**2. Diffraction efficiency**

#### **2.1. Theory**

and the number of pixels was about one hundred at that time. Because of the large pixel size, LCWFC not only loses the advantage of high spatial resolution but also mismatches the microlens array of the detector, which leads to additional spatial filtering in order to decrease the effect of the undetectable pixel for correction [3]. Moreover the small modu‐ lation amplitude makes it unavailable for many conditions. The thickness and Δn can be increased in order to increase the modulation amplitude. However, this will slow down

Along with the development of LCWFC, an increasing number of commercial LC TVs are used directly for wavefront correction. Due to the high pixel density, the capacity for wave‐ front correction has been understood gradually by the researchers and the use of kinoform to increase the modulation amplitude is also possible [4-8]. A kinoform is a kind of early bi‐ nary optical element which can be utilized in a high pixel density LCWFC. The wavefront distortion can be compressed into one wavelength with a 2π modulus of a large magnitude distortion wavefront. The modulated wavefront is quantified according to the pixel position of LCWFC. As discussed above, LCWFC only needs one wavelength intrinsic modulation

Many domestic and international researchers have devoted themselves to exploring LCWFCs from th 1970s onwards. In 1977, a LCWFC was used for beam shaping by I. N. Kompanets et al. [9]. S. T. Kowel et al. used a parallel alignment LC cell to fabricate a adap‐ tive focal length plano-convex cylindrical lens in 1981 [10]. In 1984, he also realized a spheri‐ cal lens by using two perpendicularly placed LC cells[11]. A LCWFC with 16 actuators was achieved in 1986 by A. A. Vasilev et al. and a one dimensional wavefront correction was re‐ alized [12]. Three years later, he realized beam adaptive shaping through 1296 actuators of

As a result, the LC AOS is becoming increasingly developed. In order to overcome the dis‐ advantages of a traditional deformable mirror, such as a small number of actuators and high cost, D. Bonaccini et al. discussed the possibility of using LCWFC in a large aperture tele‐ scope [14, 15]. In 1995, D. Rensheng et al. used an Epson LC TV to perform a closed-loop adaptive correction experiment [16]. Although the twisted aligned LCWFC with the re‐ sponse time of 30ms was used, the feasibility of the LC AOS for wavefront correction was verified. Hence, many American [17-23], European [24-28] and Japanese [29] groups were devoted to the study of LC AOS. In 2002, the breakthrough for LC AOS was achieved and the International Space Station and various satellites were clearly observed [30]. In recent years, Prof. Xuan's group has completed series of valuable studies [31-42]. Recently, the ap‐ plications of LCWFC have been extended to other fields, such as retina imaging [43-45], beam control [46-50], optical testing [41], optical tweezers [51-53], dynamic optical diffrac‐ tion elements [54-57], tuneable photonic crystal fibre [58, 59], turbulence simulation [60, 61]

The basic characteristics of a diffractive LCWFC are introduced in this chapter. The dif‐ fractive efficiency and the fitting error of the LCWFC are described first. For practical ap‐ plications, the effects of tilt incidence and the chromatism on the LCWFC are

the speed of the LCWFC.

68 Adaptive Optics Progress

amplitude to correct a highly distorted wavefront.

an optical addressed LCWFC [13].

and free space optical communications [62, 63].

A Fresnel phase lens model is used to approximately calculate the diffraction efficiency of the LCWFC. According to the rotational symmetry and periodicity along the *r2* direction, when the Fresnel phase lens is illuminated with a plane wave of unit amplitude, the com‐ plex amplitude of the light can be expressed as [64]:

$$f(r^2) = f(r^2 + jr\_p^2) \tag{1}$$

where *j* is an integer and the period is*rp* 2 . Also, it can be expressed by the Fourier series:

$$f(r^2) = \sum\_{n=-\infty}^{+\infty} A\_n \exp[i2\pi nr^2/r\_p^2] \tag{2}$$

The distribution of the complex amplitude at the diffraction order n can be obtained [65]:

$$A\_n = 1/\left.r\_p^2\right\|\_0^{r\_p^2} f(r^2) \exp[i2\pi nr^2/r\_p^2] \text{d}r^2 \tag{3}$$

For the Fresnel phase lens, the light is mainly concentrated on the first order (*n*=1). The dif‐ fraction efficiency of the Fresnel phase lens is defined as the intensity of the first order at its primary focus:

$$\eta = I(n=1) = \left| A\_1 \right|^2 \tag{4}$$

If the phase distribution function *f (r2 )* of the Fresnel phase lens can be achieved, the diffrac‐ tion efficiency can be calculated by Eq. (3) and Eq. (4).

To correct the distorted wavefront, the 2π modulus should be performed first to wrap the phase distribution into one wavelength. Then, the modulated wavefront will be quantized. For a example, the wrapped phase distribution of a Fresnel phase lens is shown in Fig. 1(a). To a Fresnel phase lens, the 2π phase is always quantized with equal intervals. Assuming the height before quantization is *h*, the quantization level is *N* and the height of each quan‐ tized step is *h*/*N*. Figure 1(b) is a Fresnel phase lens quantized with 8 levels.

20 **Fig.** 1**.** Phase distribution of a Fresnel phase lens: (a) 2 **Figure 1.** Phase distribution of a Fresnel phase lens: (a) 2π modulus; (b)π quantized. modulus; (b) quantized.

2 22 () ( ) *<sup>p</sup>* <sup>1</sup>*f r f r jr* , (1)

2

11 diffraction efficiency can be calculated by Eq. (3) and Eq. (4).

2 *<sup>p</sup>* <sup>2</sup>*r* . Also, it can be expressed by the Fourier series:

4 The distribution of the complex amplitude at the diffraction order n can be obtained [65]:

2 2 22 2

2

If the phase distribution function *f (r2* 10 *)* of the Fresnel phase lens can be achieved, the

12 To correct the distorted wavefront, the 2π modulus should be performed first to wrap the 13 phase distribution into one wavelength. Then, the modulated wavefront will be quantized. 14 For a example, the wrapped phase distribution of a Fresnel phase lens is shown in Fig. 1(a).

17 quantized step is *h*/*N*. Figure 1(b) is a Fresnel phase lens quantized with 8 levels.

*n*

<sup>0</sup> 1/ ( ) exp[ 2 / ] *pr*

*n p <sup>p</sup> r f r i nr r dr*

<sup>3</sup> (2)

<sup>2</sup> 2 2 ( ) exp[ 2 / ] *n p*

*I*( 1) *n A* . (4)

*f r A i nr r*

5 . (3)

6 For the Fresnel phase lens, the light is mainly concentrated on the first order (*n*=1). The 7 diffraction efficiency of the Fresnel phase lens is defined as the intensity of the first order at

where *j* is an integer and the period is

*A*

<sup>1</sup> 9 

8 its primary focus:

For a quantized Fresnel phase lens, the diffraction efficiency can be expressed as [66]:

$$
\eta = \sin c^2 \left( 1/N \right) \tag{5}
$$

**2.2. Effects of black matrix**

the diffraction efficiency is about 10%.

**Figure 3.** A Fresnel phase lens quantified by the pixel with a Black Matrix.

**Figure 4.** The diffraction efficiency as functions of the pixel interval for different quantified levels.

A LCWFC always has a Black Matrix, which will cause a small interval between each pixel, as shown in Fig. 3. At the interval area, the liquid crystal molecule cannot be driven and then the phase modulation is different to the adjacent area. This will affect the diffraction efficiency of the LCWFC, as shown in Fig. 4. It is seen that the diffraction efficiency decreas‐ es by 6.4%, 8.8%, 9.5% and 9.7%, respectively for 4, 8, 16 and 32 levels, while the pixel inter‐ val is 1µm and the pixel pitch is 20µm. Consequently, the effect magnitude of the diffraction efficiency increases for a larger number quantified levels while the maximum decrease of

Liquid Crystal Wavefront Correctors http://dx.doi.org/10.5772/54265 71

Figure 2 shows the diffraction efficiency as a function of the quantization level for a Fresnel phase lens.

**Figure 2.** Diffraction efficiency as a function of the quantization level.
