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, t_{1}, t_{2},
and t_{3} are prime numbers. M(odd) is an odd number. L(2^{
n
}) and
L[
M(odd)] are the numbers of phase levels that
correspond to M = 2^{
n
} and
M(odd),
respectively. L(2^{
n
}) 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/M_{
I
}) of the
Generated Arraya

t, t_{1}, t_{2},
and t_{3} are prime numbers. M_{
I
}(odd) is an odd number. L[
M_{
I
}(odd)] is the
number of phase levels that corresponds to
M_{
I
}(odd). M_{
I
}(odd) =
M_{
I
}(1) … M_{
I
}(n) and no common
divisor among M_{
I
}(1), … ,
M_{
I
}(n). L[
M_{
I
}(1)], … ,
L[
M_{
I
}(n)] are the numbers of phase levels that
correspond to M_{
I
}(1), … ,
M_{
I
}(n).

Table 4

Numerical Examples of the Number (L) of
Phase Levels of the Talbot Array Illuminator and the Intensity-Opening
Ratio (1/M_{
I
}) of the Illumination Array in
the π-Phase Modulated Case up to M_{
I
} = 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, t_{1}, t_{2},
and t_{3} are prime numbers. M(odd) is an odd number. L(2^{
n
}) and
L[
M(odd)] are the numbers of phase levels that
correspond to M = 2^{
n
} and
M(odd),
respectively. L(2^{
n
}) 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/M_{
I
}) of the
Generated Arraya

t, t_{1}, t_{2},
and t_{3} are prime numbers. M_{
I
}(odd) is an odd number. L[
M_{
I
}(odd)] is the
number of phase levels that corresponds to
M_{
I
}(odd). M_{
I
}(odd) =
M_{
I
}(1) … M_{
I
}(n) and no common
divisor among M_{
I
}(1), … ,
M_{
I
}(n). L[
M_{
I
}(1)], … ,
L[
M_{
I
}(n)] are the numbers of phase levels that
correspond to M_{
I
}(1), … ,
M_{
I
}(n).

Table 4

Numerical Examples of the Number (L) of
Phase Levels of the Talbot Array Illuminator and the Intensity-Opening
Ratio (1/M_{
I
}) of the Illumination Array in
the π-Phase Modulated Case up to M_{
I
} = 16