**1. Introduction**

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1210, (1974).

66 Adaptive Optics Progress

(2011).

14160-14171, (2011)

Liquid crystal (LC) was first discovered by the Austrian botanical physiologist Friedrich Re‐ initzer in 1888 [1]. It was a new state of matter beyond solid and liquid materials, having properties between those of a conventional liquid and those of a solid crystal. LC molecules usually have a stick shape. The average direction of molecular orientation is given by the director *n* ^. When light propagates along the director *<sup>n</sup>* ^, the refractive index is noted as the extraordinary index *ne*, no matter the polarization direction (in the plane perpendicular to the long axis). However, the refractive index is different depending upon the polarization direction when light moves perpendicular to the director. When an electric field is em‐ ployed, the LC molecule will be rotated so that the director *n* ^ is parallel to the electric field. Due to the applied electric field, the LC molecular can be rotated from 0° to 90° and the ef‐ fective refractive index is changed from *ne* to *no*. As a result, the effective refractive index of LC can be controlled by controlling the strength of the electric field applied on the LC. The maximum change amplitude of the refractive index is birefringence △n= *ne* - *no*.

The properties discussed above allow LC to become a potential candidate for optical wavefront correction. A liquid crystal wavefront corrector (LCWFC) modulates the wave‐ front by the controllable effective refractive index, which is dependent on the electric field. As distinct from the traditional deformable mirrors, the LCWFC has the advantag‐ es of no mechanical motion, low cost, high spatial resolution, a short fabrication period, compactness and a low driving voltage. Therefore, many researchers have investigated LCWFCs to correct the distortions.

Initially, a piston-only correction method was used in LC adaptive optics (LC AOS) to correct the distortion. The maximum phase modulation equals △n multiplying the thick‐ ness of the LC layer, and it is about 1µm. As reported [2], the pixel size was over 1mm

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 the speed of the LCWFC.

expounded. Finally, the fast response liquid crystal material is demonstrated as obtaining

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‐

2

*n*

+¥

=-¥

2

h

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

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

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

2 2 222 <sup>0</sup> 1 / ( )exp[ 2 / ] *pr A r f r i nr r dr n p <sup>p</sup>* =

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

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‐

p

2 1

p

*f r A i nr r*

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

Liquid Crystal Wavefront Correctors http://dx.doi.org/10.5772/54265 69

. Also, it can be expressed by the Fourier series:

<sup>=</sup> å (2)

ò (3)

= == *In A* ( 1) (4)

*)* of the Fresnel phase lens can be achieved, the diffrac‐

a high correction speed.

**2.1. Theory**

primary focus:

**2. Diffraction efficiency**

plex amplitude of the light can be expressed as [64]:

where *j* is an integer and the period is*rp*

If the phase distribution function *f (r2*

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

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 amplitude to correct a highly distorted wavefront.

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 an optical addressed LCWFC [13].

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] and free space optical communications [62, 63].

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 expounded. Finally, the fast response liquid crystal material is demonstrated as obtaining a high correction speed.
