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Self-referenced multiple-beam interferometric method for robust phase calibration of spatial light modulator

Yunhui Gao, Rujia Li, and Liangcai Cao

Open Access Open Access

Abstract

Phase-only liquid crystal spatial light modulator has wide ranging applications that require accurate phase retardance. The phase calibration of the spatial light modulator is therefore of vital importance. Available self-referenced calibration methods face the challenges of high time consumption, low efficiency, and low stability against the conditions. A self-referenced multiple-beam interferometric method is proposed to derive the global grayscale-phase response. As is presented theoretically and experimentally, the proposed method reduces the measuring time and improves the calibration efficiency by encoding multiple fringes in a single hologram. Results also show that the method is equally accurate when compared with traditional two-beam interferometric method, whereas providing a greater robustness against measuring errors since the standard deviation is only 56% of that of the traditional method.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Phase-only liquid crystal spatial light modulator (LC-SLM) is capable of phase modulation based on the birefringence property of liquid crystal, which is widely applied in many fields, including digital holography [1,2], pulse shaping [3], adaptive optics [4,5], optical encryption [6,7], and metrology [8], etc. The response of the phase retardance to the input grayscale value can be derived from the look-up-table (LUT) presented in the manual of the SLM and is supposed to be the same for each pixel. But the performance of a phase modulator can be affected by the undesirable issues. There may be individual differences in terms of the global grayscale-phase response due to the manufacturing defects of the SLM substrates or the nonlinear dynamic phase response depending on the address voltage [9]. As a result, phase calibration in advance or in situ is necessary. Otherwise the SLM may suffer from a loss in diffraction efficiency, which can further cause a higher noise level or a lower quality of the reconstructed image in digital holography [10].

Current methods for phase calibration of LC-SLM can be categorized into two types: the interferometric methods and the diffractive methods. The interferometric methods mainly include the self-referenced method [11–13] and the out-referenced method [14–16], while the diffractive methods comprise the diffraction pattern analyzed method [17], optical elements generated method [18] and the polarization method [19,20]. Each approach has its distinguished features and advantages and thus is suitable for different applications [21]. Among the aforementioned calibration approaches, the self-referenced interferometric method is favored for the simplicity of the overall system, since it only requires the least number of optical components in the light path [22]. Recently, a new method based on diffraction phase gratings was proposed by Fuentes et al., making it possible for a straightforward measurement of the global grayscale-phase response [12]. Essential improvements were then realized by Zhao et al. to enhance the robustness against turbulence and vibrations [13]. Currently, the accuracy of the measurement can be susceptible to the disturbance of the conditions, since it requires the precise location of the interferometric fringes. Repeated measurements are also required to improve the accuracy at the cost of the time consumption, which reduce the efficiency of the calibration. The pixels on the SLM could not be fully utilized in these traditional methods, as only one impulse is derived from an image for computation.

In this work, we propose a self-referenced multiple-beam interferometric method (SeRMI) for phase calibration of the LC-SLM by generating an interferometric pattern of multiple beams. To reduce the time consumption and increase calibration efficiency, the uploaded grayscale pattern forms a multiple-beam interference and thus the phase modulation can be derived from multiple frequency terms in a single image. With multiplexed fringes, the robustness of the measurement against the experimental conditions is significantly enhanced. Theoretical and experimental analyses have both proved that the proposed method can yield more stable calibration results in a more efficient way.

2. Phase calibration method

2.1 Model

We consider the situation of plane wave interferometry. The coherent superposition of plane waves requires interference of at least two beams to form a periodic structure. The intensity distribution of the coherent superposition of N (N ≥ 2) plane waves on the image plane is given by:

$$\begin{aligned}I({x,y} ) &= {|{{\boldsymbol U}({x,y} )} |^2} = {\left|{\sum\limits_{i = 1}^N {{{\boldsymbol U}_i}({x,y} )} } \right|^2} = {\left|{\sum\limits_{i = 1}^N {{A_i}\exp [{j{\varphi_i}({x,y} )} ]} \textrm{ }} \right|^2}\\ &= \sum\limits_{i = 1}^N {{A_i}^2} + 2\sum\limits_{1 \le i < k \le N} {{A_i}{A_k}\cos [{{\varphi_i}({x,y} )- {\varphi_k}({x,y} )} ]} \\ &= \sum\limits_{i = 1}^N {{A_i}^2} + 2\sum\limits_{1 \le i < k \le N} {{A_i}{A_k}\cos [{2\pi ({{{\overrightarrow f }_i} - {{\overrightarrow f }_k}} )\cdot \overrightarrow r + ({{\varphi_{i0}} - {\varphi_{k0}}} )} ]} \\ &= \sum\limits_{i = 1}^N {{A_i}^2} + 2\sum\limits_{1 \le i < k \le N} {{A_i}{A_k}\cos ({2\pi {{\overrightarrow f }_{ik}} \cdot \overrightarrow r + {\varphi_{ik}}} )} \end{aligned}$$
where $I({x,y} )$ and ${\boldsymbol U}({x,y} )$ refer to the overall intensity and complex amplitude distribution, respectively, at $({x,y} )$ on the image plane, and ${{\boldsymbol U}_i}({x,y} )= {A_i}\exp [{j{\varphi_i}({x,y} )} ]$ refers to the complex amplitude of the ${i^{th}}$ plane wave. For plane wave, the phase term can be further expressed as ${\varphi _i}({x,y} )= 2\pi ({{f_{ix}}x + {f_{iy}}y} )+ {\varphi _{i0}}$, where ${f_{ix}}$ and ${f_{iy}}$ denote the spatial frequency along x and y axis, respectively. It could be rewritten as ${\varphi _i}({x,y} )= 2\pi {\overrightarrow f _i} \cdot \overrightarrow r + {\varphi _{i0}}$, where ${\overrightarrow f _i} = ({{f_{ix}},{f_{iy}}} )$ is defined as the frequency vector and $\overrightarrow r = ({x,y} )$ is the vector in the spatial domain.

Equation (1) implies that the intensity distribution of the interference pattern of N plane waves consists of $C_N^2 = N({N - 1} )/2$ frequency impulses in the Fourier domain. In other words, more frequency terms, and consequently more calibration data, can be derived from the interferometric pattern of more beams, and thus hereby we propose the self-referenced multiple-beam interference method (SeRMI). In a self-referenced calibration method, the interference between the reflected test beam and the diffraction beam can yield a useful hologram. The phase shift is visualized by the movement of the fringe pattern. Both the test beam and the diffraction beam can be simplified as plane waves, but their phases are determined by different factors. The test beam ${{\boldsymbol U}_{test}}({x,y} )$ contains the phase modulation term $\Delta {\varphi _g}$ which is a function of the uploaded gray level. Meanwhile, the diffraction beam ${{\boldsymbol U}_{diff}}({x,y} )$ can form detectable fringe by interfering with the beam from the tested zone. Thus their complex amplitude at $({x,y} )$ on the image plane can be expressed as:

$$\left\{ {\begin{array}{{l}} {{{\boldsymbol U}_{test}}({x,y} )= {A_{test}}\exp [{j({\Delta {\varphi_g} + {\varphi_{tes{t_0}}}} )} ]}\\ {{{\boldsymbol U}_{diff}}({x,y} )= {A_{diff}}\exp [{j({2\pi {{\overrightarrow f }_{diff}} \cdot \overrightarrow r + {\varphi_{dif{f_0}}}} )} ]} \end{array}} \right.$$
where ${A_{diff}}$ and ${A_{test}}$ denote the amplitude of the diffraction beam and the test beam respectively. ${\varphi _{dif{f_0}}}$ and ${\varphi _{tes{t_0}}}$ stand for the initial phase term of the diffraction beam and the test beam respectively. ${{\mathop f\limits^\rightharpoonup}_{diff}}$ refers to the frequency vector of the diffraction beam. The initial phase term ${\varphi _{i0}}$ can be omitted since it remains constant and does not affect the measurement. Then, based on the plane wave interference model in Eq. (1), the interference pattern is given by:
$$\begin{aligned}I({x,y,g} )&= A_{test}^2 + \sum\limits_{i = 1}^{N - 1} {A_{dif{f_i}}^2} \\ &\quad+ 2\sum\limits_{i = 1}^{N - 1} {{A_{test}}{A_{dif{f_i}}}\cos [{2\pi ({{{\overrightarrow f }_{test}} - {{\overrightarrow f }_{dif{f_i}}}} )\cdot \overrightarrow r + \Delta {\varphi_g}} ]} \\ &\quad+ 2\sum\limits_{1 \le i < k \le N - 1} {{A_{dif{f_i}}}{A_{dif{f_k}}}\cos [{2\pi ({{{\overrightarrow f }_{dif{f_i}}} - {{\overrightarrow f }_{dif{f_k}}}} )\cdot \overrightarrow r } ]} \\ &= A_{test}^2 + \sum\limits_{i = 1}^{N - 1} {A_{dif{f_i}}^2} + 2\sum\limits_{i = 1}^{N - 1} {{A_{test}}{A_{dif{f_i}}}\cos {\varphi _{cor{r_i}}}} + 2\sum\limits_{1 \le i < k \le N - 1} {{A_{dif{f_i}}}{A_{dif{f_k}}}\cos {\varphi _{uncor{r_{i,k}}}}} \end{aligned}$$
If one of the beams is used as a test beam while the other N-1 are diffraction beams, then N-1 useful frequency terms, namely the correlated phase terms between the test beam and each of the N-1 diffraction beams ${\varphi _{cor{r_i}}}$, can be derived in a single interferometric pattern. The frequency terms ${\varphi _{uncor{r_{i,k}}}}$ between two diffraction beams ${{\boldsymbol U}_{dif{f_i}}}$ and ${{\boldsymbol U}_{dif{f_k}}}$ do not contain the phase shift related to the grayscale value and are therefore called the uncorrelated terms. The measurement data of such multiple-beam interference is equivalent to that of N-1 repeated measurements of two-beam interference. Here the SeRMI method could acquire the fringes with just one shot. In the meanwhile, the size of the interference area becomes smaller since the pixels on the SLM are divided into multiple zones, but it does not severely affect the measurement of a high-resolution SLM when N is not too large. In a word, SeRMI aims to encode as much calibration data as possible in a single shot while ensuring the size of the interference area.

2.2 Experimental setup

The layout of the calibration system is presented in Fig. 1. A HOLOEYE GAEA-2 phase-only reflective LC-SLM (3840×2160 pixels, pixel pitch 3.74 µm, fill factor 90%) is used for calibration. A laser at the wavelength of 532 nm is employed as the light source, producing a beam incident upon the SLM along the normal direction after a set of spatial filter and beam expander. Apertures are added in the light path to enhance the overall quality of the interferometric hologram which is captured by a CMOS camera (Canon EOS 550D).

 figure: Fig. 1.

Fig. 1. (a) Schematic of the SeRMI system. OBJ is a microscope objective, PH is a pinhole, A is an aperture, L is a collimating lens, M is a mirror, P is a polarizer and BS is a beam splitter. (b) Flow chart of the calibration procedure.

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The grayscale pattern uploaded to the SLM is shown in Fig. 2(b). The traditional two-beam interference method [13] is shown in Fig. 2(a) for comparison. The pattern is horizontally trisected into three parts. Similar to the traditional method, the middle part is divided into two subsections, namely the test zone and the reference zone, both of which are uploaded with uniform grayscale values. The test zone is the area being calibrated, whose grayscale value gradually changes from 0 to 255 at a given step during the calibration, and at each grayscale value the interference pattern is taken by the camera for further analysis. The grayscale value of the reference zone is set to 0 and remains unchanged throughout the experiment, serving as a reference for comparison. The left and right part of the pattern are uploaded with different phase gratings, respectively. The 1st order diffraction beams interfere with the reflected beam from the middle part, forming an interferometric pattern. Overall three beams interfere at the hologram plane in SeRMI (N = 3 in this work), while only two beams interfere in the traditional method.

 figure: Fig. 2.

Fig. 2. A comparison between (a) the traditional two-beam interference and (b) the proposed SeRMI.

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Blazed gratings are applied for the phase gratings because of their better performance compared with binary gratings. When the phase shift of the test zone changes, the intensity and contrast of the fringe remains almost constant for blazed gratings, while for binary gratings the fringe is subject to be indistinct. In the proposed method, the spatial frequencies of the two grating zones are carefully chosen. There is a vertical blazed grating and a slightly tilted blazed grating on the left part and the right part, respectively. Both gratings have the same spatial frequency along the horizontal axis. As a result, two key variables, namely the horizontal grating period and the tilting angle of the right part, should be determined before calibration. Grating period is set to 8 pixels as a balance between the imaging distance and the measuring accuracy, since a longer imaging distance may lead to a reduced image quality due to the effect of diffraction and air turbulence while a shorter imaging distance may set higher demand to the detecting device and optical alignment. The tilting angle is set to $\pi /18$, considering the trade-off between the size of the interfering zone and the ability to differentiate the two impulses in the Fourier domain. The imaging distance D, namely the distance between the SLM and the hologram plane, is given by $D = L/({3\tan \alpha } )$, where L is the width of the SLM and α is the diffraction angle of the 1st order beam, which is given by $\sin \alpha = \lambda /P$, where λ is the wavelength of the light source and P stands for the grating period.

Although a 2π blazed grating seems to meet the requirements well, the maximum phase shift of the SLM is unknown and the grayscale-phase response can be nonlinear. Nevertheless, as long as the interference yields a periodic pattern with the same spatial frequency, the previously discussed plane wave model can still serve as an approximate solution, because the other minor frequency terms do not significantly affect the measurement.

2.3 Calibration procedure

Based on the plane wave interference model, the spatial frequencies of the two diffraction beams are:

$${f_{dif{f_L}}} = 1/P\textrm{, }{f_{dif{f_R}}} ={-} 1/P + \tan \alpha /P$$
where P and $\alpha $ stand for the horizontal grating period and the tilting angle, respectively. Subscripts $L$ and R denote the diffraction beam from the left and right part of the SLM, respectively. The correlated phase terms in Eq. (3) are given by:
$$\left\{ {\begin{array}{{l}} {{\varphi_{cor{r_L}}} = 2\pi x/P - \Delta {\varphi_g} + {\varphi_{{L_0}}} - {\varphi_{tes{t_0}}}}\\ {{\varphi_{cor{r_R}}} ={-} 2\pi x/P + 2\pi y\tan \alpha /P - \Delta {\varphi_g} + {\varphi_{{R_0}}} - {\varphi_{tes{t_0}}}} \end{array}} \right.$$
Hence, we can derive the intensity distribution of the resulting pattern of the uploaded grayscale pattern in Fig. 2(b) as:
$$\begin{array}{l} I({x,y} ) = {|{{{\boldsymbol U}_{test}}({x,y} )+ {{\boldsymbol U}_L}({x,y} )+ {{\boldsymbol U}_R}({x,y} )} |^2}\\ \textrm{ } = A_{test}^2 + A_L^2 + A_R^2 + 2{A_L}{A_{test}}\cos {\varphi _{cor{r_L}}} + 2{A_R}{A_{test}}\cos {\varphi _{cor{r_R}}} + 2{A_R}{A_L}\cos {\varphi _{uncorr}} \end{array}$$
When $\Delta {\varphi _g}$ changes from $0$ to $2\pi $ for the test zone, each of the correlated terms in Eq. (6), namely the left correlated term $2{A_L}{A_{test}}\cos {\varphi _{cor{r_L}}}$ and the right correlated term $2{A_R}{A_{test}}\cos {\varphi _{cor{r_R}}}$, shifts a distance equal to its grating period, which is illustrated in Fig. 3. The interference pattern between the two diffracted beams and the reflected beam from the reference zone remains unchanged during this process.

 figure: Fig. 3.

Fig. 3. The fringe patterns for phase shift of 0, π/2, π and 3π/2. Red dotted lines are added to visualize the shift of the fringe. The left correlated term ((e)–(h)) and the right correlated term ((i)–(l)) are derived from the overall intensity distribution on the hologram plane ((a)–(d)).

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As is shown in Fig. 4, the frequency impulses are the local maxima and thus their corresponding complex angle can be easily located in the Fourier spectrum. Based on the fact that the changes in the complex angle of the first two frequency terms (the correlated frequencies ${\overrightarrow f _1}$ and ${\overrightarrow f _2}$ in Fig. 4(c)) are determined only by the phase modulation term $\Delta {\varphi _g}$, the phase shift can be derived by:

$$\Delta \varphi _g^{(i )} = \varphi (g )- \varphi (0 )= \arg {\mathcal F}\{{I({x,y,g} )} \}|{_{\overrightarrow f = {{\overrightarrow f }_i}}} - \arg {\mathcal F}\{{I({x,y,0} )} \}|{_{\overrightarrow f = {{\overrightarrow f }_i}}}$$
where arg denotes the complex angle, and the frequency vector ${\overrightarrow f _i}$ can refer to either ${\overrightarrow f _1}$ or ${\overrightarrow f _2}$. The average phase shift is given by:
$$\Delta {\varphi _g} = \frac{1}{2}({\Delta \varphi_g^{(1 )} + \Delta \varphi_g^{(2 )}} )$$
By gradually changing the grayscale value and measuring the corresponding phase shift, a global look-up table can be obtained.

 figure: Fig. 4.

Fig. 4. (a)–(c) The analysis of the SeRMI pattern. (a) The intensity distribution of the interferometric pattern in the image domain. (b) The corresponding frequency domain. (c) The correlated frequency vectors (solid lines) and the complex conjugate terms (dashed lines). (d) – (f) are the same analysis on the traditional two-beam interference for comparison.

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3. Results and discussion

3.1 Intensity distribution analysis

To obtain the intensity distribution of the hologram plane, a preliminary experiment to measure the grayscale-intensity transmission character of the detecting sensor was conducted, which is presented in Fig. 5. Based on the assumption that the camera pixel responds to the number of photons captured, we changed the time of exposure each time instead of measuring the intensity of the incident beam directly. In other words, it is regarded that changing the time of exposure is equivalent to changing the intensity. The same light source (a 532nm laser) was used to make sure that the transmission from RGB to grayscale value remained the same. As is shown in Figs. 5(b) and 5(c), the effect of nonlinearity resulted in other frequency terms in in the Fourier spectrum, which can be eliminated after modification.

 figure: Fig. 5.

Fig. 5. (a) The intensity-grayscale response of Canon EOS 550D. The Fourier spectrum of the interference pattern (b) before and (c) after modification. The red impulses belong to the expected intensity distribution while the others are formed by the effect of nonlinearity.

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3.2 Deriving the global look-up table

Holograms were captured as the grayscale value of the test zone was increased from 0 to 255 at the step of 16 (both endpoint 0 and 255 were included) and thus overall 17 images were required for a single turn of calibration. Overall six groups of calibration were conducted for both the proposed SeRMI method and the traditional two-beam interference method [13] to test and compare the stability of the measurements. The measured phase values for the grayscale levels of 16, 32, 48, 64, 128, 192 and 255 are shown in Table 1.

Tables Icon

Table 1. Measured phase values for selected grayscale levels of GAEA-2 LC-SLM

View Table

The averaged look-up table derived by the two methods is presented in Fig. 6. The maximum gap between the phase modulation measured by the two methods is 0.118 rad, indicating that the proposed method yielded similarly accurate results. The maximum phase shift is approximately 3.5π, which is in accordance with the value of 3.6π in the manual book [23]. The standard deviation of the six groups of data for each method was calculated and shown in Fig. 7. The SeRMI method shows better robustness, since most of the measuring points (except when grayscale was 255) have smaller variations. The averaged standard deviation of the proposed SeRMI method is nearly 56% of that of the traditional two-beam interference method.

 figure: Fig. 6.

Fig. 6. The look-up table derived by the traditional two-beam interference method and the proposed SeRMI method. The captured interferometric patterns and the retrieved phase shifts for grayscale 64, 128 and 192 are shown on the left.

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 figure: Fig. 7.

Fig. 7. The standard deviation of the traditional two-beam interference method and the proposed SeRMI method.

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4. Conclusion

We propose a robust, low cost and self- referenced method for the phase calibration of an LC-SLM based on the interference of multiple beams. By encoding multiple-beam interferometric pattern the image domain and decoding in the frequency domain, effective calibration data can be obtained from each image. The robustness of the calibration against the influence of the conditions is improved significantly, as is proved by a decreased standard deviation. The time consumption for exposures remains unchanged and therefore a more efficient calibration is accomplished. The proposed method has the advantage of requiring no additional optical items and is thus easy for optical alignments. The proposed SeRMI method could also be applied for the fast and accurate metrology of phase objects. In the future work, more than three beams will be considered to further improve the calibration robustness.

Funding

National Key R & D Program of China (2018YFF0212302).

References

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Figures (7)
Fig. 1.
Fig. 1. (a) Schematic of the SeRMI system. OBJ is a microscope objective, PH is a pinhole, A is an aperture, L is a collimating lens, M is a mirror, P is a polarizer and BS is a beam splitter. (b) Flow chart of the calibration procedure.

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Fig. 2.
Fig. 2. A comparison between (a) the traditional two-beam interference and (b) the proposed SeRMI.

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Fig. 3.
Fig. 3. The fringe patterns for phase shift of 0, π/2, π and 3π/2. Red dotted lines are added to visualize the shift of the fringe. The left correlated term ((e)–(h)) and the right correlated term ((i)–(l)) are derived from the overall intensity distribution on the hologram plane ((a)–(d)).

View in Article | Download Full Size | PDF

Fig. 4.
Fig. 4. (a)–(c) The analysis of the SeRMI pattern. (a) The intensity distribution of the interferometric pattern in the image domain. (b) The corresponding frequency domain. (c) The correlated frequency vectors (solid lines) and the complex conjugate terms (dashed lines). (d) – (f) are the same analysis on the traditional two-beam interference for comparison.

View in Article | Download Full Size | PDF

Fig. 5.
Fig. 5. (a) The intensity-grayscale response of Canon EOS 550D. The Fourier spectrum of the interference pattern (b) before and (c) after modification. The red impulses belong to the expected intensity distribution while the others are formed by the effect of nonlinearity.

View in Article | Download Full Size | PDF

Fig. 6.
Fig. 6. The look-up table derived by the traditional two-beam interference method and the proposed SeRMI method. The captured interferometric patterns and the retrieved phase shifts for grayscale 64, 128 and 192 are shown on the left.

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Fig. 7.
Fig. 7. The standard deviation of the traditional two-beam interference method and the proposed SeRMI method.

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Tables (1)

Tables Icon

Table 1. Measured phase values for selected grayscale levels of GAEA-2 LC-SLM

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Equations (8)

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I ( x , y ) = | U ( x , y ) | 2 = | ∑ i = 1 N U i ( x , y ) | 2 = | ∑ i = 1 N A i exp ⁡ [ j φ i ( x , y ) ]   | 2 = ∑ i = 1 N A i 2 + 2 ∑ 1 ≤ i < k ≤ N A i A k cos ⁡ [ φ i ( x , y ) − φ k ( x , y ) ] = ∑ i = 1 N A i 2 + 2 ∑ 1 ≤ i < k ≤ N A i A k cos ⁡ [ 2 π ( f → i − f → k ) ⋅ r → + ( φ i 0 − φ k 0 ) ] = ∑ i = 1 N A i 2 + 2 ∑ 1 ≤ i < k ≤ N A i A k cos ⁡ ( 2 π f → i k ⋅ r → + φ i k )
{ U t e s t ( x , y ) = A t e s t exp ⁡ [ j ( Δ φ g + φ t e s t 0 ) ] U d i f f ( x , y ) = A d i f f exp ⁡ [ j ( 2 π f → d i f f ⋅ r → + φ d i f f 0 ) ]
I ( x , y , g ) = A t e s t 2 + ∑ i = 1 N − 1 A d i f f i 2 + 2 ∑ i = 1 N − 1 A t e s t A d i f f i cos ⁡ [ 2 π ( f → t e s t − f → d i f f i ) ⋅ r → + Δ φ g ] + 2 ∑ 1 ≤ i < k ≤ N − 1 A d i f f i A d i f f k cos ⁡ [ 2 π ( f → d i f f i − f → d i f f k ) ⋅ r → ] = A t e s t 2 + ∑ i = 1 N − 1 A d i f f i 2 + 2 ∑ i = 1 N − 1 A t e s t A d i f f i cos ⁡ φ c o r r i + 2 ∑ 1 ≤ i < k ≤ N − 1 A d i f f i A d i f f k cos ⁡ φ u n c o r r i , k
f d i f f L = 1 / P ,  f d i f f R = − 1 / P + tan ⁡ α / P
{ φ c o r r L = 2 π x / P − Δ φ g + φ L 0 − φ t e s t 0 φ c o r r R = − 2 π x / P + 2 π y tan ⁡ α / P − Δ φ g + φ R 0 − φ t e s t 0
I ( x , y ) = | U t e s t ( x , y ) + U L ( x , y ) + U R ( x , y ) | 2   = A t e s t 2 + A L 2 + A R 2 + 2 A L A t e s t cos ⁡ φ c o r r L + 2 A R A t e s t cos ⁡ φ c o r r R + 2 A R A L cos ⁡ φ u n c o r r
Δ φ g ( i ) = φ ( g ) − φ ( 0 ) = arg ⁡ F { I ( x , y , g ) } | f → = f → i − arg ⁡ F { I ( x , y , 0 ) } | f → = f → i
Δ φ g = 1 2 ( Δ φ g ( 1 ) + Δ φ g ( 2 ) )
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