Dispersion, in the context of optical devices, refers to the temporal broadening or compression of optical pulse signals during transmission [1,2]. It is a fundamental characteristic of photonic devices and a critical parameter impacting the bandwidth of high-speed optical transmission systems due to nonlinear optical effects and pulse spreading [3–5]. In spectral measurements, unknown system dispersion can broaden the interference spectrum in the time domain, thereby affecting measurement range and accuracy [6–9]. In optical fiber communication and distributed fiber sensing, dispersion plays a significant role in signal degradation and distortion during long-distance, high-speed data transmission, directly influencing transmission performance [10–14].

Dispersion is not inherently detrimental; rather, it serves as a performance parameter to be managed in most cases within optical devices. In certain instances, it can even be leveraged alongside other optical modules to enhance system performance. For instance, time-stretch imaging technology exploits dispersion broadening in the time domain to achieve higher-resolution imaging [15–18]. Dispersion interference ranging technology, through precise control of system dispersion, enables high-accuracy, long-distance measurements without dead zones [19–22]. Furthermore, emerging fields like ultrafast dynamic observation [23–25], frequency modulation signal generation [26–29], generation of high-performance optical frequency combs [30–33], and dispersion compensation [34–37] necessitate precise measurement and control of optical signal dispersion in the medium.

In general, in media with minimal dispersion (such as air or short single-mode fibers) or when operating at a zero-dispersion wavelength away from optical devices, the impact of minor group velocity dispersion or higher-order dispersion is typically negligible and can be disregarded in many scenarios. However, as the demand for high-speed optical communication and precise optical measurements has escalated in the last decade, technologies reliant on ultrafast pulses have assumed paramount importance. When employing an ultrafast laser system, the extensive spectrum width of ultrafast pulses mandates careful consideration of micro and higher-order-dispersion effects, rendering accurate measurement of these dispersions across a broad spectral range crucial in contemporary optical measurement and fiber optic communication fields.

Dispersion is commonly measured through interference, phase shift, or time delay methods. While the interference method offers high precision, its drawback lies in the requirement for exceptional mechanical stability [38–40]. Moreover, this technique necessitates wavelength scanning of the light source and reference arm length, resulting in a time-consuming process to generate the dispersion curve. In the phase-shift method, dispersion is determined based on the phase difference incurred by the signal passing through the medium under examination. However, the need for wavelength scanning complicates and prolongs the measurement procedure, particularly for high-order-dispersion analysis that demands precise control over the light source’s wavelength adjustment [41,42]. Compared with the above two methods, the time delay method, known for its simplicity, is widely employed in industrial measurements. Yet, its accuracy is constrained by the detector resolution, rendering it unsuitable for objects with micro-dispersion [43–45].

In addition to the above traditional measurement methods, many novel and effective measurement methods have emerged in recent years, such as adjusting the chirp intensity of the system to improve the measurement range and accuracy for optical fiber dispersion [46]. Using this method, the measurement accuracy has been significantly improved, but the measurement range is only increased to 30 nm, and the measurement object is the dispersion generated by tens of meters of single-mode fiber, which is not suitable for wide range precision measurement of micro-dispersion. By obtaining the time domain and spectral domain information of the interference spectrum, the dispersion of single-mode fiber at different wavelengths is solved according to the corresponding relationship between delay and wavelength difference [47]. However, because the light source used in the measurement system is an ordinary mode-locked laser, the measurement range of dispersion is only 50 nm, and the measurement object is more than one thousand meters of single-mode fiber, so it is not suitable for wide range precision measurement of micro-dispersion. Fourier-plane imaging and a Kretschmann–Raether configuration were used to measure the dispersion curve of the sample in the whole visible spectral range [9]. This method can measure a wide range, and the measure object has micro-dispersion, but the measurement accuracy is not good enough. And the band that can be measured is visible light, which is not suitable for the infrared band in the field of optical fiber communication and optical measurement. Therefore, there is currently no high-precision wide range measurement method for micro and high-order dispersion.

In this paper, based on the optical frequency comb, we aim to design a universal micro-dispersion high-precision measurement method from the refractive index of a measured medium. By changing the arm length of our interference structure, we realized the phase-change velocity turning points on the interference spectrum, then obtained the dispersion information at each turn point by phase demodulation, and finally solved the dispersion curve of the measured medium by using a spectral-interferometry-based diff-iteration (SiDi) method. We verified the validity and accuracy of the model through simulation. Then we set up a corresponding measurement system to measure the dispersion curve of 14 m single-mode fiber and 0.05 m glass. Compared to a laser interferometer or the theoretical value given by manufacturers, the average relative error of refractive index measurement for single-mode fiber (SMF) reached 2.8×10−6 and for glass reached 3.8×10−6. The approach ensures high precision, while maintaining a simple system structure, with realizing adjustable resolution. We believe that our method for precise measurement of micro-dispersion holds universal significance and can enhance the performance of related systems, such as dispersion interferometric ranging systems.

Figure 1(a) shows a micro-dispersion measurement system utilizing a Michelson interferometer, with an optical frequency comb (OFC) serving as the light source. The OFC beam is directed through a coupler, splitting it equally into two paths via a beam splitter: one designated as the reference arm and the other as the measuring arm. The measuring arm incorporates the measured medium, where the optical path difference between the two arms is determined by the length of measured medium (L) and the movable air length (Lk). Upon reflection from their respective mirrors, the two light signals converge at the beam splitter, giving rise to the interference signal: I2(f)=I12(f)(α2+β2+2αβcos(φ)), where α and β are the optical spectra and amplitude coefficients of the reference and signal combs, I12(f) represents the optical signal following the comb’s division, and φ represents the phase difference between two arms. The expression for φ should be revised as follows: φ=4πLfn(f)/c+4πLkf/c, where n(f) represents the refractive index of the measured medium for light frequency f, and c is the vacuum light speed. With a second-order Taylor expansion of the refractive index n(f) at frequency f, the phase difference φ can be expressed as φ=4πLf(nc+∂n∂f(f−fc)+12∂2n∂f2(f−fc)2)c+4πLkfc.(1)

Due to the frequency domain characteristics of the comb’s output light and the oscillatory nature of the triangular wave, the interference phase of two arms undergoes periodic fluctuations, as shown in Fig. 3(a). When f=fc, φ=4πLfcn(fc)c+4πLkfcc.(2)

Near frequency fc, corresponding to the repetition frequency frep at every interval of a comb, the phase change can be expressed as dφ=4πLfrepn(fc)/c+4πLkfrep/c. According to the nature of phase cycle change, dφ≤π. For each cycle, dφ=π=4πLfrepncc+4πLkfrepc,(3)the propagation length of an optical signal in a dispersive medium should be expressed as follows: Ln=lpp/4n(fc)−Lk/n(fc), where lpp=c/frep; as shown in Fig. 1(b), the propagation of light pulses in a dispersive medium is schematic. The residual portion of the dispersive medium length relative to the pulse propagation period length is expressed as Ls=mod(L,Ln). Then the length of the entire dispersive medium is L=mLn+Ls, where m is the number of pulse cycles. Therefore, the real acquired phase can be expressed as φz=2πLsfn(fc)+4πLfn(fc)(ζ1(f−fc)+12ζ2(f−fc)2)c,(4)where ζ1=∂n/∂f and ζ2=∂2n/∂f2. The derivative of this phase can be illustrated by dφzdf=2πLsn(fc)+4πLfn(fc)ζ1+4πLn(fc)(ζ1Δf+12ζ2Δf2)+4πLfn(fc)ζ2Δfc.(5)

When f=fc, its derivative is φz′(fc)=2πn(fc)⋅(Ls+2fcLζ1)/c. When meeting φz′(fc)=2kπ/frep, the frequency of the corresponding point is defined as the phase-change velocity turning point (PCVTP), as shown in Fig. 2, where k is a natural number. As can be seen from the above derivation, we can change Ln by controlling the change of air length Lk, thus changing Ls, and finally making the PCVTP move on the broadband spectrum provided by an OFC. By analyzing the phase information at each PCVTP, the dispersion characteristics of the measured dispersive medium at that particular frequency can be derived.

The dispersion interference model was simulated using MATLAB. The pertinent parameters were configured as follows. The light source utilized was an OFC with a repetition frequency of 50 GHz and a bandwidth of 20 THz, operating within the frequency range of 177 THz to 197 THz. The length of measured single-mode optical fiber (SMF) L* is 12.51 m, and the initial air length Lk is 0 m. First, we changed the air length Lk with step size (d=200μm) to collect 25 groups corresponding to interference spectra of different air lengths, part of which are shown in Fig. 1(d). Figure 1(d) shows the process of making the PCVTP move across the broad spectrum by controlling air length Lk.

Figure 2 illustrates the process of interference spectral phase demodulation, with a specific example of Lk=0.8mm. In Fig. 2(a), the interference curve of the interference spectrum is shown after the removal of the envelope and extraction of the peak value. Subsequently, Fig. 2(b) depicts the phase curve of Fig. 2(a) following Hilbert transformation, while Fig. 2(c) displays the phase curve of Fig. 2(b) after phase unwrapping. To precisely determine the PCVTPs, the phase curve depicted in Fig. 2(c) was partitioned into left and right segments, followed by individual polynomial fitting procedures. Finally, the derivative zeros of the left and right segments were separately computed. The convergence of the two derivative zeros [illustrated by the blue line in Fig. 2(d)] signifies the location of the PCVTP. The phase demodulation method shown in Fig. 2 was used to solve the PCVTPs fturnk∼fturnk(24) corresponding to each Lk, as shown in Fig. 4(a) (blue circles, described later). When Lk=0m, we obtain the corresponding fturnk=192.65THz, according to the Sellmeier formula of the SMF core: n2(λ)=1+0.49211λ2λ2−0.04807+0.62925λ2λ2−0.11275+0.59202λ2λ2−8.29299.(6)

We can obtain the refractive index n0=1.450347534 at the frequency 192.65 THz of SMF core, and the first-order conductor ζ1(fturnk)=1.000000000027756×10−16. We used the model shown in Fig. 1(b) to precisely calculate the length of measured SMF. We took L*=12.51m as the coarse measurement length of measured SMF. According to the equation φz′(fturnk)=2kπ/frep, k=67.57 is determined initially. Since k is a natural number, then k=68. Then Ls(fturnk)=1776.736μm was calculated by the equation n0Ls(fturnk)frep/c=68−67.57. And Ln(fturnk)=lpp2n0=2068.470μm; the natural number m=7014 can be obtained in the same way. The final measurement of the SMF length is L=mLn(fturnk)+Ls(fturnk)=14.510023562m; the absolute error in the measurement outcome is 23.562 μm, demonstrating a measurement resolution capability at the sub-micron level.

Figure 3 shows the schematic diagram of the spectral-interferometry-based diff-iteration (SiDi) method and the corresponding calculation flow for solving the dispersion curve. First, we calculated the corresponding Ls and Ln according to the PCVTPs. Then, based on an initial iteration point (fturnk, n0) and the corresponding first derivative ζ1(fturnk), the refractive index of the next PCVTP can be calculated, and the first derivative of this point can be calculated through the equation 2πni+1(Ls+fturnk(i+1)Lζ1(fturnk(i+1)))/c=2kπ/frep, as shown in Fig. 3(c). According to the iterative calculation in this way, the corresponding dispersion curve can be obtained, as shown in Fig. 3(b).

Taking (fturnk=192.65THz, n0=1.450347534) as the initial iteration point, based on the measured length L of the measured fiber, the SiDi method shown in Figs. 3(b) and 3(c) was used to solve its dispersion curve. The measurement results obtained are depicted in Fig. 4(b) (green triangles). To investigate the impact of the incremental movement step d in the air length on the measurement outcomes, we varied the air length Lk with a step d=100μm increment, resulting in 49 sets of PCVTPs fturnk∼fturnk(48), as shown in Fig. 4(a) (red triangles). Subsequently, employing the same methodology, the dispersion curve of the measured SMF was computed, as illustrated in Fig. 4(b) (orange circles). By utilizing the Sellmeier formula to calculate the core refractive index of the SMF as a reference value, the error curve between the two measurement results was determined and presented in Fig. 4(c). The maximum absolute error was identified as 2.03×10−5, with an average relative error of 1.01×10−5. Notably, through an analysis of the error bands from the two measurement results, reducing the step size of the air length movement can significantly enhance the measurement accuracy.

To validate the capabilities of our micro-dispersion measurement system, we demonstrate experimental implementations of dispersion measurements for 14 m SMF and 0.05 m glass.

The proposed micro-dispersion measurement system is first used to measure the length and dispersion of 14 m SMF. The experimental system diagram is shown in Fig. 5(d). The proposed experimental system uses a chip-scaled soliton microcomb with a frequency band of about 187–197 THz and a repetition frequency of about 48.97 GHz. Its typical optical spectrum is shown in Fig. 5(e), and the typical interference spectrum of the micro-dispersion measurement system is shown in Fig. 5(f). Figure 5(a) shows the image of the butterfly-packaged MRR, and Fig. 5(b) shows the image of the micro-ring resonator with a free spectral range of 48.97 GHz. For some key parameters of our comb, the Q of the MRR is about 1.7×106, second-order dispersion of the soliton mode is about 6.5×10−26s2/m, and the external power dissipation for reaching the soliton is approximately 600–1200 mW. The optical comb is split into measurement light and reference light using a 50:50 beam splitting ratio coupler. The reference light travels through a collimator, circulates through a ring, undergoes reflection following an air path, is recollimated by the collimator, and is then directed back to the coupler after passing through the ring. It interferes with the measurement light, which has traversed the SMF, resulting in the formation of interference light. Subsequently, this light enters the optical spectrum analyzer (OSA) for the measurement of the interference spectrum.

We placed the 14 m measured SMF in the measuring arm, and solved the dispersion information at each PCVTP by adjusting the air length of the reference arm. Throughout the experiment, an electronically controlled displacement platform with a precision of 0.1 μm over the full range (100 mm) was utilized, as depicted in Fig. 5(c), moving in increments of 200 μm to capture 10 sets of interference spectra. Subsequently, the approach illustrated in Fig. 3 was employed to derive the PCVTPs corresponding to each set of interference spectra, as illustrated in Fig. 6(b). For the SMF, the refractive index at a wavelength of 1554.5 nm can be measured by a single-frequency laser as 1.4696. When fturnk=193.60THz and λturnk=1554.5nm; the ζ1(fturnk)=8.596×10−17, and n0=1.4696. According to the above model, the length of measured SMF is 13.998667689 m, the absolute error of the measurement results is 1332.311 μm, and the measured resolution can reach sub-micron level. The dispersion curve of measured SMF is shown in Fig. 6(c), and the mean relative error compared to a laser interferometer is 2.8×10−6.

Following the dispersion measurement of SMF, the proposed measurement method was experimentally validated using a 5 cm segment of K9 glass. As illustrated in Fig. 5(c), a 5 cm glass was introduced into the reference arm, chirped using a 14 m SMF in the measuring arm, and the air length Lk in the reference arm was adjusted by displacing the mirror to calculate the corresponding dispersion at each PCVTP. During the experiment, the displacement platform was incrementally moved with a step size of 20 μm to capture 100 sets of interference spectra. Among these, Fig. 6(a) displays nine sets of interference spectra at intervals of 200 μm. It is observed that as the air length increases, the PCVTP gradually shifts from right to left.

According to the SiDi method shown in Fig. 3, the PCVTPs for 100 groups of interference spectra are obtained, as shown in Fig. 6(b). In the first experiment, we measured the phase generated by SMF in the measuring arm, as shown in Fig. 6(e). In the second experiment, we measured the phase of the reference arm after incorporating the measured glass, as shown in Fig. 6(d). Therefore, the interference phase obtained by the two experiments can be subtracted to obtain the phase produced by the measured glass: φg=φ(SMF)−φ(SMF+glass), as shown in Fig. 6(f). Similarly, the dispersion curve of measured glass is obtained by the SiDi method, as shown in Fig. 6(g). Compared to the dispersion curve of the model K9 glass given by the manufacturer (the corresponding Sellmeier equation coefficients: k1=6.14555251×10−1, k2=1.45987884×10−2, k3=6.56775017×10−1, k4=2.87769588×10−3, k5=1.02699346, k6=1.07653051×102), the relative error of our measurement results is shown in Fig. 6(g), and the mean relative error is 3.8×10−6.

In this study, we aim to propose a method for precise micro-dispersion measurement. Leveraging the refractive index of the medium under examination, we conduct a Taylor expansion at the measured frequency point, deriving the true phase expression based on the periodicity of the triangular phase. Subsequently, we formulate a comprehensive dispersion interference model capable of point-to-point refractive index measurement through adjustment of the air length in one arm of the interferometer. This approach allows for the precise fitting of a dispersion curve for the medium being analyzed. The smaller the repetition frequency of the light source, the denser the frequency comb appears on the spectrum, making it easier to locate the inflection points of the phase variation to achieve higher measurement resolution. However, an excessively low repetition rate can compromise the modes recognition of the OFC, thus impeding the effective implementation of measurement algorithms. Therefore, when selecting the repetition rate of the light source, a balance between high resolution and effective measurement should be pursued. And the precision of dispersion curve measurements depends on the magnitude of iterative differentials, which in turn relies on the step size of the air length displacement in the measurement arm. Smaller step sizes in displacement lead to smaller iterative differentials and thus higher precision in dispersion curve measurements. Furthermore, the broad bandwidth of the OFC enables a wide measurement range for dispersion analysis.

Initially, we simulated the dispersion measurement model and verified the validity and accuracy of the SiDi method. Subsequently, to measure the dispersion curve of a micro-dispersion medium akin to glass, based on a Michelson interferometer, a dispersive medium with slightly higher dispersion (e.g., SMF) was introduced in the measurement arm to induce chirp. Then, the glass sample under test was incorporated in the reference arm, and the respective interference spectra were captured before and after these adjustments. Following these experiments, the phase corresponding to glass was obtained by subtracting the phase. Finally, the established SiDi method was utilized to solve the dispersion curve. In comparison to measurements obtained from a laser interferometer or theoretical values provided by the manufacturer, the average relative error for refractive index measurements was found to be 2.8×10−6 for SMF and 3.8×10−6 for glass. The approach ensures high precision, while maintaining a simple system structure, with realizing adjustable resolution, thereby propelling the practical implementation of precise measurement and control-dispersion. In addition, the measurement range of the approach is constrained by the frequency bandwidth of the OFC, with the theoretical measurement scope extending to thousands of nanometers. To enhance measurement accuracy further, employing a dual-comb with a stabilized repetition rate difference, or diminishing the air length movement step during experiments can reduce the differential length of the dispersion curve, enabling controlled resolution and ultimately yielding a more precise dispersion curve.

Acknowledgment. We thank the National Natural Science Foundation of China for help identifying collaborators for this work.