## Abstract

Recently developed methods [
J. Opt. Soc. Am. A **11**,
3292–
3307 (
1994)] permit definitive studies of two-mirror systems for a variety of applications. As an illustration of such a study, unobstructed, telecentric, plane-symmetric systems of two conic mirrors that are intended for distortion-free projection are investigated here. The magnification, speed, and field size in the examples are arbitrarily chosen to correspond to values that are appropriate for full-field soft-x-ray projection lithography. Low-order imaging constraints are applied to eliminate most of the 13 parameters that specify the configuration of such a system. It is shown that there are just two three-parameter families of systems and a single four-parameter family. The associated three-and four-dimensional merit function spaces are mapped, and selected systems from various regions of the spaces are discussed.

© 1994 Optical Society of America

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### Equations (62)

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(1)
$$s(\mathbf{y})=\frac{1}{2}({S}_{11}{y}^{2}+{S}_{22}{z}^{2})+\frac{y}{6}({\mathbb{S}}_{111}{y}^{2}+3{\mathbb{S}}_{122}{z}^{2})+O(4).$$
(2)
$$c={S}_{22}\sqrt{{S}_{22}/{S}_{11}},$$
(3)
$$\kappa =\frac{{\mathbb{S}}_{122}^{2}+{S}_{22}^{2}{({S}_{11}-{S}_{22})}^{2}}{{S}_{22}^{3}({S}_{11}-{S}_{22})},$$
(4)
$$h=\text{sgn}({\mathbb{S}}_{122})\frac{({S}_{11}-{S}_{22})}{{[{\mathbb{S}}_{122}^{2}+{S}_{22}^{2}{({S}_{11}-{S}_{22})}^{2}]}^{1/2}}.$$
(5)
$${\mathcal{S}}_{1}=\frac{[{d}_{2}-M({d}_{0}+{d}_{1})]}{2M{d}_{0}{d}_{1}\hspace{0.17em}\text{cos}({\theta}_{1})}\left[\begin{array}{cc}{\text{cos}}^{2}(\theta {}_{1})& 0\\ 0& 1\end{array}\right],$$
(6)
$${\mathcal{S}}_{2}=\frac{[M{d}_{0}-{d}_{1}-{d}_{2})]}{2{d}_{1}{d}_{2}\hspace{0.17em}\text{cos}({\theta}_{2})}\left[\begin{array}{cc}{\text{cos}}^{2}(\theta {}_{2})& 0\\ 0& 1\end{array}\right],$$
(7)
$${\mathbb{S}}_{122}^{\text{I}}=\frac{[M({d}_{0}+{d}_{1})-d{}_{2}][M({d}_{0}-d{}_{1})+d{}_{2}]\text{tan}({\theta}_{1})+2{d}_{2}(M{d}_{0}-{d}_{1}-{d}_{2})\text{tan}({\theta}_{2})}{4{M}^{2}{d}_{0}^{2}{d}_{1}^{2}},$$
(8)
$${\mathbb{S}}_{122}^{\text{II}}=\frac{2M{d}_{0}[M({d}_{0}+{d}_{1})-d{}_{2}]\text{tan}({\theta}_{1})+(M{d}_{0}-d{}_{1}-d{}_{2})(M{d}_{0}-d{}_{1}+{d}_{2})\text{tan}({\theta}_{2})}{4{d}_{1}^{2}{d}_{2}^{2}},$$
(9)
$$M\hspace{0.17em}\text{tan}({\theta}_{\text{obj}})+\text{tan}({\theta}_{\text{im}})=\frac{2}{{d}_{1}}\{[M({d}_{0}+{d}_{1})-{d}_{2}]\text{tan}({\theta}_{1})+(M{d}_{0}-{d}_{1}-{d}_{2})\text{tan}({\theta}_{2})\},$$
(10)
$${S}_{1}:=\left[\begin{array}{cc}{S}_{11}^{\text{I}}& 0\\ 0& {S}_{22}^{\text{I}}\end{array}\right],\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}{S}_{2}:=\left[\begin{array}{cc}{S}_{11}^{\text{II}}& 0\\ 0& {S}_{22}^{\text{II}}\end{array}\right].$$
(11)
$${y}_{\text{im}}=-({\tilde{M}}_{11}{y}_{\text{obj}}+\xbd{\tilde{\mathbb{L}}}_{111}{y}_{\text{obj}}^{2}+\xbd{\tilde{\mathbb{L}}}_{221}{z}_{\text{obj}}^{2})+O(3),$$
(12)
$${z}_{im}=-({\tilde{M}}_{22}{z}_{\text{obj}})+{\tilde{\mathbb{L}}}_{122}{y}_{\text{obj}}{z}_{\text{obj}})+O(3),$$
(13)
$${\tilde{M}}_{11}=\frac{\text{cos}({\theta}_{\text{obj}})}{\text{cos}({\theta}_{\text{im}})}M,$$
(14)
$${\tilde{M}}_{22}=M,$$
(15)
$${\tilde{\mathbb{L}}}_{111}=\frac{M\hspace{0.17em}{\text{cos}}^{2}({\theta}_{\text{obj}})}{2{d}_{0}{d}_{2}\hspace{0.17em}\text{cos}({\theta}_{\text{im}})}[4({d}_{1}+{d}_{2}-M{d}_{0})\text{tan}({\theta}_{2})+(3M{d}_{1}-4{d}_{2})\text{tan}({\theta}_{\text{obj}})+(4M{d}_{0}-{d}_{1})\text{tan}({\theta}_{\text{im}})],$$
(16)
$${\tilde{\mathbb{L}}}_{221}=\frac{M[4({d}_{1}+{d}_{2}-M{d}_{0})\text{tan}({\theta}_{2})+M{d}_{1}\hspace{0.17em}\text{tan}({\theta}_{\text{obj}})+{d}_{1}\hspace{0.17em}\text{tan}({\theta}_{\text{im}})]}{2{d}_{0}{d}_{2}\hspace{0.17em}\text{cos}({\theta}_{\text{im}})},$$
(17)
$${\tilde{\mathbb{L}}}_{122}=\frac{M\hspace{0.17em}\text{cos}({\theta}_{\text{obj}})[M{d}_{1}-2{d}_{2})\text{tan}({\theta}_{\text{obj}})+(2M{d}_{0}-{d}_{1})\text{tan}({\theta}_{\text{im}})]}{2{d}_{0}{d}_{2}}.$$
(18)
$$(M{d}_{1}-2{d}_{2})\text{tan}({\theta}_{\text{obj}})=({d}_{1}-2M{d}_{0})\text{tan}({\theta}_{\text{im}}),$$
(19)
$$M{d}_{1}\hspace{0.17em}\text{tan}({\theta}_{\text{obj}})+{d}_{1}\hspace{0.17em}\text{tan}({\theta}_{\text{im}})=4(M{d}_{0}-{d}_{1}-{d}_{2})\text{tan}({\theta}_{2}),$$
(20)
$$[M({d}_{0}+{d}_{1})-{d}_{2}]\hspace{0.17em}\text{tan}({\theta}_{\text{1}})=(M{d}_{0}-{d}_{1}-{d}_{2})\text{tan}({\theta}_{2}).$$
(21)
$$\text{tan}({\theta}_{\text{im}})=\frac{(M{d}_{1}-2{d}_{2})}{({d}_{1}-2M{d}_{0})}\text{tan}({\theta}_{\text{obj}}),$$
(22)
$$\text{tan}({\theta}_{1})=\frac{{d}_{1}[{d}_{2}-M({d}_{1}-M{d}_{0})]}{2({d}_{1}-2M{d}_{0})[{d}_{2}-M({d}_{0}+{d}_{1})]}\text{tan}({\theta}_{\text{obj}}),$$
(23)
$$\text{tan}({\theta}_{2})=\frac{{d}_{1}[{d}_{2}-M({d}_{1}-M{d}_{0})]}{2({d}_{1}-2M{d}_{0})[{d}_{1}+{d}_{2}-M{d}_{0})]}\text{tan}({\theta}_{\text{obj}}).$$
(24)
$${\mathcal{S}}_{1}=\frac{[{d}_{2}-M({d}_{0}+{d}_{1})]}{2M{d}_{0}{d}_{1}\hspace{0.17em}\text{cos}({\theta}_{1})}\left[\begin{array}{cc}{\text{cos}}^{2}({\theta}_{1})& 0\\ 0& 1\end{array}\right],$$
(25)
$${\mathcal{S}}_{2}=\frac{(M{d}_{0}-{d}_{1}-{d}_{2})]}{2{d}_{1}{d}_{2}\hspace{0.17em}\text{cos}({\theta}_{2})}\left[\begin{array}{cc}{\text{cos}}^{2}({\theta}_{2})& 0\\ 0& 1\end{array}\right],$$
(26)
$${\mathbb{S}}_{122}^{\text{I}}=\frac{[3{d}_{2}+M({d}_{0}-{d}_{1})][{d}_{2}-M({d}_{1}-M{d}_{0})]}{8{M}^{2}{d}_{0}^{2}{d}_{1}(2M{d}_{0}-{d}_{1})}\times \text{tan}({\theta}_{\text{obj}}),$$
(27)
$${\mathbb{S}}_{122}^{\text{II}}=\frac{(3M{d}_{0}-{d}_{1}+{d}_{2})[{d}_{2}-M({d}_{1}-M{d}_{0})]}{8{d}_{1}{d}_{2}^{2}(2M{d}_{0}-{d}_{1})}\text{tan}({\theta}_{\text{obj}}).$$
(28)
$${\tilde{M}}_{11}={\left[{\text{cos}}^{2}({\theta}_{\text{obj}})+\frac{{(M{d}_{1}-2{d}_{2})}^{2}}{{({d}_{1}-2M{d}_{0})}^{2}}{\text{sin}}^{2}({\theta}_{\text{obj}})\right]}^{1/2}M.$$
(29)
$${c}_{1}=\frac{{d}_{2}-M({d}_{0}+{d}_{1})}{2M{d}_{0}{d}_{1}\hspace{0.17em}{\text{cos}}^{2}({\theta}_{1})},$$
(30)
$${c}_{2}=\frac{M{d}_{0}-{d}_{1}-{d}_{2}}{2{d}_{1}{d}_{2}\hspace{0.17em}{\text{cos}}^{2}({\theta}_{1})},$$
(31)
$${\kappa}_{1}=\frac{4(M{d}_{0}+{d}_{2})(M{d}_{1}-2{d}_{2})}{{[{d}_{2}-M({d}_{0}+{d}_{1})]}^{2}}{\text{cos}}^{2}({\theta}_{1})-1,$$
(32)
$${\kappa}_{2}=\frac{4(M{d}_{0}+{d}_{2})({d}_{1}-2M{d}_{0})}{{(M{d}_{0}-{d}_{1}-{d}_{0})}^{2}}{\text{cos}}^{2}({\theta}_{2})-1,$$
(33)
$${d}_{1}=2M{d}_{0}.$$
(34)
$$\text{tan}({\theta}_{\text{im}})=\frac{2[M{d}_{0}(1+2M)-{d}_{2}]}{M{d}_{0}}\text{tan}({\theta}_{1}),$$
(35)
$$\text{tan}({\theta}_{2})=\frac{{d}_{2}-M{d}_{0}(1+2M)}{{d}_{2}+M{d}_{0}}\text{tan}({\theta}_{1}).$$
(36)
$${\mathcal{S}}_{1}=\frac{[{d}_{2}-M{d}_{0}(1+2M)]}{4{M}^{2}{d}_{0}^{2}\hspace{0.17em}\text{cos}({\theta}_{1})}\left[\begin{array}{cc}{\text{cos}}^{2}({\theta}_{1})& 0\\ 0& 1\end{array}\right],$$
(37)
$${\mathcal{S}}_{2}=\frac{-(M{d}_{0}+{d}_{2})}{4M{d}_{0}{d}_{2}\hspace{0.17em}\text{cos}({\theta}_{2})}\left[\begin{array}{cc}{\text{cos}}^{2}({\theta}_{2})& 0\\ 0& 1\end{array}\right],$$
(38)
$${\mathbb{S}}_{122}^{\text{I}}=\frac{[M{d}_{0}(1+2M)-{d}_{2}][3{d}_{2}-M{d}_{0}(1-2M)]}{16{M}^{4}{d}_{0}^{4}}\times \text{tan}({\theta}_{1}),$$
(39)
$${\mathbb{S}}_{122}^{\text{II}}=\frac{(M{d}_{0}+{d}_{2})[M{d}_{0}(1+2M)-{d}_{2}]}{16{M}^{2}{d}_{0}^{2}{d}_{2}^{2}}\text{tan}({\theta}_{1}).$$
(40)
$${c}_{1}=\frac{{d}_{2}-M{d}_{0}(1+2M)}{4{M}^{2}{d}_{0}^{2}\hspace{0.17em}{\text{cos}}^{2}({\theta}_{1})},$$
(41)
$${c}_{2}=\frac{-(M{d}_{0}+{d}_{2})}{4M{d}_{0}{d}_{2}\hspace{0.17em}{\text{cos}}^{2}({\theta}_{2})},$$
(42)
$${\kappa}_{1}=\frac{8(M{d}_{0}+{d}_{2})({M}^{2}{d}_{0}-{d}_{2})}{{[{d}_{2}-M{d}_{0}(1+2M)]}^{2}}{\text{cos}}^{2}({\theta}_{1})-1,$$
(43)
$${\kappa}_{2}=-1.$$
(44)
$${\tilde{M}}_{11}={\left\{1+\frac{4{[M{d}_{0}(1+2M)-{d}_{2}]}^{2}}{{(M{d}_{0})}^{2}}\hspace{0.17em}{\text{tan}}^{2}({\theta}_{1})\right\}}^{1/2}M.$$
(45)
$${d}_{2}={M}^{2}{d}_{0}.$$
(46)
$$2M{d}_{0}[M\hspace{0.17em}\text{tan}({\theta}_{\text{obj}})+\text{tan}({\theta}_{\text{im}})+2(1+M)\text{tan}({\theta}_{2})]=0,$$
(47)
$$M{d}_{0}(1+M)[\text{tan}({\theta}_{1})+\text{tan}({\theta}_{2})]=0.$$
(48)
$$\text{tan}({\theta}_{2})=-\text{tan}({\theta}_{1}),$$
(49)
$$\text{tan}({\theta}_{\text{im}})=2(1+M)\text{tan}({\theta}_{1})-M\hspace{0.17em}\text{tan}({\theta}_{\text{obj}}).$$
(50)
$${\mathcal{S}}_{1}=\frac{-(1+M)}{4M{d}_{0}\hspace{0.17em}\text{cos}({\theta}_{1})}\left[\begin{array}{cc}{\text{cos}}^{2}({\theta}_{1})& 0\\ 0& 1\end{array}\right],$$
(51)
$${\mathcal{S}}_{2}=\frac{1}{M}{\mathcal{S}}_{1},$$
(52)
$${\mathbb{S}}_{122}^{\text{I}}=\frac{{(1+M)}^{2}}{16{M}^{2}{d}_{0}^{2}}\hspace{0.17em}\text{tan}({\theta}_{1}),$$
(53)
$${\mathbb{S}}_{122}^{\text{II}}=\frac{1}{{M}^{2}}{\mathbb{S}}_{122}^{\text{I}}.$$
(54)
$${c}_{1}=\frac{-(1+M)}{4M{d}_{0}\hspace{0.17em}{\text{cos}}^{2}({\theta}_{1})},$$
(55)
$${c}_{2}={c}_{1}/M,$$
(56)
$${\kappa}_{1}={\kappa}_{2}=-1.$$
(57)
$${\tilde{M}}_{11}=\text{cos}({\theta}_{\text{obj}}){\{1+{[2(1+M)\text{tan}({\theta}_{1})-M\hspace{0.17em}\text{tan}({\theta}_{\text{obj}})]}^{2}\}}^{1/2}M.$$
(58)
$${L}_{y}=\frac{\text{cos}({\theta}_{\text{im}})\times 2\hspace{0.17em}\text{cm}}{\text{cos}({\theta}_{\text{obj}})\mid M\mid},$$
(59)
$${L}_{z}=\frac{2\hspace{0.17em}\text{cm}}{\mid M\mid}.$$
(60)
$$\begin{array}{l}{\tilde{\mathbb{L}}}_{111}=\frac{2{M}_{11}\hspace{0.17em}{\text{cos}}^{2}({\theta}_{\text{obj}})\text{tan}({\theta}_{2})}{{d}_{0}{d}_{2}({M}_{11}+{M}_{22})[{M}_{22}({d}_{0}+{d}_{1})-{d}_{2}]\text{cos}({\theta}_{\text{im}})}\\ \times \hspace{0.17em}\{{M}_{11}{d}_{0}{d}_{2}{(1+{M}_{22})}^{2}+{M}_{11}{M}_{22}{d}_{1}({d}_{0}+{d}_{1}+{d}_{2})\\ +\hspace{0.17em}{M}_{22}{d}_{2}({M}_{22}{d}_{0}-{d}_{1}-{d}_{2})\\ -\hspace{0.17em}{M}_{11}^{2}{d}_{0}[{M}_{22}({d}_{0}+{d}_{1})-{d}_{2}]\},\end{array}$$
(61)
$${\tilde{\mathbb{L}}}_{221}=-\frac{({M}_{11}+{M}_{22})({M}_{22}{d}_{0}-{d}_{1}-{d}_{2})\text{tan}({\theta}_{2})}{{d}_{0}{d}_{2}\hspace{0.17em}\text{cos}({\theta}_{\text{im}})},$$
(62)
$${\tilde{\mathbb{L}}}_{122}=\frac{{M}_{22}({M}_{22}{d}_{1}-{d}_{2}-{M}_{22}^{2}{d}_{0})\text{sin}({\theta}_{\text{obj}})}{{d}_{0}{d}_{2}}.$$