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Dispersed fringe sensor (DFS) is an important phasing sensor of next-generation optical astronomical telescopes. The measurement errors induced by the measurement noise of three piston estimation methods for the DFS including least-squared fitting (LSF) method, frequency peak location (FPL) method and main peak position (MPP) method, are analyzed theoretically and validated experimentally in this paper. The experimental results coincide well with the theoretical analyses. The MPP, FPL, LSF are used respectively when the DFS operates with broadband light (central wavelength: 706 nm, bandwidth: 23 nm). The corresponding root mean square (RMS) value of estimated piston error can be achieved to be 1 nm, 3 nm, 26 nm, respectively. Additionally, the range of DFS with the FPL can be more than 100 μm at the same time. The FPL method can work well both in coarse and fine phasing stages with acceptable accuracy, compared with LSF method and MPP method.

The next-generation optical astronomical telescopes mostly use segmented primary mirrors. Therefore, the piston errors (optical path difference) between different segmented mirrors will exist, and the assemble errors is usually larger than one wavelength; this will remarkably reduce image quality and should be less than 1/20*λ*.^{[1–4]} In addition, the piston errors caused by the working environment must be measured online in real time. The dispersed fringe sensor (DFS) was proposed for these factors,^{[5]} the experiments demonstrated that the DFS is an efficient and robust method for coarse phasing of segmented primary mirrors.^{[1]} The NIRCam of James Webb Space Telescope (JWST) measured larger piston error by using the DFS, the range of detection is about ±300 μm. The piston errors of segmented mirrors can be reduced to less than about 300 nm for the JWST by the DFS.^{[6]} The phasing camera of the Giant Magellan Telescope (GMT) also uses the DFS to measure the piston errors for coarse phasing, the phasing camera has a ±50 μm capture range and can reduce piston error to smaller than a root mean square (RMS) value of 120 nm.^{[7]}

The accuracy of dispersed fringe sensor is determined by internal optical calibration error, wavefront error, and image quality of detector.^{[8]} Spechle *et al.* presented an advanced DFS; an algorithm allows the overall method to be less sensitive to calibration errors.^{[8]} Van *et al.* proposed an improved method to reduce the requirement for wavefront aberration, and simulation shows that this method can improve sky coverage.^{[9]} There is no discussion about the image quality of the detector and estimation method is more applicable and practical, which are the focuses of this paper.

The DFS is a modified structure of Young’s Double-Slit Interference.^{[7]} It uses dispersed element to modulate the interference light with piston error, and dispersed element spreads the light according to its wavelength, forming a spectrum on the camera. The piston errors of the segment mirrors will result in the intensity modulation in the dispersion direction.^{[5]} The intensity is approximately described as a sine curve,^{[10]} and the interference pattern along the interference direction is also affected by the piston error.^{[9,11,12]} At present, several methods have been proposed to extract the piston error from a dispersed fringe image. According to the intensity in the dispersion direction, Shi *et al.* used least-squared fitting (LSF) to fit the frequency of the curve of the intensity versus frequency to determine the piston error (least-squared fitting method, LSF),^{[1]} and the dispersed fringe sensor of the JWST used this method.^{[10]} The dispersed fringe sensors in the phasing camera of the GMT obtains obtained estimation piston errors with frequency peak location method (FPL),^{[9]} this method is also bases based on the frequency of the curve of the intensity versus frequency on in the dispersion direction, but it estimates the piston error form frequency information of the curve in the two-dimensional frequency domain of the dispersed fringe. Zhao and Cao,^{[11]} and Zhang *et al.*^{[12]} used the main peak position (MPP) method to estimate the piston error. Young’s double-slit interference is the foundation of this method, the main peak position in the interference direction has approximately linear relationship with the piston error.^{[9]}

Therefore, different piston error estimation methods for dispersed fringe sensor have different error transfer processes of the noise in the dispersed fringe image. There are no analyses on the piston measurement error induced by the measurement noise. This paper concentrates on the estimation error caused by noise, when using different estimation methods to estimate piston error. The relation between the estimation error and noise is analyzed and experimental results accord well with the theoretical analyses. At the same time, the experiment shows that the first moment centroid calculation method can promise the accuracy of estimation result by using the FPL method, whether in coarse phasing or fine phasing. According to these results, the researchers or engineers can choose appropriate estimation method for the dispersed fringe sensor and optimize parameters of estimation method of dispersed fringe sensor.

This paper is divided into four parts. In Section

As shown in Fig.

As the general descriptions of the dispersed fringe by different phenomena, many methods can be used to calculate piston error,^{[5,9,11,12]} which include the LSF, FPL, and MPP.

For the dispersed fringe image with piston error, the intensity along the dispersed axis is periodically changed, and the cycle has monotonic relationship with the piston error as illustrated in Fig.

The intensity at the central line along the dispersion can be written as the following fundamental equation for convenience:^{[1]}

(1) |

*γ*is the fringe visibility,

*δ*is the piston error, and

^{[5,10]}Therefore, the piston error

*δ*is calculated by the LSF method.

The change of intensity cycle at the dispersed axis also influences absolute value of the Fourier transform of the dispersed fringe,^{[9]} another estimation method (FPL) is based on this point; the process of the FPL method for estimating the piston error is illustrated in Fig.

As shown in Fig. *y* axis.^{[9]} So, the piston error *δ* can be calculated by

(2) |

Along the interferential direction which is perpendicular to the dispersed axis, the change of the intensity accords with double-slit interference regulation as shown in Fig.

As shown in Fig. *y*(*λ*) and piston error (*δ*) are related by

(3) |

*T*(

*λ*) is the spacial cycle of the interference pattern along the

*y*axis in Fig.

^{[9]}

This method based on the intensity along the dispersion is approximately described as a sine curve, and the frequency of the curve has monotonic relationship with piston error. According to this phenomenon, the piston error can be estimated from the frequency of the curve which is fitted by the LSF algorithm. The photon noise is a Poisson distribution in the condition of small photon counts, while the DFS works in the environment with larger counts, the photon noise can be considered as Gaussian noise.^{[14]} At the same time, this will make analysis easier. All of the following analysis is based on this prerequisite. There are two different least square methods to calculate the parameter of signal, one is the three-parameter sine-fitting algorithm, and the other is four-parameter sine-fitting algorithm.^{[15]} When the sampling rate is high enough, the fitting errors of two different algorithms can be ignored.^{[16]} When signal data is contaminated by Gaussian noise, the maximum and minimum values of fitting error have the relationship with noise^{[16]}

(4) |

*A*is the amplitude of light intensity, and

*M*is the sampling rate.

For the dispersed fringe sensor, when the bandwidth is far less than the central wavelength, equation (

(5) |

Therefore, piston error can be calculated by

(6) |

Then, the frequency *f* has the following relationship:^{[15]}

(7) |

*s*,

*A*,

*M*, and

*φ*are the estimated sine-wave parameters.

From Eqs. (

(8) |

Therefore, the bounds of variance for estimation error are shown, respectively, as follows:

(9) |

In absolute value of the two-dimensional Fourier transform of the dispersed fringe, the location of the side lobe changes linearly with piston error in the *y* ordinate. Therefore, the error caused by noise would be introduced into the process of extracting the location or position of the side peak in the frequency domain.

When the centroid position (^{[17]}

(10) |

*M*and

*L*are the numbers of pixels in the two-dimensional direction, respectively,

^{[17]}

(11) |

*i*,

*j*)th pixel. Since the Fourier transform of a Gaussian is also a Gaussian,

^{[18]}the noise in the frequency domain of an image with Gaussian noise is of Gaussian distribution. The variance (

^{[18]}

(12) |

(13) |

When extracting the centroid position of side peak in frequency domain, one-dimensional intensity along the locomotive direction is enough to estimate the location, because it contains all the information about the piston error. Hence, according to Eq. (

(14) |

Substituting Eq. (

(15) |

When the piston error is calculated from Eq. (

The MPP method only uses intensity in the interferential direction, in which the process of extracting the centroid position is the same as that of the FPL, and we have centroid detection error, which can be expressed as

(16) |

According to Eq. (

(17) |

The experimental layout is shown in Fig. ^{[7]} to generate relative optical path difference. The PA can easily generate controllable static piston error of segmented telescope and is used in prototype phasing camera for the GMT.^{[7]} Therefore, there is no need to use segmented mirrors nor telescope to generate piston errors. Finally, the beam splitter prism (BS1) reflects the beams into the dispersed fringe sensor and dispersed fringe image is captured by the CCD2 camera (Baumer TXG12).

The dispersed fringe image with a resolution of 500×40, the pixel size of the camera is 3.75 μm. The bandwidth of light is 23 nm with a central wavelength of 706 nm, and these can be changed by light source (NKT Superk). Residual surface error of plane wave is *λ*/10 peak value (PV), the dual-beam generated by the double stilt, therefore the wavefront distortion has a limited effect on dispersed fringe image. The angular calibration error between the camera and the dispersed element is less than ±5 (see Fig.

In the experiment, several groups of dispersed fringe images are generated by the DFS in different noise conditions by changing the mean ADU (the minimum unit of digital readout from A/D converter) and standard deviation of background noise. And then, for each group different estimation methods are used to calculate piston errors of the dispersed fringe images in the same noise condition. The noise effects on the accuracy of different estimation methods can be compared with the standard deviation of these piston errors. The result of the experimental error accords well with the theoretical calculation, and the correctness of the theoretical expressions are verified.

The LSF is suitable for calculating large piston error (larger than one wavelength), and the MPP can only estimate the fraction of piston error (less than one wavelength),^{[11,12]} so two kinds dispersed fringe images with different piston errors are used to calculate the errors. As shown in Fig.

There are 20 groups of dispersed fringe image set with the same light signal intensity, but their signal-to-noise ratios (SNRs) are different. In each group of dispersed fringe images, the standard deviation of the noise in the dispersed fringe imageis approximate. The standard deviation of noise increases with the serial number of image groups linearly. Consequently, the SNR is increased. The standard deviation of piston error is increased from 1 to 20 ADU, and the step size is 1 ADU.

Figure ^{[19]}

(18) |

The characteristic curves of the error for the LSF in different noise conditions are shown in Fig. *M* = 500, and

Figure *L* = 10 for the FPL.

The result of the MPP method is illustrated in Fig. *L* = 8 for the MPP.

In the casewhere there are two kinds of dispersed fringe images with different piston errors, the FPL is compared with the LSF by using dispersed fringe image with large piston error and a resolution of 500 in dispersed direction. The theoretical values of the FPL are calculated from Eq. (*L* = 10 for the FPL.

As seen in Fig.

In this paper, we analyze the accuracy provided by the LSF, the FPL, and the MPP methods when dispersed fringe image is corrupted by noise. The theoretical analyses are confirmed by experimental results. When dispersed fringe images with small piston error are used and the RMS of estimation piston error is less than *λ*/20 (35.3 nm), the FPL method requires that the PSNR should be larger than 23 dB, the MPP method can meet the requirement with the data shown in Fig.

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