**2.9. Diffraction efficiency with tilt incidence**

**2.8. The effect of pixel crossover on the phase modulation**

82 Adaptive Optics Progress

LCWFC may be used at the tilt incidence condition with a little tilt angle.

**Figure 16.** Illustration of the pixel crossover - P1, P2 and P2 are pixels, *d* is the thickness of the cell.

**Figure 17.** The pixel crossover *W* as a function of the incident angle.

For the tilt incidence, shown in Fig.16, the incident light in one pixel could transmit through an adjacent pixel, which is called pixel crossover. The maximum error of the pixel crossover is *W*. The pixel crossover will also affect the phase modulation of the LCWFC. Because each pixel is an actuator with a corresponding phase modulation, the light should go through just one pixel so as to control the phase modulation accurately. For a 19µm pixel size and *d*=1.6µm, *W* as a function of the incident angle is shown in Fig.17. The results show that *W*=0.33µm for a tilt incident angle of 6°. For a pixel with a size of *P*, the ratio of the light which transmits through adjacent pixels can be expressed with *W*/*P*. If the ratio equals to zero, it illustrates that the light is the vertical incidence and that it goes through just one pix‐ el. For an incident angle of 6°, the ratio is only 1.77% and it may be ignored. As such, the

Because the phase of each pixel changes with the tilt incidence, the diffraction efficiency will decrease [64]. The Fresnel phase lens model [71] is used to calculate the change of the dif‐ fraction efficiency and 16 quantified levels are selected. The simulated results show that at an incident angle of 6°, the diffraction efficiency is reduced by 3% (Fig.18). For the incident angles less than 3°, the reduction in diffraction efficiency is less than 1% - a negligible loss for most applications.

**Figure 18.** Diffraction efficiency as a function of the incident angle.

#### *2.9.1. Chromatism*

The chromatism of the LCWFC includes refractive index chromatism and quantization chro‐ matism. Refractive index chromatism is caused by the LC material, and is generally called dispersion. Meanwhile, quantization chromatism is caused by the modulo 2π of the phase wrapping. Theoretically, the LCWFC is only suitable for use in wavefront correction for a single wavelength and not on a waveband due to chromatism. However, if a minor error is allowed, LCWFC can be used to correct distortion within a narrow spectral range.

#### **2.10. Effects of chromatism on the diffraction efficiency of LCWFC**

The measured birefringence dispersion of a nematic LC material (RDP-92975, DIC) is shown in Fig. 19. It can be seen that the birefringence Δ*n* is dependent on the wavelength and the dispersion of the LC material is particularly severe when the wavelength is less than 500 nm. Since a phase wrapping technique is used, the phase distribution should be modulo 2π, and it should then be quantized [71]. Assuming that the quantization wavelength is *λ0*, the thickness of the LC layer is *d*, and *Vmax* denotes the voltage needed to obtain a 2π phase mod‐ ulation, such that the maximum phase modulation of the LCWFC can be expressed as:

$$
\Delta\varphi\_{\text{max}}(\lambda\_0) = 2\pi \frac{\Delta n(\lambda\_0, V\_{\text{max}})d}{\lambda\_0} = 2\pi \tag{20}
$$

the LC AOS. Although only one kind of LC material is measured and analysed, the re‐ sults are helpful in the use of LCWFCs because almost all the nematic LC materials have

Liquid Crystal Wavefront Correctors http://dx.doi.org/10.5772/54265 85

**Figure 20.** The phase modulation as a function of the wavelength for λ*<sup>0</sup>* = 550 nm, 633 nm and 750 nm, respectively the two horizontal dashed lines indicate the phase deviation range while the four vertical dashed lines illustrate three

**Figure 21.** The diffraction efficiency as a function of wavelength for λ*0* = 550 nm, 633 nm and 750 nm, respectively.

The above calculated results show that it is only possible to correct the distortion in a nar‐ row waveband using only one LCWFC. Therefore, to realize the distortion correction in a broadband - such as 520–810 nm - multi-LCWFCs are necessary; each LCWFC is responsible for the correction of different wavebands and then the corrected beams are combined to re‐

sub-wavebands of 520–590 nm, 590–690 nm and 690–820 nm, respectively.

**2.11. Broadband correction with multi-LCWFCs**

similar dispersion characteristics.

For any other wavelength λ, it can be rewritten as:

$$
\Delta\phi\_{\text{max}}(\lambda) = 2\pi \frac{\Delta n(\lambda, V\_{\text{max}})d}{\lambda} \tag{21}
$$

**Figure 19.** The birefringence ∆n as a function of the wavelength.

For a quantization wavelength of 550 nm, 633 nm and 750 nm, the variation of the maxi‐ mum phase modulation as a function of wavelength is shown in Fig. 20. Assuming that the deviation of the phase modulation is 0.1, for *λ0*=550, 633 and 750 nm, the corresponding spectral ranges are calculated as 520–590 nm, 590–690 nm and 690–810 nm, respectively. If a 10% phase modulation error is acceptable, then the LCWFC can only be used to correct the distortion for a finite spectral range.

The variation of Δ*n* and *λ* affects the diffraction efficiency of the LCWFC. Using the Fresnel phase lens model, the diffraction efficiency for any other wavelength *λ* can be described as [80]:

$$\eta = \left| \frac{\sin(\pi d \Delta n(\mathcal{X}, V\_{\text{max}}) / \mathcal{A})}{\pi (d \Delta n(\mathcal{X}, V\_{\text{max}}) / \mathcal{A} - 1)} \right|^2 \tag{22}$$

The effects of Δ*n* and *λ* on the diffraction efficiency are shown in Fig. 21. For *λ0* = 550 nm, 633 nm and 750 nm, and their respective corresponding wavebands of 520–590 nm, 590–690 nm and 690–810 nm, the maximum energy loss is 3%, which is acceptable for the LC AOS. Although only one kind of LC material is measured and analysed, the re‐ sults are helpful in the use of LCWFCs because almost all the nematic LC materials have similar dispersion characteristics.

0 max

max

l

p

D= = (20)

D = (21)

0

l

(, ) ()2 <sup>2</sup> *nVd* l

> (, ) () 2 *nV d* l

For a quantization wavelength of 550 nm, 633 nm and 750 nm, the variation of the maxi‐ mum phase modulation as a function of wavelength is shown in Fig. 20. Assuming that the deviation of the phase modulation is 0.1, for *λ0*=550, 633 and 750 nm, the corresponding spectral ranges are calculated as 520–590 nm, 590–690 nm and 690–810 nm, respectively. If a 10% phase modulation error is acceptable, then the LCWFC can only be used to correct the

The variation of Δ*n* and *λ* affects the diffraction efficiency of the LCWFC. Using the Fresnel phase lens model, the diffraction efficiency for any other wavelength *λ* can be

> max max sin( ( , ) / ) ( ( , ) / 1) *dn V dn V* pl

The effects of Δ*n* and *λ* on the diffraction efficiency are shown in Fig. 21. For *λ0* = 550 nm, 633 nm and 750 nm, and their respective corresponding wavebands of 520–590 nm, 590–690 nm and 690–810 nm, the maximum energy loss is 3%, which is acceptable for

 l 2

<sup>D</sup> <sup>=</sup> D - (22)

 l

D

 p D

max 0

max

jlp

jl

For any other wavelength λ, it can be rewritten as:

84 Adaptive Optics Progress

**Figure 19.** The birefringence ∆n as a function of the wavelength.

h

pl

distortion for a finite spectral range.

described as [80]:

**Figure 20.** The phase modulation as a function of the wavelength for λ*<sup>0</sup>* = 550 nm, 633 nm and 750 nm, respectively the two horizontal dashed lines indicate the phase deviation range while the four vertical dashed lines illustrate three sub-wavebands of 520–590 nm, 590–690 nm and 690–820 nm, respectively.

**Figure 21.** The diffraction efficiency as a function of wavelength for λ*0* = 550 nm, 633 nm and 750 nm, respectively.

#### **2.11. Broadband correction with multi-LCWFCs**

The above calculated results show that it is only possible to correct the distortion in a nar‐ row waveband using only one LCWFC. Therefore, to realize the distortion correction in a broadband - such as 520–810 nm - multi-LCWFCs are necessary; each LCWFC is responsible for the correction of different wavebands and then the corrected beams are combined to re‐ alize the correction in the whole waveband. The proposed optical set-up is shown in Fig. 22, where a polarized beam splitter (PBS) is used to split the unpolarized light into two linear polarized beams. An unpolarized light can be looked upon as two beams with cross polar‐ ized states. Because the LCWFC can only correct linear polarized light, an unpolarized inci‐ dent light can only be corrected in one polarization direction while the other polarized beam will not be corrected. Therefore, if a PBS is placed following the LCWFC, the light will be split into two linear polarized beams: one corrected beam goes to a camera; the other uncor‐ rected beam is used to measure the distorted wavefront by using a wavefront sensor (WFS). This optical set-up looks like a closed loop AOS, but it is actually an open-loop optical lay‐ out. This LC adaptive optics system must be controlled through the open-loop method [31, 81]. Three dichroic beam splitters (DBSs) are used to acquire different wavebands. A 520– 810 nm waveband is acquired by using a band-pass filter (DBS1). This broadband beam is then divided into two beams by a long-wave pass filter (DBS2). Since DBS2 has a cut-off point of 590 nm, the reflected and transmitted beams of the DBS2 have wavebands of 520– 590 nm and 590–820 nm, respectively. The transmitted beam is then split once more by an‐ other long-wave pass filter (DBS3) whose cut-off point is 690 nm. Through DBS3, the reflect‐ ed and transmitted beams acquire wavebands of 590–690 nm and 690–810 nm, respectively. Thus, the broadband beam of 520–810 nm is divided into three sub-wavebands, each of which can be corrected by an LCWFC. After the correction, three beams are reflected back and received by a camera as a combined beam. Using this method, the light with a wave‐ band of 520–810 nm can be corrected in the whole spectral range with multi-LCWFCs.

fraction-limited resolution has been achieved. Figure 23(c) shows the resolving ability for a waveband of 590–690 nm. The first element of the fifth group is resolved and the resolution is 31.25 µm, which is near the diffraction-limited resolution of 30.4 µm for a 633 nm wave‐ length. The corrected result for 520–690 nm is shown in Fig. 23(d). The first element of the fifth group can also be resolved. These results show that a near diffraction-limited resolution

Liquid Crystal Wavefront Correctors http://dx.doi.org/10.5772/54265 87

**Figure 23.** Images of the resolution target for different wavebands: (a) no correction; (b) 520–590 nm; (c) 590–690

In applications of LCWFCs, the response speed is a key parameter. A slow response will sig‐ nificantly decrease the bandwidth of LC AOS. To improve the response speed, dual-fre‐ quency and ferroelectric LCs have been utilized to fabricate the LCWFC [82, 83]. However, there are some shortcomings with these fast materials. The driving voltage of the dual-fre‐ quency LCWFC is high and it is incompatible with the very large scale integrated circuit. The phase modulation of the ferroelectric LCWFC is very slight and it is hard to correct the distortions. Nematic LCs have no such problems. However, its response speed is slow. In

this section, we introduce how to improve the response speed of nematic LCs.

nm; (d) 520–690 nm - the circular area represent the resolving limitation.

*2.11.1. Fast response liquid crystal material*

of an optical system can be obtained by using multi-LCWFCs.

**Figure 22.** Optical set-up for a broadband correction - PLS represents a point light source, PBS is a polarized beam splitter, DBS means dichroic beam splitter, DBS1 is a band-pass filter, and DBS2 and DBS3 are long-pass filters.

The broadband correction experimental results are shown in Fig. 23. A US Air Force (USAF) resolution target is utilized to evaluate the correction effects in a broad waveband. Firstly, the waveband of 520–590 nm is selected to perform the adaptive correction. After the correc‐ tion, the second element of the fifth group of the USAF target is resolved, with a resolution of 27.9 µm (Fig.23(b)). Considering that the entrance pupil of the optical set-up is 7.7 mm, the diffraction-limited resolution is 26.4 µm for a wavelength of 550 nm. Thus, a near dif‐ fraction-limited resolution has been achieved. Figure 23(c) shows the resolving ability for a waveband of 590–690 nm. The first element of the fifth group is resolved and the resolution is 31.25 µm, which is near the diffraction-limited resolution of 30.4 µm for a 633 nm wave‐ length. The corrected result for 520–690 nm is shown in Fig. 23(d). The first element of the fifth group can also be resolved. These results show that a near diffraction-limited resolution of an optical system can be obtained by using multi-LCWFCs.

**Figure 23.** Images of the resolution target for different wavebands: (a) no correction; (b) 520–590 nm; (c) 590–690 nm; (d) 520–690 nm - the circular area represent the resolving limitation.
