or negative integer here u03b8 m n is the angle of m nth diffracted order u03b8 i is the angle of incidence u03bb is the incident wavelength and d1 and d2 are the periods of the two gratings the theoretical results calculated from equation 3 are listed in table 1 together with measured values it will be seen that the two sets of data agree with each other quite well in the calculations for table 1 u03b8 i 0 normal incidence d 1 0 36 u03bcm d 2 0 45 u03bcm u03bb 0 6328 u03bcm and sin u03b8 mn m 1 7578 n 1 4062 table 1______________________________________measured and calculated diffraction anglesfrom a doubly periodic structurem n calc u2296 sub m n meas u2296 sub m n______________________________________ 0 0 0 trans 1 1 20 58 21 8 1 1 20 58 21 5 2 2 44 7 45 9 2 2 44 7 45 9______________________________________ the appearance of various diffracted orders from device 25 may be explained as follows though invisible the electromagnetic fields of the higher order space harmonics which are diffracted individually by each grating are evanescent because of the small separation between the two gratings the evanescent fields diffracted from one grating may interact with the other resulting in many higher harmonics radiation modes that propagate out of the structure therefore higher order beams appear if the separation is large the evanescent fields from one grating will be too weak to interact with the other grating more specifically although the higher order beams may still be in existence their intensities are too low to be observable thus the interaction of the two gratings of different periods through the evanescent fields results in the.