Computationally image-corrected dual-comb microscopy with a free-running single-cavity dual-comb fiber laser

Dual-comb microscopy (DCM), an interesting imaging modality based on the optical-frequency-comb (OFC) mode and image pixel one-to-one correspondence, benefits from scan-less full-field imaging and simultaneous confocal amplitude and phase imaging. However, the two fully frequency-stabilized OFC sources requirement hampers DCM practicality due to the complexity and costs. Here, a bidirectional single-cavity dual-comb fiber laser (SCDCFL) is adopted as a DCM low-complexity OFC source. Computational image correction reduces the image blur caused by the SCDCFL residual timing jitter. Nanometer-order step surface profilometry with a 14.0 nm uncertainty highlights the image-corrected DCM effectiveness. The proposed method enhances the DCM versality and practicality.


Introduction
The ability to act as an optical carrier of amplitude and phase with a vast number of discrete, regularly spaced frequency channels is an interesting aspect of an optical frequency comb (OFC). While this aspect has been effectively applied for optical frequency rulers in optical frequency metrology and spectroscopy with the help of laser control [1][2][3], the combination of this aspect with space-to-wavelength conversion opens a new door to imaging applications for OFCs, namely, dual-comb imaging (DCI) [4][5][6][7][8][9][10][11]. In DCI, the image pixels to be measured are spectrally encoded into OFC modes by space-to-wavelength conversion or spectral encoding (SE) [13][14][15][16]. Then, the entire image is decoded at the same time from the mode-resolved spectrum of the image-encoded OFC acquired by dual-comb spectroscopy (DCS) [17][18][19][20] based on the one-to-one correspondence between image pixels and OFC modes.
Due to the scan-less imaging capability in SE and the capability for simultaneous acquisition of amplitude and phase spectra in DCS, the combination of DCI with confocal laser microscopy, namely, dual-comb microscopy (DCM) [4,5,8,11], enables scan-less confocal one-dimensional (1D) or two-dimensional (2D) imaging of amplitude and/or phase. For example, DCM has been effectively applied for surface topography of a nanometer-scale step-structured sample and nonstaining imaging of standing culture fixed cells [5]. Furthermore, DCM has been further expanded to confocal fluorescence microscopy featuring the scan-less fluorescence lifetime imaging [12], which is an important imaging modality in life sciences. However, the need for two fully frequency-stabilized OFC sources hampers the practical use of DCI and DCM due to their costs and complexity.
Recently, low-complexity OFC sources have been developed for versatile DCS. A quantum cascade laser (QCL) [21] is a chip-scale, high-power OFC source.
While a pair of QCLs has been applied for DCS [22], this approach often suffers from poor mutual coherence between them. The microresonator soliton comb (microcomb) [23] is another chip-scale OFC source with better mutual coherence between the microcombs. A pair of microcombs has been used for DCS [24] and even DCI [9].
However, the relatively large repetition rate frep corresponding to the frequency spacing significantly reduces the number of sampling points in the spectrum or the image. For example, the total number of image pixels was only several hundred in 2D images [9].
The ideal OFC source for versatile DCI and DCM has high mutual coherence to suppress image blur and moderate frep to enable a sufficient number of 2D image pixels without the need for any frequency stabilization. One promising OFC source is a single-cavity dual-comb fiber laser (SCDCFL) [25][26][27][28][29][30][31][32][33]. In an SCDCFL, a pair of OFCs with slightly different repetition frequencies (frep1, frep2 = frep1 + ∆frep) is generated from a single fiber cavity by multiplexing mode-locking oscillation in wavelength [25][26][27], polarization [28][29][30], or propagation direction [31][32][33]. Since the dual OFCs propagate through the same cavity, they experience almost the same cavity disturbances, and the resulting common-mode fluctuations prevent the decline in the mutual coherence between them under no active frequency stabilization. Furthermore, an frep of approximately 100 MHz leads to tens-to-hundreds of thousands of OFC modes within the range of the optical spectrum, which is sufficient for the number of 2D image pixels.
Although these SCDCFLs have been extensively applied for DCS, no attempts to apply them for DCI and DCM have been made.
In this article, we adopted a bidirectional SCDCFL [33] for versatile DCM.
This bidirectional SCDCFL benefits from good spectral overlap over a wide spectral range, high stability and wide tunability of ∆frep, and passive cancelation of commonmode noise. The image blur resulting from the residual timing jitter between the dual OFCs in the SCDCFL was computationally corrected by use of a self-reference image or external-reference image. Figure 1 shows a schematic drawing of the experimental setup for DCM. A bidirectional SCDCFL was used as a DCM light source. As the details of the bidirectional SCDCFL are given elsewhere [33], we briefly describe the laser here.

Experimental setup
Two independent mode-locking oscillations were achieved in clockwise-circulating light and counterclockwise-circulating light in a fiber ring cavity by nonlinear polarization rotation and two saturable absorber mirrors. Part of the noncommon optical path in the cavity suppresses the competition of the two mode-locking oscillations and enables independent tunability of frep1, frep2, and ∆frep. The temperature of the fiber cavity was actively controlled by a combination of a thermistor and a Peltier heater. After optical amplification with a pair of erbium-doped fiber amplifiers (EDFAs), two counterpropagating output light beams from the SCDCFL, namely, CCW-OFC (center wavelength = 1550 nm, mean power = 190 mW, frep_CCW = 43,037,370 Hz) and CW-OFC (center wavelength = 1550 nm, mean power = 7 mW, frep_CW = 43,038,493 Hz, ∆frep = frep_CCW -frep_CW = 1,123 Hz), were used as a signal OFC and a local OFC, respectively.

Computational image correction
The residual timing jitter between the dual OFCs in the SCDCFL leads to a blurred image in DCM because the image corresponds to the 2D spectrograph of OFC modes. To compensate for this image blur, we proposed two kinds of image correction based on an image autocorrelation analysis [35]. The first method is self-reference image correction without the use of the reference arm. In this method, we calculated an autocorrelation function of each confocal amplitude image and extracted the center of the function as an error signal of image blur. Then, we corrected the position of each acquired image by comparing the center of the autocorrelation function in the acquired image with that in the first image and compensating for the difference between them. This method benefits from high robustness to external disturbance due to the use of the common optical path and is effective for imaging static objects with high image contrast. However, it is not suitable for dynamic objects because acquired images are always corrected by comparing images consecutively acquired at different positions with the first image acquired at a fixed position.
To extend the image correction to dynamic objects, we proposed the second method, namely, external-reference image correction using the reference arm. In this case, we simultaneously acquired two confocal amplitude images decoded from the signal-image-encoded CCW-OFC and the reference-image-encoded CCW-OFC.
Then, we calculated their autocorrelation functions and extracted their centers as an error signal of image blur. Then, we corrected the position of each acquired signal image by comparing the centers of the autocorrelation functions. While the second image correction is applicable to dynamic samples due to the temporal synchronization between the signal and reference images, the use of a noncommon optical path in the reference arm makes the image correction less robust to external disturbances.

Basic performance of the bidirectional SCDCFL
We first evaluated the basic performance of the bidirectional SCDCFL.

Evaluation of image accumulation
Image accumulation is often required to improve the signal-to-noise ratio  Fig. 4(d).
We also calculated the image contrast of the confocal amplitude images when the image contrast was defined as the ratio of the difference to the sum of the maximum and minimum amplitudes across a chart pattern. Since no differences exist in the image contrast along the horizontal direction (not shown), we show here the image contrast along the vertical direction. Figure 5

Quantitativeness of confocal phase imaging
We next evaluated the quantitativeness of the confocal phase imaging with self-reference image correction. Although the test chart has surface unevenness corresponding to the presence or absence of reflective film, its reflectivity also depends on the presence or absence of the reflective film. To obtain a reflective sample with surface unevenness and constant reflectivity, we formed a thin gold coating on the test chart and used this chart as a confocal phase imaging sample.  Figure 6(b) shows a cross-sectional profile of the confocal phase image along the white vertical line in Fig. 6(a). The step profile was determined to be a phase difference of 0.782 rad, corresponding to a step height of 97.1 nm. For comparison, we beforehand determined this step height to be 90 nm by atomic force microscopy (AFM, Hitachi High-Tech, AFM5500M, axial repeatability ≤ 1 nm). In Fig. 5(f), the phase stability of the self-reference image correction was 0.057 rad for 1000 consecutive images, corresponding to an uncertainty of 14.0 nm in the step height.
The difference in the step height between DCM and AFM was within the range of this uncertainty. Importantly, axial precision to nanometer order was achieved with the help of the self-reference image correction even though the free-running SCDCFL was used for DCM.

Confocal amplitude and phase imaging of dynamic objects
We extended the computationally image-corrected DCM to a dynamic object.
Here, the test chart was laterally and axially moved by a translation stage. Figure 7 compares snapshots of the (i) confocal amplitude and (ii) phase images for a static test chart (image size = 252 µm by 294 µm, pixel size = 50 pixels by 348 pixels, image acquisition time = 890 µs) among (a) no image correction, (b) self-reference image correction, and (c) external-reference image correction. Visualization 2 shows the movie of confocal amplitude and phase images when the test chart was laterally moved. In the no image correction, movement of the test chart was observed with image blur resulting from the residual timing jitter in the SCDCFL. In the self-reference image correction, the image blur was not suppressed because the image correction was performed through comparison with the first image and was hence unsuitable for a moving object. In the external-reference image correction, the image blur was moderately reduced, and the resulting image visualized the movement of the test chart.
Although the background of the confocal phase image largely fluctuated in each frame, the relative phase distribution reflects the surface profile of the sample. Visualization 3 shows the movie of confocal amplitude and phase images when the test chart was axially moved. Since the test chart is static in the lateral dimensions, the suppression effect of image blur was more clearly confirmed than in Visualization 2. More importantly, the confocality of the DCM was confirmed in the confocal amplitude images.

Discussion
We The external-reference image correction effectively reduces the slow image blur, as shown by the blue plot in Fig. 4(d), revealing its applicability to both static and dynamic objects; however, the fast image blur remains compared with the selfreference image correction. We next discuss the possibility of further reducing the residual fast image blur. To cancel the effect of the residual timing jitter in the SCDCFL, we used a reference image for image correction. Such use of a reference image should suppress the fluctuation caused by d∆fceo and/or md∆frep; however, it remained.
One reason for the residual fluctuation is the noncommon optical path between the signal-image-encoded CCW-OFC and the reference-image-encoded CCW-OFC.
Such spatial noncommoness makes image acquisition sensitive to environmental disturbances because air disturbances or vibrations independently influence the optical paths of these two arms. However, the air disturbances or vibrations are considerably slow compared to the data acquisition speed of the interferogram.
Actually, a noncommon reference arm worked well in compensating for the phase fluctuation in previous research on DCM [6,11]. Another reason is the time separation in the interferogram (= 2.67 ns) between the signal-image-encoded CCW-OFC and the reference-image-encoded CCW-OFC. If this time separation is within the range of the mutual coherence length between the dual OFCs, then its influence is negligible.
However, the mutual coherence length of the free-running SCDCFL is shorter than that of the fully stabilized dual OFC sources due to the residual timing jitter. We consider that the time separation of 2.67 ns is outside of the mutual coherence length of the SCDCFL. We consider that the fast image blur remaining in the externalreference image correction result is due to the temporal noncommonness rather than to the spatial noncommonness. To achieve temporal commonness, the signal-imageencoded CCW-OFC and the reference-image-encoded CCW-OFC have to be multiplexed with no time separation. Polarization multiplexing is one possible method, although one has to consider the polarization dependence of the 2D-SE.

Conclusion
We introduced the SCDCFL into DCM to generalize DCM from the viewpoint of reduced complexity of the light source. To compensate for the image blur caused by the residual timing jitter in the SCDCFL, two kinds of computational image correction were applied for confocal amplitude and phase imaging. The self-reference image correction completely suppressed both the slow and fast image blur in the static sample, and its high phase quantitativeness was highlighted by the surface profilometry of a nanometer-order step surface with an uncertainty of 14.0 nm. The external-reference image correction could compensate for the slow image blur and shows high applicability to both static and moving samples. We discussed the potential to further reduce the fast image blur remaining in the external-reference image correction result. This DCM featuring reduced complexity of the light source will expand the application field of DCM in life sciences and industry.

Disclosures
The authors declare no conflicts of interest.