The number of phase levels of a Talbot array illuminator is an
important factor in the estimation of practical fabrication complexity
and cost. We show that the number (L) of phase
levels of a Talbot array illuminator has a simple relationship to the
prime number. When there is an alternative π-phase modulation in
the output array, the relations are similar.
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t, t1, t2,
and t3 are prime numbers. M(odd) is an odd number. L(2n) and
L[
M(odd)] are the numbers of phase levels that
correspond to M = 2n and
M(odd),
respectively. L(2n) can be obtained from
Eq. (22). M can be decomposed into
M(1), … , M(n) without common divisors among
them. L[
M(1)], … ,
L[
M(n)] are the numbers of phase levels that correspond to
M(1), … , M(n).
Table 2
Numerical Examples of the Number (L) of
Phase Levels of Talbot Array Illuminators and the Opening Ratio
(1/M) of the Illumination Array up to
M = 16
M
3
4
5
6
7
8
9
10
11
12
13
14
15
16
L
2
3
3
4
4
4
4
6
6
6
7
8
6
7
Table 3
Simple Relations among the Number (L) of
Phase Levels of π-Phase-Modulated Talbot Array Illuminators and the
Intensity-Opening Ratio (1/MI) of the
Generated Arraya
t, t1, t2,
and t3 are prime numbers. MI(odd) is an odd number. L[
MI(odd)] is the
number of phase levels that corresponds to
MI(odd). MI(odd) =
MI(1) … MI(n) and no common
divisor among MI(1), … ,
MI(n). L[
MI(1)], … ,
L[
MI(n)] are the numbers of phase levels that
correspond to MI(1), … ,
MI(n).
Table 4
Numerical Examples of the Number (L) of
Phase Levels of the Talbot Array Illuminator and the Intensity-Opening
Ratio (1/MI) of the Illumination Array in
the π-Phase Modulated Case up to MI = 16
M
3
4
5
6
7
8
9
10
11
12
13
14
15
16
L
4
4
6
4
8
8
8
6
12
8
14
8
12
16
Tables (4)
Table 1
Simple Relations between the Number (L) of
Phase Levels of Talbot Array Illuminators and the Opening Ratio
(1/M) of the Generated Arraya
t, t1, t2,
and t3 are prime numbers. M(odd) is an odd number. L(2n) and
L[
M(odd)] are the numbers of phase levels that
correspond to M = 2n and
M(odd),
respectively. L(2n) can be obtained from
Eq. (22). M can be decomposed into
M(1), … , M(n) without common divisors among
them. L[
M(1)], … ,
L[
M(n)] are the numbers of phase levels that correspond to
M(1), … , M(n).
Table 2
Numerical Examples of the Number (L) of
Phase Levels of Talbot Array Illuminators and the Opening Ratio
(1/M) of the Illumination Array up to
M = 16
M
3
4
5
6
7
8
9
10
11
12
13
14
15
16
L
2
3
3
4
4
4
4
6
6
6
7
8
6
7
Table 3
Simple Relations among the Number (L) of
Phase Levels of π-Phase-Modulated Talbot Array Illuminators and the
Intensity-Opening Ratio (1/MI) of the
Generated Arraya
t, t1, t2,
and t3 are prime numbers. MI(odd) is an odd number. L[
MI(odd)] is the
number of phase levels that corresponds to
MI(odd). MI(odd) =
MI(1) … MI(n) and no common
divisor among MI(1), … ,
MI(n). L[
MI(1)], … ,
L[
MI(n)] are the numbers of phase levels that
correspond to MI(1), … ,
MI(n).
Table 4
Numerical Examples of the Number (L) of
Phase Levels of the Talbot Array Illuminator and the Intensity-Opening
Ratio (1/MI) of the Illumination Array in
the π-Phase Modulated Case up to MI = 16