Scan-less confocal phase microscopy based on dual comb spectroscopy of two-dimensional-image-encoding optical frequency comb

Confocal imaging and phase imaging are powerful tools in life science research and industrial inspection. To coherently link the two techniques with different depth resolutions, we introduce an optical frequency comb (OFC) to microscopy. Two-dimensional (2D) image pixels of a sample were encoded onto OFC modes via 2D spectral encoding, in which OFC acted as an optical carrier with a vast number of discrete frequency channels. Then, a scan-less full-field confocal image with a depth resolution of 62.4 um was decoded from a mode-resolved OFC amplitude spectrum obtained by dual-comb spectroscopy. Furthermore, a phase image with a depth resolution of 13.7 nm was decoded from a mode-resolved OFC phase spectrum under the above confocality. The phase wrapping ambiguity can be removed by the match between the confocal depth resolution and the phase wrapping period. The proposed hybrid microscopy approach will be a powerful tool for a variety of applications.


Introduction (less than 500 words→ →467 words)
Optical microscopy has been widely used as a powerful tool in life science research and industrial inspection. Therefore, the development of optical microscopy has been a principal driving force in these fields. Among various types of optical microscopy, scanning confocal laser microscopy (CLM) [1][2][3] has attracted attention due to its three-dimensional (3D) imaging capability with confocality, enabling a confocal depth resolution of wavelength λ , spray light elimination, and a wide range of depths. Recently, scan-less CLM was achieved by a combination of one-dimensional (1D) spectral encoding CLM with line-field CLM [4] or two-dimensional (2D) spectral encoding [5,6], enabling rapid image acquisition and robustness to external disturbances. If the depth resolution in these scan-less CLM images can be further decreased beyond the limit of the confocality, CLM will become an even more powerful tool in a variety of applications and fields.
Another type of optical microscopy with interesting features is digital holographic microscopy (DHM) [7][8][9], which enables 3D imaging with sub-wavelength (sub-λ) depth resolution using phase images reconstructed from a digital hologram.
Although such 3D images can be acquired without the need for mechanical scanning, the range of depth is limited to less than the phase wrapping period λ/2 due to the phase wrapping ambiguity. Although the range of depth can be increased by the use of synthesized wavelengths between two laser lights at different wavelengths, it -4-remains less than 10 µm [10,11].
Thus, CLM and DHM both have advantages and disadvantages and are complementary to each other in depth measurement. If the confocality is given the phase image to match the confocal depth resolution with the phase wrapping period, a sub-λ depth resolution can be achieved with a wide range of depths without the phase wrapping ambiguity. As such, scan-less confocal phase imaging will largely expand the applications of optical microscopy.
In this article, we applied an optical frequency comb (OFC) [12][13][14] to scan-less confocal phase imaging. Although OFCs have been widely used as an optical frequency ruler that is traceable to a frequency standard in optical frequency metrology, we propose another use of OFCs, namely an optical carrier having a vast number of discrete frequency channels with a frequency spacing equal to the repetition frequency f rep . After encoding 2D image pixels of a sample on each OFC mode via the bidirectional conversion between wavelength and 2D space (namely, 2D spectral encoding) [5,6,15,16], the 2D image can be decoded from a single mode-resolved OFC spectrum due to the one-to-one correspondence between image pixels and OFC modes. Fortunately, current state-of-the-art dual-comb spectroscopy, namely DCS, enables us to rapidly, precisely, and accurately acquire the mode-resolved OFC amplitude and phase spectra without the need for mechanical scanning [17][18][19][20]. We demonstrate a proof-of-principle experiment of scan-less confocal phase imaging by DCS of a 2D-image-encoding OFC. -5-

Results (2196 words)
Principle of operation Figure 1(a) illustrates the principle of scan-less CLM based on 2D spectral encoding [5,6]. After passing through a light-source pinhole, each OFC mode is individually diffracted at different solid angles by a 2D spectral disperser for a wavelength-to-space conversion. The spatially dispersed OFC modes form a 2D spectrograph of the OFC and are then focused at different positions of a sample as a 2D array of focal spots by an objective lens combined with a relay lens (not shown).
After being reflected by the sample, the 2D spectrograph is spatially overlapped again for the space-to-wavelength conversion. As such, the bidirectional conversion between the wavelength and 2D space modulates the spectral shape of the OFC depending on the 2D information of the sample. In this way, a one-to-one correspondence is established between the 2D image pixels and the OFC modes.
The spectrally modulated OFC light passes through the detection pinhole, giving confocality. The mode-resolved amplitude and phase spectra of the 2D-image-encoded OFC are acquired by DCS. Finally, the confocal 2D image of the sample is decoded from the mode-resolved OFC spectra. Figure 1(b) presents the principle of the confocal phase imaging. In CLM, only volume information in the vicinity of the focal point, namely the confocal volume (depth = ∆z), can be obtained due to the confocality. DCS provides the mode-resolved amplitude and phase spectra for this confocal volume. The 2D -6-amplitude image, decoded from the mode-resolved amplitude spectrum, gives the 2D mapping of reflection, absorption, or scattering of the sample for the entire confocal volume. Therefore, the 2D amplitude image is equivalent to the confocal 2D image acquired by CLM. In contrast, the 2D phase image, decoded from the mode-resolved phase spectrum, gives the 2D phase mapping in the confocal volume. While this 2D phase image is similar to that in DHM where sub-λ depth resolution can be achieved [7][8][9], a clear difference between them is the existence of confocality. Therefore, we call it the confocal phase image. Here, if the confocal depth resolution ∆z is set equal to the phase wrapping period λ/2, the confocality can be coherently linked with the phase image. In other words, the confocality not only resolves the phase wrapping ambiguity but also achieves a wide range of depths, similar to CLM. Therefore, the use of both amplitude and phase images enables a sub-λ-resolved phase image with the confocality and a λ-resolved confocal image with a wide range of depths.  [5,6,15,16] composed of a virtually imaged phased array (VIPA) [21] and a diffraction grating, relay lenses, an objective lens, a confocal pinhole, and DCS optical systems. The OFC1 (signal comb) light forms a 2D array of focal spots, corresponding to a 2D -7-spectrograph of the OFC1 light, on a sample, by passing through the 2D spectral disperser, the relay lenses, and the objective lens. The reflected OFC1 light inversely propagates the same optics for the spatial overlapping of each OFC mode, resulting in encoding of a 2D image on the OFC1 spectrum. After passing through the detection pinhole for confocality, the OFC1 light interferes with the OFC2 (local comb) light to generate an interferogram. Finally, the mode-resolved amplitude and phase spectra for the OFC1 light are obtained by Fourier transform of the acquired interferogram.

Scan-less confocal phase imaging
We first measured the mode-resolved amplitude and phase spectra when a 1951 USAF resolution test chart (positive type) was placed at the focal position (d = 0 µm) as a sample. We accumulated 100 temporal waveforms of interferogram We next decoded a 2D amplitude image of the test chart from the mode-resolved amplitude spectra in Fig. 2(c). To eliminate the influence of the spectral shape in the OFC1 light, a normalized amplitude spectrum was obtained by using the spectrum in Fig. 2(a) as a reference. Figure 3 Although the image distortion in the 2D spectral encoding was compensated when decoding the image from the spectrum, it was somewhat remained. To confirm the confocality, we move the position of the test chart before, at, and after the focal position. Figure 3 from the signal phase spectrum. Figure 3(c) shows a 2D phase image (image size = 760 × 168 µm, pixel size = 82 × 151 pixels, image acquisition time = 81 ms) when we set the test chart in focus (d = 0 µm). This is corresponding to the confocal phase image in Fig. 1(b). Although the obtained phase image was similar to the amplitude image in Fig. 3(a), its image contrast arises from the phase difference between the pattern region and the surrounding region, which are corresponding to reflection-coated region and the uncoated region, respectively. The thickness of the reflection-coated layer was determined to be 116±7.4 nm from the phase difference Finally, we demonstrated 3D shape measurement based on the confocal phase image. As a sample, we made a three-step structure with nm order on a silicon substrate, whose surface was covered by a gold thin film. shows the 3D shape of the sample calculated from the phase image in Fig. 4(b). We determined the step difference to be 86.0±13.6 nm for the 1st/2nd step-difference and 139.9 ± 14.3 nm for the 2nd/3rd step-difference. These values were in good agreement with them measured by the atomic force microscope: 86.0±0.1 nm for the 1st/2nd step-difference and 140±0.3 nm for the 2nd/3rd step-difference.

Evaluation of spatial resolution
We next considered the spatial resolution of the proposed system. A detailed spatial distribution of the 2D-spectral-encoding OFC1 light on a sample is -11-given in the Methods. Figure 5(a) is a schematic drawing of a 2D array of focal spots on a sample when the transmission resonance spectrum of the VIPA includes a single OFC1 mode [16]. In contrast, if multiple modes of OFC1 are contained in the transmission resonance spectrum of the VIPA, a series of focal spots is spatially overlapped along the vertical direction and is still discrete along the horizontal direction, as shown in Fig. 5(b) [5,6]. In the present setup with a VIPA resonance Because the spatial resolution for this condition is similar to that in the scan-less CLM based on 2D-image-encoding [5,6], the spatial resolution is respectively estimated to be 4.7 µm for the horizontal dimension and 2.0 µm for the vertical dimension from the spectral resolution of the grating and VIPA combined with the OL. We determined the actual spatial resolution from the 2D amplitude image of a knife edge to be 9.28 µm for the horizontal direction and 1.61±0.13 µm for the vertical direction. The former resolution was limited by discrete distribution of horizontal spots whereas the latter resolution was limited by the instrumental resolution of the 2D spectral disperser.
These values are in reasonable agreement with the expected values.
Although one-to-one correspondence could not be established between the image pixels and OFC modes in the present setup, this condition did not increase the spatial resolution because the instrumental resolution is larger than the comb-mode -12-linewidth and the comb spacing in OFC1. It is not very difficult to achieve one-to-one correspondence in the form of Fig. 5(a) using a commercially available OFC with a higher f rep (for example, f rep = 250 MHz). The discrete 2D distribution of focal spots in Fig. 5(a) has two benefits for imaging. First, crosstalk between adjacent pixels can be suppressed for both the horizontal and vertical dimensions. Second, image deconvolution [22] can be applied by the use of spectral interleaving in OFC [23][24][25].
These benefits can be achieved only using an OFC with discrete channels as a light source.
We next considered the depth resolution of the proposed system. The depth resolution in the 2D amplitude image is limited by the confocality in this system, in the same manner as traditional CLM [3]. We confirmed the confocal depth resolution of 61.4 µm in the present system (not shown), which is in reasonable agreement with the theoretical value (=43.8 µm). Although this confocal depth resolution was still larger than the phase wrapping period λ/2 (= 0.775 µm) in the present setup, it will be possible to match the period by using tight confocal optics and higher-NA OL. In contrast, the depth resolution in the 2D phase image is limited by the phase noise of the signal in the same manner as DHM. Because the phase noise in the present system was approximately λ/226, the depth resolution was estimated to be 6.9 nm, which is reasonably consistent with the step measurement in the test chart and the three-step-structure sample. If the confocal depth resolution is set equal to the phase wrapping period λ/2, a depth resolution of 13.7 nm and a range of depth up to the -13-working distance of the OL (≈ several mm) can be achieved simultaneously without the phase wrapping ambiguity.

Discussion (758 words)
We demonstrated that full use of the mode-resolved amplitude and phase spectra enables the fusion of confocal imaging and phase imaging as complementary to each other; to the best of our knowledge, this is the first demonstration of this fusion. Although it is well known that DCS provides both amplitude and phase spectra due to Fourier transform spectroscopy, these two full spectra have not been so actively used in previous applications of DCS. Recently, a similar approach was effectively applied for polarization-modulation-free DCS ellipsometry [26]. These demonstrations clearly indicate that simultaneous use of amplitude and phase spectra has the potential to enhance the utility of DCS and may lead to novel applications of DCS.
One may consider the similarity of the proposed method to 2D-spectral-encoding scan-less CLM [5,6]. Although 2D-spectral-encoding was used for the scan-less measurement in both methods, the proposed method has several significant differences from 2D-spectral-encoding CLM. The main difference is the simultaneous acquisition of the confocal image and phase image as demonstrated above. Another difference is the ultra-discrete sampling in the OFC mode. Such ultra-discrete sampling is beneficial because of the improved signal-to-noise ratio -14-(SNR) in this imaging method in addition to the reduction of pixel crosstalk and the adoption of image deconvolution. The OFC has a discretely localized distribution of optical energy with a constant frequency spacing in the optical frequency region, and DCS enables one to acquire the entire optical energy in the radio-frequency (RF) region without loss, as shown in Fig. 6(a). In contrast, the noise component is not  Fig. 6(a) should be better than that in Fig. 6(b). Such an enhanced SNR will directly lead to enhanced image contrast or decreased image acquisition time.
Unfortunately, the limited dynamic range of the photodetector used in DCS blurs -15-these effects in the presented results. However, if the optical comb spectrum is optimized to the limited dynamic range by spectral shaping [27] or mode filtering [28], the enhanced SNR obtained by noise rejection will be clearer.
Finally, we discuss the versatility of the proposed method. In this article, we achieved hybrid confocal and phase imaging under scan-less conditions by imposing 2D image pixels on an OFC mode via 2D spectral encoding. If another physical quantity to be measured is superimposed on each OFC mode by another dimensional conversion, a vast amount of data for the measured quantity can be simultaneously obtained from a single mode-resolved OFC spectrum. Such an approach, which we refer to as dimensional-conversion OFC, will offer the possibility of expanding the application fields of OFC beyond optical frequency metrology because this method can be applied for other optical metrologies by using a variety of dimensional conversions.
For example, time-to-wavelength conversion [29] and polarization-to-wavelength conversion [30] are promising methods for dimensional-conversion OFC. The ultra-narrow OFC mode linewidth enables an infinitesimal resolution of the physical quantity, whereas an ultra-constant sampling interval can be achieved by the frequency spacing of the OFC. An overly discrete sampling interval can be reduced by use of a spectral interleaving OFC [23][24][25].
Therefore, dimensional-conversion OFC has the potential to exceed the limit in the direct measurement of the physical quantity.
In summary, we have successfully demonstrated scan-less confocal phase includes a single OFC1 mode (namely, ∆f VIPA < f rep1 ) [16]. Although the horizontal and vertical dimensions of the focal spot should depend on the linewidth of each OFC mode (= ∆ν mode ) under an infinite spectral resolution, the actual dimensions were respectively limited by the finite spectral resolution of the 2D spectral disperser [5,6].
In contrast, if multiple modes of OFC1 are contained in the transmission resonance spectrum of the VIPA (∆f VIPA < f rep1 ), a series of focal spots along the vertical direction is spatially overlapped although the series is still discrete along the horizontal direction, as shown in Fig. 5(b). In other words, a one-to-one correspondence was established between the image pixels and the OFC mode only along the horizontal direction. Because the horizontal and vertical dimensions of the focal spot in Fig. 5(b) are similar to those in Fig. 5(a), the only difference between the results is whether the image pixels along the vertical direction are continuous or discrete.