**1. Introduction**

The term "diffraction" has been defined by Sommerfeld as follows (E. Hecht, 2002): any deviation of light rays from rectilinear paths which cannot be interpreted as reflection or refraction. Diffraction is caused by the confinement of the lateral extent of a wave and is most importantly when that confinement is comparable in size to the wavelength of the radiation being used. The first accurate report and description of such a phenomenon was made by Grimaldi and was published in the year 1665. Later, in 1678, Christian Huygens expressed the intuitive conviction that of each point on the wavefront of a disturbance were considered to be a new source of a secondary spherical disturbance. This technique, however, ignores most of each secondary wavelet and retains only that portion common to the envelope. As a result of this inadequacy, this principle is unable to account for the details of the diffraction process. The difficulty was resolved by Fresnel and Kirchhoff with his addition of the concept of interference in late 18th century. These types of diffraction are known for two centuries in the form of diffraction gratings which periodically modulate the incident wave-front. An ideal grating generates a set of waves, called diffraction orders that propagate into discrete directions. The diffraction angles θ *<sup>m</sup>* are given by the well-known grating equation sinθ *<sup>m</sup>* = *m D* λ , where λ is the wavelength of light and D is the grating period, is shown in Fig. 1. The amplitudes of the diffraction orders are determined by the structure of the periodic modulation. The demand for electromagnetic analysis is arising together with the advance of the fabrication technology. A significant step was taken as well since the development of computers. It can compute such a complicated wave field numerical analysis from which the design a desired grating is much easier.

The DOE is designed for splitting the input beam into M diffraction orders. For array illuminators, equal power intensity with high uniformity is necessary. To achieve this special feature, a periodic nature together with binary phase structure, which was proposed by Dammann in the early 70's, is one of the solutions (H. Dammann & K. Gortler, 1971; H. Dammann & E. Klotz, 1977). To analyze the performance of the Dammann grating, we employ the Burckhardt, Kaspar and Knop (BKK) method and TE-polarized dependent mode is normally assumed (C. B. Burckhardt, 1966). In order to have an easier understanding, the entire Dammann grating has the identities in terms of periodic, symmetric and binary structure. The total normalized diffraction efficiency is just the sum of all required

Design of Circular Dammann Grating: Fabrication and Analysis 121

The concept of generating CDG using this method is based on the theory of conventional Dammann Grating (C. Zhou & L. Liu, 1995). The cross section and first order spectrum of CDG is shown in Fig. 2 and 3 respectively. We assumed that if the diffraction spots rotate 360 degrees continuously, circular rings will be formed. Fig. 4 and 5 illustrate these ideas. Table 1 shows some numerical results. For the formula of Circular Dammann Grating, the

: *th n order* <sup>2</sup> 2 2

0

*n* η *nP* ∞

=−∞

The feature size, which is the minimum distance between two different continuous

max( ) min( ) max( ) min( ) *n n n n*

*I I*

*I I uni*

The radius (*R*) of each ring with the focal length (*f*) of converging lens will then be

Normalized Transition Points in Half

*n <sup>n</sup> <sup>R</sup> <sup>f</sup> <sup>D</sup>* λ

1 0.5 0.81 0 0.5 2 0.20525 0.29067 0.5 0.72 0.00006 0.08542 3 0.11649 0.24024 0.26741 0.38396 0.74 0.0002 0.02717

4 0.099104 0.18382 0.26295 0.32925 0.8 0.00014 0.00508

The numerical solutions with near optimum efficiency and uniformity of CDG are listed in

1 1 <sup>1</sup> ( ) {[ ( 1) cos(2 )] [ ( 1) sin(2 )] } *N N*

⎡ ⎤

*k k*

π

*n k k k k P Mn nr nr*

= =

*N*

*k P r* =

1 2 ( 1) 1

2 1 *n*

−

*k k*

2

= −+ <sup>⎢</sup> <sup>⎥</sup> <sup>⎣</sup> <sup>⎦</sup> <sup>∑</sup> (2.2)

<sup>=</sup> ∑ (2.3)

min *k k* <sup>1</sup> *r r* Δ = − <sup>+</sup> (2.4)

<sup>−</sup> <sup>=</sup> <sup>+</sup> (2.5)

= (2.6)

Efficiency Uniformity Feature

Size

= = − +− ∑ ∑ (2.1)

 π

**2.1 Circular spot rotation method** 

transition points, is

Table 1.

Circle number

0.5

The uniformity is defined as,

of Period\*

0.4196 0.49492 0.5

\*The other half period could be calculated using 0.5 *ki i x x* <sup>+</sup> = +

Table 1. Some Numerical Solutions of CDG by Spot Rotation Method.

intensities among different orders will then be as follows:

The overall normalized efficiency is therefore given as,

2 2

π

0 : *th order*

*n*

normalized diffraction power. Normally, for reducing the zero-order intensity and polarization dependent loss, the ratio of period/wavelength should be as large as possible. We have also employed Dammann Grating as a beam splitter (J. F. Wen & P. S. Chung, 2007; J. F. Wen & P. S. Chung, 2007).

Fig. 1. Diffraction Angle for DOE.
