Digital holography (DH)^[1–4] utilizes light interference to reconstruct the complex light field of objects. It has emerged as a powerful tool across diverse domains, such as biological imaging^[5,6], industrial detection^[7], and chip inspection^[8,9]. Over the years, hyperspectral digital holography (HSDH)^[10–16] has gained significant attention. It is an extension of the hyperspectral imaging (HSI)^[17,18] technology within the field of DH. In contrast to conventional DH, HSDH obtains holographic images of objects at multiple wavelengths, effectively addressing the 2π phase ambiguity through multi-wavelength hierarchical phase unwrapping^[19]. This advancement substantially enhances the maximum unambiguous depth range, facilitating the 3D imaging of macroscopic objects. Furthermore, HSDH enables the retrieval of object spectra, broadening its utility in areas such as object classification and recognition.

In recent demonstrations, it has been established that HSDH can experience significant improvement through the utilization of optical frequency combs (OFCs)^{[10–13,20–22]}. The OFC^[23] is capable of generating numerous narrow laser lines with precisely defined repetition frequencies simultaneously. Compared to traditional tunable lasers, it offers heightened frequency stability, improved coherence, and fixed frequency intervals, which are of vital importance to HSDH.

Two methodologies are generally adopted for integrating OFCs into HSDH systems. The first is mode selection^[13,20–22]. This approach involves the sequential selection of each longitudinal mode of the OFC through optical filtering or mode locking. The second is dual-comb interferometry (DCI)^[10–12]. In this technique, the interference of two OFCs with slightly different repetition frequencies is employed to transfer the desired information from high-frequency lightwaves to the beat signals that can be directly obtained by the photodetector. The different orders of comb lines are separated according to their correspondingly different beat frequencies^[24]. DCI enables the parallel detection of multiple comb lines without the need for spectral scanning and strict timing control, offering advantages in measurement speed and robustness in comparison with mode selection.

An electro-optic frequency comb (EOFC) refers to an OFC generated using an electro-optic modulator (EOM). The repetition frequency of the EOFC is determined by the radio frequency (RF) signal supplied to the EOM. This type of frequency comb is well-suited as a light source for DC-HSDH systems, as the coherence between dual EOFCs can be reliably ensured by employing the same seed laser and synchronizing the clocks of the RF sources. In a notable development, Vicentini et al. constructed an in-line DC-HSDH system based on the EOFC in 2021^[10], achieving the resolution of 100 comb lines using a camera with a frame rate of 320 Hz. However, because the in-line DC-HSDH adopts the traditional DCI configuration^[25], it demands a temporal frequency offset between the two EOFCs to distinguish the positive and negative side bands. In achieving this, two acousto-optic modulators (AOMs) are typically required. To drive the two AOMs, two paths of high-power single-frequency electrical signals are necessary, significantly limiting the system’s simplicity and electric power efficiency. Moreover, the insertion loss of AOMs inevitably affects the optical power efficiency.

In this paper, we propose a simplified DC-HSDH system based on spatial heterodyne DCI. The utilization of AOMs is rendered unnecessary. Compared to traditional in-line DC-HSDH, the proposed system enables a simplified system structure, reduced electric power consumption, and enhanced optical power efficiency. Furthermore, it effectively improves the space-time bandwidth product (STBP) of the detector matrix by doubling the temporal bandwidth efficiency. The principle of the proposed system is illustrated in the following section.

Two EOFCs with an identical central frequency f0 are used as the laser source, with one acting as the probe comb and the other as the reference comb. The two EOFCs possess slightly different repetition frequencies denoted as frep1 and frep2 (=frep1+Δfrep), respectively. In spatial heterodyne DCI, the probe light beam and the reference light beam are incident on the detector matrix at distinct angles. Assuming the probe beam reaches the detector matrix with no off-axis angle, and the reference beam reaches the detector matrix at a small off-axis angle of (a,b), the light-field functions of the nth-order comb line pair can be expressed as {Epro,n=Apro,nexp[2πj(f0t+nfrep1t+ϕpro,n)]Eref,n=Aref,nexp[2πj(f0t+nfrep2t+kax+kby+ϕref,n)],(1)where A and ϕ represent the amplitude and phase, respectively, k=1/λn is the wave number, λn=c/(f0+nfrep1) is the wavelength, x and y are the spatial coordinates, and t is the temporal coordinate. Then, the spatiotemporal beat signal generated by the interference between the two light fields can be expressed as $$\begin{gathered} sbeat=G·|Epro,n+Eref,n|2 \\ =G·(|Epro,n|2+|Eref,n|2+Epro,n*Eref,n+Epro,nEref,n*), \end{gathered}$$(2)where G represents the conversion gain. The first two terms are the single-signal-beating interference (SSBI), and the last two are the signal of interest (SOI) and its complex conjugation. By substituting Eq. (1) into Eq. (2), it can be known that Epro,n*Eref,n and Epro,nEref,n* have carrier frequencies of (ka,kb,nΔfrep) and (−ka,−kb,−nΔfrep), respectively, occupying a unique pair of centrosymmetric bands in the spatiotemporal Fourier domain of sbeat. The temporal carrier frequency of the SOI is a function of the comb line order n. Therefore, the SOI corresponding to the nth-order comb line can be extracted using a spatiotemporal band-pass filter centered at the frequency of (ka,kb,nΔfrep). The extracted SOI can subsequently be employed for holographic imaging through a process similar to the conventional off-axis DH. Details of the holographic reconstruction algorithm are presented in the Supplement 1.

It can be known that the off-axis angle between the probe light and the reference light leads to the generation of a spatial carrier frequency, thereby distinguishing the positive and negative side bands. Without this spatial carrier frequency (when a=b=0), the SOI of the nth-order comb line and the conjugate SOI of the −nth-order comb line would be overlapped, since they have the same carrier frequencies of (0,0,nΔfrep).

It is noteworthy that, to resolve 2N+1 comb lines, in-line DC-HSDH systems require a temporal bandwidth of (4N+2)Δfrep, whereas spatial heterodyne DC-HSDH systems only require a temporal bandwidth of (2N+1)Δfrep, exhibiting a twofold temporal bandwidth efficiency.

However, the spatial heterodyne DC-HSDH system performs filtering in the spatial domain, which consequently impacts its spatial bandwidth efficiency. The STBP is introduced as a comprehensive metric that evaluates the efficiency of both temporal and spatial bandwidths. The STBPs of in-line DC-HSDH systems and the proposed spatial heterodyne DC-HSDH system can be expressed as STBPi=fs2ΔfrepXY·min[BxBy,1](3)and STBPs=fsΔfrepXY·min[BxBy,1/2],(4)respectively. Here, fs is the sampling rate of the detector matrix, XY is the total pixel number of the detector matrix, and BxBy is the digital spatial bandwidth product of the SOI. It can be observed that STBPs and STBPi satisfy the relation of STBPs≥STBPi. Furthermore, when the surface of the imaging object is relatively smooth and BxBy is less than 1/2, the relation STBPs=2STBPi holds, indicating that the information density within the space-time bandwidth of the detector matrix is effectively enhanced. Compared to the in-line DC-HSDH, the spatial heterodyne DC-HSDH exhibits a disadvantage in terms of reduced space-bandwidth product (SBP). This limitation impacts its performance in applications requiring high-spatial-resolution holographic imaging. To mitigate this problem, the synthetic aperture method^[27] can be employed as a compensatory approach to address the reduction in SBP.

Proof-of-concept experiments were performed to verify the above principle. The setup of the spatial heterodyne DC-HSDH system is shown in Fig. 1(a). A dual-drive EOFC^[26] generation configuration was adopted. A narrow-linewidth fiber laser (FL, NKT Adjustik E15) operating at 1550 nm was used as the seed laser. This laser output continuous lightwave was split into a probe path and a reference path. In the probe path, an electric signal with an RF of frep1=6GHz, generated by a microwave generator (Agilent E8257D) with low phase noise, was divided into two paths. Each path was amplified to 3 W by an electrical amplifier (Mini-Circuits, ZVE-3 W-183+) before being input into the DD-MZM to modulate the laser. The phase delays of the two electric input paths were adjusted to satisfy the flat spectrum condition. In the reference path, a similar configuration was adopted, with the exception that the frequency of the input electric signal fed into the DD-MZM was set to be frep2=frep1+Δfrep=6GHz+2Hz. The two microwave generators were referenced to the same clock. The spectra of the generated EOFCs are presented in Fig. 2.

Two collimators (Thorlabs, F280FC-1550) were utilized to emit the light beams. A lensless camera (Xenics, Bobcat-320-star) with a frame rate of 30 Hz, a resolution of 320×256, a pitch size of 20μm×20μm, and an analog/digital (AD) conversion resolution of 14 bits was employed to detect the light. The imaging object was a reflector with a 1 mm step, positioned 65 mm away from the camera plane. Additionally, a cell filled with HCN gas (Wavelength References, HCN-13-100) was coupled in the probe path to modulate the transmission spectrum. The probe light was reflected by the object before reaching the camera. The reference light was directly reflected by a mirror and reached the camera at a slight off-axis angle. The probe light and the reference light interfered with each other on the camera plane, generating interferograms with fringe patterns.

The data acquisition time of the camera spanned 10 s (see Visualization 1). Each interferogram was acquired in one frame without accumulation. By arranging the interferograms captured by the camera in temporal order, the interferogram hypercube (IHC) can be obtained as shown in Fig. 1(b).

IHC is a three-dimensional (3D) spatiotemporal signal. By applying a 3D fast Fourier transform (FFT) to the IHC, a Fourier hypercube (FHC) can be obtained as shown in Fig. 1(c). FHC is a representation of the IHC in the Fourier domain, with fx and fy representing the spatial frequency coordinates and ft representing the temporal frequency coordinates. As shown in Fig. 1(h), which is a cross-section of the FHC at ft=mΔfrep=2m(Hz), two peaks exhibiting central symmetry about the zero-spatial-frequency point (fx=0,fy=0) correspond to the SOI of the mth-order comb line and the conjugate SOI of the −mth-order comb line, respectively. Moreover, as shown in Fig. 1(g), since the energy of the SSBI signal is concentrated near the zero-spatiotemporal-frequency point (ft=0,fx=0,fy=0), the SOIs corresponding to the majority of comb lines remain unaffected by the SSBI signal. This characteristic results in the proposed system having higher spatial resolution in comparison with the conventional off-axis DH system.

The FHC was spatially band-pass filtered and unrolled on the ft-axis to obtain the temporal Fourier spectrum as shown in Fig. 1(d). A total of 15 comb lines uniformly spaced at a temporal frequency interval of 2 Hz were resolved.

Subsequently, temporal band-pass filtering was applied to the spatially filtered FHC to extract each comb line. Following this, an inverse fast Fourier transform (IFFT) was utilized to derive the phase map and the amplitude map of the light field on the camera plane, as depicted in Fig. 1(e). After that, the obtained maps were back-propagated to the object plane using ASA, as shown in Fig. 1(f)^[28].

The final hyperspectral holographic imaging results of the object are shown in Fig. 3. Figure 3(a) presents the phase map measured at a single wavelength, exhibiting a strongly wrapped appearance. A multi-wavelength hierarchical phase unwrapping (HPU) method was used to unwrap the phase map. The unwrapped phase map was then converted to the depth map by a factor of d/2π, where d=c/(2frep1)≈25mm (this value is verified in the following subsection) is the maximum unambiguous depth range. According to the resulting depth map as shown in Fig. 3(b), the two object surfaces can be clearly distinguished. The root mean square error (RMSE) was calculated to be 121 μm, which corresponds to an unwrapped phase error of 0.03 rad.

To suppress the depth errors associated with wavelength-dependent phase aberrations, local phase unwrapping (LPU)^[13] was utilized, and the outcome is presented in Fig. 3(c). Details of LPU are presented in the Supplement 1. The cross-sections along the dashed lines in Figs. 3(b) and 3(c) are depicted in Fig. 3(d). The suppression of the depth errors is evident, and the depth step between the two object surfaces was observed to be close to 1 mm, which is in agreement with expectations.

The uniformly normalized amplitude maps measured at three different wavelengths are shown in Figs. 3(e)–3(g). The brightness of the amplitude maps is inversely proportionate to the gas absorption intensity for each corresponding wavelength. The dividing line between the two object surfaces exhibits a low intensity due to the light scattering associated with the depth step. By computing the average pixel intensity values from the amplitude maps measured at different wavelengths, the transmission spectrum of the gas sample can be derived, as shown in Fig. 3(h). It can be seen that the experimental measurement results closely align with the true data obtained through high-resolution spectral scans using a tunable laser.

When the accumulation time was reduced to 1 s, the corresponding imaging results are presented in Fig. 4. The temporal spectrum, obtained by applying a spatial filter to the FHC, is illustrated in Fig. 4(a). The depth map, obtained by using multi-wavelength HPU, is shown in Fig. 4(b), while the depth error map is depicted in Fig. 4(c). The RMSE was calculated to be 152 μm, corresponding to an unwrapped phase error of 0.04 rad. Additionally, the measured transmission spectrum is displayed in Fig. 4(d). Although reducing the accumulation time leads to a degradation in imaging quality due to a lower signal-to-noise ratio, the increased imaging speed enhances the system’s applicability in dynamic imaging scenarios.

The physical basis of HPU is to measure the phase of an object using lasers of different wavelengths, after which the difference between the measured phases can be used to achieve phase unwrapping. The maximum unambiguous depth range d is expanded from λ/2 to Δλ/2, where Δλ is the synthesized wavelength formed by beating the lasers of different wavelengths.

In the context of our proposed system, the optical frequency difference between every two adjacent comb lines equals the repetition frequency frep1=6GHz. Therefore, the maximum unambiguous depth range of the proposed system can be calculated as d=Δλ/2=c/(2frep1)≈25mm.

An experiment was conducted to verify this value. As shown in Fig. 5(a), the distance between the two surfaces of the stepped reflector was enlarged to dreal=35mm, making it greater than the maximum unambiguous depth range of the system. Figure 5(b) presents the amplitude map when focusing on the front surface. Figure 5(c) demonstrates the depth map obtained by hierarchically unwrapping the phase maps measured at different wavelengths. The cross-section of the depth map along X=3.6mm is shown in Fig. 5(d). It can be seen that the measured distance between the two surfaces dmea is close to 10 mm, which equals drealMODd=35MOD25(mm), thus verifying the system’s maximum unambiguous depth range of 25 mm.

The maximum number of resolvable comb lines M serves as a crucial performance metric for DC-HSDH systems, as it determines the unwrapped phase error and the spectroscopic measurement bandwidth. In the context of the proposed system, it can be calculated as M=fs/Δfrep, where fs represents the frame rate of the camera. By reducing Δfrep, an increase in M can be achieved. This augmentation is ultimately constrained by the system’s phase noise. In practical applications, the coherent integration time of the system is limited due to the presence of phase noise, rendering it unfeasible to resolve beat signals with excessively small frequency intervals. Experimental analysis revealed that the primary source of phase noise within the system stemmed from the fiber. The influence of ambient noise, such as temperature changes and air vibrations, introduced time-varing deviations in the optical path difference between the two paths of fiber, consequently leading to phase noise. By stabilizing the experimental environment and minimizing the length of the fiber, the phase noise within the system can be significantly reduced.

In conclusion, we have proposed a simplified DC-HSDH system based on spatial heterodyne DCI. In the verification experiments, holographic imaging with a maximum unambiguous depth range of 25 mm was realized by hierarchical phase unwrapping using 15 wavelengths. Simultaneously, the spectroscopic measurement over a wide band of 90 GHz was realized at a frequency interval of 6 GHz. Compared to conventional in-line DC-HSDH systems, the proposed system demonstrates an improvement in the STBP. Moreover, the removal of frequency-shifting devices significantly reduces system complexity, electrical power costs, and optical insertion loss, showcasing its potential for highly compact and cost-effective DC-HSDH systems.