Polarization encoded optical N-valued inverter

Tanay Chattopadhyay; Panchatapa Bhowmik; Jitendra Nath Roy

doi:10.1364/JOSAB.29.002852

Journal of the Optical Society of America B
Vol. 29,
Issue 10,
pp. 2852-2860
(2012)
•https://doi.org/10.1364/JOSAB.29.002852

Polarization encoded optical $N$ -valued inverter

Tanay Chattopadhyay, Panchatapa Bhowmik, and Jitendra Nath Roy

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Tanay Chattopadhyay, Panchatapa Bhowmik, and Jitendra Nath Roy, "Polarization encoded optical N-valued inverter," J. Opt. Soc. Am. B 29, 2852-2860 (2012)

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Abstract

An inverter, in wider sense, may be defined as a transformation of an incoming state into an outgoing state in a logic-based information processing system. In this paper, a simple circuit for inverters two-valued (binary), three-valued (ternary), and four-valued (quaternary) logical systems has been designed. An outline has been given for designing of all possible 27 three-valued inverters. The circuit is designed with the help of a polarization converter and a polarization isolator only. Mathematical analysis of the polarization converters is established with Jones matrix. This circuit will be useful in a future all-optical multivalued logic-based information processing system.

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Inverter		PCM Block Used
Number	Output	First Stage PCM-1	Second Stage PCM-2	Name of the Logic Functions and Its Mathematical Expressions
$F_{1}$	$[\begin{matrix} 1 & 1 & 2 \end{matrix}]$	PC-1	—	${}^{\underset{\neg}{1}}x = {\begin{matrix} (x + 1) & if x = 0 \\ x & otherwise \end{matrix}$
		—	—
		—	—
$F_{2}$	$[\begin{matrix} 1 & 2 & 1 \end{matrix}]$	PC-1	—	${}^{\underset{\neg}{2}}x = {\begin{matrix} (x + 1) & if x \geq 0 \\ (x + 2) & otherwise \end{matrix}$
		PC-3	—
		PC-3	PC-1
$F_{3}$	$[\begin{matrix} 2 & 0 & 2 \end{matrix}]$	PC-2	—	${}^{\underset{\neg}{3}}x = {\begin{matrix} (x + 2) & if x \leq 1 \\ x & otherwise \end{matrix}$
		PC-1	—
		—	—
$F_{4}$	$[\begin{matrix} 1 & 0 & 0 \end{matrix}]$	PC-1	—	Delta interval type-1, $\overset{a}{x} = {\begin{matrix} 1 & if x = a \\ 0 & otherwise \end{matrix}$ $\overset{0}{x} = [\begin{matrix} 1 & 0 & 0 \end{matrix}]$ $\overset{1}{x} = [\begin{matrix} 0 & 1 & 0 \end{matrix}]$ $\overset{2}{x} = [\begin{matrix} 0 & 0 & 1 \end{matrix}]$
		PC-1	—
		PC-3	—
$F_{5}$	$[\begin{matrix} 0 & 1 & 0 \end{matrix}]$	—	—
		—	—
		PC-3	—
$F_{6}$	$[\begin{matrix} 0 & 0 & 1 \end{matrix}]$	—	—
		PC-1	—
		PC-3	PC-1
$F_{7}$ $F_{8}$	$[\begin{matrix} 2 & 0 & 0 \end{matrix}]$ $[\begin{matrix} 0 & 2 & 0 \end{matrix}]$	PC-2	—	Delta interval type-2, $x^{a} = {\begin{matrix} 2 & if x = a \\ 0 & otherwise \end{matrix}$ $x^{0} = [\begin{matrix} 2 & 0 & 0 \end{matrix}]$ , (NTI) $x^{1} = [\begin{matrix} 0 & 2 & 0 \end{matrix}]$ $x^{2} = [\begin{matrix} 0 & 0 & 2 \end{matrix}]$ $NTI = negative ternary inverter$
		PC-1	—
		PC-3	—
		—	—
		PC-3	—
		PC-3	—
$F_{9}$	$[\begin{matrix} 0 & 0 & 2 \end{matrix}]$	—	—
		PC-1	—
		—	—
$F_{10}$	$[\begin{matrix} 0 & 0 & 0 \end{matrix}]$	—	—	${}^{\underset{\neg}{10}}x = 0$
		PC-1	—
		PC-3	—
$F_{11}$	$[\begin{matrix} 0 & 1 & 2 \end{matrix}]$	—	—	Identity reversible inverter, ${}^{\underset{\neg}{11}}x = x$
		—	—
		—	—
$F_{12}$	$[\begin{matrix} 0 & 2 & 1 \end{matrix}]$	—	—	Reversible inverters, ${}^{\underset{\neg}{12}}x = (2 - x) + 1$ ${}^{\underset{\neg}{13}}x = 2 - (1 + x)$
		PC-3	—
		PC-3	PC-1
$F_{13}$	$[\begin{matrix} 1 & 0 & 2 \end{matrix}]$	PC-1	—
		PC-1	—
		—	—
$F_{14}$	$[\begin{matrix} 2 & 1 & 0 \end{matrix}]$	PC-2	—	Reversible inverter/simple ternary inverter (STI), ${}^{\underset{\neg}{14}}x = \bar{x} = (2 - x)$
		—	—
		PC-3	—
$F_{15}$	$[\begin{matrix} 1 & 2 & 0 \end{matrix}]$	PC-1	—	Cycle/successor/universal inverter of type-1/reversible inverter ${}^{\underset{\neg}{15}}x = Suc (x) = (x + 1)$
		PC-3	—
		PC-3	—
$F_{16}$	$[\begin{matrix} 2 & 0 & 1 \end{matrix}]$	PC-2	—	Universal Inverter of type-2/reversible inverter ${}^{\underset{\neg}{16}}x = (x + 2)$
		PC-1	—
		PC-3	PC-1
$F_{17}$	$[\begin{matrix} 1 & 1 & 1 \end{matrix}]$	PC-1	—	${}^{\underset{\neg}{17}}x = 1$
		—	—
		PC-3	PC-1
$F_{18}$	$[\begin{matrix} 2 & 1 & 1 \end{matrix}]$	PC-2	—	${}^{\underset{\neg}{18}}x = {\begin{matrix} (2 - x) & if x < 2 \\ (2 + x) & if x = 2 \end{matrix}$
		—	—
		PC-3	PC-1
$F_{19}$	$[\begin{matrix} 1 & 1 & 0 \end{matrix}]$	PC-1	—	Lower open semi-interval ${}^{\underset{\neg}{19}}x = {}^{a [}x = {\begin{matrix} 1 & if x < a \\ 0 & otherwise \end{matrix}$ ${}^{2 [}x = [\begin{matrix} 1 & 1 & 0 \end{matrix}]$
		—	—
		PC-3	—
$F_{20}$	$[\begin{matrix} 1 & 0 & 1 \end{matrix}]$	PC-1	—	${}^{\underset{\neg}{20}}x = {\begin{matrix} (1 - x) & if x \leq 1 \\ (x - 1) & if x > 1 \end{matrix}$
		PC-1	—
		PC-3	PC-1
$F_{21}$	$[\begin{matrix} 0 & 1 & 1 \end{matrix}]$	—	—	Upper open semi-interval ${{}^{]}x}^{a} = {\begin{matrix} 1 & if a < x \\ 0 & otherwise \end{matrix}$ ${{}^{]}x}^{0} = [\begin{matrix} 0 & 1 & 1 \end{matrix}]$
		—	—
		PC-3	PC-1
$F_{22}$	$[\begin{matrix} 2 & 2 & 0 \end{matrix}]$	PC-2	—	Literals, ${{}^{a}x}^{b} = {\begin{matrix} 2 & if a \leq x \geq b \\ 0 & otherwise \end{matrix}$ ${{}^{0}x}^{1} = [\begin{matrix} 2 & 2 & 0 \end{matrix}]$ , (PTI) ${{}^{0}x}^{2} = [\begin{matrix} 2 & 2 & 2 \end{matrix}]$ ${{}^{1}x}^{2} = [\begin{matrix} 0 & 2 & 2 \end{matrix}]$ $PTI = positive ternary inverter$
		PC-3	—
		PC-3	—
$F_{23}$	$[\begin{matrix} 2 & 2 & 2 \end{matrix}]$	PC-2	—
		PC-3	—
		—	—
$F_{24}$	$[\begin{matrix} 0 & 2 & 2 \end{matrix}]$	—	—
		PC-3	—
		—	—
$F_{25}$	$[\begin{matrix} 2 & 2 & 1 \end{matrix}]$	PC-2	—	${}^{\underset{\neg}{25}}x = {\begin{matrix} (x - 1) & if x > 1 \\ 2 & otherwise \end{matrix}$
		PC-3	—
		PC-3	PC-1
$F_{26}$	$[\begin{matrix} 2 & 1 & 2 \end{matrix}]$	PC-2	—	${}^{\underset{\neg}{26}}x = {\begin{matrix} (2 - x) & if x < 2 \\ x & otherwise \end{matrix}$
		—	—
		—	—
$F_{27}$	$[\begin{matrix} 1 & 2 & 2 \end{matrix}]$	PC-1	—	${}^{\underset{\neg}{27}}x = {\begin{matrix} (x + 1) & if x = 0 \\ 2 & otherwise \end{matrix}$
		PC-3	—
		—	—

Inverter		PCM Block Used
Number	Output	First Stage PCM-1	Second Stage PCM-2	Name of the Logic Functions and Its Mathematical Expressions
$F_{1}$	$[\begin{matrix} 1 & 1 & 2 \end{matrix}]$	PC-1	—	${}^{\underset{\neg}{1}}x = {\begin{matrix} (x + 1) & if x = 0 \\ x & otherwise \end{matrix}$
		—	—
		—	—
$F_{2}$	$[\begin{matrix} 1 & 2 & 1 \end{matrix}]$	PC-1	—	${}^{\underset{\neg}{2}}x = {\begin{matrix} (x + 1) & if x \geq 0 \\ (x + 2) & otherwise \end{matrix}$
		PC-3	—
		PC-3	PC-1
$F_{3}$	$[\begin{matrix} 2 & 0 & 2 \end{matrix}]$	PC-2	—	${}^{\underset{\neg}{3}}x = {\begin{matrix} (x + 2) & if x \leq 1 \\ x & otherwise \end{matrix}$
		PC-1	—
		—	—
$F_{4}$	$[\begin{matrix} 1 & 0 & 0 \end{matrix}]$	PC-1	—	Delta interval type-1, $\overset{a}{x} = {\begin{matrix} 1 & if x = a \\ 0 & otherwise \end{matrix}$ $\overset{0}{x} = [\begin{matrix} 1 & 0 & 0 \end{matrix}]$ $\overset{1}{x} = [\begin{matrix} 0 & 1 & 0 \end{matrix}]$ $\overset{2}{x} = [\begin{matrix} 0 & 0 & 1 \end{matrix}]$
		PC-1	—
		PC-3	—
$F_{5}$	$[\begin{matrix} 0 & 1 & 0 \end{matrix}]$	—	—
		—	—
		PC-3	—
$F_{6}$	$[\begin{matrix} 0 & 0 & 1 \end{matrix}]$	—	—
		PC-1	—
		PC-3	PC-1
$F_{7}$ $F_{8}$	$[\begin{matrix} 2 & 0 & 0 \end{matrix}]$ $[\begin{matrix} 0 & 2 & 0 \end{matrix}]$	PC-2	—	Delta interval type-2, $x^{a} = {\begin{matrix} 2 & if x = a \\ 0 & otherwise \end{matrix}$ $x^{0} = [\begin{matrix} 2 & 0 & 0 \end{matrix}]$ , (NTI) $x^{1} = [\begin{matrix} 0 & 2 & 0 \end{matrix}]$ $x^{2} = [\begin{matrix} 0 & 0 & 2 \end{matrix}]$ $NTI = negative ternary inverter$
		PC-1	—
		PC-3	—
		—	—
		PC-3	—
		PC-3	—
$F_{9}$	$[\begin{matrix} 0 & 0 & 2 \end{matrix}]$	—	—
		PC-1	—
		—	—
$F_{10}$	$[\begin{matrix} 0 & 0 & 0 \end{matrix}]$	—	—	${}^{\underset{\neg}{10}}x = 0$
		PC-1	—
		PC-3	—
$F_{11}$	$[\begin{matrix} 0 & 1 & 2 \end{matrix}]$	—	—	Identity reversible inverter, ${}^{\underset{\neg}{11}}x = x$
		—	—
		—	—
$F_{12}$	$[\begin{matrix} 0 & 2 & 1 \end{matrix}]$	—	—	Reversible inverters, ${}^{\underset{\neg}{12}}x = (2 - x) + 1$ ${}^{\underset{\neg}{13}}x = 2 - (1 + x)$
		PC-3	—
		PC-3	PC-1
$F_{13}$	$[\begin{matrix} 1 & 0 & 2 \end{matrix}]$	PC-1	—
		PC-1	—
		—	—
$F_{14}$	$[\begin{matrix} 2 & 1 & 0 \end{matrix}]$	PC-2	—	Reversible inverter/simple ternary inverter (STI), ${}^{\underset{\neg}{14}}x = \bar{x} = (2 - x)$
		—	—
		PC-3	—
$F_{15}$	$[\begin{matrix} 1 & 2 & 0 \end{matrix}]$	PC-1	—	Cycle/successor/universal inverter of type-1/reversible inverter ${}^{\underset{\neg}{15}}x = Suc (x) = (x + 1)$
		PC-3	—
		PC-3	—
$F_{16}$	$[\begin{matrix} 2 & 0 & 1 \end{matrix}]$	PC-2	—	Universal Inverter of type-2/reversible inverter ${}^{\underset{\neg}{16}}x = (x + 2)$
		PC-1	—
		PC-3	PC-1
$F_{17}$	$[\begin{matrix} 1 & 1 & 1 \end{matrix}]$	PC-1	—	${}^{\underset{\neg}{17}}x = 1$
		—	—
		PC-3	PC-1
$F_{18}$	$[\begin{matrix} 2 & 1 & 1 \end{matrix}]$	PC-2	—	${}^{\underset{\neg}{18}}x = {\begin{matrix} (2 - x) & if x < 2 \\ (2 + x) & if x = 2 \end{matrix}$
		—	—
		PC-3	PC-1
$F_{19}$	$[\begin{matrix} 1 & 1 & 0 \end{matrix}]$	PC-1	—	Lower open semi-interval ${}^{\underset{\neg}{19}}x = {}^{a [}x = {\begin{matrix} 1 & if x < a \\ 0 & otherwise \end{matrix}$ ${}^{2 [}x = [\begin{matrix} 1 & 1 & 0 \end{matrix}]$
		—	—
		PC-3	—
$F_{20}$	$[\begin{matrix} 1 & 0 & 1 \end{matrix}]$	PC-1	—	${}^{\underset{\neg}{20}}x = {\begin{matrix} (1 - x) & if x \leq 1 \\ (x - 1) & if x > 1 \end{matrix}$
		PC-1	—
		PC-3	PC-1
$F_{21}$	$[\begin{matrix} 0 & 1 & 1 \end{matrix}]$	—	—	Upper open semi-interval ${{}^{]}x}^{a} = {\begin{matrix} 1 & if a < x \\ 0 & otherwise \end{matrix}$ ${{}^{]}x}^{0} = [\begin{matrix} 0 & 1 & 1 \end{matrix}]$
		—	—
		PC-3	PC-1
$F_{22}$	$[\begin{matrix} 2 & 2 & 0 \end{matrix}]$	PC-2	—	Literals, ${{}^{a}x}^{b} = {\begin{matrix} 2 & if a \leq x \geq b \\ 0 & otherwise \end{matrix}$ ${{}^{0}x}^{1} = [\begin{matrix} 2 & 2 & 0 \end{matrix}]$ , (PTI) ${{}^{0}x}^{2} = [\begin{matrix} 2 & 2 & 2 \end{matrix}]$ ${{}^{1}x}^{2} = [\begin{matrix} 0 & 2 & 2 \end{matrix}]$ $PTI = positive ternary inverter$
		PC-3	—
		PC-3	—
$F_{23}$	$[\begin{matrix} 2 & 2 & 2 \end{matrix}]$	PC-2	—
		PC-3	—
		—	—
$F_{24}$	$[\begin{matrix} 0 & 2 & 2 \end{matrix}]$	—	—
		PC-3	—
		—	—
$F_{25}$	$[\begin{matrix} 2 & 2 & 1 \end{matrix}]$	PC-2	—	${}^{\underset{\neg}{25}}x = {\begin{matrix} (x - 1) & if x > 1 \\ 2 & otherwise \end{matrix}$
		PC-3	—
		PC-3	PC-1
$F_{26}$	$[\begin{matrix} 2 & 1 & 2 \end{matrix}]$	PC-2	—	${}^{\underset{\neg}{26}}x = {\begin{matrix} (2 - x) & if x < 2 \\ x & otherwise \end{matrix}$
		—	—
		—	—
$F_{27}$	$[\begin{matrix} 1 & 2 & 2 \end{matrix}]$	PC-1	—	${}^{\underset{\neg}{27}}x = {\begin{matrix} (x + 1) & if x = 0 \\ 2 & otherwise \end{matrix}$
		PC-3	—
		—	—

Inputs	TUI First Type Outputs	TUI Second Type Outputs
0	1	2
1	2	0
2	0	1

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Equations (22)

Journal of the Optical Society of America B