}_{x}\left(x,y\right),$$
(9)
$${p}_{y}\left(x,y\right)={\mathrm{\omega}}_{0}y+K\left({x}^{2}+{y}^{2}\right)y+{\mathrm{\varphi}}_{y}\left(x,y\right),$$
(10)
$$\mathrm{\varphi}_{x}{}^{k+1}\left(x,y\right)=\mathrm{\varphi}_{x}{}^{k}\left(x,y\right)-\mathrm{\tau}\frac{\partial U\left(x,y\right)}{\partial {\mathrm{\varphi}}_{x}\left(x,y\right)},$$
(11)
$${T}_{x}\left(x,y\right)=\left[K\left({x}^{2}+{y}^{2}\right)x+{\mathrm{\varphi}}_{x}\left(x,y\right)\right]P\left(x,y\right),{T}_{y}\left(x,y\right)=\left[K\left({x}^{2}+{y}^{2}\right)y+{\mathrm{\varphi}}_{y}\left(x,y\right)\right]P\left(x,y\right).$$
(12)
$${I}_{1}\left(x,y\right)=\left\{1+cos\left[{\mathrm{\omega}}_{0}x+K\left({x}^{2}+{y}^{2}\right)x\right]\right\}P\left(x,y\right).$$
(13)
$${I}_{2}\left(x,y\right)=\left\{1+cos\left[{\mathrm{\omega}}_{0}x+K\left({x}^{2}+{y}^{2}\right)x+{\mathrm{\varphi}}_{x}\left(x,y\right)\right]\right\}P\left(x,y\right).$$
(14)
$${I}_{F}\left(x,y\right)=1+cos\left[{T}_{\mathit{Fx}}\left(x,y\right)\right],$$
(15)
$${T}_{\mathit{Sx}}\left(x,y\right)=\mathrm{arctan}\left\{\frac{\mathrm{LPF}\left[H\left(x,y\right)sin\left({\mathrm{\omega}}_{0}x\right)\right]}{\mathrm{LPF}\left[H\left(x,y\right)cos\left({\mathrm{\omega}}_{0}x\right)\right]}\right\},$$
(16)
$${I}_{S}\left(x,y\right)=1+cos\left[{\mathrm{\omega}}_{0}x+{T}_{\mathit{Sx}}\left(x,y\right)\right],$$