## Abstract

The feasibility of using null screens for testing the segments of a
parabolic segmented telescope mirror for the Large Millimeter Telescope
(LMT) is analyzed. An algorithm for designing the null screen
for testing the off-axis segments of conic surfaces is
described. Actual screen designs for the different classes of
segments of the LMT are presented. The sensitivity of the test and
the required accuracies for the fabrication and positioning of the
screen are analyzed. A measuring accuracy of ∼12 µm in
surface sagitta is within the reach of the technique.

© 2000 Optical Society of America

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### Equations (35)

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(1)
$$\mathbf{I}=\frac{\left(-{x}_{0},-{y}_{0},a-b\right)}{{\left[x_{0}{}^{2}+y_{0}{}^{2}+{\left(a-b\right)}^{2}\right]}^{1/2}}.$$
(2)
$$x=\frac{\left(z-a\right)}{\left(b-a\right)}{x}_{0},$$
(3)
$$y=\frac{\left(z-a\right)}{\left(b-a\right)}{y}_{0},$$
(4)
$$z\left(x,y\right)=\frac{D}{B+{\left({B}^{2}-\mathit{AD}\right)}^{1/2}},$$
(5)
$$A=c\left(1+k{cos}^{2}\mathrm{\theta}\right),$$
(6)
$$B=\frac{1}{{\left(1+k{sin}^{2}\mathrm{\theta}\right)}^{1/2}}-\mathit{ck}sin\mathrm{\theta}cos\mathrm{\theta}x,$$
(7)
$$D=c\left(1+k{sin}^{2}\mathrm{\theta}\right){x}^{2}+{\mathit{cy}}^{2},$$
(8)
$$tan\mathrm{\theta}=\frac{{\mathit{cx}}_{c}}{{\left[1-\left(k+1\right){c}^{2}x_{c}{}^{2}\right]}^{1/2}},$$
(9)
$$y=\frac{-\mathrm{\beta}\pm {\left({\mathrm{\beta}}^{2}-4\mathrm{\alpha}\mathrm{\gamma}\right)}^{1/2}}{2\mathrm{\alpha}},$$
(10)
$$\mathrm{\alpha}=\frac{A{\left(a-b\right)}^{2}+\mathit{Ex}_{0}{}^{2}}{y_{0}{}^{2}}+c-2S\frac{\left(a-b\right){x}_{0}}{y_{0}{}^{2}},$$
(11)
$$\mathrm{\beta}=\frac{2}{{y}_{0}}\left[{\mathit{aSx}}_{0}+\left(a-b\right)\left(\sqrt{c/E}-\mathit{aA}\right)\right],$$
(12)
$$\mathrm{\gamma}={a}^{2}A-2a\sqrt{c/E}.$$
(13)
$$S=\mathit{ck}\mathrm{sin}\mathrm{\theta}\mathrm{cos}\mathrm{\theta},$$
(14)
$$E=c\left(1+k{sin}^{2}\mathrm{\theta}\right).$$
(15)
$$x=\frac{{x}_{0}}{{y}_{0}}y.$$
(16)
$$\mathbf{R}=\mathbf{I}-2\left(\mathbf{I}\xb7\mathbf{N}\right)\mathbf{N},$$
(17)
$$\mathbf{N}=\frac{\left(-{z}_{x},-{z}_{y},1\right)}{{\left(1+z_{x}{}^{2}+z_{y}{}^{2}\right)}^{1/2}}.$$
(18)
$${z}_{x}=\frac{\left(\mathit{DT}+\mathit{BD}\right)S-\left(\mathit{BT}+{B}^{2}-\mathit{AD}/2\right)2\mathit{Ex}}{{\left(B+S\right)}^{2}S},$$
(19)
$${z}_{y}=\frac{\left(\mathit{BT}+{B}^{2}-\mathit{AD}/2\right)2\mathit{cy}}{{\left(B+S\right)}^{2}S},$$
(20)
$$X=x+\left(a-z\right)\frac{{x}_{0}\left(z_{x}{}^{2}-z_{y}{}^{2}-1\right)+2{z}_{x}\left({y}_{0}{z}_{y}+a-b\right)}{\left(a-b\right)\left(z_{x}{}^{2}+z_{y}{}^{2}+1\right)-2\left({x}_{0}{z}_{x}+{y}_{0}{z}_{y}\right)},$$
(21)
$$Y=y+\left(a-z\right)\frac{{y}_{0}\left(z_{y}{}^{2}-z_{x}{}^{2}-1\right)+2{z}_{y}\left({x}_{0}{z}_{x}+a-b\right)}{\left(a-b\right)\left(z_{x}{}^{2}+z_{y}{}^{2}+1\right)-2\left({x}_{0}{z}_{x}+{y}_{0}{z}_{y}\right)}.$$
(22)
$${x}_{0}=\mathit{nd}-L/2,$$
(23)
$${y}_{0}=\mathit{md}-L/2,$$
(24)
$$b=\left(1-\frac{L}{{D}_{0}}\right)a,$$
(25)
$${D}_{s}=\left(2-\frac{a}{f}\right){D}_{0},$$
(26)
$${f}_{t}=\frac{{\left(1-{\mathit{kc}}^{2}x_{c}{}^{2}\right)}^{3/2}}{2c},$$
(27)
$${f}_{s}=\frac{{\left(1-{\mathit{kc}}^{2}x_{c}{}^{2}\right)}^{1/2}}{2c}.$$
(28)
$$\u3008\mathrm{\rho}\u3009=\frac{1}{{N}^{2}}\sum _{i=1}^{N}\sum _{j=1}^{N}{\left(\mathrm{d}x_{\mathit{ij}}{}^{2}+\mathrm{d}y_{\mathit{ij}}{}^{2}\right)}^{1/2}.$$
(29)
$$\mathrm{\Delta}x=\frac{d}{{\mathit{lx}}_{\mathit{ij}}}\mathrm{\Delta}X.$$
(30)
$$\mathrm{distortion}\left(\mathrm{percent}\right)=\frac{\mathrm{\xi}}{\mathrm{\beta}}\times \frac{{\mathrm{\alpha}}^{2}+{\mathrm{\beta}}^{2}}{\mathrm{\alpha}\mathrm{\beta}},$$
(31)
$$h>2.44\frac{\mathrm{\lambda}\left(b-a\right)}{{A}_{max}}.$$
(32)
$$\mathbf{N}\xb7\mathbf{S}=0.$$
(33)
$$z-{z}_{i}=-\int \left(\frac{{N}_{x}}{{N}_{z}}\mathrm{d}x+\frac{{N}_{y}}{{N}_{z}}\mathrm{d}y\right),$$
(34)
$$\mathbf{R}=\frac{\left(X-{x}_{s},Y-{y}_{s},a-{z}_{s}\right)}{{\left[{\left(X-{x}_{s}\right)}^{2}+{\left(Y-{y}_{s}\right)}^{2}+{\left(a-{z}_{s}\right)}^{2}\right]}^{1/2}},$$
(35)
$$\mathbf{N}=\frac{\mathbf{R}-\mathbf{I}}{|\mathbf{R}-\mathbf{I}|}.$$