5. Experimental measurement of point-diffraction wavefront

The numerical simulation based on the vector diffraction theory such as FDTD method provides an easy and efficient way to estimate the point-diffraction wavefront error. However, both the computational accuracy and complex practical factors (such as environmental disturbance and performances of various optical parts) can introduce significant deviation from an ideal case. Due to the accuracy limitation of standard optics, the traditional interferometers fail to measure the point-diffraction wavefront error, which is expected to be in the order of subnanometer or even smaller. Various experimental testing methods have been proposed to measure the point-diffraction wavefront error, the majority of which are based on the hybrid method [15] and null test [28]. Typically, the hybrid method requires several measurements with the rotation and displacement of the optics under test, it is sensitive to environmental disturbance and cannot completely separate the systematic error. The null test is a self-reference method, and it is widely applied to reconstruct the point-diffraction wavefront and calibrate the PDI. However, it requires the foreknowledge about the system configuration to remove the high-order aberrations, which are introduced by the point-source separation and cannot be negligible in the case of high-NA wavefront. In this section, a high-precision measurement method, which is based on shearing interferometry, is presented to analyze the point-diffraction wavefront from SMA fiber [21].

Figure 15 shows the schematic diagram of SMA fiber projector used to evaluate the sphericity of point-diffraction wavefront. The SMA fiber can be applied to obtain both high diffraction light power and high-NA spherical wavefront. Four parallel SMA single-mode fibers S1, S2, S3, and S4 with coplanar exit ends are integrated in a metal tube. To obtain the identical wavefront parameters, the four SMA fibers with same aperture size and cone angle are carefully chosen in the experiment to minimize the measurement error. The coherent beams are coupled into the SMA fibers, and the point-diffraction waves from the fibers interfere on a detection plane, as shown in Figure 15(b). By alternatively switching on the waves from the fiber pairs G1 (S1 and S2) and G2

Figure 15. Schematic diagram of SMA fiber point-diffraction wavefront measurement system. (a) Structure of SMA fiber projector and (b) schematic diagram of SMA fiber point-diffraction wavefront measurement [21].

(S3 and S4), the shearing interferograms in x and y directions can be obtained and the corresponding shearing wavefronts can be measured with a phase-shifting method, respectively.

### 5.1. Point-diffraction wavefront retrieval method

With the shearing wavefronts obtained from the projector shown in Figure 15, the differential Zernike polynomials fitting method can be applied to retrieve the SMA fiber point-diffraction wavefront. Denoting the point-diffraction wavefront under test as Wðx, yÞ, we have the shearing wavefronts ΔWxðx, yÞ and ΔWyðx, yÞ in x and y directions,

$$\begin{cases} \Delta W\_x(\mathbf{x}, y) = \mathcal{W}(\mathbf{x}, y) \text{-} \mathcal{W}(\mathbf{x} + \mathbf{s}, y) \\ \Delta W\_y(\mathbf{x}, y) = \mathcal{W}(\mathbf{x}, y) \text{-} \mathcal{W}(\mathbf{x}, y + \mathbf{s}) \end{cases} \tag{6}$$

where s is the shearing amount as shown in Figure 15(b). The wavefront Wðx, yÞ can be described with Zernike polynomials and can be decomposed into a series (N terms) of orthonormal polynomials fZiðx, yÞg with the corresponding coefficients faig,

$$\mathcal{W}(\mathbf{x}, \mathbf{y}) = \sum\_{i=1}^{N} a\_i \mathbf{Z}\_i(\mathbf{x}, \mathbf{y}). \tag{7}$$

According to Eqs. (6) and (7), the shearing wavefronts ΔWxðx, yÞ and ΔWyðx, yÞ can be expressed as follows:

$$\begin{cases} \Delta W\_{\mathbf{x}}(\mathbf{x}, \mathbf{y}) = \sum\_{i=1}^{N} a\_{i} \Delta Z\_{i, \mathbf{x}}(\mathbf{x}, \mathbf{y}) \\ \Delta W\_{\mathbf{y}}(\mathbf{x}, \mathbf{y}) = \sum\_{i=1}^{N} a\_{i} \Delta Z\_{i, \mathbf{y}}(\mathbf{x}, \mathbf{y}) \end{cases}, \tag{8}$$

where the differential Zernike polynomials ΔZi, <sup>x</sup>ðx, yÞ and ΔZi,yðx, yÞ can be written as follows:

$$\begin{cases} \Delta Z\_{i,x}(\mathbf{x},y) = Z\_i(\mathbf{x},y) - Z\_i(\mathbf{x}+s,y) \\ \Delta Z\_{i,y}(\mathbf{x},y) = Z\_i(\mathbf{x},y) - Z\_i(\mathbf{x},y+s) \end{cases} \tag{9}$$

Denoting the shearing wavefronts, differential Zernike polynomials and the coefficients in Eq. (8) as ΔW ¼ ðΔWx,ΔWyÞ <sup>T</sup>, <sup>Δ</sup><sup>Z</sup> ¼ ðΔZi, <sup>x</sup>,ΔZi, <sup>y</sup><sup>Þ</sup> <sup>T</sup> and a, respectively, Eq. (8) can be transformed into a matrix form,

$$
\Delta \mathbf{W} = \Delta \mathbf{Z} \mathbf{a}.\tag{10}
$$

Thus, the coefficients {ai} of Zernike polynomials in Eq. (7) can be obtained from the leastsquares solution of Eq. (10),

$$\mathbf{a} = (\Delta \mathbf{Z}^T \Delta \mathbf{Z})^{-1} \Delta \mathbf{Z}^T \Delta \mathbf{W}. \tag{11}$$

According to Eq. (11), the retrieval of point-diffraction wavefront Wðx, yÞ depends on the measurement precision of shearing wavefronts ΔWxðx, yÞ and ΔWyðx, yÞ. Traditionally, the systematic error introduced by lateral displacement between SMA fibers can be calibrated by removing Zernike tilt and power terms. However, the residual high-order aberrations can significantly influence the measurement precision, especially in the case of high NA and large lateral displacement. Thus, a general and rigorous method for geometric error removal is required to realize the high-precision measurement of point-diffraction wavefront.

### 5.2. High-precision method for systematic error calibration

(S3 and S4), the shearing interferograms in x and y directions can be obtained and the corresponding shearing wavefronts can be measured with a phase-shifting method, respectively.

Figure 15. Schematic diagram of SMA fiber point-diffraction wavefront measurement system. (a) Structure of SMA fiber

projector and (b) schematic diagram of SMA fiber point-diffraction wavefront measurement [21].

With the shearing wavefronts obtained from the projector shown in Figure 15, the differential Zernike polynomials fitting method can be applied to retrieve the SMA fiber point-diffraction wavefront. Denoting the point-diffraction wavefront under test as Wðx, yÞ, we have the shear-

> ΔWxðx, yÞ ¼ Wðx, yÞ−Wðx þ s, yÞ <sup>Δ</sup>Wyðx, <sup>y</sup>Þ ¼ <sup>W</sup>ðx, <sup>y</sup>Þ−Wðx, <sup>y</sup> <sup>þ</sup> <sup>s</sup>Þ,

where s is the shearing amount as shown in Figure 15(b). The wavefront Wðx, yÞ can be described with Zernike polynomials and can be decomposed into a series (N terms) of ortho-

> N i¼1

According to Eqs. (6) and (7), the shearing wavefronts ΔWxðx, yÞ and ΔWyðx, yÞ can be

N i¼1

N i¼1

where the differential Zernike polynomials ΔZi, <sup>x</sup>ðx, yÞ and ΔZi,yðx, yÞ can be written as follows:

aiΔZi,xðx, yÞ

,

aiΔZi,yðx, yÞ

aiZiðx, yÞ: (7)

Wðx, yÞ ¼ ∑

ΔWxðx, yÞ ¼ ∑

ΔWyðx, yÞ ¼ ∑

(6)

(8)

5.1. Point-diffraction wavefront retrieval method

expressed as follows:

200 Optical Interferometry

ing wavefronts ΔWxðx, yÞ and ΔWyðx, yÞ in x and y directions,

�

normal polynomials fZiðx, yÞg with the corresponding coefficients faig,

8 >><

>>:

In the null test of pinhole diffraction wavefront and single-mode fiber diffraction wavefront, the high wavefront NA in a pinhole PDI and large lateral displacement between fibers in fiber PDI could introduce some high-order geometric aberrations, respectively. Different from traditional pinhole PDI and single-mode fiber PDI, the null test of SMA fiber diffraction wavefront involves both high NA and large lateral displacement, placing much higher requirement on the calibration of the systematic error introduced by lateral displacement between SMA fibers. A double-step calibration method based on three-dimensional coordinate reconstruction and symmetric lateral displacement compensation can be applied to completely remove systematic error. It should be noted that the possible longitudinal displacement between SMA fibers may also introduce certain systematic error; however, it can be well minimized with the fine adjusting mechanism. Besides, the error introduced by the longitudinal displacement is low-order aberration, and it can be well calibrated with traditional misalignment calibration method by subtracting the Zernike piston, tilt and power terms.

#### 5.2.1. First-step calibration based on three-dimensional coordinate reconstruction

Without loss of generality, we take the displacement in x direction between SMA fibers as the geometric error calibration model to be analyzed, as shown in Figure 16.

Supposing that the distance between exit apertures of fiber pairs G1 (S1 and S2) and an arbitrary point Pðx, y, zÞ on the detection plane are R<sup>1</sup> and R2, we have the corresponding optical path difference OPD,

$$\text{OPD} = R\_1(\mathbf{x}\_1, y\_1, z\_1; \mathbf{x}, y, z) - R\_2(\mathbf{x}\_2, y\_2, z\_2; \mathbf{x}, y, z), \tag{12}$$

where ðx1, y1, z1Þ and ðx2, y2, z2Þ are the three-dimensional coordinates of S1 and S2, ðx, y, zÞ is that of the arbitrary known point P. To simplify the analysis, the origin of the coordinate system is located at S1, the distance between the SMA fiber projector and CCD detector is D. Thus, the OPD in Eq. (12) can be simplified as follows:

$$\text{OPD} = \sqrt{\mathbf{x}^2 + \mathbf{y}^2 + D^2} \cdot \sqrt{\left(\mathbf{x} + \mathbf{s}\right)^2 + \mathbf{y}^2 + D^2}. \tag{13}$$

According to the one-to-one correspondence of the OPD distribution on the detection plane and the coordinate of fiber apertures, the 3D coordinate measurement method with PDI introduced in Section 3 can be applied to reconstruct the 3D coordinates in Eq. (12) that is the global minimum Φ� of the residual function FðΦÞ,

$$F(\mathbf{OP}) = \sum\_{k} f\_k^2(\mathbf{OP}) = \sum\_{k} (\mathbf{OPD}\_k - \widehat{\xi}\_k)^2,\tag{14}$$

where the vector Φ ¼ {ðx1, y1, z1Þ;ðx2, y2, z2Þ} is the coordinates of SMA fibers under measure-

ment, the subscript k indicates the point number on the detection plane, ξ \_ <sup>k</sup> is the measured OPD and OPD<sup>k</sup> is the OPD reconstructed from coordinates Φ according to Eq. (13). With the reconstructed coordinates Φ� , the systematic error can be preliminarily calibrated,

$$\text{OPD}\_1 = \text{ OPD}(\mathbf{OP})\text{-OPD}(\mathbf{OP}^\*). \tag{15}$$

However, the reconstruction accuracy of fiber coordinates can only reach the order of submicron in the practical case, resulting in obvious residual error in the calibration result. The root mean square (RMS) value of the residual error in the preliminary calibration is 0:0077λ corresponding to the 0.5 μm coordinate reconstruction error for 0.60 NA fibers and 250 μm lateral displacement between two fibers. To further remove the residual systematic error, a second-step calibration, which is based on symmetric lateral displacement compensation, needs to be carried out.

Figure 16. Geometry for systematic error analysis [21].

#### 5.2.2. Second-step calibration based on symmetric lateral displacement compensation

The expression for OPD in Eq. (12) can be simplified as follows:

OPD ¼ R1ðx1, y1, z1; x, y, zÞ−R2ðx2, y2, z2; x, y, zÞ, (12)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>2</sup> <sup>þ</sup> <sup>y</sup><sup>2</sup> <sup>þ</sup> <sup>D</sup><sup>2</sup>

: (13)

, (14)

<sup>k</sup> is the measured

\_

Þ: (15)

ðx þ sÞ

ðOPDk−ξ \_ kÞ 2

, the systematic error can be preliminarily calibrated,

where ðx1, y1, z1Þ and ðx2, y2, z2Þ are the three-dimensional coordinates of S1 and S2, ðx, y, zÞ is that of the arbitrary known point P. To simplify the analysis, the origin of the coordinate system is located at S1, the distance between the SMA fiber projector and CCD detector is D.

−

According to the one-to-one correspondence of the OPD distribution on the detection plane and the coordinate of fiber apertures, the 3D coordinate measurement method with PDI introduced in Section 3 can be applied to reconstruct the 3D coordinates in Eq. (12) that is the

where the vector Φ ¼ {ðx1, y1, z1Þ;ðx2, y2, z2Þ} is the coordinates of SMA fibers under measure-

OPD and OPD<sup>k</sup> is the OPD reconstructed from coordinates Φ according to Eq. (13). With the

OPD1 ¼ OPDðΦÞ−OPDðΦ�

However, the reconstruction accuracy of fiber coordinates can only reach the order of submicron in the practical case, resulting in obvious residual error in the calibration result. The root mean square (RMS) value of the residual error in the preliminary calibration is 0:0077λ corresponding to the 0.5 μm coordinate reconstruction error for 0.60 NA fibers and 250 μm lateral displacement between two fibers. To further remove the residual systematic error, a second-step calibration, which is based on symmetric lateral displacement compensation,

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>x</sup><sup>2</sup> <sup>þ</sup> <sup>y</sup><sup>2</sup> <sup>þ</sup> <sup>D</sup><sup>2</sup>

Thus, the OPD in Eq. (12) can be simplified as follows:

OPD ¼

global minimum Φ� of the residual function FðΦÞ,

reconstructed coordinates Φ�

202 Optical Interferometry

needs to be carried out.

Figure 16. Geometry for systematic error analysis [21].

q

FðΦÞ ¼ ∑ k f 2 <sup>k</sup> ðΦÞ ¼ ∑ k

ment, the subscript k indicates the point number on the detection plane, ξ

$$\text{OPD\*}\!a\_2\!\!Z\_2 + a\_2\!\!Z\_2 + a\_1\!\!a\_2\!\!I\_1\!\!\!o + a\_3\!\!Z\_3\!\!a\_3\!\!\!R\_1\!\!\!a\_1\!\!\!R\_2\!\!\!R\_3$$

where Z<sup>2</sup> refers to x tilt terms, Z9, Z19, and Z<sup>33</sup> are Zernike primary, secondary, and tertiary x coma terms, respectively, a2, a9, a19, and a<sup>33</sup> are the corresponding coefficients, and

$$\begin{cases} a\_2 = s[-t\text{-}ts^2/(2D^2) + t^3/3 + 3t^3s^2/(4D^2) - 3t^5/16 - 15t^5s^2/(16D^2) + t^7/8] \\ a\_9 = s[t^3/6 + 3t^3s^2/(8D^2) - 3t^5/20 - 15t^5s^2/(20D^2) + t^7/8] \\ a\_{10} = s[-3t^5/80 - 3t^5s^2/(16D^2) + 3t^7/56] \\ a\_{33} = s(t^7/112) \end{cases} . \tag{17}$$

Similarly, the reconstructed optical path difference OPDðΦ� Þ (though with certain coordinate reconstruction error) follows the relationship given by Eq. (16). Thus, the residual systematic error OPD1 after first-step calibration, according to Eqs. (15) and (16), can be written as follows:

$$\text{OPD}\_1 \mathbb{1} \Delta a\_2 \mathbb{Z}\_2 + \Delta a\_2 \mathbb{Q}\_2 + \Delta a\_1 \mathbb{1} g \mathbb{Z}\_{19} + \Delta a\_{33} \mathbb{Z}\_{33},\tag{18}$$

where {Δai} (i = 2, 9, 19, 33) are the corresponding Zernike coefficients. With the implementation of the first-step calibration, the coordinate reconstruction error reaches the order of submicron, and the corresponding approximation error in Eq. (18) can be well restricted and is negligible. According to Eq. (16), the major systematic error introduced by the lateral displacement includes tilt and coma terms, they cannot be completely removed with traditional misalignment calibration method by subtracting the Zernike piston, tilt and power terms. From Eq. (16), the residual coma aberration due to lateral displacement depends on the lateral displacement s, NA, and distance D, the corresponding Zernike coefficients are odd functions about s. Thus, the superposition of geometric errors OPD<sup>ð</sup>s<sup>Þ</sup> and OPD<sup>ð</sup>−s<sup>Þ</sup> corresponding to opposite shear directions can be expressed as follows:

$$\text{OPD}^{(s)} + \text{OPD}^{(-s)} \bowtie 0. \tag{19}$$

According to the analysis above, the geometric error can be further reduced by superposing two measurements with opposite shear directions, corresponding to lateral displacement s and −s, respectively. Denoting the preliminarily calibrated wavefront data corresponding to lateral displacement s and −s as W<sup>ð</sup>s<sup>Þ</sup> <sup>m</sup><sup>1</sup> and <sup>W</sup><sup>ð</sup>−s<sup>Þ</sup> <sup>m</sup><sup>1</sup> , and the true shearing wavefront under test ΔW, we have

$$\begin{cases} \boldsymbol{W}\_{m1}^{(s)} = \Delta \boldsymbol{W} + \mathbf{OPD}\_{1}{}^{(s)} \\ \boldsymbol{W}\_{m1}^{(\neg s)} = \Delta \boldsymbol{W} + \mathbf{OPD}\_{1}{}^{(\neg s)} \end{cases} \tag{20}$$

According to Eqs. (19) and (20), the shearing wavefront ΔW can be obtained with

$$
\Delta \mathsf{W} \models (\mathsf{W}\_{m1}^{(s)} + \mathsf{W}\_{m1}^{(-s)})/2. \tag{21}
$$

To get the measurement data with the lateral displacement −s, one may rotate the SMA fiber projector by 180°. Figure 17(a) and (b) shows the residual systematic errors corresponding to various lateral displacement amounts with different NAs with traditional method and the double-step calibration method. The traditional method is not valid when the lateral displacement is over 50 μm because the residual error RMS is larger than 2:0 · 10<sup>−</sup><sup>3</sup> λ with the 0.10 NA fiber. The residual error also increases significantly with the lateral displacement, especially for the case with high NA fiber. In Figure 17(b), the residual error is less than 1:0 · 10<sup>−</sup><sup>4</sup> λ RMS within a 300 μm lateral displacement even for the 0.60 NA fiber, it confirms the feasibility of the double-step calibration in Section 5.2.

To obtain point-diffraction spherical wavefront in the experiment, the projector should be placed at the far-field zone that is the distance D >> 2ð2dÞ 2 =λ from the fibers [29], where d is the aperture size. With the unwrapped phase distribution, the systematic error calibration is carried out to obtain the shearing wavefront with the high-precision double-step calibration method introduced in Section 5.2. The experimental measurement of SMA fiber point-diffraction wavefront has been carried out with the system diagram shown in Figure 15. The unwrapped original wavefront before and after 180° rotation in x and y directions is shown in Figure 18(a)– (d), and the corresponding precalibrated shearing wavefronts after the first-step calibration are shown in Figure 18(f)–(i), respectively. Figure 18(e) and (j) is the true shearing wavefronts in x and y directions obtained with the double-step calibration method. According to Figure 18, obvious residual errors can be seen in the obtained wavefronts after first-step calibration, the precalibrated shearing wavefront error RMS in x and y directions are 0:0254λ and 0:0194λ, respectively. The residual error is further removed after the second-step calibration, and true shearing wavefront error RMS in x and y directions are 1:38 · 10<sup>−</sup><sup>4</sup> λ and 1:36 · 10<sup>−</sup><sup>4</sup> λ, respectively. Besides, there is no significant odd error in the finally retrieved wavefronts in Figure 18(e) and (j). Thus, the high-accuracy calibration is realized with the proposed method.

Figure 17. Residual errors in the calibration of systematic error under various lateral displacements and NAs in computer simulation. (a) RMS value with traditional method and (b) RMS value with the proposed double-step calibration method [21].

With the true shearing wavefronts in x and y directions after systematic error calibration, the point-diffraction wavefront can be reconstructed based on the differential Zernike polynomials fitting method introduced in Section 5.1. The measured wavefront error compared with an ideal sphere is shown in Figure 19(a), PV and RMS are 9:20 · 10<sup>−</sup><sup>4</sup> λ and 1:54 · 10<sup>−</sup><sup>4</sup> λ, respectively. Figure 19(b) and (c) shows the measured wavefronts after the 45° rotation and translation along z direction of projector, respectively. It can be seen from Figure 19 that the measured wavefront shape in the original position agrees well with those obtained with projector rotation and translation, demonstrating the measurement repeatability and accuracy. The protective glass on CCD detector could introduce additional deformation on point-diffraction wavefront, and the measurement accuracy is expected to be further improved by removing the protective glass. Due to the smoothing effect of each pixel, the relatively large pixel acts as a low-pass filter and the measured wavefront should be taken as a macroparameter. By adopting

<sup>Δ</sup>W≅ðW<sup>ð</sup>s<sup>Þ</sup>

ment is over 50 μm because the residual error RMS is larger than 2:0 · 10<sup>−</sup><sup>3</sup>

the double-step calibration in Section 5.2.

204 Optical Interferometry

[21].

at the far-field zone that is the distance D >> 2ð2dÞ

shearing wavefront error RMS in x and y directions are 1:38 · 10<sup>−</sup><sup>4</sup>

Thus, the high-accuracy calibration is realized with the proposed method.

<sup>m</sup><sup>1</sup> <sup>þ</sup> <sup>W</sup><sup>ð</sup>−s<sup>Þ</sup>

To get the measurement data with the lateral displacement −s, one may rotate the SMA fiber projector by 180°. Figure 17(a) and (b) shows the residual systematic errors corresponding to various lateral displacement amounts with different NAs with traditional method and the double-step calibration method. The traditional method is not valid when the lateral displace-

fiber. The residual error also increases significantly with the lateral displacement, especially for

within a 300 μm lateral displacement even for the 0.60 NA fiber, it confirms the feasibility of

To obtain point-diffraction spherical wavefront in the experiment, the projector should be placed

aperture size. With the unwrapped phase distribution, the systematic error calibration is carried out to obtain the shearing wavefront with the high-precision double-step calibration method introduced in Section 5.2. The experimental measurement of SMA fiber point-diffraction wavefront has been carried out with the system diagram shown in Figure 15. The unwrapped original wavefront before and after 180° rotation in x and y directions is shown in Figure 18(a)– (d), and the corresponding precalibrated shearing wavefronts after the first-step calibration are shown in Figure 18(f)–(i), respectively. Figure 18(e) and (j) is the true shearing wavefronts in x and y directions obtained with the double-step calibration method. According to Figure 18, obvious residual errors can be seen in the obtained wavefronts after first-step calibration, the precalibrated shearing wavefront error RMS in x and y directions are 0:0254λ and 0:0194λ, respectively. The residual error is further removed after the second-step calibration, and true

Besides, there is no significant odd error in the finally retrieved wavefronts in Figure 18(e) and (j).

Figure 17. Residual errors in the calibration of systematic error under various lateral displacements and NAs in computer simulation. (a) RMS value with traditional method and (b) RMS value with the proposed double-step calibration method

2

the case with high NA fiber. In Figure 17(b), the residual error is less than 1:0 · 10<sup>−</sup><sup>4</sup>

<sup>m</sup><sup>1</sup> Þ=2: (21)

=λ from the fibers [29], where d is the

λ and 1:36 · 10<sup>−</sup><sup>4</sup>

λ with the 0.10 NA

λ RMS

λ, respectively.

Figure 18. Measurement results of shearing wavefront retrieval in SMA fiber point-diffraction wavefront measurement. Unwrapped original wavefronts, including (a) x direction and (c) y direction before 180° rotation, (b) x direction and (d) y direction after 180-degree rotation; shearing wavefronts after the first-step calibration, including (f) x direction and (h) y direction before 180° rotation, (g) x direction and (i) y direction after 180-degree rotation; finally retrieved shearing wavefronts after the double-step calibration, including (e) x and (j) y directions [21].

Figure 19. Experimental result of point-diffraction wavefront measurement. (a) Measured point-diffraction wavefront in original position, (b) measured wavefront after 45° projector rotation, and (c) measured wavefront after projector translation [21].

the CCD with smaller pixel size and placing the SMA fiber projector at a position farther from CCD detector, the smoothing effect can be further decreased.
