**2.2. Effects of black matrix**

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).


Aperture of the lens (m)

19 (a) (b)

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

Aperture of lens (m) 18

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.

h

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

( ) <sup>2</sup>

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

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). 15 To a Fresnel phase lens, the 2π phase is always quantized with equal intervals. Assuming 16 the height before quantization is *h*, the quantization level is *N* and the height of each

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)

**0.1**

**0.2**

**0.3**

**0.4**

Height of quantified level (m)

**0.5**

**0.6**

**-200 -100 0 100 200**

= sin 1 / *c N* (5)

*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 

0.0

phase lens.

0.1

0.2

0.3

Height of lens (m)

0.4

0.5

0.6

70 Adaptive Optics Progress

8 its primary focus:

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 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.

#### **2.3. Mismatch between the pixel and the period**

Because the pixel has a certain size *P*, the period *T* of a Fresnel phase lens cannot be divided exact‐ ly by the pixel, as shown in Fig. 5. This error is similar to the linewidth error caused by the lithog‐ raphy technique. For one period, the integer is *n* and the remainder is *γ* after *T* modulo *P*. If γ≤0.5P, there are *n* pixels in one period; on the contrary, there are *n*+1 pixels. As such, the maxi‐ mum error is 0.5P for the first period. According to Eq. (3), the distribution of the complex ampli‐ tude of the first order can be acquired with the known phase distribution function in one period. Then, the diffraction efficiency can be obtained. As shown in Fig. 6, when the error of the first pe‐ riod changes from 0 to 0.5P, the diffraction efficiency decreases from 81% to 78.3%. The pixel number effect on the variation of the diffraction efficiency is also calculated while the error is 0.5P (Fig. 7). The decrease of the diffraction efficiency is 1% when the pixel number is 7. Accord‐ ingly, if the pixel number is not less than 7 in one period, the effect of the pixel size can be ignored.

**Figure 7.** The decrease of the diffraction efficiency as a function of the pixel number.

The wavefront compensation error always exists due to the finite number of the wavefront correction element used for the correction of the atmospheric turbulence. Hudgin gave the

> 5/3 0 ( ) *<sup>s</sup> <sup>r</sup> <sup>f</sup> <sup>r</sup>* <sup>=</sup> a

where *rs* is the actuator spacing, *r0* is the atmosphere coherence length and *α* is a con‐ stant depending on the response function of the actuator. For continuous surface deform‐ able mirrors (DMs), the response function of the actuator is a Gaussian function and *α* ranges from 0.3-0.4 [68]. For a piston-only response function, *α* is 1.26 [69]. Researchers always use a piston-only response function to evaluate a LCWFC and have proved that the actuators need to be 4-5 times as large as that of the DM's [69, 70]. However, the case is totally different when a diffractive LCWFC (DLCWFC) is used where the kino‐ form or phase wrapping technique is employed to expand the correction capability [71,

(6)

Liquid Crystal Wavefront Correctors http://dx.doi.org/10.5772/54265 73

relationship between the compensation error and the actuator size as follows [67]:

*2.3.1. Wavefront compensation error*

72]. Therefore, Eq. (6) is not suitable any more.

**Figure 5.** The mismatch between the pixel and the period of the Fresnel lens.

**Figure 6.** The diffraction efficiency as a function of the period error.

**Figure 7.** The decrease of the diffraction efficiency as a function of the pixel number.

#### *2.3.1. Wavefront compensation error*

**2.3. Mismatch between the pixel and the period**

72 Adaptive Optics Progress

**Figure 5.** The mismatch between the pixel and the period of the Fresnel lens.

**Figure 6.** The diffraction efficiency as a function of the period error.

Because the pixel has a certain size *P*, the period *T* of a Fresnel phase lens cannot be divided exact‐ ly by the pixel, as shown in Fig. 5. This error is similar to the linewidth error caused by the lithog‐ raphy technique. For one period, the integer is *n* and the remainder is *γ* after *T* modulo *P*. If γ≤0.5P, there are *n* pixels in one period; on the contrary, there are *n*+1 pixels. As such, the maxi‐ mum error is 0.5P for the first period. According to Eq. (3), the distribution of the complex ampli‐ tude of the first order can be acquired with the known phase distribution function in one period. Then, the diffraction efficiency can be obtained. As shown in Fig. 6, when the error of the first pe‐ riod changes from 0 to 0.5P, the diffraction efficiency decreases from 81% to 78.3%. The pixel number effect on the variation of the diffraction efficiency is also calculated while the error is 0.5P (Fig. 7). The decrease of the diffraction efficiency is 1% when the pixel number is 7. Accord‐ ingly, if the pixel number is not less than 7 in one period, the effect of the pixel size can be ignored.

> The wavefront compensation error always exists due to the finite number of the wavefront correction element used for the correction of the atmospheric turbulence. Hudgin gave the relationship between the compensation error and the actuator size as follows [67]:

$$f = a(\frac{r\_s}{r\_0})^{5/3} \tag{6}$$

where *rs* is the actuator spacing, *r0* is the atmosphere coherence length and *α* is a con‐ stant depending on the response function of the actuator. For continuous surface deform‐ able mirrors (DMs), the response function of the actuator is a Gaussian function and *α* ranges from 0.3-0.4 [68]. For a piston-only response function, *α* is 1.26 [69]. Researchers always use a piston-only response function to evaluate a LCWFC and have proved that the actuators need to be 4-5 times as large as that of the DM's [69, 70]. However, the case is totally different when a diffractive LCWFC (DLCWFC) is used where the kino‐ form or phase wrapping technique is employed to expand the correction capability [71, 72]. Therefore, Eq. (6) is not suitable any more.

#### **2.4. The effect of quantization on the wavefront error**

Firstly, the wavefront error generated during the phase wrapping due to quantization is considered. Since a LCWFC is a two-dimensional device, the quantification is performed along the x and y axes by taking the pixel as the unit. According to the diffraction theory, the correction precision as a function of the quantization level can be deduced [73]. If the pixel size is not considered, the root mean square (RMS) error of the diffracted wavefront as a function of the quantization level can be simplified as [73]:

$$
\Delta W = \frac{\lambda}{2\sqrt{3}N} \tag{7}
$$

**2.5. Zernike polynomials for atmospheric turbulence**

*even odd*

Where:

Kolmogorov turbulence theory is employed to analyse the distribution of the quantization level across an atmospheric turbulence wavefront. Noll described Kolmogorov turbulence by using Zernike polynomials [74]. According to him, Zernike polynomials are redefined as:

> ( ) ( ) ( ) ( ) ( ) ( )

r

r

= + ¹ = + ¹

m

Z 2 n 1 R cos m , m 0 Z 2 n 1 R sin m , m 0

m

=

( ) ( )

The parameters *n* and *m* are integral and have the relationship *m*≤*n* and*n* − |*m*| =*even*. An atmospheric turbulence wavefront can be described by using a Kolmogorov phase structure



R r

( ) 6.88 *<sup>r</sup> D r*

1 n s!

nm nm s! s ! s ! 2 2

5/3

5/3

<sup>−</sup>2*m*)/2 (*n* + 1)(*n* ′ + 1) and *D* is the telescope diameter. *δmm′* is the Kro‐

0

*r* æ ö <sup>=</sup> ç ÷ ç ÷ è ø

By combining the phase structure function and the Zernike polynomials, the covariance be‐

and *Zj′* with amplitudes *aj*

<sup>0</sup> 5/3 /2 / 17 / 3 / 2 17 / 3 / 2 23 / 3 / 2

necker delta function. By using Eq. (11), the coefficients of the Zernike polynomials can be easily computed. If the first *J* modes of the Zernike polynomials are selected, the atmospher‐

1

*J t jj j* f *a Z* =

[( ) ]( ) [( ) ] [( ) ] [( ) ]


*K nn D r j j even a a n n n n n n*

G +-¢

å (9)

n 2s

and *aj′* can be deduced as [75]:

*j j odd*


<sup>=</sup> å (12)

(8)

75

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

(10)

(11)

 q

 q

( ) ( )

( ) ( ) ( )

Z n 1R , m 0

r

j n

n m <sup>s</sup> <sup>2</sup> <sup>m</sup>

= +

( )


s=0

*zz mm*

¢ ¢ d

n

tween the Zernike polynomials *Zj*

ì ï í ï î

0,

ic turbulence wavefront is represented as:

function, as below [74]:

*j j*

¢

where*Kz <sup>z</sup>* ′=2.698(−1)(*<sup>n</sup>*+*<sup>n</sup>* ′

r

j n 0

j n

Where *N* is the quantization level and λ is the wavelength*.* If *N*=30, then the RMS error can be as small as λ/100. For *N*=8, RMS=0.036λ and the corresponding Strehl ratio is 0.95. Figure 8 shows the diffracted wavefront RMS error as a function of the quantization level *N*. As can be seen, the wavefront RMS error reduces drastically at first, and then approaches to a con‐ stant gradually when the quantization level becomes greater than 10. The wavefront RMS error can be calculated for a known quantization level on a wavefront. For a DLCWFC, the wavefront compensation error is directly determined by the quantization level without any need to consider the pixel number. Therefore, the distribution of the quantization level on the atmospheric turbulence should be calculated first for a given pixel number, telescope aperture and atmospheric coherence length, and then the wavefront compensation error can be calculated by using Eq. (7).

**Figure 8.** The wavefront RMS error as a function of the quantization level.

#### **2.5. Zernike polynomials for atmospheric turbulence**

Kolmogorov turbulence theory is employed to analyse the distribution of the quantization level across an atmospheric turbulence wavefront. Noll described Kolmogorov turbulence by using Zernike polynomials [74]. According to him, Zernike polynomials are redefined as:

$$\begin{array}{ll} \mathbf{Z}\_{even} & \mathbf{j} = \sqrt{\mathbf{2}\left(\mathbf{n} + 1\right)} \mathbf{R}\_{n}^{\text{m}}\left(\boldsymbol{\rho}\right) \cos\left(\mathbf{m}\boldsymbol{\theta}\right) \text{, } \mathbf{m} \neq \mathbf{0} \\\\ \mathbf{Z}\_{odd} & \mathbf{j} = \sqrt{\mathbf{2}\left(\mathbf{n} + 1\right)} \mathbf{R}\_{n}^{\text{m}}\left(\boldsymbol{\rho}\right) \sin\left(\mathbf{m}\boldsymbol{\theta}\right) \text{, } \mathbf{m} \neq \mathbf{0} \\\\ \mathbf{Z}\_{j} & \mathbf{j} = \sqrt{\left(\mathbf{n} + 1\right)} \mathbf{R}\_{n}^{0}\left(\boldsymbol{\rho}\right) \text{, } & \mathbf{m} = \mathbf{0} \end{array} \tag{8}$$

Where:

**2.4. The effect of quantization on the wavefront error**

74 Adaptive Optics Progress

a function of the quantization level can be simplified as [73]:

**Figure 8.** The wavefront RMS error as a function of the quantization level.

be calculated by using Eq. (7).

Firstly, the wavefront error generated during the phase wrapping due to quantization is considered. Since a LCWFC is a two-dimensional device, the quantification is performed along the x and y axes by taking the pixel as the unit. According to the diffraction theory, the correction precision as a function of the quantization level can be deduced [73]. If the pixel size is not considered, the root mean square (RMS) error of the diffracted wavefront as

2 3

*N* l

Where *N* is the quantization level and λ is the wavelength*.* If *N*=30, then the RMS error can be as small as λ/100. For *N*=8, RMS=0.036λ and the corresponding Strehl ratio is 0.95. Figure 8 shows the diffracted wavefront RMS error as a function of the quantization level *N*. As can be seen, the wavefront RMS error reduces drastically at first, and then approaches to a con‐ stant gradually when the quantization level becomes greater than 10. The wavefront RMS error can be calculated for a known quantization level on a wavefront. For a DLCWFC, the wavefront compensation error is directly determined by the quantization level without any need to consider the pixel number. Therefore, the distribution of the quantization level on the atmospheric turbulence should be calculated first for a given pixel number, telescope aperture and atmospheric coherence length, and then the wavefront compensation error can

D = (7)

*W*

$$\mathbf{R}\_{\mathbf{n}}^{\mathrm{m}}\left(\boldsymbol{\rho}\right) = \sum\_{\mathbf{s}=0}^{\left(\mathbf{n}-\mathbf{m}\right)} \frac{\mathbf{\hat{s}}^{\mathrm{s}} \mathbf{\hat{n}}^{\mathrm{s}} \left(\mathbf{n}-\mathbf{s}\right)!}{\mathbf{s}! \left[\left(\mathbf{n}+\mathbf{m}\right)\bigvee\_{2}^{\mathrm{s}} - \mathbf{s}\right]! \left[\left(\mathbf{n}-\mathbf{m}\right)\bigvee\_{2}^{\mathrm{s}-\mathbf{s}} - \mathbf{s}\right]!} \cdot \mathbf{r}^{\mathrm{n}-\mathbf{s}} \tag{9}$$

The parameters *n* and *m* are integral and have the relationship *m*≤*n* and*n* − |*m*| =*even*. An atmospheric turbulence wavefront can be described by using a Kolmogorov phase structure function, as below [74]:

$$D(r) = 6.88 \left(\frac{r}{r\_0}\right)^{5/3} \tag{10}$$

By combining the phase structure function and the Zernike polynomials, the covariance be‐ tween the Zernike polynomials *Zj* and *Zj′* with amplitudes *aj* and *aj′* can be deduced as [75]:

$$\Gamma\left\{a\_{\boldsymbol{j}},a\_{\boldsymbol{j}'}\right\}=\left\{\frac{K\_{\text{n}'}\delta\_{nm}\Gamma\left[\left(n+n'-5/3\right)/2\right]\left(\boldsymbol{D}\,\boldsymbol{\upbeta}\,\boldsymbol{r}\_{o}\right)^{\otimes 3}}{\Gamma\left[\left(n-n'+17/3\right)/2\right]\Gamma\left[\left(n'-n+17/3\right)/2\right]\Gamma\left[\left(n+n'+23/3\right)/2\right]}\,\boldsymbol{j}-\boldsymbol{j}'=\text{even}\tag{11}$$

where*Kz <sup>z</sup>* ′=2.698(−1)(*<sup>n</sup>*+*<sup>n</sup>* ′ <sup>−</sup>2*m*)/2 (*n* + 1)(*n* ′ + 1) and *D* is the telescope diameter. *δmm′* is the Kro‐ necker delta function. By using Eq. (11), the coefficients of the Zernike polynomials can be easily computed. If the first *J* modes of the Zernike polynomials are selected, the atmospher‐ ic turbulence wavefront is represented as:

$$\phi\_t = \sum\_{j=1}^{l} a\_j Z\_j \tag{12}$$

Therefore, the atmospheric turbulence wavefront *Ф<sup>t</sup>* can be calculated by using Eqs. (11) and (12). As the phase wrapping technique is employed, the atmospheric turbulence wavefront can be wrapped into 2π and quantized and thus the distribution of the quantization level across a telescope aperture *D* can be determined.

the pixel number *PN* and the telescope aperture *D* is calculated for *r0*=10cm and *N*=16, as shown in Fig. 10. It can be seen that <*PN*> is a linear function of *D* when *N* and *r0* are fixed. That is to say, the larger the aperture of the telescope, the more the pixel number will be needed if a DLCWFC is used to correct the atmospheric turbulence. Specifically, for a tele‐ scope with a diameter of 2 metres, the total pixel number will be 96×96, while for a telescope with a diameter of 4 metres the total pixel number will be 168×168. *PN* as a function of *N* is also computed for *r0*=10cm and *D* =2 m, as shown in Fig.11. It illustrates that when *D* and *r0* are fixed, <*PN*> is a linear function of *N*. This means that the more that the quantization level

Liquid Crystal Wavefront Correctors http://dx.doi.org/10.5772/54265 77

**Figure 10.** as a function of the telescope aperture *D* - ■ represents the calculated data for *r0*=10 cm and *N*=16, and

**Figure 11.** as a function of the quantization level *N* - ▲ represents the calculated data for *D*=2 m and *r0*=10 cm, and

is used, the more the pixel number will be needed.

the solid curve represents the fitted data.

the solid curve represents the fitted data.

**Figure 9.** The field of the DLCWFC - the circle represents the wavefront of atmospheric turbulence and P1…PN are the pixel numbers of the DLCWFC.

#### **2.6. Calculation of the required pixel number of DLCWFCs**

In practice, people hope to calculate the desired pixel number of a DLCWFC expediently for a given telescope aperture *D*, a quantization level *N*, and an atmospheric coherence length *r0*. Therefore, it is necessary to deduce the relation between the pixel number of the DLCWFC and *D*, *N* and *r0*. As shown in Fig. 9, the DLCWFC aperture can be represented by the pixel number across the aperture, which is called *PN*. The circle represents the atmos‐ pheric turbulence wavefront *Ф<sup>t</sup>* . Since the atmospheric turbulence wavefront is random, the ensemble average <*Ф<sup>t</sup>* > should be used in the calculation. The modulated and quantized at‐ mospheric turbulence wavefront can be expressed as:

$$\mod(\{\phi\_t\}) = f(N, D, r\_0, \{P\_N\}) \tag{13}$$

where mod( ) denotes the modulo 2π. If <*Ф<sup>t</sup>* > is known, <*PN*> can be expressed as a function of the telescope aperture *D*, the quantization level *N*, and the atmospheric coherence length *r0*. By using Eqs. (11) and (12), <*Ф<sup>t</sup>* > can be calculated and the first 136 modes of the Zernike polynomials are used in the calculation. For the randomness of the atmospheric turbulence wavefront, different quantization levels are used during the quantization, depending upon the fluctuation degree of the wavefront. Here, *N* is defined as the minimum quantization level so that the sum of those quantization levels greater than *N* should occupy 95% of the quantization levels included in the atmospheric turbulence wavefront. Fifty atmospheric turbulence wavefronts are used to achieve the statistical results. First, the relation between the pixel number *PN* and the telescope aperture *D* is calculated for *r0*=10cm and *N*=16, as shown in Fig. 10. It can be seen that <*PN*> is a linear function of *D* when *N* and *r0* are fixed. That is to say, the larger the aperture of the telescope, the more the pixel number will be needed if a DLCWFC is used to correct the atmospheric turbulence. Specifically, for a tele‐ scope with a diameter of 2 metres, the total pixel number will be 96×96, while for a telescope with a diameter of 4 metres the total pixel number will be 168×168. *PN* as a function of *N* is also computed for *r0*=10cm and *D* =2 m, as shown in Fig.11. It illustrates that when *D* and *r0* are fixed, <*PN*> is a linear function of *N*. This means that the more that the quantization level is used, the more the pixel number will be needed.

Therefore, the atmospheric turbulence wavefront *Ф<sup>t</sup>* can be calculated by using Eqs. (11) and (12). As the phase wrapping technique is employed, the atmospheric turbulence wavefront can be wrapped into 2π and quantized and thus the distribution of the quantization level

**Figure 9.** The field of the DLCWFC - the circle represents the wavefront of atmospheric turbulence and P1…PN are the

In practice, people hope to calculate the desired pixel number of a DLCWFC expediently for a given telescope aperture *D*, a quantization level *N*, and an atmospheric coherence length *r0*. Therefore, it is necessary to deduce the relation between the pixel number of the DLCWFC and *D*, *N* and *r0*. As shown in Fig. 9, the DLCWFC aperture can be represented by the pixel number across the aperture, which is called *PN*. The circle represents the atmos‐

mod( ) ( , , , ) *t N* <sup>0</sup>

of the telescope aperture *D*, the quantization level *N*, and the atmospheric coherence length

polynomials are used in the calculation. For the randomness of the atmospheric turbulence wavefront, different quantization levels are used during the quantization, depending upon the fluctuation degree of the wavefront. Here, *N* is defined as the minimum quantization level so that the sum of those quantization levels greater than *N* should occupy 95% of the quantization levels included in the atmospheric turbulence wavefront. Fifty atmospheric turbulence wavefronts are used to achieve the statistical results. First, the relation between

f

. Since the atmospheric turbulence wavefront is random, the

= *f NDr P* (13)

> can be calculated and the first 136 modes of the Zernike

> is known, <*PN*> can be expressed as a function

> should be used in the calculation. The modulated and quantized at‐

across a telescope aperture *D* can be determined.

**2.6. Calculation of the required pixel number of DLCWFCs**

mospheric turbulence wavefront can be expressed as:

where mod( ) denotes the modulo 2π. If <*Ф<sup>t</sup>*

*r0*. By using Eqs. (11) and (12), <*Ф<sup>t</sup>*

pixel numbers of the DLCWFC.

76 Adaptive Optics Progress

pheric turbulence wavefront *Ф<sup>t</sup>*

ensemble average <*Ф<sup>t</sup>*

**Figure 10.** as a function of the telescope aperture *D* - ■ represents the calculated data for *r0*=10 cm and *N*=16, and the solid curve represents the fitted data.

**Figure 11.** as a function of the quantization level *N* - ▲ represents the calculated data for *D*=2 m and *r0*=10 cm, and the solid curve represents the fitted data.

The relationship between <*PN*> and *r0* is also calculated with the variables *N* and *D.* Figure 12 shows only three curves with three pairs of fixed *N* and *D.* This time, the relationship is not a linear function anymore but an exponential function. With more pairs of *N* and *D* fixed, more curves can be obtained, but these are not shown in the figure. The relationship be‐ tween <*PN*> and *r0* can be expressed by the following formula:

$$
\left\langle P\_N \right\rangle = A + B r\_0^{-6/5} \tag{14}
$$

Normally, the quantization level of 8 is suitable for the atmospheric turbulence correction for three reasons. Firstly, a higher correction accuracy can be obtained. Whe*n N*=8, the RMS error can be reduced down to 0.035λ and the Strehl ratio can be increased to 95%. Secondly, a higher diffraction efficiency can be obtained. According to the diffractive optics theory [73], the diffraction efficiency is as large as 95% for *N*=8. Finally, the total pixel number can be controlled within a reasonable range. Of course, a smaller wavefront RMS error and a higher diffraction efficiency can be achieved with a larger quantization level. But, in that case, the required pixel number of the DLCWFC will be increased drastically, which will lead to a significantly slower computation and data transformation rate of the LCAOS. Fig. 13 shows the relation between *PN*, D and *r0* for *N*=8. As can be seen, the desired pixel num‐ ber apparently increases when the atmospheric coherence length becomes smaller and the telescope aperture becomes larger. For instance, the total pixel number of the DLCWFC is 1700×1700 when *r0*=5 cm and *D*=20 m. However, if *r0*=10 cm, the total pixel number can be reduced down to 768×768. Therefore, the strength of the atmosphere turbulence is a key fac‐ tor which must be considered when designing the LCAOS for a ground-based telescope.

Liquid Crystal Wavefront Correctors http://dx.doi.org/10.5772/54265 79

**Figure 13.** as functions of the atmosphere coherence length *r0* and the telescope aperture *D* for *N*=8.

Currently, reflective LCWFC devices [77-79], such as liquid crystal on silicon (LCOS) de‐ vices, are especially attractive because of their small fill factor, high reflectivity and short response time. To separate the incident beam from the reflected beam for a reflective LCWFC, the incident light should go to the LCWFC with a tilt angle. Alternatively, the

*2.6.1. Tilt incidence*

where *A* and *B* are the coefficients. *A* is only related to *N* and can be expressed as *A*=6.25*N*. As <*PN*> is a linear function of *D* and *N*, the coefficient *B* can be expressed as:

$$B = a + bN + cD + dND\tag{15}$$

where *a*, *b*, *c* and *d* are the coefficients. By substituting the known value of *N*, *D* and the cal‐ culated coefficient *B*, the value of *a*, *b*, *c* and *d* is determined to be 15, -23, -150 and 91, respec‐ tively, by using the least-square method. Thus, <*PN*> can be expressed as:

$$\left\langle P\_N \right\rangle = 6.25N + \left(15 - 1.5D - 23N + 0.91ND\right)r\_0^{-6/5} \tag{16}$$

where the units of *D* and *r0* is centimetres. The total pixel number of the DLCWFC can be calculated by using *PN* ×*PN*. By combining Eqs. (7) and (16), the compensation error of the DLCWFC can be evaluated for the atmospheric turbulence correction. These two formulas are not suitable for modal types of LCWFCs [76] or other types that do not use the diffrac‐ tion method to correct the atmospheric turbulence.

**Figure 12.** as a function of the atmosphere coherence length *r0* - the line is the fitted curve and ■, ● and ∗ represent the computed data with *N*=16 and *D*=4 m, N=8 and *D*=4 m, and *N*=8 and *D*=2 m, respectively.

Normally, the quantization level of 8 is suitable for the atmospheric turbulence correction for three reasons. Firstly, a higher correction accuracy can be obtained. Whe*n N*=8, the RMS error can be reduced down to 0.035λ and the Strehl ratio can be increased to 95%. Secondly, a higher diffraction efficiency can be obtained. According to the diffractive optics theory [73], the diffraction efficiency is as large as 95% for *N*=8. Finally, the total pixel number can be controlled within a reasonable range. Of course, a smaller wavefront RMS error and a higher diffraction efficiency can be achieved with a larger quantization level. But, in that case, the required pixel number of the DLCWFC will be increased drastically, which will lead to a significantly slower computation and data transformation rate of the LCAOS. Fig. 13 shows the relation between *PN*, D and *r0* for *N*=8. As can be seen, the desired pixel num‐ ber apparently increases when the atmospheric coherence length becomes smaller and the telescope aperture becomes larger. For instance, the total pixel number of the DLCWFC is 1700×1700 when *r0*=5 cm and *D*=20 m. However, if *r0*=10 cm, the total pixel number can be reduced down to 768×768. Therefore, the strength of the atmosphere turbulence is a key fac‐ tor which must be considered when designing the LCAOS for a ground-based telescope.

**Figure 13.** as functions of the atmosphere coherence length *r0* and the telescope aperture *D* for *N*=8.

#### *2.6.1. Tilt incidence*

The relationship between <*PN*> and *r0* is also calculated with the variables *N* and *D.* Figure 12 shows only three curves with three pairs of fixed *N* and *D.* This time, the relationship is not a linear function anymore but an exponential function. With more pairs of *N* and *D* fixed, more curves can be obtained, but these are not shown in the figure. The relationship be‐

6/5

where *A* and *B* are the coefficients. *A* is only related to *N* and can be expressed as *A*=6.25*N*.

where *a*, *b*, *c* and *d* are the coefficients. By substituting the known value of *N*, *D* and the cal‐ culated coefficient *B*, the value of *a*, *b*, *c* and *d* is determined to be 15, -23, -150 and 91, respec‐

where the units of *D* and *r0* is centimetres. The total pixel number of the DLCWFC can be calculated by using *PN* ×*PN*. By combining Eqs. (7) and (16), the compensation error of the DLCWFC can be evaluated for the atmospheric turbulence correction. These two formulas are not suitable for modal types of LCWFCs [76] or other types that do not use the diffrac‐

**Figure 12.** as a function of the atmosphere coherence length *r0* - the line is the fitted curve and ■, ● and ∗ represent

the computed data with *N*=16 and *D*=4 m, N=8 and *D*=4 m, and *N*=8 and *D*=2 m, respectively.

( ) 6/5

<sup>0</sup> 6.25 15 1.5 23 0.91 *NP N D N ND r*- = +- - + (16)

As <*PN*> is a linear function of *D* and *N*, the coefficient *B* can be expressed as:

tively, by using the least-square method. Thus, <*PN*> can be expressed as:

tion method to correct the atmospheric turbulence.

*<sup>N</sup>* <sup>0</sup> *P A Br*- = + (14)

*B a bN cD dND* =+ + + (15)

tween <*PN*> and *r0* can be expressed by the following formula:

78 Adaptive Optics Progress

Currently, reflective LCWFC devices [77-79], such as liquid crystal on silicon (LCOS) de‐ vices, are especially attractive because of their small fill factor, high reflectivity and short response time. To separate the incident beam from the reflected beam for a reflective LCWFC, the incident light should go to the LCWFC with a tilt angle. Alternatively, the incident light is perpendicular to the LCWFC and a beam splitter is placed before the LCWFC to separate the reflected and incident beams. However, the second method will result in a 50% loss in each direction, reducing output power to 25% of the input. To avoid the energy loss, the tilt incidence is a suitable method for a LCWFC. However, the tilt incidence will affect the phase modulation and the diffraction efficiency of the LCWFC. A reflective LCWFC model is selected to perform the analysis and the acquired results are suitable for the transmitted LCWFC.

#### **2.7. Effect of the tilt incidence on the phase modulation of the LCWFC**

In order to simplify the model of the reflective LCWFC, the border effect is neglected and all of the molecules have the same tilt angle. The simplified model is shown in Fig.14. The for‐ mer board is glass and the back is silicon. The liquid crystal molecule is aligned parallel to the board. The tilt incident angle is θ′. The liquid crystal material is a uniaxial birefringence material - it has an ordinary index *no* and the extraordinary index *ne(θ)*. *ne(θ)* can be obtained with the index ellipsoid equation [41]:

$$n\_{\rm e} \left( \theta \right) = \frac{n\_{\rm o} n\_{\rm e}}{\left( n\_{\rm o} \, ^2 \cos^2 \theta + n\_{\rm e} ^2 \sin^2 \theta \right)^{1/2}},\tag{17}$$

**Figure 14.** Simplified model of the reflective LCWFC with tilt incidence.

that the LCWFC may be used with a small tilt angle.

mulated data.

For *ne*=1.714, *no*=1.516, λ=633nm and *d*=1.6µm, the phase modulation as a function of the inci‐ dent angle is shown in Fig.15. The simulated results show that the phase modulation is re‐ duced by at most 1% for incident angles under 6°. The measured result is also shown in the figure. The trends of the simulated and measured curves are similar. The difference of in phase shift might be caused by the border effect. For the actual liquid crystal cell, a rubbing polyimide (PI) film is used to align the liquid crystal molecules. The PI layer will anchor the liquid crystal molecules at the border; this causes the tilt angle of the liquid crystal molecule at the interface to be different from the centre. The simulated and measured results indicate

Liquid Crystal Wavefront Correctors http://dx.doi.org/10.5772/54265 81

**Figure 15.** Phase shift as a function of the incident angle - -●- is the measured curve and -∗- represents the si‐

where *θ* represents the tilt angle of the molecule and *ne* is the off-state extraordinary refrac‐ tive index. Assume that the LCWFC without the applied voltages and the polarization direc‐ tion is the same as the LC director. For the tilt incidence as shown in Fig.14, it is equivalent to the rotation of the LC director with an angle *θ′.* Hence, although the tilt angle of the mole‐ cule is zero, the extraordinary refractive index is changed to *ne(θ′)* with the tilt incidence. Furthermore, the tilt incidence will change the transmission distance of the light in the liq‐ uid crystal cell with a factor of 1/cos*θ′*. Consequently, the phase modulation with the tilt in‐ cidence and no applied voltages can be expressed as:

$$P\_{\rm tilt} = \frac{2\pi (n\_e(\theta') - n\_o)d}{\lambda \cos \theta'}.\tag{18}$$

If the pre-tilt angle of the liquid crystal molecule is considered, Eq.(18) can be rewritten as:

$$P\_{\rm tilt} = \frac{2\pi (n\_e(\theta' + \theta\_0) - n\_o)d}{\lambda \cos \theta'},\tag{19}$$

where *θ0* is the pre-tilt angle, *d* is the thickness of the liquid crystal cell and λ is the relevant wavelength.

**Figure 14.** Simplified model of the reflective LCWFC with tilt incidence.

incident light is perpendicular to the LCWFC and a beam splitter is placed before the LCWFC to separate the reflected and incident beams. However, the second method will result in a 50% loss in each direction, reducing output power to 25% of the input. To avoid the energy loss, the tilt incidence is a suitable method for a LCWFC. However, the tilt incidence will affect the phase modulation and the diffraction efficiency of the LCWFC. A reflective LCWFC model is selected to perform the analysis and the acquired

In order to simplify the model of the reflective LCWFC, the border effect is neglected and all of the molecules have the same tilt angle. The simplified model is shown in Fig.14. The for‐ mer board is glass and the back is silicon. The liquid crystal molecule is aligned parallel to the board. The tilt incident angle is θ′. The liquid crystal material is a uniaxial birefringence material - it has an ordinary index *no* and the extraordinary index *ne(θ)*. *ne(θ)* can be obtained

( )

cos sin

where *θ* represents the tilt angle of the molecule and *ne* is the off-state extraordinary refrac‐ tive index. Assume that the LCWFC without the applied voltages and the polarization direc‐ tion is the same as the LC director. For the tilt incidence as shown in Fig.14, it is equivalent to the rotation of the LC director with an angle *θ′.* Hence, although the tilt angle of the mole‐ cule is zero, the extraordinary refractive index is changed to *ne(θ′)* with the tilt incidence. Furthermore, the tilt incidence will change the transmission distance of the light in the liq‐ uid crystal cell with a factor of 1/cos*θ′*. Consequently, the phase modulation with the tilt in‐

> 2( ( ) ) . cos *e o*

If the pre-tilt angle of the liquid crystal molecule is considered, Eq.(18) can be rewritten as:

<sup>0</sup> 2( ( ) ) , cos *e o*

where *θ0* is the pre-tilt angle, *d* is the thickness of the liquid crystal cell and λ is the relevant

*n nd <sup>P</sup>*

l q

p qq

*n nd <sup>P</sup>* p q

l q

+

e 1

o e

q

*n n*

o e

*n n*

2 2 22 2

,

¢ - <sup>=</sup> ¢ (18)

¢ + - <sup>=</sup> ¢ (19)

(17)

 q

results are suitable for the transmitted LCWFC.

80 Adaptive Optics Progress

with the index ellipsoid equation [41]:

**2.7. Effect of the tilt incidence on the phase modulation of the LCWFC**

( )

q

=

*n*

cidence and no applied voltages can be expressed as:

*tilt*

*tilt*

wavelength.

For *ne*=1.714, *no*=1.516, λ=633nm and *d*=1.6µm, the phase modulation as a function of the inci‐ dent angle is shown in Fig.15. The simulated results show that the phase modulation is re‐ duced by at most 1% for incident angles under 6°. The measured result is also shown in the figure. The trends of the simulated and measured curves are similar. The difference of in phase shift might be caused by the border effect. For the actual liquid crystal cell, a rubbing polyimide (PI) film is used to align the liquid crystal molecules. The PI layer will anchor the liquid crystal molecules at the border; this causes the tilt angle of the liquid crystal molecule at the interface to be different from the centre. The simulated and measured results indicate that the LCWFC may be used with a small tilt angle.

**Figure 15.** Phase shift as a function of the incident angle - -●- is the measured curve and -∗- represents the si‐ mulated data.

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

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

**2.9. Diffraction efficiency with tilt incidence**

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

for most applications.

*2.9.1. Chromatism*

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

Liquid Crystal Wavefront Correctors http://dx.doi.org/10.5772/54265 83

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

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:

allowed, LCWFC can be used to correct distortion within a narrow spectral range.

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

**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.
