Preamble

Vol. 43, No. 3

September 2025

PROGRESS IN ASTRONOMY Vol. 43, No. 3 Sept., 2025
doi: 10.3969/j.issn.1000-8349.2025.03.09

Scrambling Performance Evaluation Method Based on Fourier Spectrum
PANG Yuchen¹,²,³, PENG Zifu¹,²,³, GENG Tao¹,²,⁴, SUN Weimin¹,²,⁴, YAN Yunxiang¹,²,³,⁴
(1. Key Lab of In-fibre Integrated Optics, Ministry of Education, Harbin Engineering University, Harbin 150001, China;
2. Key Laboratory of Photonic Materials and Devices Physics for Oceanic Applications, Ministry of Industry and Information Technology, Harbin Engineering University, Harbin 150001, China;
3. Qingdao Innovation and Development Center of Harbin Engineering University, Qingdao 266000, China;
4. Advanced Photonics Institute, College of Physics and Optoelectronic Engineering, Harbin Engineering University, Harbin 150001, China)

Abstract

Fiber scrambling technology applied to high-resolution spectroscopic observations can effectively improve radial velocity measurement accuracy, providing a valuable tool for frontier scientific research such as the search for Earth-like planets. Currently, the impact of mode noise generated by fiber transmission on high-resolution spectrometers cannot be ignored, and analyzing fiber scrambling effects directly from the modal speckle patterns in the output field has become an important research topic. To address this challenge, we have developed a scrambling performance evaluation method based on Fourier spectrum analysis. By employing two-dimensional Fourier spectrum analysis combined with calculations of image contrast and visibility, we conducted comparative analyses of output speckle patterns from different optical fibers. The results confirm that this method is more universal, unified, and non-discriminatory, providing important methodological support for evaluating the accuracy and effectiveness of fiber scrambling performance.

Keywords: spectrograph; telescope; imaging; radial velocity; image processing

1 Introduction

In modern astronomical detection, searching for Earth-like planets in habitable zones and studying their atmospheric composition to find extraterrestrial life represents a hot topic in current astronomical research. In recent years, the number of planets discovered using the radial velocity method based on the Doppler shift principle has been second only to the transit method. Current short-term (10¹ min) radial velocity measurement precision has reached the 1 cm/s level [1], while Earth-like planets in habitable zones around Sun-like stars induce radial velocity signals of approximately 9 cm/s [2]. Although mode-locked laser astro-comb systems can achieve the required measurement precision, they cannot maintain measurement stability for over a year. Moreover, due to their manufacturing complexity, such high-precision astro-comb systems are currently only commercially available from Menlo Systems in Germany, and the actual measurement precision achieved varies when the comb is used with different spectrometers. Therefore, it is essential to improve spectrometer measurement precision to enhance radial velocity measurement accuracy.

To ensure the accuracy of spectral measurements and radial velocity precision, optical fibers are used in astronomical spectrometers to transmit light collected by telescopes. However, when coherent light is transmitted through multimode fibers, interference between different modes produces bright and dark laser speckle patterns in the near-field output [3, 4]. During our experiments, we also discovered that similar speckle-like patterns appear in the near-field under incoherent white light illumination [5], exhibiting the same fiber length dependence. Fiber scrambling technology can effectively suppress the generation of both types of speckle patterns, thereby improving radial velocity measurement precision.

Since the scrambling performance of single circular fibers is limited, Chazelas et al. [6] tested a polygonal cross-section fiber in 2010, and two years later, Spronck et al. [7] and Feger et al. [8] conducted comparative experiments on the intensity distributions in the near-field and far-field of polygonal and circular fibers, demonstrating the superior scrambling characteristics of polygonal fibers.

Generally, fiber scrambling performance is evaluated comprehensively from both near-field and far-field perspectives. Near-field evaluation methods can be categorized into three main approaches. The first is the scrambling gain coefficient (SG), a classical metric for near-field scrambling performance. Initially defined as the ratio of starlight lateral displacement at the detector input to the displacement of the point spread function (PSF) in the spectrometer [9], SG is expressed as $SG=(d/D)/(f/F)$, where $d$ is the starlight displacement, $D$ is the fiber diameter, $f$ is the PSF displacement, and $F$ is the full width at half maximum of the PSF. However, since the incident displacement $d$ in actual observations constantly changes, researchers believe that a series of continuous results is more convincing than a single SG value, and thus provide centroid drift diagrams alongside SG for comprehensive evaluation [6, 10, 11]. The second common method involves plotting intensity distribution curves, which are one-dimensional intensity profiles along single-pixel slices through the image centroid [12, 13]. Scrambling performance is assessed by analyzing the smoothness and fluctuations of these curves. The third approach is an emerging method based on Fourier power spectrum image analysis, which involves obtaining the power spectral density of near-field output speckle patterns through two-dimensional Fourier transform, then drawing radially averaged power spectra through the image center for analysis [14, 15].

However, in experiments using these methods to verify the scrambling performance of non-circular fibers, we discovered asymmetries in scrambling gain and centroid shifts. Inspired by the third method, this paper proposes a fiber scrambling performance evaluation method based on Fourier spectrum. By performing two-dimensional Fourier transform on near-field output speckle patterns, then obtaining normalized spectra through rotational averaging, we analyze and compare spectral curves between different fibers. Simultaneously, we calculate contrast and visibility after low-pass filtering for quantitative evaluation. Compared with previous methods, our proposed approach provides more unified and comprehensive image processing, effectively avoids randomness, and ensures the accuracy and effectiveness of fiber scrambling performance evaluation.

2 Experiments

Non-circular fiber scrambling methods [16–19] achieve mode redistribution by altering the fiber core shape to disrupt light propagation paths within the core, thereby achieving scrambling effects. To verify the enhanced scrambling performance of non-circular fibers, we constructed a scrambling test optical system that can both simulate and detect incident spots entering at different positions on the fiber end face (simulating telescope system incident offsets) and collect near-field speckle images from the fiber output.

2.1 Tested Fibers

We tested three fiber geometries: circular, square, and octagonal fibers with the following parameters:

(1) Circular-core step-index multimode fiber (denoted as SI): core diameter of 105 µm, cladding diameter of 125 µm, numerical aperture (NA) of 0.22.
(2) Square-core step-index multimode fiber (denoted as SQ): core dimensions of 100 µm × 100 µm, cladding dimensions of 330 µm × 330 µm, NA of 0.22.
(3) Octagonal-core step-index multimode fiber (denoted as OCT): core diameter of 200 µm, cladding diameter of 660 µm, NA of 0.22.

2.2 Test Platform

To evaluate the scrambling performance of different fiber types, we designed and constructed a scrambling test optical system, as shown in the optical path diagram in [FIGURE:1]. The system can use various laser or LED sources as incident sources to simulate starlight signals. After reflecting off the measurement fiber end face, the incident spot passes through a beam splitter into an end-face observation imaging system for real-time monitoring of incident spot offset. The near-field detection device consists of a variable-magnification telecentric lens and a high-precision camera (4024 × 3036 pixels) to collect near-field speckle images.

2.3 Experimental Results

Using the optical system shown in [FIGURE:1], we obtained speckle patterns with both a 650 nm red laser and an incoherent LED source. [FIGURE:2] shows the output speckle patterns obtained with the 650 nm laser source, while [FIGURE:3] shows the patterns obtained with the incoherent LED source. Both figures only display results for central incidence. In each figure, the rows from top to bottom correspond to circular, square, and octagonal fibers, while the columns from left to right represent fiber lengths of 0.5 m, 1 m, 2 m, and 5 m.

From [FIGURE:2], pronounced laser speckle mode noise is evident, with speckle size and intensity decreasing as fiber length increases. Polygonal fibers exhibit smaller and weaker speckles. [FIGURE:3] reveals a new phenomenon we discovered under incoherent LED illumination. Due to its similarity to laser speckle mode noise, we term this "mode pattern" [5], which also shows fiber length dependence. The special patterns observed in the central regions of the 1 m and 2 m circular fibers in [FIGURE:3] are attributed to the fiber end-face polishing process.

2.4 Analysis Methods

2.4.1 Centroid Drift

To simulate starlight signal offset effects, we used a five-dimensional translation stage at the fiber input to control incident spot displacement along both x and y axes on the fiber end face. The step size was 10% of the fiber diameter, with seven positions (including the origin) measured at each step, while collecting near-field output images. Due to manual adjustment of the translation stage, small errors occurred with each movement. To reduce random noise, we continuously acquired 10 images at each position for averaging, and performed three independent measurements in each direction while maintaining the same fixed position to improve result accuracy.

We first processed the experimental near-field speckle images using the classical analysis method—centroid drift diagrams and scrambling gain calculations. The results are shown in [FIGURE:4] and [FIGURE:5], where the top three subplots in each figure show results from x-axis offset, and the bottom three show y-axis offset results. The horizontal axis represents relative incident offset (ratio of source displacement to test fiber diameter). The vertical axis in [FIGURE:4] shows centroid drift of the fiber near-field output, while [FIGURE:5] shows the scrambling gain coefficient SG. Error bars represent variations among the three repeated measurements.

The line graphs in [FIGURE:4] clearly show that near-field centroid drift is not linearly related to relative incident offset, meaning SG distribution is asymmetric relative to the origin, as shown in [FIGURE:5]. In fact, Avila et al. [20] observed incomplete symmetry in output patterns as early as 2010, and more explicitly stated in 2022 [21] that SG distribution is asymmetric relative to central incidence, with near-field patterns changing differently along incident offset directions and depending on fiber path geometry. Ye et al. [16] also found during measurements that centroid drift was not linearly related to incident position offset. Earlier studies [22] attributed this to coma generation and concluded that incident angle affects results more than offset position. Based on these findings and our experimental results, we confirm that classical evaluation methods using centroid drift diagrams and SG coefficients produce asymmetric and uncertain measurement results.

This outcome directly affects spectrometers where the fiber near-field forms a mirror image relationship with spectral lines. In high-resolution spectrometers requiring long-term stability and high radial velocity precision, the fiber's inherent scrambling characteristics can decouple various changing environmental conditions (such as temperature, pressure, and telescope position) between the spectrometer and telescope, while ensuring stable and uniform illumination at the spectrometer input. Fiber-fed spectrometers image the fiber output directly onto the detector, where small changes in fiber output appear as small changes in the image plane, manifesting as spectral line displacement or shape changes. Therefore, when the source undergoes symmetric offset at the fiber input with equal distances in opposite directions, the fiber output image should theoretically produce similarly symmetric centroid drift and yield nearly identical scrambling gain coefficients. However, our verification experiments produced asymmetric results, causing spectral lines to shift and deform to varying degrees. This leads to non-uniform error magnitudes for spectral signals in different regions during calibration, making calibration difficult, or causing spectral measurement results to be poorly reproducible across measurements, increasing measurement instability.

Beyond the asymmetry issue, our calculations also revealed that polygonal fibers showed no apparent improvement in scrambling performance compared to circular fibers when evaluated by centroid drift alone. This contradicts direct morphological observations from the images. We speculate this may result from non-uniformity in fiber surfaces and camera sensitivity, indicating that evaluating fiber scrambling effects solely through centroid drift and SG, and their resulting spectral line shifts and deformations, is incomplete.

2.4.2 Visibility Analysis

To reduce error accumulation from multiple processing steps, we directly analyzed the collected speckle images and proposed another quantitative scrambling evaluation method—calculating contrast and visibility. Since our research focuses on large-area mode patterns and to avoid influence from special central patterns caused by polishing, we calculated contrast and visibility across subsets of the entire speckle distribution, as shown in the sampling method in [FIGURE:6a]. In [FIGURE:6b], $C$ represents contrast and $V$ represents visibility. Contrast is the ratio of standard deviation to mean value of image grayscale data, while visibility is the difference-to-sum ratio of maximum to minimum grayscale values. Both metrics represent brightness variations in an image; smaller variation ranges indicate lower contrast between bright and dark regions, weaker modal speckles, and better fiber scrambling performance.

[FIGURE:6b] shows calculated contrast and visibility results for the three fiber types. Polygonal fibers exhibit lower contrast and visibility than circular fibers, proving they have fewer modal speckles and better scrambling performance. Additionally, contrast and visibility decrease with increasing fiber length, confirming the length dependence of mode patterns. However, large error bars at some points indicate that this manual random sampling method still contains certain randomness.

3 Speckle Analysis Method Based on Fourier Spectrum

The two scrambling evaluation methods used in Chapter 2 both suffer from significant randomness and uncertainty. We sought a more universal processing method focused on analyzing modal variations, combining qualitative analysis through two-dimensional image Fourier transform spatial frequency spectra with quantitative calculations of contrast and visibility.

3.1 Fourier Spectrum Analysis

We first normalized the intensity of experimental images to ensure each image had equal energy in the time domain, then performed two-dimensional fast Fourier transformation (2D FFT). The resulting two-dimensional spectrum decomposes energy intensity into contributions from different spatial frequencies [23], where bright spots of varying intensity represent the degree of grayscale difference between a pixel and its neighbors (the magnitude of grayscale gradient), i.e., the spatial frequency magnitude. To clearly demonstrate differences between fiber spectra, we rotationally averaged the two-dimensional spatial frequency images to obtain normalized frequencies in one-dimensional space. Using bilinear interpolation, we rotated each spectrum by 1° for 179 rotations (spectrum images are axisymmetric, so 180° covers all spatial frequencies), then superimposed these 180 images and plotted data from any direction as a one-dimensional spectrum. [FIGURE:7] shows one-dimensional spectra for the three fiber types at four different lengths, with the vertical axis representing normalized frequency and the horizontal axis representing one-dimensional spatial frequency. The high-precision camera used for near-field image acquisition has 1.85 µm pixels, so spatial frequency is simply expressed in µm⁻¹ and plotted up to the Nyquist frequency (270 µm⁻¹).

In Fourier spectra, high-frequency components characterize regions with dramatic grayscale variations, with their intensity representing speckle strength. Low-frequency components correspond to slowly varying grayscale regions; larger proportions indicate fewer mode patterns and better fiber scrambling performance. The intensity proportions at different frequencies for each fiber at four lengths are shown in [FIGURE:8].

For spatial frequencies below 10 µm⁻¹, the curves essentially overlap, indicating stable speckle sizes across fiber types and lengths. Octagonal fibers show significantly larger proportions of low-frequency components than the other two fibers, with more power distributed at lower spatial frequencies, demonstrating fewer speckle patterns and superior scrambling performance. The flat portions at the rear of the curves can be considered high-frequency noise; we suspect differences in noise frequency among the three fibers result from minor exposure time variations.

Applying the same image processing to the offset incidence experimental images from Section 2.4.1, [FIGURE:9] shows results for 5 m fiber length, with circular (SI), square (SQ), and octagonal (OCT) fibers from top to bottom. The left vertical axis in each plot represents normalized frequency after rotational averaging of 2D Fourier spectra, while the right vertical axis shows intensity proportion at each spatial frequency. The left three plots show results from x-axis offset, the right three from y-axis offset, with legend percentages representing incident offset distance as a percentage of test fiber diameter.

The offset incidence curves do not completely coincide with the central incidence curves. To accurately compare curve dispersion differences between fibers, we calculated root mean square (RMS) differences. RMS values between each offset curve and the central curve in [FIGURE:9] were computed, with average RMS values for each offset direction shown in Table 1, where RMS is dimensionless like normalized frequency.

Data in Table 1 clearly show that circular fibers exhibit larger differences between offset and central curves, meaning incident offset affects circular fiber Fourier spectra more significantly, while polygonal fibers maintain more stable performance, again confirming their superior scrambling characteristics.

3.2 Contrast and Visibility

Fourier spectra provide valuable qualitative analysis of scrambling performance. However, Fourier spectra are more suitable for fields with numerous speckles and large intensity fluctuations [23], whereas the mode patterns generated by our LED source are relatively sparse, resulting in small curve differences between the three fiber types. Therefore, we introduced contrast and visibility calculations for quantitative analysis. Before this, considering the special central patterns from polishing shown in [FIGURE:3], we needed an image processing method to remove or attenuate these patterns, enabling direct calculation across the entire core region without manually avoiding central patterns, thus eliminating the randomness of three sampling attempts described in Section 2.4.2.

To filter out high-frequency special patterns at fiber centers, we selected the Butterworth low-pass filter [24], which lies between the sharply filtering ideal low-pass filter and the smoothly filtering Gaussian low-pass filter. This filter has two adjustable parameters: cutoff frequency $D_0$ and order $n$ [25]. For two-dimensional images, $D_0$ is the distance from a frequency domain point to the center, and larger $n$ produces a steeper filter shape with more pronounced ringing. We used the frequency at which intensity proportion differences were maximized among the three fiber types at the same length as the cutoff frequency $D_0$, then compared our Fourier spectrum curves with Butterworth low-pass filter functions of different orders, finding that orders of 1–5 were suitable. We then processed experimental images repeatedly using low-pass filters with different orders at 0.2 intervals, ultimately determining that a filter with $n=2.2$ provided optimal filtering while maintaining image clarity. This Butterworth low-pass filter was applied to all centrally incident experimental images, with calculated contrast and visibility results shown in [FIGURE:10]. Before filtering, a 1000 pixel × 1000 pixel region at the speckle pattern center was selected for contrast and visibility calculation, shown in [FIGURE:10a]. [FIGURE:10b] shows contrast and visibility after filtering, and [FIGURE:10c] shows the improvement degree from filtering. Legend symbols: $C$ for contrast, $V$ for visibility.

[FIGURE:10] demonstrates that polygonal fibers have lower contrast and visibility than circular fibers, longer fibers have lower values than shorter fibers, and filtering produces the most significant improvement for polygonal fibers. All results confirm that polygonal fibers exhibit significantly superior scrambling performance compared to circular fibers. Applying the same method to the offset incidence experimental results from Section 2.4.1, [FIGURE:11] shows contrast statistics and [FIGURE:12] shows visibility statistics, with horizontal axes representing relative incident offset and vertical axes being dimensionless ratios.

Combined results from [FIGURE:11] and [FIGURE:12] show that polygonal fibers have significantly lower overall contrast and visibility values than circular fibers, with smaller variations between different lengths, again confirming better scrambling performance. Furthermore, compared with previous centroid drift ([FIGURE:4]) and scrambling gain ([FIGURE:5]), contrast and visibility exhibit better symmetry, matching our experimental offset results more closely and providing greater persuasiveness.

These experiments demonstrate that both qualitative Fourier spectrum analysis and quantitative contrast and visibility calculations can accurately reflect actual optical field variations and scrambling performance differences, enabling more accurate and reasonable scrambling evaluation and validating the feasibility of our method. Stürmer et al. [10] noted that near-field measurement methods are not directly related to radial velocity errors but only serve for comparing different fibers or scrambling methods. Our method can only compare scrambling performance between different fibers or methods and cannot directly correlate with radial velocity errors or directly improve radial velocity measurement precision like centroid drift and SG coefficient. We propose applying this method to assist centroid offset calculations before spectral data processing, by eliminating images with distorted and offset spectral lines caused by external influences (such as fiber surface uniformity, illumination dependence of optical components, detector sensitivity, etc.), and ultimately selecting images most suitable for spectral analysis and calibration. These selected images should guarantee stable centroid offset and high scrambling gain coefficients while maintaining low contrast and visibility with good symmetry and uniformity. This pre-screening of large datasets can reduce processing of abnormal spectral images caused by external influences, ensure data quality and result reliability, and effectively improve spectral processing efficiency.

4 Summary

Fiber scrambling is a crucial technical means for improving radial velocity measurement precision. Addressing the asymmetry and uncertainty issues in single-performance-index SG evaluation, this paper proposes a comprehensive fiber scrambling performance evaluation method based on Fourier spectrum analysis combined with contrast and visibility analysis. This method integrates multiple characteristics including fiber near-field centroid drift SG measurement, spectral analysis, near-field symmetry, and uniformity for comprehensive evaluation, compensating for the insufficient stability of SG measurement as a single performance index. It can effectively avoid random errors from human selection in fiber near-field scrambling analysis, reduce interference from external environments on scrambling performance evaluation, and achieve more universal, unified, and non-discriminatory comprehensive scrambling performance evaluation. This enhances the reliability of comprehensive scrambling performance comparisons between different methods and provides support for developing high-performance fiber scrambling technologies for high-resolution spectral analysis.

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Scrambling Performance Evaluation Method Based on Fourier Spectrum
PANG Yuchen¹,²,³, PENG Zifu¹,²,³, GENG Tao¹,²,⁴, SUN Weimin¹,²,⁴, YAN Yunxiang¹,²,³,⁴
(1. Key Lab of In-fibre Integrated Optics, Ministry of Education, Harbin Engineering University, Harbin 150001, China;
2. Key Laboratory of Photonic Materials and Devices Physics for Oceanic Applications, Ministry of Industry and Information Technology, Harbin Engineering University, Harbin 150001, China;
3. Qingdao Innovation and Development Center of Harbin Engineering University, Qingdao 266000, China;
4. Advanced Photonics Institute, College of Physics and Optoelectronic Engineering, Harbin Engineering University, Harbin 150001, China)

Abstract: The application of optical fiber scrambling technology in high-resolution spectral observation can effectively improve the accuracy of radial velocity measurement, and provide a favorable tool for cutting-edge scientific problems such as searching for terrestrial planets. At present, the influence of mode noise generated by optical fiber transmission on high-resolution spectrometers is becoming more and more prominent, and how to directly analyze the scrambling effect of optical fiber through the output speckle field has become an important topic. In order to solve this problem, a scrambling performance evaluation method based on Fourier spectrum is constructed, and the output speckle field of different fibers are compared and analyzed by combining 2D Fourier spectrum analysis and calculation of image contrast and visibility. This more universal, unified and non-differentiated method provides an important method support for evaluating the accuracy and effectiveness of optical fiber scrambling effect.

Key words: spectrograph; telescope; imaging; radial velocity; image processing