Fiber Modal Scrambling Performance Evaluation Method Based on Fourier Spectrum

Yuchen Pang 1,2,3, Zifu Peng 1,2,3, Tao Geng 1,2,4, Weimin Sun 1,2,4, Yunxiang Yan 1,2,3,4

(1. Key Laboratory of In-Fiber Integrated Optics, Ministry of Education, Harbin Engineering University, Harbin 150001, China; 2. Key Laboratory of Marine Photonics Materials and Devices Physics, Ministry of Industry and Information Technology, Harbin Engineering University, Harbin 150001, China; 3. Qingdao Innovation and Development Center, Harbin Engineering University, Qingdao 266000, China; 4. Institute of Advanced Photonics, College of Physics and Optoelectronic Engineering, Harbin Engineering University, Harbin 150001, China)

Keywords: Spectrometer; Telescope; Imaging; Radial Velocity; Image Processing

Chinese Library Classification: P111.4 Document Code: A

1 Introduction

In modern astronomical exploration, the search for extraterrestrial life is conducted by searching for Earth-like planets located in the habitable zone and studying their atmospheric compositions. Exoplanets are currently a hot topic in astronomical research. In recent years, the number of planets discovered using the radial velocity method based on the Doppler shift principle is second only to the transit method. The current short-term (101 min) measurement precision of radial velocity has reached the order of 1 cm/s[1]. Compared to the radial velocity signal of approximately 9 cm/s[2] induced by an Earth-like planet orbiting a Sun-like star within the habitable zone, the measurement precision of the mode-locked laser astronomical frequency comb system can meet the requirements, but it cannot maintain the measurement duration for more than one year.

Moreover, such high-precision astronomical frequency comb systems, due to their manufacturing complexity, are currently only commercially available from Menlo Systems GmbH in Germany and the actual measurement accuracy achieved by using this optical frequency comb with different spectrometers varies, making it highly necessary to improve the optical measurement accuracy of the spectrometer is enhanced to improve the precision of radial velocity measurements.

To improve the accuracy of spectral measurements and ensure the precision of radial velocity determinations, optical fibers are employed in astronomical spectrographs transmitting the light collected by the telescope. However, when using multimode fibers to transmit coherent light, due to the interference between different modes, due to mutual interference, bright and dark laser speckle patterns appear in the near-field image of the fiber output [3, 4]. During our experiments, we also observed that under incoherent white light illumination, similar speckle-like patterns emerge in the near-field image of the fiber [5], which similarly exhibit a dependence on fiber length. Fiber scrambling technology can effectively suppress the generation of both types of speckle patterns, thereby improving radial velocity measurement accuracy. Since the scrambling performance of a single circular fiber is limited, Chazelas et al. [6] tested a fiber with a polygonal cross-section in 2010. Two years later, Spronck et al. [7] and Feger et al. [8] also conducted comparative experiments on the intensity distribution in the near and far fields of polygonal and circular fibers, demonstrating that polygonal fibers possess superior scrambling characteristics.

Generally, we comprehensively evaluate the fiber's scrambling performance from both near-field and far-field perspectives. Among these, near-field evaluation methods for scrambling performance can be broadly categorized into the following three types. The first method is the scrambling gain (SG) coefficient, a classic metric for near-field scrambling performance. Its initial definition is the ratio of the transverse displacement of starlight at the detector input to the displacement of the point spread function (PSF) at the spectrograph [9], given by SG=(d/D)/(f/F) (where d is the displacement of starlight, D is the fiber diameter, f is the displacement of the PSF, and F is the full width at half maximum of the PSF). However, in actual observations, the incident displacement d continuously varies. Therefore, researchers argue that a series of continuously varying results is more convincing than a single SG value, and thus often provide centroid drift plots alongside to jointly evaluate the scrambling effect [6, 10, 11]. The second common method involves plotting intensity distribution curves, specifically generating one-dimensional intensity distribution curves based on single-pixel slices through the image centroid [12, 13]. The scrambling performance is then assessed by analyzing the smoothness and fluctuations of these curves. The final method is an emerging approach based on the Fourier power spectrum of the image.

The analysis method involves performing a two-dimensional Fourier transform on the output near-field spot pattern to obtain the power spectral density of the image, and then evaluating the scrambling effect by analyzing the radially averaged power spectrum drawn through the center point of the image [14, 15].

However, in experiments using the aforementioned methods to verify the mode scrambling performance of specialty optical fibers, we discovered the mode scrambling gain and centroid the asymmetry of the offset. Inspired by the third method, this paper proposes a fiber mode scrambling approach based on Fourier spectrum performance evaluation method. By performing a two-dimensional Fourier transform on the output near-field spot pattern and then applying rotational averaging, the normalized spectrum is obtained. Figure; then analyze and compare the spectral curves among different optical fibers, while calculating the image contrast after low-pass filtering processing visibility is quantified and evaluated. Compared to previous methods, the approach proposed in this paper provides more unified processing of images refinement and integration can effectively avoid randomness, ensuring the accuracy and validity of fiber mode scrambling performance evaluation.

2 Experiments

The deformed fiber scrambling method [16–19] achieves modal redistribution by altering the fiber core shape and disrupting the propagation paths of light rays within the core, thereby realizing the scrambling effect. To verify the scrambling enhancement effect of the deformed fiber, this paper establishes a set of the disturbance mode testing optical system can simulate and detect the incident light spot entering at different positions on the fiber end face to simulate the telescope. The system can simultaneously adjust the incident offset and capture the spot pattern of the fiber's output near field.

2.1 Test Fiber

The experiment tested three shapes of optical fibers, namely circular, square, and octagonal fibers, with the following parameters.

(1) The core of a circular-core step-index multimode fiber (hereinafter referred to as SI) the core diameter is 105 µm, the cladding diameter is 125 µm, and the numerical aperture (NA) is 0.22.

(2) Square-core step-index multimode fiber (hereinafter referred to as SQ) has a core the core size is 100 µm × 100 µm, the cladding size is 330 µm × 330 µm, and the NA is 0.22.

(3) Octagonal-core step-index multimode fiber (hereinafter referred to as OCT) the fiber core diameter is 200 µm, the cladding diameter is 660 µm, and the NA is 0.22.

2.2 Test Platform

To test the scrambling effects of different types of optical fibers, we designed and constructed an optical test system for fiber scrambling, with its optical path shown in Figure 1. This system can employ various types of lasers or LED light sources as incident sources to simulate starlight signals. The incident light spot is reflected by the measured fiber end face and then enters the end-face observation imaging system through a beam splitter cube, enabling real-time monitoring of the incident spot's positional drift. The near-field detection device consists of a telecentric lens with adjustable magnification and a high-precision camera (4,024×3,036 pixels), used to capture images of the output near-field light spot.

Figure 1 Experimental test optical path diagram

2.3 Experimental Results

The optical path system we employed is shown in Fig. 1, utilizing a red laser with a wavelength of 650 nm and an incoherent LED light source. Two types of speckle patterns were obtained. Figure 2 shows the output speckle pattern obtained with a 650 nm laser source, while Figure 3 displays the speckle pattern obtained when an incoherent LED source is incident. Both figures only present results when the light source is incident at the center of the fiber. In each figure, from top to bottom are circular, square, and octagonal fibers, and from left to right are fiber lengths of 0.5 m, 1 m, 2 m, and 5 m, respectively.

Figure 2 Near-field pattern of optical fiber under 650 nm laser incidence

Figure 3 Near-field pattern of optical fiber under LED illumination

From Figure 2, the presence of distinct laser speckle pattern noise can be observed, with the speckle size and intensity varying along the fiber length. The degree of increase is reduced, and the speckle patterns of polygonal fibers are smaller and weaker.

Figure 3 shows a new phenomenon we observed under illumination from an incoherent LED light source, which resembles laser speckle pattern noise. Hence, it is referred to as a "mode pattern" [5], and it can be observed that this phenomenon also exhibits a similar dependence on fiber length. As for the special patterns in the central region of the circular fiber for 1 m and 2 m in Figure 3, we attribute them to the polishing process of the fiber end face.

2.4 Data Processing

2.4.1 Centroid Shift

To simulate the offset effect of starlight signals, we employed a five-dimensional translation stage at the fiber input end to achieve displacement control of the incident light spot along the 𝑥-axis and 𝑦-axis directions on the fiber end face. The movement step size was set to 10% of the fiber diameter, with 7 positions (including the origin) traversed while collecting the output near-field patterns of the fiber. Due to manual adjustment of the translation stage, minor manual errors existed in each movement. To reduce random noise, we continuously captured 10 images at each position for averaging, and performed three independent measurements in each direction while maintaining the same operator position to enhance result accuracy.

First, we employed the most classical analytical methods—centroid drift plots and mode scrambling gain calculations—to process the experimentally obtained near-field spot patterns. The results are shown in Figure 4 and Figure 5, respectively, where the upper three subplots in each figure correspond to results obtained from offsets along the 𝑥-axis, and the lower three subplots correspond to results from offsets along the 𝑦-axis. The horizontal coordinate uniformly represents the relative incident offset, defined as the ratio of the light source incident offset to the test fiber diameter. The vertical coordinate in Figure 4 denotes the centroid drift of the fiber's near-field output pattern, while that in Figure 5 represents the mode scrambling gain coefficient SG. The error bars indicate the variability among three repeated measurements.

From Figure 4, it is evident that the centroid offset of the output near-field is not linearly related to the relative incident offset, indicating that the SG distribution is asymmetric with respect to the origin, as shown in Figure 5. In fact, as early as 2010, Avila et al. [20] experimentally observed that the output pattern distribution is not completely symmetric, and in 2022 [21], they more explicitly stated that the SG distribution is asymmetric under on-axis incidence conditions, and that the near-field spot pattern undergoes different changes along the incident offset direction, which are related to the shape of the fiber path. Ye Huiqi et al. [16] also found during measurements that the centroid drift of the output near-field is not linearly related to the incident position offset. Earlier research [22] attributed this to the generation of coma aberration and concluded that the influence of the incident light angle is greater than that of the offset position. Based on the findings of the above scholars and our experimental results, it can be confirmed that using the classical evaluation methods of centroid drift maps and the scrambling gain coefficient SG leads to asymmetry and uncertainty in the measurement results.

Figure 4 Centroid drift diagrams of three types of optical fibers

Figure 5 Scatter plots of SG for three types of optical fibers

This outcome will directly affect spectrometers where the optical fiber near-field and spectral lines form a mirror-image relationship. In high-resolution spectrographs with long-term stability and high radial velocity measurement accuracy, the scrambling properties of the optical fiber itself can decouple the spectrograph from various changing external environmental conditions (such as temperature, pressure, and telescope position) during transmission, while simultaneously ensuring the stability and homogenization of the spectrograph's input light source. In fiber-fed spectrographs, the output end of the transmission fiber is directly imaged onto the detector, where minute changes in the fiber output correspond to minute changes in the image plane, thus manifesting as the shift or change in the shape of spectral lines. Therefore, when the light source undergoes symmetric, opposite-direction, and equal-distance movements at the input end of the optical fiber, when offset occurs, the image at the fiber output end should theoretically produce a centroid offset with similar symmetry and obtain nearly identical perturbations the mode gain coefficient. However, our verification experiments yielded asymmetric results, which would cause spectral lines to exhibit varying degrees of the displacement and deformation of the detector result in varying degrees of error in spectral signals from different regions during spectral calibration, making the calibration process this leads to difficulties in operation, or causes spectral measurement results to be difficult to reproduce across multiple measurements, increasing measurement instability and other issues.

In addition to the issue of asymmetric results, we also found from the computational results that, compared to the centroid shift of circular optical fibers, the mode scrambling performance of polygonal optical fibers does not appear to show significant improvement, and such calculation results are consistent with the direct morphological characteristics observed in the images. This discrepancy does not align. We speculate that this may be caused by the non-uniformity of the fiber surface and the non-uniform sensitivity of the imaging camera. Therefore, evaluating the mode scrambling effect of the fiber solely based on centroid shift and scrambling gain SG, as well as the resulting shift and deformation of spectral lines in the spectrometer, is not comprehensive.

2.4.2 Visibility Analysis

To reduce the error accumulation caused by multiple processing steps, we directly process and analyze the collected speckle patterns, thus proposing another method for quantifying the disturbance effect is to calculate contrast and visibility. Considering that our research focus is primarily on large- on the patterned area, to avoid the influence of the special pattern at the center caused by the end-face polishing process on the calculation results, we computed the contrast and visibility within a subset of the entire spot distribution, with the sampling method shown in Fig. 6a). In Fig. 6b), 𝐶 represents contrast and 𝑉 represents visibility. Contrast is the ratio of the standard deviation to the mean of the image grayscale data, while visibility is the ratio of the difference between the maximum and minimum grayscale values in the image to their sum; both can represent the brightness variation range of an image.

A smaller perimeter indicates lower contrast between bright and dark regions in the speckle pattern, weaker modal speckle, and better fiber mode scrambling performance.

Figure 6 a) Sampling method for the subset of speckle patterns; b) Contrast and visibility of the speckle patterns in Figure 3

Figure 6b) presents the calculation results of contrast and visibility for three different optical fibers. It can be observed that the polygonal fiber exhibits lower contrast and visibility compared to the circular fiber, demonstrating that the polygonal fiber generates fewer modal speckles and achieves better scrambling performance. Furthermore, as as the fiber length increases, both contrast and visibility decrease, once again demonstrating the length dependence of the pattern. However, the error bars in the figure are relatively large at certain points, indicating that this artificial random sampling method still exhibits a certain degree of randomness.

3 Speckle Analysis Method Based on Fourier Spectrum

The two scrambling evaluation methods used in Chapter 2 both exhibit significant randomness and contingency. We aim to find a more universal processing approach and focus on the analysis of modal changes. Therefore, we first combine the two-dimensional image Fourier trans a qualitative analysis is performed on the exchanged spatial frequency spectral lines, followed by quantitative calculations using contrast and visibility.

3.1 Fourier Spectrum Analysis

First, we perform intensity normalization on the experimental images to ensure that each image has the same energy in the time domain; then we apply two-dimensional fast Fourier transformation (2D FFT). The resulting two-dimensional spectrum decomposes the energy intensity into individual contributions from different spatial frequencies in two dimensions [23], where bright and dark spots indicate the degree of difference in grayscale values between a pixel and its adjacent pixels in the original image (i.e., the magnitude of the grayscale gradient), representing the magnitude of spatial frequency at that point. To more clearly and intuitively display the differences in the spectra of different optical fibers, we perform rotational averaging on the two-dimensional spatial frequency images.

To obtain the normalized frequency in the one-dimensional spatial frequency domain, we employ a bilinear interpolation method. Each spectrum image is rotated by 1°, for a total of 179 rotations (since the spectrum exhibits axial symmetry, rotating through 180° suffices to cover all spatial frequencies). These 180 images are then superimposed, and data from any direction is extracted to plot the one-dimensional spectrum. Figure 7 displays the one-dimensional spectra of three optical fibers at four different lengths, where the vertical axis represents normalized frequency and the horizontal axis denotes one-dimensional spatial frequency. The high-precision camera used in the experiment for capturing near-field images has a pixel size of 1.85 µm; thus, spatial frequency is simply expressed in units of µm⁻¹ and plotted up to the Nyquist frequency (270 µm⁻¹).

Figure 7 One-dimensional spectra of three types of optical fibers at four different lengths

In the Fourier spectrum, high-frequency components characterize regions where the image's grayscale values change abruptly, corresponding to the intensity of these high-frequency components the intensity represents the speckle pattern; the low-frequency components correspond to regions with gradual grayscale variations, where a higher proportion indicates fewer modal patterns and better fiber scrambling performance. The intensity proportions of different frequencies for each fiber at four lengths are shown in Figure 8.

Figure 8 Normalized frequency spectra of three types of optical fibers at four different lengths and the intensity proportions corresponding to each spatial frequency.

As shown in Figure 8, in the region where the spatial frequency is less than 10 µm⁻¹, the curves essentially coincide, indicating that the spot sizes of different types and lengths remain basically stable. The proportion of low-frequency components in octagonal fibers is significantly greater than that in the other two types of fibers, with more the power intensity distribution is concentrated at lower spatial frequencies, indicating that the octagonal fiber speckle pattern is less complex and possesses superior mode scrambling performance. As for the plateau region in the latter part of the curve, it can be regarded as high-frequency noise. The differences in noise frequencies among the three fibers are speculated to result from minor variations in exposure time.

Next, we apply the same image processing method to the experimental images obtained after incident offset in Section 2.4.1.

Figure 9 presents processing results using a 5 m fiber length as an example, showing from top to bottom: circular fiber (SI), square fiber (SQ), and octagonal fiber (OCT). In each subplot, the left vertical axis represents the normalized frequency after rotational averaging of the two-dimensional Fourier spectrum, while the right vertical axis indicates the intensity proportion of each spatial frequency. The left three subplots correspond to incident beam offsets along the 𝑥-axis, while the right three subplots correspond to offsets along the 𝑦-axis. The percentages in the legend represent the ratio of incident offset distance to the test fiber diameter.

As shown in Figure 9, the off-center incidence curves do not completely coincide with the central incidence curve. To more accurately compare the dispersion differences among the curves of various fibers, we calculated the root mean square (RMS) between the curves. For each fiber in Figure 9, the RMS was computed between the curves of each offset direction and the central curve. The average RMS values for each offset direction are presented in Table 1, where RMS, like the normalized frequency, is a dimensionless quantity.

The data in Table 1 clearly show that the two curves of circular optical fibers exhibit greater differences, meaning that incident offsets have a more significant impact on the Fourier spectrum curves of circular optical fibers, whereas polygonal optical fibers can maintain a relatively stable state, once again verifying their superiority modal properties of circular optical fibers.

Figure 9. Normalized frequency spectra of three optical fibers under incident offset and the corresponding intensity proportions at each spatial frequency.

Table 1 RMS Mean Values of 3 Types of Fiber Incident Offset Curves and Central Curves

3.2 Contrast and Visibility

Spectrograms are of significant value for the qualitative analysis of mode scrambling effects. However, Fourier spectrograms are more suitable for fields with a large number of speckles and significant intensity fluctuations [23], whereas the mode patterns generated by the LED light source used in this experiment are relatively sparse, resulting in smaller differences in the curves of the three types of fibers. Therefore, subsequent analysis will introduce calculations of contrast and visibility for quantitative evaluation. Prior to this, considering the special patterns in the central regions of certain fibers in Figure 3, which are produced during the polishing process, we need to find an image processing method to remove or attenuate these patterns. This would eliminate the need to manually extract subset regions while avoiding the central patterns for analysis, allowing direct calculation over the entire core area and avoiding the randomness associated with the three samplings described in Section 2.4.2.

To filter out the high-frequency special patterns at the center of the optical fiber, we selected a low-pass filter with moderate filtering sharpness among the three options. The Butterworth low-pass filter lies between the ideal low-pass filter and the smooth Gaussian low-pass filter [24]. This filter has two adjustable variables: the cutoff frequency 𝐷₀ and the order 𝑛 [25]; for two-dimensional images, 𝐷₀ is the distance from a point in the frequency domain to the center point, and a larger 𝑛 results in a steeper filter shape, meaning more pronounced ringing artifacts. We took the frequency at which the maximum difference in intensity proportion among the three types of optical fibers occurs at the same length as the cutoff frequency 𝐷₀ for the low-pass filter. Then, by comparing our Fourier spectrum curve with the function curves of the Butterworth low-pass filter at different orders, we found that the order can be chosen between 1 and 5. Subsequently, we processed the experimental images repeatedly using low-pass filters of different orders at intervals of 0.2, and ultimately found that the filter with 𝑛=2.2 achieved the best filtering effect while maintaining image clarity. Next, we applied this Butterworth low-pass filter to all experimental images with central incidence and calculated the contrast and visibility results for various optical fibers, as shown in Fig. 10. Before filtering, a 1,000 pixel × 1,000 pixel region was selected at the center of the speckle pattern, and calculations were performed to obtain the contrast and visibility, with results shown in Fig. 10a); Fig. 10b) presents the contrast and visibility of the speckle pattern after filtering; Fig. 10c) shows the degree of optimization of the two data sets by the filtering process. In the legend, 𝐶 represents contrast, and 𝑉 represents visibility.

Figure 10. Contrast and visibility of the speckle pattern before and after filtering.

As shown in Figure 10, the contrast and visibility of polygonal fibers are both lower than those of circular fibers, while the contrast and visibility of long fibers are both lower than those of short fibers. Moreover, filtering processing provides the most significant improvement for polygonal fibers. All results demonstrate that the scrambling performance of polygonal fibers is significantly superior to that of circular fibers. Using the same method, the results from the incident offset experiment in Section 2.4.1 were calculated. Figure 11 shows the contrast statistics, and Figure 12 shows the visibility statistics, where the horizontal coordinate is uniformly the relative incident offset, and the vertical coordinates are both dimensionless ratios.

Figure 11 Contrast statistics of 3 types of optical fibers under 4 different lengths and different offsets

Figure 12 Visibility statistics of three types of optical fibers under four different lengths and various offsets.

Based on the combined results of Figures 11 and 12, the overall values of contrast and visibility for polygonal optical fibers are significantly smaller than those for circular optical fibers. Furthermore, the numerical differences between different lengths of the hexagonal fiber are smaller, which once again confirms the superior mode scrambling performance of the hexagonal fiber. Additionally, compared to the previous centroid drift (Fig. 4) and scrambling gain (Fig. 5), the contrast and visibility exhibit better symmetry, aligning more closely with our experimental offset results and providing greater persuasiveness.

From the above experiments, it can be seen that both the qualitative analysis results of the Fourier spectrum diagrams and the quantitative measurements of contrast and visibility the calculation results can accurately reflect the actual light field variations and differences in mode scrambling effects, enabling effective evaluation of mode scrambling performance. More accurate and reasonable, verifying the feasibility of the method proposed in this paper. Stürmer et al. [10] mentioned that the near-field measurement method is not directly related to radial velocity errors and only functions for comparisons between different fibers or scrambling methods; that is, this method can only be used to compare the scrambling performance between different fibers or different scrambling methods, and cannot replace the centroid offset and scrambling gain coefficient SG to establish a direct relationship with radial velocity errors, nor can it directly improve the accuracy of radial velocity measurements. We propose applying this method to the calculation of auxiliary centroid offsets before spectral data processing, by eliminating images where spectral lines are distorted and shifted due to external influences (such as fiber surface uniformity, illumination dependence of optical components, detector sensitivity, etc.), ultimately selecting the images most suitable for spectral analysis and calibration. These images can ensure stable centroid offsets and high scrambling gains coefficient can also ensure low contrast and visibility while maintaining good symmetry and uniformity. This characteristic, present in a large number of screening conducted prior to data processing can reduce the handling of spectral anomaly images caused by external influences, ensuring the quality of processed data the quality of the data and the reliability of the results effectively improve the efficiency of spectral processing.

4 Conclusion

Fiber scrambling is a crucial technical means for improving the precision of radial velocity measurements. To address the asymmetry and uncertainty issues in single-performance metric evaluation using the scrambling gain (SG), this paper proposes a comprehensive evaluation method for fiber scrambling performance based on Fourier spectrum analysis, combined with contrast and visibility analysis. This method integrates multiple characteristics—including SG measurements of near-field centroid shifts in optical fibers, spectral analysis, and near-field symmetry and uniformity—to comprehensively assess fiber scrambling performance. It compensates for the instability of SG measurements as a single-performance metric, effectively avoids the random selection errors caused by human intervention in near-field scrambling analysis, and reduces external... to eliminate the interference of boundary environments on the evaluation of disturbance model performance, achieving a more universal, unified, and indiscriminate comprehensive evaluation of disturbance model performance, to enhance the reliability of comprehensive modulation performance comparisons among different modulation methods, for high-performance applications in high-resolution spectral analysis, it provides support for the research and development of optical fiber mode scrambling technology.

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Received: 2024-09-08; Revised: 2024-10-22

Funding Project: National Natural Science Foundation of China (12103015)

Corresponding author: Yan Yunxiang, yxyan@nao.cas.cn

This version posted 2025-10-10.