Diffraction Orders

Generally, several integers m will satisfy the grating equation – we call each of these values a diffraction order.

Existence of Diffraction Orders

For a particular groove spacing d, wavelength λ and incidence angle α, the grating equation is generally satisfied by more than one diffraction angle β. In fact, subject to restrictions discussed below, there will be several discrete angles at which the condition for constructive interference is satisfied. The physical significance of this is that the constructive reinforcement of wavelets diffracted by successive grooves merely requires that each ray be retarded (or advanced) in phase with every other; this phase difference must therefore correspond to a real distance (path difference) which equals an integral multiple of the wavelength. This happens, for example, when the path difference is one wavelength, in which case we speak of the positive first diffraction order (m = 1) or the negative first diffraction order (m = –1), depending on whether the rays are advanced or retarded as we move from groove to groove. Similarly, the second order (m = 2) and negative second order (m = –2) are those for which the path difference between rays diffracted from adjacent grooves equals two wavelengths.

The grating equation reveals that only those spectral orders for which |mλ/d| < 2 can exist; otherwise, |sinα + sinβ| > 2, which is physically meaningless. This restriction prevents light of wavelength λ from being diffracted in more than a finite number of orders. Specular reflection, for which m = 0, is always possible; that is, the zero order always exists (it simply requires β = –α). In most cases, the grating equation allows light of wavelength λ to be diffracted into both negative and positive orders as well. Explicitly, spectra of all orders m exist for which

–2d < < 2d, m an integer   (2-11)

For λ/d << 1, a large number of diffracted orders will exist. As seen from the grating equation mλ= d (sinα + sinβ), the distinction between negative and positive spectral orders is that

  • β > –α for positive orders (m > 0)
  • β < –α for negative orders (m < 0) 
  • β = –α for specular reflection (m = 0)

This sign convention requires that m > 0 if the diffracted ray lies to the left (the counter-clockwise side) of the zero order (m = 0), and m < 0 if the diffracted ray lies to the right (the clockwise side) of the zero order. This convention is shown graphically in Figure 2-4.

Figure 2-4. Sign convention for the spectral order m. In this example α is positive.

Overlapping of Diffracted Spectra

The most troublesome aspect of multiple order behavior is that suc-cessive spectra overlap, as shown in Figure 2-5. It is evident from the grating equation that light of wavelength λ diffracted by a grating along direction β will be accompanied by integral fractions λ/2, λ/3, etc.; that is, for any grating instrument configuration, the light of wavelength λ diffracted in the m = 1 order will coincide with the light of wavelength λ/2 diffracted in the m = 2 order, etc. In this example, the red light (600 nm) in the first spectral order will overlap the ultraviolet light (300 nm) in the second order. A detector sensitive at both wavelengths would see both simultaneously. This superposition of wavelengths, which would lead to ambiguous spectroscopic data, is inherent in the grating equation itself and must be prevented by suitable filtering (called order sorting), since the detector cannot generally distinguish between light of different wavelengths incident on it (within its range of sensitivity). [See also Section 2.7 below.]

Figure 2-5. Overlapping of spectral orders. The light for wavelengths 100, 200 and 300 nm in the 2nd order is diffracted in the same direction as the light for wavelengths 200, 400 and 600 nm in the 1st order. In this diagram, light is incident from the right, so α < 0.

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