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Abstract
Dispersion plays a great role in ultrafast laser oscillators, ultrashort pulse amplifiers, and many other nonlinear optical dynamics. Therefore, dispersion measurement is crucial for device characterization, system design and nonlinear dynamics investigation therein. In this work, we demonstrate a versatile approach, i.e., Kalman filtering-aided white-light interferometry, for group delay dispersion (GDD) characterization. Extended Kalman filter is adopted to track the cosine-like interferogram, and to eliminate the unintended bias and the envelope, providing a nearly ideal phase retrieval and GDD estimation. The measurement range could span from tens of fs2 to tens of ps2, with an uncertainty of about 0.1%, enabling precise GDD measurement for diverse optical components, ranging from a millimeter-thick glass slide to highly dispersive chirped fiber Bragg gratings. Benefited by the simplicity, convenient setup, and easy operation as well as relatively low cost, this approach would help photonic device characterization, dispersion management and nonlinear dynamics investigation in the laboratory and work plant.
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1. Introduction
Dispersion is a ubiquitous phenomenon occurring in optical media showing itself as the frequency-dependent phase velocity of the light, and thereby a short optical pulse would change its shape as it propagates. This is crucial for many nonlinear optical processes and ultrashort pulse propagation [1,2]. For example, dispersion plays a great role in pulse shaping dynamics in ultrafast laser cavities [3,4], and it is also responsible for pulse stretching and compression in chirped pulse amplification [5,6]. To fully harness dispersion during ultrashort pulse propagation, diverse optical components are employed, including prisms, gratings, fiber Bragg gratings (FBG), dispersion-shifted fibers, dispersion compensation fibers, photonic crystal fibers and waveguides, just to name a few. They have unveiled many otherwise unobservable physics [7–9] and find wide applications in optical communications [10], ultrafast optics [11], nonlinear wave generation [12–14], and quantum optics [15–17]. In these applications, precise dispersion measurement is a prerequisite.
White-light interferometry is widely used for dispersion measurement due to its simplicity and phase sensitive nature [18,19]. In this approach, dispersion is encoded in the interference fringe chirp, and phase retrieval algorithms have been developed, such as nonlinear curve fitting [20], Fourier Transform [21], wavelet transform [22], Hilbert transform [23] among many others. In all of these approaches, the unintended slow-varying bias and envelope of the interferogram would intertwine with the slow-varying chirp that stands for dispersion. Moreover, the interferometer is always susceptible to environmental perturbations such as vibrations, airflow or temperature fluctuations. All of these issues would hinder precise phase retrieval and hence accurate dispersion measurement.
As a famous optimal parameter estimation approach, Kalman filtering is an elegant algorithm based on a linear unbiased minimal-variance estimate principle [24]. It essentially combines noisy observations and inaccurate dynamics optimally to give a better sequential estimation. To cope with nonlinear dynamics such as wave tracking, extended Kalman filtering (EKF) has been well developed, and has found many important real-time applications in guidance, navigation and phase tracking [24–26]. For the case of phase retrieval, its recursive nature enables gradual adaption to the low slow-varying bias and envelope, based on a model that is hard or even impossible to be too accurate. Meanwhile, EKF could filter out the fluctuations due to vibrations or acoustic noise at high frequency, therefore would enable accurate phase estimation.
Here in this work, we would combine the best of the two techniques, i. e., easy setup and operation of white-light interferometry and high reliability of EKF, to demonstrate a versatile approach for dispersion measurement. The most important feature is that our approach enables a large measurement range and guarantees high precision. In the following, we will illustrate this with the GDD measurement of optical fiber, thin glass slide, and chirped fiber Bragg gratings (FBGs) to show that the EKF-aided white-light interferometry is able to measure tiny and huge GDD measurements, ranging from tens of fs$^{2}$ to tens of ps$^{2}$ and with a precision of $0.1{\%}$.
2. Principle and methodology
We used Michelson interferometer for white-light interferometry, and the setup is shown in Fig. 1(a). Broadband light is split into two arms, and passes the device under test (DUT) and the matching fiber, respectively. The reflected beams are combined at the fiber coupler, and the interferogram is recorded by the optical spectrum analyzer (OSA). Note that the matching fiber with known dispersion is used for rough optical path balance, while the movable mirror is for precise optical path balance. Although highly coherent tunable lasers or supercontinuum sources are applicable, amplified spontaneous emission light sources and other low coherence sources are preferred, which is important for excess interference suppression coming from unintended reflections. In the setup, ambient airflow and temperature control have been taken to alleviate the slow drift of the interferogram.
Fig. 1. (a) Schematic optical setup for white-light interferometry based dispersion measurement. Broadband light is split into two arms by the fiber coupler and passes the device under test (DUT) and matching fiber, respectively. After collimated by the fiber collimators (col1 and col2) and reflected by the mirrors, it returns back into the two fiber arms. The interferogram is recorded by the optical spectrum analyzer. The matching fiber and the movable mirror (mounted on a moving stage) are used for roughly and precisely balancing the optical path length of the two arms. Three cases for DUT are displayed, and will be used in Part 3. (b) Protocols for dispersion measurement via EKF-aided white-light interferometry.
By using the recorded interferogram and the optical spectra of the two arms, we follow the protocols in Fig. 1(b) to estimate the dispersion of the DUT. Mainly, there are three steps involved:
- (1) Data preparation: This gives the roughly normalized interferogram, where unintended bias and envelop might still persist. Although unnormalized interferogram may still work here because of the bias and envelop extracting capability of EKF, a roughly normalized interferogram can greatly improve the accuracy of the phase retrieval.
- (2) EKF-aided phase retrieval: This is the main point of the present work. Here we adopt EKF for wave-tracking of the cosine-like interferogram and thus the residual bias and envelope (or amplitude modulation) can be eliminated. Consequently, interferometric phase can be inferred, and the details will be shown later.
- (3) Dispersion estimation: We use a polynomial fitting of the retrieved frequency-dependent interferometric phase to give the GDD.
To provide more details of the protocols, we are going into the equations involved. The interferometric fringe can be given by
where $I_1(\omega )$, $I_2(\omega )$ and $\phi (\omega )$ are the optical spectra for the two arms and the spectral phase difference between them, respectively. After retrieving the relative phase delay $\phi (\omega )$ from the interferogram, the group delay dispersion (GDD) for transmissive devices (such as fibers and glass slide in this work) can be inferred by in which the factor of 1/2 is due to the double pass in the Michelson interferometer here, and the GDD for reflective devices (such as chirped FBGs or reflecting mirrors) is inferred byIn this work, we propose EKF-based wave tracking for phase retrieval ($\phi (\omega )$) and hence the GDD calculation. Now we are going to formulate the observation model, state vector and dynamic model necessary for EKF.
According to Eq. (1), the cosine-like interferogram to track can be described as
where $A_0$, $A_c$, and $\Omega _d$ are the bias, amplitude and local "angular frequency" of the cosine-like interferogram, respectively, and $n$ is the data index of the interferogram and $w(n)$ is the additive noise introduced during the measurement. Equation (3) is the observation model for EKF in this work.The state vector we are going to observe in EKF is $x=[x_1, x_2, x_3, x_4]$, whose components are cosine wave, sine wave, the "angular frequency" and bias, respectively. Specifically, $x_1=A_0+A_c cos(\Omega _{d}n)$, $x_2=A_0+A_c sin(\Omega _{d}n)$, $x_3=\Omega _d$, $x_4=A_0$.
It is straightforward to show that the dynamical model is,
Now we are ready for the wave tracking with EKF. All the equations for the EKF are listed as Eqs. (1–(12). The detailed description for dynamic model and EKF equations can be found elsewhere such as in [27], and here is the brief description: Eq. (5) updates x(n|n) according to the previous state x(n-1|n-1) and the observation of y(n), K(n) is the Kalman gain, P(n) is the process covariance matrix, f(x) is the transfer function (i. e., Eq. (4)) with its Jacobian matrix of F(n), H is the observation matrix (obtained from the observation model), Q and r stands for the process noise matrix and measurement noise, respectively.
Notice that the amplitude and bias of the cosine wave can be calculated from
which, as well as the bias ($x_4$), can help to assess the tracking performance. The interferometric phase can be calculated byConsequently, the GDD could be obtained according to Eq. (2a) or Eq. (2b) by polynomial fitting of $\phi (\nu |\nu )$,
3. Dispersion measurement results
3.1 Case 1: a piece of optical fiber
We first illustrate our approach with conventional optical fiber dispersion measurement. We built the optical setup by all polarization-maintaining fibers and devices (as shown in Fig. 1), and now the amplified spontaneous emission in Yb doped fiber was used as the broadband light source and the DUT was a piece of optical fiber (PM 980, Corning) (Case 1 in Fig. 1(a)). We balanced the interferometer by adjusting the position of the moving mirror until observing the interferogram shown in Fig. 2(a). This interferogram and the optical spectrum for each arm in Fig. 2(a) are used to roughly normalize the interferogram shown in Fig. 2(b) by virtue of Eq. (1).
Fig. 2. Fiber dispersion measurement via EKF-aided white-light interferometry. (a) The interferogram (red) and the optical spectra for each arm (black and blue). (b) The roughly normalized measured interferogram (black dashed) and the corresponding EKF retrieved trace (red solid). Also were shown the EKF retrieved cosine-like interferogram, the adjoint sine-like wave, the bias and calculated amplitude of the interferogram. (c) The EKF-retrieved interferometric phase (square) and its polynomial fitting (red solid). (d) The GDD calculated by a fourth-order polynomial fitting.
Subsequently, we used EKF to retrieve the interferometric phase following Eqs. (4)–(14). The retrieved cosine-like interferogram, the adjoint sine-like wave and the bias (i.e., $x_1$, $x_2$ and $x_4$ in the state vector, respectively) are shown in Fig. 2(b). Accordingly, the interferometric phase could be obtained by Eq. (14), and is shown in Fig. 2(c). The nearly ideal polynomial fitting indicates that the GDD was about 5293.9 fs2 at 1040 nm in Fig. 2(d), corresponding to a fiber length of about 13 cm. Note that the GDD in Fig. 2(d) seems like a straight line despite the fourth-order polynomial fitting. This can be explained by the fact that a relatively small fourth-order dispersion (953 fs$^4$) has a negligible contribution to the GDD.
3.2 Case 2: fiber + fused silica glass slide
To show the capability of precise dispersion measurement, we would measure the dispersion of a fused silica glass slide (measured thickness: 1.1 mm), which can be calculated according to Sellmeier equation. This was achieved by calculating the GDD difference between Case 2 and Case 1.
We first measured the GDD for the fiber plus fused silica glass slide shown in Case 2. The normalized interferogram was shown in Fig. 3(a), and that for the case of fiber only (Fig. 2(b)) has also been displayed in Fig. 3(a) for comparison. Despite the slight difference, EKF can nicely track the interferogram (black dashed trace is the measured one, while the red solid trace is the EKF tracked one). Accordingly, the interferometric phase can be retrieved, shown in Fig. 3(b). The slight difference between the GDDs for fiber plus glass slide and that for fiber only (grey line in Fig. 3(c)) (i.e., the GDD for the glass slide) is noticeable, and the GDD@1040 nm for the former is 5334.8 fs2 (compared to the GDD@1040 nm of 5293.9 fs2 in Fig. 2(d)).
Fig. 3. Dispersion measurement for fiber plus a fused silica glass slide (thickness: 1.1 mm). (a) The normalized measured interferogram (black dashed) and the corresponding EKF retrieval (red solid). Also are shown the EKF retrievals of the cosine-like interferogram, the adjoint sine-like trace, the bias and the envelope. The interferogram for fiber only in Fig. 2 (grey dashed) was also shown as a comparison. (b) The EKF retrieved group delay phase (square) and its polynomial fitting (red solid). That for fiber only in Fig. 2(c) is also shown for comparison. (c) Calculated GDDs for the cases of fiber+glass slide and fiber only in Fig. 2(c) according to the polynomial fitting. (d) The GDDs for fiber only in Fig. 2 and fiber plus glass slide, calculated by 10 interferogram records for each case. The mean values are 5293.4 fs2 and 5333.8 fs2, respectively, indicating the GDD for glass slide is 40.4 fs2=2*1.1 mm*18.4 fs2/mm. The factor of 2 comes from the double pass in the Michelson interferometer.
To demonstrate the precision of our approach, we recorded 10 interferograms for the cases of fiber only and another 10 for the fiber plus glass slide, then calculated the GDD by polynomial fitting of the EKF retrieved interferometric phase accordingly, and finally obtained the mean values of 5293.4 fs2 and 5333.8 fs2 at 1040 nm, respectively (shown in Fig. 3(d)). The standard deviations of the 10 measurements in Fig. 2(d) were calculated to be about 5 fs2 in both cases, giving an uncertainty of about $0.1{\%}$. The difference, accounting for the 2*GDD of the glass slide, is calculated to be 40.4 fs2=2*1.1 mm*18.4 fs2/mm, close to the value calculated based on the Sellmeier equation for fused silica, i.e., 2*1.1 mm*18.24 fs2/mm=40.13 fs2.
3.3 Case 3: chirped fiber Bragg gratings
Chirped FBGs can show huge GDDs, and are widely used for dispersion management in optical communications and ultrafast fiber laser amplifiers. The usual dispersion measurement approaches, such as differential phase-shift method [28], modulated phase-shift method [29] and pulse delay method [30] involve complicated setup and expensive facilities. Here, we are going to show that the approach in this work is capable of eliminating the unintended bias and envelope (or amplitude modulation) that is due to the short coherence length.
We recorded the interferograms at different central wavelengths by relocating the mirror in the reference arm. The typical interferograms and the reflection spectrum of the chirped FBG are displayed in Fig. 4(a). Despite the normalization by using Eq. (1), the unintended bias remains and is clearly observed in the normalized interferogram in Fig. 4(b). This residual bias and the interferogram envelope need to be eliminated for accurate phase retrieval. By using EKF, we can track the measured cosine-like interferogram, as well as the adjoint sine-like trace and the bias, as shown in Fig. 4(b). Accordingly, the envelope (or amplitude modulation) and the interferometric phase can be calculated according to Eq. (13) and Eq. (14), respectively. The calculated interferometric phase is displayed in Fig. 4(c), whose polynomial fitting gives the GDD of −27.12 ps2 @ 1040 nm. Similarly, according to the recorded interferograms with a central wavelength ranging from 1016 nm and 1038 nm (the typical ones have been shown in Fig. 4(a)), the GDDs at other wavelengths can be calculated, shown in Fig. 4(d). Following the same way to Fig. 3(d), we can evaluate the measurement uncertainties to be about 0.1${\%}$ by use of tens of interferogram recorded at each central wavelength.
Fig. 4. Dispersion measurement for chirped FBGs. (a) Typical interferograms for the chirped FBGs (blue, red and green for typical interferograms with different central wavelengths), along with the reflection spectrum (black). They are shifted vertically for clarity. The zoomed-in inset shows the details of the white-light interferogram. (b) Normalized interferogram (black dashed) and the EKF-retrievals of cosine-like interferogram, adjoint sine-like trace, the bias and the envelope (or amplitude modulation) of the white-light interferogram, as indicated in the plot legends. (c) The retrieved interferometric phase (black square) according to Fig. 4(b) and its polynomial fitting (red solid). The GDD at 1028 nm is estimated to be −27.12 ps2. (d) GDD estimated from the interferograms recorded around each wavelength, the typical ones of which are shown in Fig. 4(a).
All of these results in Fig. 4 indicate that our approach is able to measure huge GDDs, e.g., equivalent to that of several kilometers long optical fibers (about 20 ps2/km @1040 nm). We believe that it is possible to measure even higher GDDs, limited by our available phase mask for chirped FBG fabrication at present.
4. Disscussion and conclusion
We have demonstrated a versatile GDD measuring approach via Kalman filtering-aided white-light interferometry. No more facilities other than optical spectrum analyzer, low cost broadband light source (such as amplified spontaneous emission) and conventional fiber components are involved in our approach, all of which are common in the laboratory and work plant. Extended Kalman filtering is used to retrieve the measured cosine-like interferogram, and the unintended bias and envelope can be eliminated, enabling accurate interferometric phase retrieval and GDD estimation. The measured GDD can range from tens of fs2 to tens of ps$^{2}$, spanning six orders of magnitude, and the measurement uncertainty is about $0.1{\%}$, compared to a few hundreds or thousands of fs$^{2}$ usually demonstrated [20,31].
Benefited from the easy setup, simple operation of white-light interferometry and highly sophisticated Kalman filtering, this approach would find wide applications in dispersion measurement with large dynamic range and high precision, both for optical laboratory and work plant, helping optical devices and system design, and nonlinear dynamics diagnostics. Approaches in this work would also be promising for the highly accurate phase retrieval for interferometry and many other scenarios of fringe analysis.
Funding
National Key Research and Development Program of China (2020YFC2200403); National Natural Science Foundation of China (U22A6003, 62175122, 62301418); Beijing Municipal Natural Science Foundation (4212048).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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