Criteria of suitability of the 0/-1 order principle

This criteria can be translated to a necessary relationship between phase mask period D and illumination wavelength using the grating equation:

m·λ = (sinΘ₁ + sinΘ₂)·D

(Grating equation)

The phase mask is used at Bragg incidence, where the diffracted first order (m=1) equals the incidence angle. Hence the grating equation for Bragg condition becomes:

λ = 2·sinΘ·D,

equivalent to:

λ/(2·D) = sin(Θ)

(Grating equation in Bragg condition)

The criteria for suitability of the 0/-1 order principle can be expressed as 2 situations which must be fulfilled:

1. Diffraction must occur, i.e. a first order must exist.
2. The second order must not exist.

Mathematically, this means that the sin term of the grating equation in Bragg condition must be less than unity i.e.:

λ/(2·D) ≤ 1, equivalent to λ ≤ 2·D

The incidence angle is given by the grating equation in Bragg condition: sinΘ1 = λ/(2D).

Now, using the grating equation once more, the criteria for non-existence of the 2nd order can be found, given that m=2, together with the condition that the equation must be unsolvable, i.e. the sin term must be larger than unity:

m·λ = (sinΘ₁ + sinΘ₂)·D

Replacing sinΘ₁ = λ/2·D and m=2 now gives:

(2λ-½)/D = sinΘ₂

As the left term should be larger than unity, the equation reduces to:

(2/3)·D ≤ λ

The complete criteria for suitability of the 0/-1 order principle is therefore expressed as:

(2/3)·D ≤ λ ≤ 2D