**2. Introduction to Circular Dammann Grating (CDG)**

In previous section, we know that light could be diffracted into multiple orders with equal intensities and high uniformity with the theory of Dammann grating. The diffracted spots could be one-dimensional or two dimensional depending on the applications. For some applications, optical systems need circular images, e.g. laser free space communication system (J. Jia, C. Zhou & L. Liu, 2003), fast focal length measurement (S. Zhao, J. F. Wen & P. S. Chung, 2007; S. Zhao & P. S. Chung, 2007) and usage in DFB laser (C. Wu *et al,* 1991; T. Erdogan *et al,* 1992). We have then further extended the Dammann grating into CDG. The CDG is one of the possible candidates which can produce circular beams in ring-shape at the image plane.

Zhou et al (C. Zhou, J. Jia & L. Liu, 2003) first proposed the concept of CDG in 2003 based on the modulation of the Bessel function using a binary phase annulus mask. The phase and radius of each annulus can be modified so that the intensity at the far field can be manipulated. However the CDG does not have the periodic nature as most of the gratings required and therefore, it is only a DOE and equal separation cannot also be achieved. Recently Zhao and Chung (S. Zhao & P. S. Chung, 2007) proposed a new design method for the periodic CDG using the coefficients of the circular sine series for generating equalintensity and equal-spacing of optical rings, which means those infinite circular periods can be repeated. We have also presented another two novel approaches based on the concept of circular spot rotation (J. F. Wen, S. Y. Law & P. S. Chung, 2007), Hankel transform (J. F. Wen & P. S. Chung, 2008) to achieve the same objectives as mentioned above with higher efficiency and uniformity. In our research, we have employed the Circular Dammann Grating into angle, area and distance measurements respectively (J. F. Wen & P. S. Chung, 2008; J. F. Wen, Z. Y. Chen & P. S. Chung, 2008; J. F. Wen, Z. Y. Chen, & P. S. Chung, 2010).

## **2.1 Circular spot rotation method**

120 Advances in Unconventional Lithography

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;

In previous section, we know that light could be diffracted into multiple orders with equal intensities and high uniformity with the theory of Dammann grating. The diffracted spots could be one-dimensional or two dimensional depending on the applications. For some applications, optical systems need circular images, e.g. laser free space communication system (J. Jia, C. Zhou & L. Liu, 2003), fast focal length measurement (S. Zhao, J. F. Wen & P. S. Chung, 2007; S. Zhao & P. S. Chung, 2007) and usage in DFB laser (C. Wu *et al,* 1991; T. Erdogan *et al,* 1992). We have then further extended the Dammann grating into CDG. The CDG is one of the possible candidates which can produce circular beams in ring-shape at the

Zhou et al (C. Zhou, J. Jia & L. Liu, 2003) first proposed the concept of CDG in 2003 based on the modulation of the Bessel function using a binary phase annulus mask. The phase and radius of each annulus can be modified so that the intensity at the far field can be manipulated. However the CDG does not have the periodic nature as most of the gratings required and therefore, it is only a DOE and equal separation cannot also be achieved. Recently Zhao and Chung (S. Zhao & P. S. Chung, 2007) proposed a new design method for the periodic CDG using the coefficients of the circular sine series for generating equalintensity and equal-spacing of optical rings, which means those infinite circular periods can be repeated. We have also presented another two novel approaches based on the concept of circular spot rotation (J. F. Wen, S. Y. Law & P. S. Chung, 2007), Hankel transform (J. F. Wen & P. S. Chung, 2008) to achieve the same objectives as mentioned above with higher efficiency and uniformity. In our research, we have employed the Circular Dammann Grating into angle, area and distance measurements respectively (J. F. Wen & P. S. Chung, 2008; J. F. Wen, Z. Y. Chen & P. S. Chung, 2008; J. F. Wen, Z. Y. Chen, & P. S. Chung, 2010).

J. F. Wen & P. S. Chung, 2007).

Fig. 1. Diffraction Angle for DOE.

image plane.

**2. Introduction to Circular Dammann Grating (CDG)** 

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 intensities among different orders will then be as follows:

$$n^{\text{th}} order: \ P\_n = \left| M(n) \right|^2 = \frac{1}{n^2 \pi^2} \{ \sum\_{k=1}^N (-1)^k \cos(2\pi n r\_k) \}^2 + \{ \sum\_{k=1}^N (-1)^k \sin(2\pi n r\_k) \}^2 \tag{2.1}$$

$$\mathbf{0}^{\text{fl}}\,\text{order}:\quad P\_0 = \left[\mathbf{2}\sum\_{k=1}^{N}(-1)^k r\_k + \mathbf{1}\right]^2\tag{2.2}$$

The overall normalized efficiency is therefore given as,

$$\eta = \sum\_{n=-\infty}^{\infty} nP\_{2n-1} \tag{2.3}$$

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

$$
\Delta = \min \left| r\_{k+1} - r\_k \right| \tag{2.4}
$$

The uniformity is defined as,

$$
\mu mi = \frac{\max(I\_n) - \min(I\_n)}{\max(I\_n) + \min(I\_n)} \tag{2.5}
$$

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

$$R\_n = \frac{n\mathcal{X}}{D}f \tag{2.6}$$

The numerical solutions with near optimum efficiency and uniformity of CDG are listed in Table 1.


\*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.

Design of Circular Dammann Grating: Fabrication and Analysis 123

The concept of this grating is based on the theory of Hankel transform. It is a two dimensional Fourier transform with a radially symmetric integral and is also called the Fourier-Bessel transform (F. Bowman, 1958). The cross section of CDG is still same as Fig. 2. The problem of side lobe effect is one of the major concerns in both Zhao and our circular rotation methods. The effect is shown in Fig. 6. In this section, we present another novel approach based on the Bessel function together with Hankel transform in symmetry structure to achieve the same objectives with better performance in terms of efficiency and uniformity while the lower side lobe power could be obtained. Table 2 shows some

2

Overall

Efficiency3 Uniformity

 π

= −+ <sup>⎢</sup> <sup>⎥</sup> <sup>⎣</sup> <sup>⎦</sup> <sup>∑</sup> (2.8)

Fig. 5. Circular Spot Rotation Method's type CDG Profile Design.

numerical results. The comparisons are shown in Fig. 7-9.

Normalized Transition Points in

0.32894 0.41929 0.49616

2: It describes the sum of powers in terms of main lobes only.

1: Feature is the minimum distance between two subsequent transition points

3: It describes the sum of all powers including the main and side lobes. \*: The other half period could be calculated using 0.5 *ki i x x* <sup>+</sup> = +

Table 2. Some Numerical Solutions of CDG by Hankel Transform.

: *th n order*

0.38396 0.5

0.5

The intensities equations for each diffraction order are given as follows:

0 : *th order*

2 1

*k*

0

1 0.39763 0.4907 0.5 0.0093 0.665 0.931

*q*

*N*

1

=

2 1 1

*k k*

⎡ ⎤

π

<sup>+</sup>

2

(Main Lobe)

== − + ∑ (2.7)

<sup>1</sup> ( ) 2 ( 1) (2 ) (2 )

*k q k k N N*

*I Mq r J qr r J qr*

1 2 ( 1) 1

*N*

*k I r* =

Half of Period\* Feature1 Efficiency2

2 0.091023 0.3266 0.5 0.0610 0.612 0.855 0.043 3 0.10036 0.23569 0.26741 0.0353 0.60 0.866 0.05

4 0.099572 0.1838 0.26314 0.0038 0.69 0.99 0.036

**2.2 Hankel transform** 

Circle Number

Note:

Fig. 2. Cross Section of CDG.

Fig. 3. First order CDG spectrum.

Fig. 4. Concept of Circular Spot Rotation Method's type CDG.

Fig. 5. Circular Spot Rotation Method's type CDG Profile Design.

## **2.2 Hankel transform**

122 Advances in Unconventional Lithography

Fig. 2. Cross Section of CDG.

Fig. 3. First order CDG spectrum.

Fig. 4. Concept of Circular Spot Rotation Method's type CDG.

The concept of this grating is based on the theory of Hankel transform. It is a two dimensional Fourier transform with a radially symmetric integral and is also called the Fourier-Bessel transform (F. Bowman, 1958). The cross section of CDG is still same as Fig. 2. The problem of side lobe effect is one of the major concerns in both Zhao and our circular rotation methods. The effect is shown in Fig. 6. In this section, we present another novel approach based on the Bessel function together with Hankel transform in symmetry structure to achieve the same objectives with better performance in terms of efficiency and uniformity while the lower side lobe power could be obtained. Table 2 shows some numerical results. The comparisons are shown in Fig. 7-9.

The intensities equations for each diffraction order are given as follows:

$$m^{\text{fl}} order: \ I\_q = \left| M(q) \right|^2 = \frac{1}{q^2} \left| 2 \sum\_{k=1}^N (-1)^{k+1} r\_k I\_1(2\pi q r\_k) + r\_N I\_1(2\pi q r\_N) \right|^2 \tag{2.7}$$

$$\mathbf{0}^{\text{th}}\,\text{order}:\ I\_0 = \left[\mathbf{2}\sum\_{k=1}^{N}(-1)^k r\_k + \mathbf{1}\right]^2\tag{2.8}$$


Note:

1: Feature is the minimum distance between two subsequent transition points

2: It describes the sum of powers in terms of main lobes only.

3: It describes the sum of all powers including the main and side lobes.

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

Table 2. Some Numerical Solutions of CDG by Hankel Transform.

Design of Circular Dammann Grating: Fabrication and Analysis 125

Fig. 8. Side Lobe Intensities Comparisons among Three Different Methods (Triangle: Zhao's symmetric method, Dot: Zhao's asymmetric method, Star: Spot Rotation Method, Cross:

Fig. 9. Uniformities Comparisons among Three Different Methods (Triangle: Zhao's symmetric method, Dot: Zhao's asymmetric method, Star: Spot Rotation Method, Cross:

Hankel Transform).

Hankel Transform).

Fig. 6. Side Lobe Effect of CDG.

Fig. 7. Main Lobe Intensities Comparisons among Four Different Methods (Triangle: Zhao's symmetric method, Dot: Zhao's asymmetric method, Star: Spot Rotation Method, Cross: Hankel Transform).

Fig. 7. Main Lobe Intensities Comparisons among Four Different Methods (Triangle: Zhao's symmetric method, Dot: Zhao's asymmetric method, Star: Spot Rotation Method, Cross:

Fig. 6. Side Lobe Effect of CDG.

Hankel Transform).

Fig. 8. Side Lobe Intensities Comparisons among Three Different Methods (Triangle: Zhao's symmetric method, Dot: Zhao's asymmetric method, Star: Spot Rotation Method, Cross: Hankel Transform).

Fig. 9. Uniformities Comparisons among Three Different Methods (Triangle: Zhao's symmetric method, Dot: Zhao's asymmetric method, Star: Spot Rotation Method, Cross: Hankel Transform).

Design of Circular Dammann Grating: Fabrication and Analysis 127

Fig. 11. Number of period against beam width (FWHM).

Fig. 12. Number of Period against separation between two lobes.

The overall normalized efficiency, feature size, uniformity and radius are defined same as equations 2.3-2.6.

Side lobe is existed around the main lobe in every diffracted order, is shown in Fig. 6, no matter which method is applied. It is because our design is not able to fully express the concept of circular phase modulation, as the circular profile can not be completely decomposed into a square pixel representation and also finite number of periods happened in practical case. Thus side lobe existed. The lobe separation is not only governed by number of period (*ND*), but also controlled by the input wavelength and the focal length of the lens. With the assistance of diffraction theory, the final equation is then defined, i.e.

$$s = \frac{2\lambda}{3ND}f\tag{2.9}$$

Fig. 10-12 shows this relationship. From these figures, we can conclude that 100 periods is the optimum solution.

Fig. 10. Number of Period against efficiency.

The overall normalized efficiency, feature size, uniformity and radius are defined same as

Side lobe is existed around the main lobe in every diffracted order, is shown in Fig. 6, no matter which method is applied. It is because our design is not able to fully express the concept of circular phase modulation, as the circular profile can not be completely decomposed into a square pixel representation and also finite number of periods happened in practical case. Thus side lobe existed. The lobe separation is not only governed by number of period (*ND*), but also controlled by the input wavelength and the focal length of the lens.

> 2 <sup>3</sup> *s f ND* λ

Fig. 10-12 shows this relationship. From these figures, we can conclude that 100 periods is

= (2.9)

With the assistance of diffraction theory, the final equation is then defined, i.e.

equations 2.3-2.6.

the optimum solution.

Fig. 10. Number of Period against efficiency.

Fig. 11. Number of period against beam width (FWHM).

Fig. 12. Number of Period against separation between two lobes.

Design of Circular Dammann Grating: Fabrication and Analysis 129

*h*

substrate in using is quartz and the refractive index is shown in Fig. 14.

where

λ

Fig. 14. Refractive Index of quartz.

( ) 1 0 2

material and the surrounding medium at the operating wavelength respectively. The

Most lithographic masks are binary transmission masks. That is, they contain alternating clear and opaque areas. The opaque areas mean the Chromium remains on top of the substrate. These masks are usually made by forming the pattern in a light-sensitive photoresist on top of thin chrome later on the glass mask. Once the photoresist is developed, the chrome, where the photoresist has been removed, was protected. The mask pattern is exposed using optical pattern generators with controllable beam size. A variety of file formats, e.g. GDSII, CIF and BMP, can be used. The machine in our lab is "Microtech LW405". The positioning accuracy is 1um and the minimum linewidth is 0.8um. These patterns are transformed into pixel forms with a given dimension and the magnified mask pattern is shown in Fig. 15. The accuracy of patterning curve is related to the wavefront error introduced by the required shape approximation. Increasing the number of pixels can help to improve the accuracy. However, as we expect, the higher degree of accuracy can result of more time spending and the amount of data is also increased. Once the mask is

fabricated, which is shown in Fig. 16, we can move on to the next step.

*n n* λ

is the input wavelength and *n*<sup>1</sup> , *n*<sup>0</sup> are the indices of refraction of the substrate

<sup>=</sup> <sup>−</sup> (3.1)
