Optimal Patterns for Alignment

W. Makous

doi:10.1364/AO.13.000659

Applied Optics
Vol. 13,
Issue 3,
pp. 659-664
(1974)
•https://doi.org/10.1364/AO.13.000659

Optimal Patterns for Alignment

W. Makous

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W. Makous, "Optimal Patterns for Alignment," Appl. Opt. 13, 659-664 (1974)

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Abstract

Patterns with optimal autocorrelation functions are sometimes optimal for optical alignment. When the finest gratings that can be resolved exceed those that actually can be used, errors of alignment can be reduced by nearly a factor of 2 by incorporating segments of a grating into the patterns. Under some conditions, resolution of the inspecting device need not limit accuracy of alignment. Bull’s-eye patterns may be optimal for visual alignment.

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x	1		2		3		4		5		6		7	E(S)
∑\|f(x)	1		1		1		0		1		0		0
− f(x − 0)\| =	1		1		1		0		1		0		0
E(S_t₌₀) =	0	+	0	+	0	+	0	+	0	+	0	+	0	= 0
∑\|f(x)	1		1		1		0		1		0		0
− f(x − 1)\| =	0		1		1		1		0		1		0
E(S_t₌₁) =	1	+	0	+	0	+	1	+	1	+	1	+	0	= 4
∑\|f(x)	1		1		1		0		1		0		0
− f(x − 2)\| =	0		0		1		1		1		0		1
E(S_t₌₂) =	1	+	1	+	0	+	1	+	0	+	0	+	1	= 4
∑\|f(x)	1		1		1		0		1		0		0
− f(x − 3)\| =	1		0		0		1		1		1		0
E(S_t₌₃) =	0	+	1	+	1	+	1	+	0	+	1	+	0	= 4
∑\|f(x)	1		1		1		0		1		0		0
− f(x − 4)\| =	0		1		0		0		1		1		1
E(S_t₌₄) =	1	+	0	+	1	+	0	+	0	+	1	+	1	= 4
∑\|f(x)	1		1		1		0		1		0		0
− f(x − 5)\| =	1		0		1		0		0		1		1
E(S_t₌₅) =	0	+	1	+	0	+	0	+	1	+	1	+	1	= 4
∑\|f(x)	1		1		1		0		1		0		0
− f(x − 6)\| =	1		1		0		1		0		0		1
E(S_t₌₆) =	0	+	0	+	1	+	1	+	1	+	0	+	1	= 4
$E (S_{t}) = \sum_{x = 1}^{7} ∣ f (x) - f (x - t) ∣$

t	E(S)_m	E(S)_m′	z_m	z_m′	p_m	p_m′	(1 − p_m)	(1 − p_m′)	tp_m	tp_m′
1	8	10	0.566	0.707	0.2857	0.2398	0.7143	0.7602	0.2857	0.2398
2	16	20	1.131	1.414	0.1290	0.0787	0.8710	0.9213	0.2580	0.1574
3	24	30	1.697	2.121	0.0449	0.0170	0.9551	0.9830	0.1347	0.0510
4	32	40	2.263	2.828	0.0118	0.0023	0.9882	0.9977	0.0472	0.0092
5	40	50	2.828	3.536	0.0023	0.0002	0.9977	0.9998	0.0115	0.0010
6	48	60	3.394	4.243	0.0004	0.0000	0.9996	1.0000	0.0024	0.0000
7	56	70	3.960	4.950	0.0000	0.0000	1.0000	1.0000	0.0000	0.0000
8	64	80	4.525	5.657	0.0000	0.0000	1.0000	1.0000	0.0000	0.0000
9	72	90	5.091	6.364	0.0000	0.0000	1.0000	1.0000	0.0000	0.0000
10	81	100	5.728	7.071	0.0000	0.0000	1.0000	1.0000	0.0000	0.0000
Probability of alignment							0.5856	0.6867
Probability of misalignment							0.4144	0.3133
Mean error = ∑tp									0.7395	0.4584

	m Sequence	m′ Sequence

		p = 2	p > 2
E(S_t₌₀)	0	0	0
E(S_t₌₁)	$\frac{n + 1}{2}$	$\frac{3 n_{2}}{4} - \frac{k}{2}$	$n_{p} (1 - \frac{1}{2 p}) - \frac{k}{2}$
E(S_t₌₂)	$\frac{n + 1}{2}$	$\frac{n_{2}}{2} + k$	$\frac{n_{p}}{p} + k$
μ^(′) = E(S_t_≠0)_min	$\frac{n + 1}{2}$	$\frac{n_{2}}{2} - k$	$\frac{n_{p}}{p} + k$
$\frac{μ^{'}}{μ}$		$\frac{3}{2} - \frac{k + 3}{n + 1}$	$2 - \frac{1}{p} - \frac{2 p + k - 1}{n + 1}$

x	1		2		3		4		5		6		7	E(S)
∑\|f(x)	1		1		1		0		1		0		0
− f(x − 0)\| =	1		1		1		0		1		0		0
E(S_t₌₀) =	0	+	0	+	0	+	0	+	0	+	0	+	0	= 0
∑\|f(x)	1		1		1		0		1		0		0
− f(x − 1)\| =	0		1		1		1		0		1		0
E(S_t₌₁) =	1	+	0	+	0	+	1	+	1	+	1	+	0	= 4
∑\|f(x)	1		1		1		0		1		0		0
− f(x − 2)\| =	0		0		1		1		1		0		1
E(S_t₌₂) =	1	+	1	+	0	+	1	+	0	+	0	+	1	= 4
∑\|f(x)	1		1		1		0		1		0		0
− f(x − 3)\| =	1		0		0		1		1		1		0
E(S_t₌₃) =	0	+	1	+	1	+	1	+	0	+	1	+	0	= 4
∑\|f(x)	1		1		1		0		1		0		0
− f(x − 4)\| =	0		1		0		0		1		1		1
E(S_t₌₄) =	1	+	0	+	1	+	0	+	0	+	1	+	1	= 4
∑\|f(x)	1		1		1		0		1		0		0
− f(x − 5)\| =	1		0		1		0		0		1		1
E(S_t₌₅) =	0	+	1	+	0	+	0	+	1	+	1	+	1	= 4
∑\|f(x)	1		1		1		0		1		0		0
− f(x − 6)\| =	1		1		0		1		0		0		1
E(S_t₌₆) =	0	+	0	+	1	+	1	+	1	+	0	+	1	= 4
$E (S_{t}) = \sum_{x = 1}^{7} ∣ f (x) - f (x - t) ∣$

t	E(S)_m	E(S)_m′	z_m	z_m′	p_m	p_m′	(1 − p_m)	(1 − p_m′)	tp_m	tp_m′
1	8	10	0.566	0.707	0.2857	0.2398	0.7143	0.7602	0.2857	0.2398
2	16	20	1.131	1.414	0.1290	0.0787	0.8710	0.9213	0.2580	0.1574
3	24	30	1.697	2.121	0.0449	0.0170	0.9551	0.9830	0.1347	0.0510
4	32	40	2.263	2.828	0.0118	0.0023	0.9882	0.9977	0.0472	0.0092
5	40	50	2.828	3.536	0.0023	0.0002	0.9977	0.9998	0.0115	0.0010
6	48	60	3.394	4.243	0.0004	0.0000	0.9996	1.0000	0.0024	0.0000
7	56	70	3.960	4.950	0.0000	0.0000	1.0000	1.0000	0.0000	0.0000
8	64	80	4.525	5.657	0.0000	0.0000	1.0000	1.0000	0.0000	0.0000
9	72	90	5.091	6.364	0.0000	0.0000	1.0000	1.0000	0.0000	0.0000
10	81	100	5.728	7.071	0.0000	0.0000	1.0000	1.0000	0.0000	0.0000
Probability of alignment							0.5856	0.6867
Probability of misalignment							0.4144	0.3133
Mean error = ∑tp									0.7395	0.4584

m sequence (n = 7):	1		1		1			1		0
m′ sequence (n = 14):	1	0	1	0	1	0	1	1	0	0	1	1
m sequence (n = 15):	1	1	1	1	0	0	1	0	0	1	1	1	0

Abstract

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Figures (6)

Tables (4)

Equations (3)

Applied Optics