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Diffraction patterns of a rotated grating are investigated from both theoretical and experimental points of view. It is shown that as the grating rotates, the angle of deviation of each diffracted line relative to the incident light goes through a minimum, and that the angle between any two successive diffracted lines goes through a minimum value which is the same for all adjacent diffracted lines. It is also shown that the angle between diffraction lines with n=1 and n=-1 is not sensitive to small variations of the grating from being normal to the incident light. Finally, a method is suggested for determining the line distance of a diffraction grating with high accuracy.

Rotating diffraction gratings have a number of applications. For example, a pair of counter rotating diffraction gratings can be used in laser beam scanners for obtaining a linear scan rate for a flat field recorder [

When a diffraction grating of line distance d is exposed to a light ray of wavelength λ , the general equation for the condition of constructive interference is given by [

d ( sin β − sin α ) = n λ (1)

where n is any integer, β is the angle of nth-order diffraction, and α is the angle of incidence. Both α and β are considered positive if measured counterclockwise from the axis (normal to the diffraction grating) and negative if measured clockwise, as shown in

Now suppose that we illuminate a diffraction grating by a monochromatic ray of light of wavelength λ and then turn the grating, keeping the direction of the incident light fixed, starting from normal incidence, the angle through which the grating turns would become the angle of incidence for light, which is α in Equation (1). However, instead of the diffraction angle β , let us study the angle of deviation θ between a diffracted beam and the incident beam, as shown in

d [ sin ( θ + α ) − sin α ] = n λ , n = 0 , ± 1 , ± 2 , ⋯ (2)

where α is negative if the grating turns counterclockwise (

The case n = 0 is trivial and leads to θ = 0 . We are, however, interested in the higher-order diffractions and the behavior of θ as a function of α . We shall only consider counterclockwise rotations of the diffraction grating ( α < 0 ) since clockwise rotations can be obtained from the former by a symmetry operation on the experimental setup. From Equation (2) we have

θ = sin − 1 ( n λ d + sin α ) − α (3)

Let us investigate the behavior of this function, in particular, find out if there are any maxima or minima in θ ,

d θ d α = cos α 1 − ( n λ d + sin α ) 2 − 1 (4)

This expression becomes zero when

1 − ( n λ d + sin α ) 2 = cos α (5)

which reduces to

sin α = − n λ 2 d (6)

and becomes infinity if

1 − ( n λ d + sin α ) 2 = 0 (7)

which reduces to

n λ d + sin α = ± 1 (8)

In the last equation, the positive root will not be considered since it gives α > 0 (up to n = 5 for the choice of λ = 632.8 nm and d = 3342 nm ), corresponding to a clockwise rotation. Therefore, the possibility of a minimum or maximum in Equation (3) exist only if one of the following conditions is satisfied:

sin α = − n λ 2 d or sin α = − 1 − n λ d (9)

We now consider each case separately.

Case 1: sin α = − n λ 2 d

For the counterclockwise rotations ( α < 0 ) that we are considering, we must have n > 0 . Furthermore, substituting the above equation into Equation (2), we obtain

sin ( θ + α ) = n λ 2 d (10)

Then adding the first of Equations (9) and (10) gives

sin ( θ + α ) + sin α = 0 (11)

or

2 sin ( θ + 2 α 2 ) cos ( θ 2 ) = 0 (12)

which holds only if θ = − 2 α . It is easily verified that under this condition, d 2 θ / d α 2 > 0 , and hence θ is a minimum.

Case 2: sin α = − 1 − n λ d

Clearly this can hold only if n < 0 which corresponds to the diffracted rays below the normal to the grating (

Therefore, our theoretical analysis shows that when the diffraction grating turns counterclockwise the diffracted rays with n > 0 go through a minimum angle of deviation. The minimum angle of deviation for the nth-order diffracted beam takes place when the grating rotates through an angle α n relative to the normal incidence, given by

α n = − sin − 1 ( n λ 2 d ) (13)

The minimum angle of deviation for the nth-order diffraction is then given by

θ n ( min ) = − 2 α n = 2 sin − 1 ( n λ 2 d ) (14)

There are no other minima or maxima in the angle of deviation.

In order to test these results, we used a diffraction grating with a line distance of d = 3342 ± 10 nm , and a 0.95 mW He-Ne laser ( λ = 632.8 nm ). The line distance of the grating was obtained by measuring it at ten different points of the grating, using the laser light.

Another interesting feature of these diffraction patterns is that the angle through which the grating has to turn for a diffraction line to go to its minimum deviation increases with the order of diffraction n. As a result, the angle between any two adjacent diffracted rays decreases first and then increases as the grating turns, resulting in a minimum value. To see this, we write Equation (3) for n and again for n + 1 , and subtract them to get

− α n | θ n ( min ) | |||
---|---|---|---|---|

n | calc | meas | calc | meas |

1 | 5.43 ± 0.02 | 5.5 ± 0.1 | 10.87 ± 0.03 | 10.93 ± 0.02 |

2 | 10.91 ± 0.03 | 11.1 ± 0.2 | 21.83 ± 0.07 | 21.89 ± 0.04 |

3 | 16.50 ± 0.05 | 16.6 ± 0.3 | 33.00 ± 0.10 | 33.14 ± 0.03 |

4 | 22.25 ± 0.07 | 22.5 ± 0.2 | 44.51 ± 0.14 | 44.68 ± 0.02 |

5 | 28.25 ± 0.09 | 28.6 ± 0.2 | 56.51 ± 0.18 | 56.75 ± 0.02 |

θ n + 1 − θ n = sin − 1 [ ( n + 1 ) λ d + sin α ] − sin − 1 [ n λ d + sin α ] (15)

Then differentiation of this equation with respect to α gives

d d α ( θ n + 1 − θ n ) = cos α 1 − [ ( n + 1 ) λ d + sin α ] 2 − cos α 1 − [ n λ d + sin α ] 2 (16)

which vanishes if and only if

α = − sin − 1 [ λ d ( n + 1 2 ) ] (17)

It can easily be verified that when this condition is met, θ n + 1 − θ n is a minimum. Therefore, when α satisfies Equation (17), the angle between diffraction lines of order n and n + 1 becomes a minimum with a value given by

( θ n + 1 − θ n ) min = 2 sin − 1 ( λ 2 d ) (18)

which is independent of n. Therefore the minimum angle reached between any two adjacent diffraction lines is the same.

It should be pointed out that some of the results discussed in this work were experimentally observed by Lock [

( θ n + 1 − θ n ) min | ||
---|---|---|

− α | calc | meas |

5.4 | 10.87 | 10.90 |

16.5 | 10.87 | 10.87 |

28.3 | 10.87 | 10.88 |

41.5 | 10.87 | 10.93 |

58.4 | 10.87 | 10.88 |

of the diffraction angles. In fact, it turns out that the angle between the diffraction maxima with n = 1 and n = − 1 is not sensitive to small deviations of the grating from being normal to the incident beam. To see this we consider the angle between the diffraction maxima with n and − n :

θ n − θ − n = sin − 1 ( n λ d + sin α ) − sin − 1 ( − n λ d + sin α ) (19)

Taking the derivative of this angle with respect to α , we find

d d α ( θ n − θ − n ) = cos α 1 − ( n λ d + sin α ) 2 − cos α 1 − ( n λ d − sin α ) 2 (20)

which vanishes if and only if α = 0 . Therefore, the angle θ n − θ − n having a stationary value at α = 0 , is not affected appreciably by small deviations of α from zero.

Finally, as an application of the above results, we rewrite Equation (14) as

Since

We investigated the diffraction patters of a rotated grating, from both theoretical and experimental points of view. The results of this investigation show several interesting behaviors of the system. First, as the grating rotates, the angle of deviation of each diffracted line relative to the incident light goes through a minimum. Although this behavior has been reported by Lock [

Second, we have shown that the rotation angle of the grating that results in a minimum angle of deviation for a given diffracted line increases with the order of diffraction. As a result, the angle between any two successive diffracted lines first decreases and then increases, resulting in a minimum value which is the same for any two adjacent diffracted lines. This result has not been reported previously.

Third, it is shown that the angle between diffraction lines with

Finally, we have suggested a new method for determining the line distance of a diffraction grating with high accuracy.

This work was supported in part by a URAP grant from the University of Wisconsin-Parkside.

The authors declare no conflicts of interest regarding the publication of this paper.

Mohazzabi, P., Mattson, D.J. and Ponce Jr., J.A. (2019) Diffraction Pattern of a Rotated Grating. Journal of Applied Mathematics and Physics, 7, 1870-1876. https://doi.org/10.4236/jamp.2019.78128