Criteria of suitability of the 0/-1 order principle

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This technical note derives the relationship between Phase mask period and Illumination wavelength that must be fulfilled for the 0/-1 order principle to work

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.

1) A first order must 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

2) The second order must exist

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

Conclusion

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

(2/3)·D ≤ λ ≤ 2D

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