Conditions for primitive-lattice-vector-direction equal contrasts in four-beam-interference lithography

Justin L. Stay; Thomas K. Gaylord

doi:10.1364/AO.48.004801

Applied Optics
Vol. 48,
Issue 24,
pp. 4801-4813
(2009)
•https://doi.org/10.1364/AO.48.004801

Conditions for primitive-lattice-vector-direction equal contrasts in four-beam-interference lithography

Justin L. Stay and Thomas K. Gaylord

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Justin L. Stay and Thomas K. Gaylord, "Conditions for primitive-lattice-vector-direction equal contrasts in four-beam-interference lithography," Appl. Opt. 48, 4801-4813 (2009)

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Abstract

Four distinct conditions for primitive-lattice-vector-direction equal contrasts in four-beam interference are introduced and described. By maximizing the absolute contrast subject to an equal contrast condition, lithographically useful interference patterns are found. Each condition is described in terms of the corresponding constraints on the plane wave wave vectors, polarizations, and intensities. The resulting locations of global intensity maxima, minima, and saddle points are presented. Subordinate conditions for unity absolute contrast are also developed. Three lattices are treated for each condition: simple cubic, face-centered cubic, and body-centered cubic.

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Lattice	Primitive Lattice Vectors			Wave Vectors
	a	b	c	$k_{1}$	$k_{2}$	$k_{3}$	$k_{4}$
Cubic	$\frac{\sqrt{3}}{2} λ [100]$	$\frac{\sqrt{3}}{2} λ [010]$	$\frac{\sqrt{3}}{2} λ [001]$	$k_{o} \frac{1}{\sqrt{3}} [111]$	$k_{o} \frac{1}{\sqrt{3}} [- 111]$	$k_{o} \frac{1}{\sqrt{3}} [1 - 11]$	$k_{o} \frac{1}{\sqrt{3}} [11 - 1]$
Face-centered cubic	$\frac{3 \sqrt{3}}{4} λ [110]$	$\frac{3 \sqrt{3}}{4} λ [011]$	$\frac{3 \sqrt{3}}{4} λ [101]$	$k_{o} \frac{1}{3 \sqrt{3}} [333]$	$k_{o} \frac{1}{3 \sqrt{3}} [115]$	$k_{o} \frac{1}{3 \sqrt{3}} [511]$	$k_{o} \frac{1}{3 \sqrt{3}} [151]$
Body-centered cubic	$\frac{\sqrt{3}}{4} λ [- 111]$	$\frac{\sqrt{3}}{4} λ [1 - 11]$	$\frac{\sqrt{3}}{4} λ [11 - 1]$	$k_{o} \frac{1}{\sqrt{3}} [111]$	$k_{o} \frac{1}{\sqrt{3}} [1 - 1 - 1]$	$k_{o} \frac{1}{\sqrt{3}} [- 11 - 1]$	$k_{o} \frac{1}{\sqrt{3}} [- 1 - 11]$

Lattice	$E_{2} / E_{1}$	$E_{3} / E_{1}$	$E_{4} / E_{1}$	${\hat{e}}_{1}$	${\hat{e}}_{2}$	${\hat{e}}_{3}$	${\hat{e}}_{4}$	V	$V_{abs}$
Simple cubic	1	$- 1$	$- 1$	$(\begin{matrix} - 0.7887 \\ 0.5774 \\ 0.2113 \end{matrix})$	$(\begin{matrix} 0.2113 \\ - 0.5774 \\ 0.7887 \end{matrix})$	$(\begin{matrix} - 0.7887 \\ - 0.5774 \\ 0.2113 \end{matrix})$	$(\begin{matrix} 0.2113 \\ 0.5774 \\ 0.7887 \end{matrix})$	$- 1 / 6$	1.0
Face-centered cubic	$- 4.4930$	4.4930	$- 1.6271$	$(\begin{matrix} 0.4083 \\ - 0.8165 \\ 0.4083 \end{matrix})$	$(\begin{matrix} - 0.7607 \\ - 0.5902 \\ 0.2702 \end{matrix})$	$(\begin{matrix} - 0.2702 \\ 0.5902 \\ 0.7607 \end{matrix})$	$(\begin{matrix} 0.6804 \\ - 0.2722 \\ 0.6804 \end{matrix})$	$- 0.0575$	0.2599
Body-centered cubic	1	$- 1$	$- 1$	$(\begin{matrix} - 0.7887 \\ 0.5774 \\ 0.2113 \end{matrix})$	$(\begin{matrix} 0.2113 \\ - 0.5774 \\ 0.7887 \end{matrix})$	$(\begin{matrix} - 0.7887 \\ - 0.5774 \\ 0.2113 \end{matrix})$	$(\begin{matrix} 0.2113 \\ 0.5774 \\ 0.7887 \end{matrix})$	$- 1 / 6$	1.0

Lattice	$E_{2} / E_{1}$	$E_{3} / E_{1}$	$E_{4} / E_{1}$	${\hat{e}}_{1}$	${\hat{e}}_{2}$	${\hat{e}}_{3}$	${\hat{e}}_{4}$	V	$V_{abs}$
Simple cubic	0.4472	1	0.4472	$(\begin{matrix} - 0.1310 \\ - 0.6325 \\ 0.7634 \end{matrix})$	$(\begin{matrix} 0.7071 \\ 0 \\ 0.7071 \end{matrix})$	$(\begin{matrix} - 0.1310 \\ 0.6325 \\ 0.7634 \end{matrix})$	$(\begin{matrix} 0.7071 \\ 0 \\ 0.7071 \end{matrix})$	$1 / 6$	$1 / 2$
Face-centered cubic	1.2186	1.2186	1.4556	$(\begin{matrix} 0.4083 \\ - 0.8165 \\ 0.4083 \end{matrix})$	$(\begin{matrix} 0.5455 \\ - 0.8361 \\ 0.0581 \end{matrix})$	$(\begin{matrix} 0.0581 \\ - 0.8361 \\ 0.5455 \end{matrix})$	$(\begin{matrix} 0.6804 \\ - 0.2722 \\ 0.6804 \end{matrix})$	0.3719	0.8531
Body-centered cubic	$- 0.4472$	$- 0.4472$	$- 1$	$(\begin{matrix} 0.1310 \\ - 0.7634 \\ 0.6325 \end{matrix})$	$(\begin{matrix} 0.7071 \\ 0.7071 \\ 0 \end{matrix})$	$(\begin{matrix} 0.7071 \\ 0.7071 \\ 0 \end{matrix})$	$(\begin{matrix} - 0.1310 \\ 0.7634 \\ 0.6325 \end{matrix})$	$1 / 6$	$1 / 2$

Lattice	$E_{2} / E_{1}$	$E_{3} / E_{1}$	$E_{4} / E_{1}$	${\hat{e}}_{1}$	${\hat{e}}_{2}$	${\hat{e}}_{3}$	${\hat{e}}_{4}$	V	$V_{abs}$
Simple cubic	$- 1.2214$	$- 1.2214$	0.8459	$(\begin{matrix} - 0.7830 \\ 0.1911 \\ 0.5919 \end{matrix})$	$(\begin{matrix} - 0.2208 \\ - 0.7912 \\ 0.5704 \end{matrix})$	$(\begin{matrix} 0.1561 \\ 0.7721 \\ 0.6160 \end{matrix})$	$(\begin{matrix} 0.8148 \\ - 0.4532 \\ 0.3616 \end{matrix})$	$- 0.1860$	0.9140
Face-centered cubic	$- 1.8251$	$- 0.53804$	1.4018	$(\begin{matrix} - 0.7873 \\ 0.2063 \\ 0.5810 \end{matrix})$	$(\begin{matrix} - 0.3307 \\ - 0.9105 \\ 0.2482 \end{matrix})$	$(\begin{matrix} - 0.2718 \\ 0.7163 \\ 0.6427 \end{matrix})$	$(\begin{matrix} 0.7405 \\ - 0.2711 \\ 0.6150 \end{matrix})$	$- 0.1201$	0.5461
Body-centered cubic	1.3265	$- 0.9333$	$- 1.2221$	$(\begin{matrix} - 0.7830 \\ 0.5919 \\ 0.1911 \end{matrix})$	$(\begin{matrix} 0.2208 \\ - 0.5704 \\ 0.7912 \end{matrix})$	$(\begin{matrix} - 0.8148 \\ - 0.3616 \\ 0.4532 \end{matrix})$	$(\begin{matrix} 0.1561 \\ 0.6160 \\ 0.7721 \end{matrix})$	$- 0.1860$	0.9140

Lattice	$E_{2} / E_{1}$	$E_{3} / E_{1}$	$E_{4} / E_{1}$	${\hat{e}}_{1}$	${\hat{e}}_{2}$	${\hat{e}}_{3}$	${\hat{e}}_{4}$	V	$V_{abs}$
Simple cubic	1	2	2	$(\begin{matrix} 0.0 \\ - 0.7071 \\ 0.7071 \end{matrix})$	$(\begin{matrix} 0.0 \\ - 0.7071 \\ 0.7071 \end{matrix})$	$(\begin{matrix} - 0.7071 \\ - 0.7071 \\ 0.0 \end{matrix})$	$(\begin{matrix} 0.7071 \\ - 0.7071 \\ 0.0 \end{matrix})$	$1 / 5$	$2 / 3$
Face-centered cubic	1	2	$- \sqrt{2}$	$(\begin{matrix} 0.7071 \\ - 0.7071 \\ 0.0 \end{matrix})$	$(\begin{matrix} 0.7071 \\ - 0.7071 \\ 0.0 \end{matrix})$	$(\begin{matrix} 0.0969 \\ - 0.9031 \\ 0.4184 \end{matrix})$	$(\begin{matrix} - 0.9031 \\ 0.0969 \\ 0.4184 \end{matrix})$	$1 / 3$	1.0
Body-centered cubic	1	2	$- 2$	$(\begin{matrix} 0.0 \\ - 0.7071 \\ 0.7071 \end{matrix})$	$(\begin{matrix} 0.0 \\ - 0.7071 \\ 0.7071 \end{matrix})$	$(\begin{matrix} - 0.7071 \\ - 0.7071 \\ 0.0 \end{matrix})$	$(\begin{matrix} - 0.7071 \\ 0.7071 \\ 0.0 \end{matrix})$	$1 / 5$	$2 / 3$

Lattice	$E_{2} / E_{1}$	$E_{3} / E_{1}$	$E_{4} / E_{1}$	${\hat{e}}_{1}$	${\hat{e}}_{2}$	${\hat{e}}_{3}$	${\hat{e}}_{4}$	V	$V_{abs}$
Simple cubic	$- 0.78901$	0.7769	$- 0.5049$	$(\begin{matrix} 0.6395 \\ - 0.7594 \\ 0.1199 \end{matrix})$	$(\begin{matrix} - 0.4298 \\ - 0.8161 \\ 0.3863 \end{matrix})$	$(\begin{matrix} - 0.6613 \\ 0.0841 \\ 0.7454 \end{matrix})$	$(\begin{matrix} 0.7417 \\ - 0.0752 \\ 0.6665 \end{matrix})$	$- 0.2488$	0.9942
Face-centered cubic	$- 0.8742$	0.8915	$- 0.3104$	$(\begin{matrix} - 0.1980 \\ - 0.5870 \\ 0.7850 \end{matrix})$	$(\begin{matrix} - 0.9810 \\ 0.0610 \\ 0.1840 \end{matrix})$	$(\begin{matrix} - 0.2392 \\ 0.9353 \\ 0.2608 \end{matrix})$	$(\begin{matrix} 0.1661 \\ - 0.2252 \\ 0.9601 \end{matrix})$	$- 0.1994$	0.7620
Body-centered cubic	0.7769	$- 0.78901$	$- 0.5049$	$(\begin{matrix} - 0.7594 \\ 0.6395 \\ 0.1199 \end{matrix})$	$(\begin{matrix} 0.0841 \\ - 0.6613 \\ 0.7454 \end{matrix})$	$(\begin{matrix} - 0.8161 \\ - 0.4298 \\ 0.3863 \end{matrix})$	$(\begin{matrix} - 0.0752 \\ 0.7417 \\ 0.6665 \end{matrix})$	$- 0.2488$	0.9942

Lattice	$E_{2} / E_{1}$	$E_{3} / E_{1}$	$E_{4} / E_{1}$	${\hat{e}}_{1}$	${\hat{e}}_{2}$	${\hat{e}}_{3}$	${\hat{e}}_{4}$	V	$V_{abs}$
Simple cubic	$- 0.5956$	0.4575	0.4782	$(\begin{matrix} - 0.7193 \\ 0.0251 \\ 0.6942 \end{matrix})$	$(\begin{matrix} 0.7588 \\ 0.6405 \\ 0.1183 \end{matrix})$	$(\begin{matrix} - 0.7473 \\ - 0.6585 \\ 0.0888 \end{matrix})$	$(\begin{matrix} - 0.6509 \\ 0.7524 \\ 0.1015 \end{matrix})$	0.2974	0.7903
Face-centered cubic	$- 0.7226$	$- 0.6966$	0.6943	$(\begin{matrix} - 0.7201 \\ 0.6934 \\ 0.0268 \end{matrix})$	$(\begin{matrix} 0.0071 \\ - 0.9808 \\ 0.1947 \end{matrix})$	$(\begin{matrix} 0.1083 \\ - 0.9195 \\ 0.3779 \end{matrix})$	$(\begin{matrix} - 0.8840 \\ 0.0849 \\ 0.4598 \end{matrix})$	0.3948	0.9899
Body-centered cubic	0.4575	$- 0.5956$	0.4782	$(\begin{matrix} 0.0251 \\ - 0.7193 \\ 0.6942 \end{matrix})$	$(\begin{matrix} - 0.6585 \\ - 0.7473 \\ 0.0888 \end{matrix})$	$(\begin{matrix} 0.6405 \\ 0.7588 \\ 0.1183 \end{matrix})$	$(\begin{matrix} 0.7524 \\ - 0.6509 \\ 0.1015 \end{matrix})$	0.2974	0.7903

Lattice	$E_{2} / E_{1}$	$E_{3} / E_{1}$	$E_{4} / E_{1}$	${\hat{e}}_{1}$	${\hat{e}}_{2}$	${\hat{e}}_{3}$	${\hat{e}}_{4}$	V	$V_{abs}$
Simple cubic	$2 / 3$	$- 2 / 3$	$1 / 3$	$(\begin{matrix} 0.5826 \\ - 0.7867 \\ 0.2041 \end{matrix})$	$(\begin{matrix} - 0.5000 \\ - 0.8090 \\ 0.3090 \end{matrix})$	$(\begin{matrix} - 0.3090 \\ 0.5000 \\ 0.8090 \end{matrix})$	$(\begin{matrix} 0.8090 \\ - 0.3090 \\ 0.5000 \end{matrix})$	$\sqrt{6} / 9$	$\sqrt{6} / 3$
Face-centered cubic	$2 / 3$	$1 / 3$	$- 2 / 3$	$(\begin{matrix} - 0.7144 \\ 0.6995 \\ 0.0149 \end{matrix})$	$(\begin{matrix} - 0.9045 \\ - 0.3455 \\ 0.2500 \end{matrix})$	$(\begin{matrix} - 0.2500 \\ 0.9045 \\ 0.3455 \end{matrix})$	$(\begin{matrix} 0.3455 \\ - 0.2500 \\ 0.9045 \end{matrix})$	$\sqrt{6} / 9$	$\sqrt{6} / 3$
Body-centered cubic	$1 / 3$	$2 / 3$	$2 / 3$	$(\begin{matrix} - 0.2041 \\ - 0.5826 \\ 0.7867 \end{matrix})$	$(\begin{matrix} - 0.5000 \\ - 0.8090 \\ 0.3090 \end{matrix})$	$(\begin{matrix} - 0.3090 \\ 0.5000 \\ 0.8090 \end{matrix})$	$(\begin{matrix} 0.8090 \\ - 0.3090 \\ 0.5000 \end{matrix})$	$\sqrt{6} / 9$	$\sqrt{6} / 3$

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Equations (47)

Applied Optics