High-resolution wavemeter based on polarization modulation of fiber speckles

When coherent light travels through an optical multimode waveguide, a granular speckle pattern can be observed at the output end face. Goodman found that the nature of the speckle depends on the frequency of the light.^1,2 Even when the input light wavelength varies slightly, the speckles appear to experience small random changes, usually at the speckle edges.³ According to the subtle differences between patterns, the wavelength or spectrum of the light can be measured precisely by processing the image data of the pattern.⁴ Cao was first to report a fiber speckle spectrometer using a piece of multimode fiber (MMF) as the diffusing media.⁵ Subsequently, with the development of machine learning, many researchers are using speckle patterns for imaging and sensing by using deep learning algorithms.^6–8 One of the essential conditions for the generation of speckles is the existence of intermodal dispersion in the multimode waveguide. The phase delay $Δφ$ between guided modes is $Δφ=β1−βm⋅L$⁠, where $β1$ and $βm$ are the propagation constants of the fundamental mode and the m_th order mode, respectively, and L is the length of the waveguide. When $Δφ$ is larger than $2π$⁠, speckles can be produced at the exit end face of the waveguide.² Furthermore, in the well-known phenomenon of chromatic dispersion (CD), the propagation constants of guiding modes are wavelength dependent.^9,10 By tuning the input wavelengths, the phase delay changes according to $dφ/dλ=(dβ1/dλ−dβm/dλ)⋅L$⁠, and the speckle pattern varies as a consequence. The phase delay’s dependence on the wavelength is the fundamental basis to measure the light’s wavelength and spectrum, while the CD (⁠$dβm/dλ$⁠) is the key factor in determining the wavelength resolution.^2,11 There are two main methods to increase $dφ/dλ=(dβ1/dλ−dβm/dλ)⋅L$⁠: exciting modes with higher CD and extending the waveguide’s length. Researchers have explored many different waveguides such as multimode fibers, fiber tapers, spiral silicon waveguides, and microspheres.^12–17 In these studies, CD is the fundamental source of speckle changes. However, there exist other mechanisms to modulate the speckles and improve the wavelength resolution and are therefore worth exploring.

In Refs. 12, 18, and 19, it had been mentioned that the polarization state of the incident light could have an impact on the speckles’ profile. However, there is currently no research on how the polarization state impacts the speckles’ wavelength sensitivity. In this study, we first design and introduce the polarization modulation of the fiber speckles and investigate its influence on the wavelength sensitivity. The pivotal component of the proposed polarization modulation system is the in-line polarization rotator (IPR). Its structure along with the entire polarization modulation system is shown in Fig. 1.

The IPR device is based on the magneto-optic effect,²⁰ which consists of a cylindrical magneto-optical crystal surrounded by a tube of permanent magnets. In this study, the radius and the length of the magneto-optical crystal are 2 mm and 6 mm, respectively. To clearly show the structure of the IPR, it is magnified and shown in Fig. 1. In order to eliminate the environmental temperature disturbance, the whole IPR device is placed under temperature control. A tunable laser is used as the light source that supplies linearly polarized light into a polarization maintaining fiber (PMF). When linearly polarized light from the output of the PMF travels through the magneto-optic crystal, the polarization state of the light will be rotated under the applied magnetic field.^21,22 The rotation angle $θ$ can be written as

where H is the magnetic field intensity along the light propagation direction, L represents the length of the magneto-optical crystal, and V is the Verdet constant. V depends on the material properties of the magneto-optical crystal and the source light’s wavelength, which can be written as follows:^22,23

where $μ0$ is the vacuum permeability, n is the refractive index of the magneto-optical crystal, c is the velocity of light in the vacuum, m and e are, respectively, the charge and the mass of an electron, $χ$ is dependent on the environmental temperature, and A and B are material constants of the crystal. When H, L, and $χ$ are fixed, Eq. (1) can be rewritten as

where C and D are material dependent constants. Based on Eq. (3), the polarization rotated angle changes with the wavelength of the incident light. The red curve in Fig. 3 is the spectrum of S-polarized light obtained by putting a polarization beam splitter (PBS) after the IPR. The periodic curve shows that the linear polarization is continuously changing and is rotated back to the initial state after a sufficient change in wavelength. It should be noted that the IPR is a common and low-cost device that closely resembles a fiber isolator. After the light polarization is modulated by the IPR, a 1-m long RCF (NA = 0.15, outer diameter = 125 μm, core = 100 × 25 μm²) is chosen as the light diffusion waveguide. The RCF has a non-circular core that is sensitive to light polarization.²⁴ 

In the experiment, as the wavelength of the tunable laser is varied from 1530 nm to 1562 nm with a step of 0.08 nm, a series of speckle pictures were captured with an infrared CCD and are shown in Fig. 2. Additionally, a cut-on wavelength filter (Edmund No. 89-664) was also added in front of the CCD to minimize the impact from the ambient environment. In the supplementary video, it can be seen that the speckle patterns are significantly changed as the wavelength is scanned. Interestingly, it is observed that similar speckle patterns appear periodically with a period of about 5.6 nm. Some of these patterns (for wavelengths of 1530.0 nm–1536.0 nm) are listed in Fig. 2. The 16 speckles have the same wavelength difference of about 0.4 nm. It can be seen that the first two pictures on the top row [(a) and (b)] look quite similar to the last two ones on the bottom row [(o) and (p)]. A structural similarity index measure (SSIM) algorithm is then introduced to quantitatively evaluate the similarity of all other acquired speckle patterns.

By using the SSIM, the structural similarity between speckle pictures is evaluated mainly from three aspects: brightness, contrast, and structure.²⁵ The SSIM was calculated between the first speckle pattern (1530.0 nm) and all other acquired patterns. The calculation results are shown in Fig. 3(a). It can be seen that the SSIM curve (red curve) has periodic behavior that coincides with the periodicity of the spectrum of the S-polarized light within the wavelength range of 1530 nm–1547 nm. Because the speckles are still under the influence of CD, the SSIM values decline and gradually deviate from the period of the S-polarized spectrum. By contrast, we also sampled patterns using the same RCF but without the IPR and calculated the SSIM, as shown in Fig. 3(b). Obviously, there is no periodicity at all, showing that the similarity between speckles is generated from the polarization modulation. In addition, the polarization modulation does not weaken the CD but produces more changes in the speckles. More importantly, the polarization modulation is an effective method for improvement of wavelength sensitivity compared with the straightforward method of extending the fiber length. In order to understand how the polarization changes the speckle pattern inside the RCF, the following simulation was performed.

Optical waveguide simulation software, MODE, was used to examine the mode coupling coefficient as the input polarization is changed.²⁶ In the simulation, we built linearly polarized Gaussian beam profiles with different polarization angles and a rectangular waveguide and then set different boundary conditions for the solver. The overlap integral between beam profiles and mode fields of the waveguide were used to obtain the mode coupling coefficient in simulations.^27,28 Figure 4 shows the simulation results of the mode fields and their corresponding coupling coefficients at different polarization angles. It can be seen that the mode coupling coefficient of the RCF has a strong polarization dependence. This means that the intensity ratio of these guided modes from intermodal interference will periodically change and finally modulate the speckle pattern. This could explain why the similarity between speckles occurs at the same periodicity as that of the polarization state, as shown in Fig. 3(a).

An experiment was then performed to investigate the influence of polarization modulation on the correlation of the fiber speckles and the wavelength resolution. An ultra-high precision laser frequency tuning system was built as follows. It consisted of a low phase noise narrow linewidth distributed feedback (DFB) laser (RIO ORION^TM Series 1550 nm, with the central wavelength of 1550.042 nm and the linewidth of 10 kHz), an electro-optic modulator (EOM) (iXblue, MPZ-LN-10), a microwave source (MWS) (HP 8341B), a power amplifier (PA), and an optical tunable filter (OTF) (Yenista Optics, XTM-50),²⁹ as shown in Fig. 5(a). When the EOM was driven by an amplified microwave with a frequency of Δν, the laser with central frequency (ν₀) can be modulated into a series of symmetrical sidebands, such as ν₀ − Δν, ν₀, ν₀ + Δν, … After filtering the sidebands (e.g., ν_0, ν₀ + Δν…) by using the OTF, only a single modulated sideband (ν₀ − Δν) is preserved. In order to eliminate the influence of useless sidebands, the modulation frequency was always less than the filter bandwidth of the OTF. According to Ref. 29, the tuning resolution of the modulated light is determined by the linewidth of the laser source and the precision of the MWS. Figure 5(b) shows the spectrum of the filtered laser sideband tuned by Δν = 8 GHz and 10 GHz, respectively. It is noted that the broadness of the laser spectrum is due to the limited optical spectrum analyzer (OSA) resolution (20 pm).

The laser frequency was then finely tuned by ν_step, with the frequency bias (ν_bias) of 8 GHz. Thus, the frequency shift is $Δν=νbias+n⋅νstep$⁠, where the number of tuning channels n = 1000. First, the correlations of speckle patterns at different modulation frequencies were examined for different Δν. In order to display the correlation intuitively, an algorithm called Arc Cosine Similarity (ACS) was introduced, which can be expressed as

The ACS describes the correlation of two vectors in a geometric perspective, a method that has been widely used in pattern recognition, machine learning, decision making, and image processing.^30,31 In this study, the ACS represents the angle between two patterns in the Euclidean space. The larger the angle, the less correlated they are. The calculation results are shown in Fig. 6. Generally, Fig. 6 shows that the angle between patterns sharply increases with polarization modulation. However, without polarization modulation, the ACS shows a gentle trend. By comparison of the two curves, the polarization modulation mechanism makes adjacent patterns tend to be uncorrelated faster. It is apparent that the polarization has a larger contribution to the speckles’ change than the CD from intermodal interference. The polarization modulation thus provides another mechanism to improve the speckles’ wavelength sensitivity without extending the waveguide length. In the previous studies, 50-m or even 100-m-long fibers have been used to increase the wavelength resolution.^3,32

Finally, a transmission matrix method was used to reconstruct the wavelength of the speckle patterns.^5,16,32 The algorithm is derived from the following equation:⁵ 

where I is the speckle pattern, T is the transmission matrix, and λ is the wavelength corresponding to I. According to Eq. (5), solving for λ is equivalent to solving a linear equation. When T and I are known, λ can be calculated by $λ=T−1×I$⁠. Generally, T is an irreversible matrix, but a pseudo-inverse matrix can be used to replace the inverse of T. The SVD (singular value decomposition) is an efficient approach to decompose the T into three invertible matrices. It can be written as $T=U×Σ×VT$⁠, where U and V are invertible matrices, each known as a semi-orthogonal matrix, and ∑ is a square diagonal matrix. Therefore, T⁻¹ can be expressed as $T−1=V×Σ×U−1$ and the unknown wavelength from a speckle pattern can be obtained from right-multiplying I by T⁻¹.

In this study, we calibrated the transmission matrix through the statistical characteristics of the denoised speckle patterns with different wavelengths.³³ The transmission matrix is calibrated based on 4000 wavelength channels starting from 1550.042 nm, as shown in Fig. 7(a). Figure 7(b) shows a reconstructed spectrum of the speckle pattern with a tuning frequency of 8.04 GHz by using the transmission matrix method. To investigate the resolution of the wavemeter, we recorded two speckle patterns at the modulation frequency of 8.040 000 GHz and 8.040 025 GHz, respectively. Figure 7(c) shows their reconstructed wavelengths. As seen in the enlarged part of Fig. 7(c), the recovered spectrum (red solid curve) has two easily distinguishable peaks that only differ by 0.2 fm. Furthermore, the wavemeter’s ability to measure multiple wavelengths was also tested. A speckle pattern that included three wavelengths (tuning frequencies of 8.005 GHz, 8.040 GHz, and 8.095 GHz) was taken as the probe signal. The reconstructed spectrum is shown in Fig. 7(d). It is clear that the wavemeter can precisely demodulate multiple wavelengths in one speckle pattern.

In summary, a new polarization modulation mechanism to improve the wavemeter wavelength resolution was introduced in this study. The pivotal component in the polarization modulation-based wavemeter is the IPR. It can rotate the linearly polarized light to different angles as the wavelength changes. The similarity among modulated speckles shows a clear polarization dependence and higher wavelength sensitivity. Further theoretical simulations indicate that the mode coupling coefficients of the RCF vary with the polarization angle and modulate the speckle patterns. The experimental results show that the polarization modulation does not weaken the existing CD but greatly increases the speckle’s wavelength sensitivity. In our demonstration experiment, we built a high precision laser tuning system to examine the measurement resolution. The smallest wavelength variation of 0.2 fm can be detected with polarization modulation. All of the results mentioned above have proved that polarization modulation is an effective mechanism to enhance the wavelength resolution. In addition, it should be noted that the method used in this study is not limited to the RCF and can be utilized in other waveguides with non-circular cores. In addition, these findings pave a new way to design high-resolution wavemeters with shorter, more compact optical waveguides.

See the supplementary material for a movie of the polarization modulated speckle patterns captured in the wavelength range of 1530 nm–1562 nm with a step of 0.08 nm.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

This work was funded by the Natural Science Foundation of Zhejiang Province (Grant Nos. LY19F050010 and LY16F050006) and the Zhejiang Provincial Department of Science and Technology (Zhejiang Xinmiao Talents Program) (Grant No. 2020R409042).