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Abstract
Fourier-transform spectral imaging captures frequency-resolved images with high spectral resolution, broad spectral range, high photon flux, and low stray light. In this technique, spectral information is resolved by taking Fourier transformation of the interference signals of two copies of the incident light at different time delays. The time delay should be scanned at a high sampling rate beyond the Nyquist limit to avoid aliasing, at the price of low measurement efficiency and stringent requirements on motion control for time delay scan. Here we propose, what we believe to be, a new perspective on Fourier-transform spectral imaging based on a generalized central slice theorem analogous to computerized tomography, using an angularly dispersive optics decouples measurements of the spectral envelope and the central frequency. Thus, as the central frequency is directly determined by the angular dispersion, the smooth spectral-spatial intensity envelope is reconstructed from interferograms measured at a sub-Nyquist time delay sampling rate. This perspective enables high-efficiency hyperspectral imaging and even spatiotemporal optical field characterization of femtosecond laser pulses without a loss of spectral and spatial resolutions.
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1. Introduction
Hyperspectral imaging produces a large series of wavelength- or frequency-resolved images of an object over a broad spectral range for applications such as remote sensing [1,2], target recognition [3], medical diagnosis [4], biomedical engineering [5], agricultural [6], and food inspection [7]. Mathematically, hyperspectral imaging obtains a three-dimensional (3D) data cube of the object including two-dimensional (2D) spatial and one-dimensional (1D) spectral information [8]. Various spectral imaging techniques have been developed.
Dispersive imaging spectrometers are applied for hyperspectral imaging, but the narrow incident slit limits the photon flux and has to be transversely shifted for 3D spectral-spatial information [9,10]. To remove the incident slit, a filtering imaging spectrometer applies narrow-bandpass filters or electronically tunable filters for imaging at limited and selected wavelength channels with a reasonable spectral resolution [11,12]. Coded aperture snapshot spectral imaging (CASSI) resolves the spectral information from the spatial one by encoding the incident beam profile with a designed mask [13–15], however, data reconstruction based on the concept of compressive sensing is complicated and still under development [16]. Without any spatial filter limiting the photon flux, computed tomography imaging spectrometers use dispersive and imaging optics to project the spectral-spatial profiles of the object onto the detector at different projection angles [17–20], and reconstruct the original 3D object with algorithms based on the central slice theorem [21], which states that any projection image Fourier transformed is one slice that passes through the origin in the Fourier space.
Fourier-transform spectral imaging is another hyperspectral imaging technique [22,23], with the advantages of high photon flux, high spectral resolution, broad wavelength coverage, high accuracy for determining the absolute wavelengths, and low stray light [24–26]. In this scheme, the incident light is split into two copies whose time delay is scanned by an interferometer, the spectrum on each imaging pixel is calculated by Fourier transforming the time-delay dependent interference signals. The time delay is scanned at a rate larger than the Nyquist limit (typically tens of steps per optical cycle for robust measurements and improved signal-to-noise ratio [27–29]) to avoid aliasing [30,31], and over a broad range to improve the spectral resolution. Thus the Fourier-transform spectral imaging measurement is time consuming. Various new designs such as the slit or lenslet-array-based imagers [32–34] and phase-controlled spectrometers [35] can partially solve the problem but at the price of degraded spectral or spatial resolutions. In addition, for Fourier-transform spectral imaging especially at short wavelengths, Nyquist sampling of the time delay requires extremely precise motion control of the interferometer, which is technically challenging [36].
For incident light with finite bandwidth, various undersampling techniques have been developed [37–39]. These techniques estimate the maximum and the minimum frequency components of the incident light, and choose the time delay sampling rate carefully to avoid aliasing [40]. So the maximum time delay sampling step size for these techniques equals one half of the inverse of the incident light bandwidth, which should be known in priori [41].
In this paper, we revisit Fourier-transform spectral imaging with a novel perspective of a generalized central slice theorem [21]. By applying the angular dispersive optics to a standard Fourier-transform spectral imager, the central frequency is directly determined and its measurement is decoupled from the smooth spectral envelope, which is reconstructed from interferograms measured at different time delays using the generalized central slice theorem analogous to computerized tomography. In this case, undersampling can be achieved without priori information about the incident light bandwidth.
2. Principle
Similar to conventional Fourier-transform spectral imaging, the incident broadband light is split into two copies by a standard Mach-Zehnder interferometer [ Fig. 1(a)]. They illuminate the object at the object plane (OP), which is imaged to the charge-coupled device (CCD) by a spherical lens (L). The object plane can also be located before the interferometer if it is relay imaged. An angularly dispersive optical grating (G) is inserted so that images of the object at different frequencies $I({x,y,\omega } )$ are spatially shifted along the dispersion direction, coupling the $I({x,y,\omega } )$ information along x and $\omega $ directions. Without a loss of generality, we express $I({x,y,\omega } )$ in the form of ${I_y}({x,\omega } )$ because information for different y can be processed independently. When the time delay between the two copies of the incident illuminating light is $\tau $, they interfere and the CCD camera sums up the interference patterns of different frequency components, i.e.
Here the parameter $a = dz/d\omega $ describes the spatial shift rate of the images at different frequencies along the $z$-direction on the CCD detection plane, the image pixel at x of the $\omega $ frequency component is mapped to $z = x + a({\omega - {\omega_0}} )$ onto the CCD detector plane. The factor $1 + \cos \omega \tau $ describes the interference between the two temporally delayed copies of the incident light. The profile ${I_y}({x,\omega } )$ can be calculated from measured ${S_y}({z,\tau } )$ data by solving a group of linear equations discretized from Eq. (1) at different time delay $\tau $ which is sufficiently sampled with a small step size. This procedure requires sampling the time delay at a rate higher than the Nyquist limit.
Fig. 1. Principle of sub-Nyquist sampling Fourier-transform spectral imaging. (a) Schematic diagram of the experimental setup, including a Mach-Zehnder interferometer with one arm controlling the time delay, a lens L imaging the object plane (OP) onto the detection plane of a CCD camera, and a well-calibrated angularly dispersive optics (grating G here) for sub-Nyquist sampling. Here the object plane is after the Mach-Zehnder interferometer, but it can also be located before the interferometer if it is relay imaged. (b) An analogy to the central slice theorem that the 2D Fourier transformation of a spectral-spatial intensity profile ${I_y}({x,\omega - {\omega_0}} )$ is sliced and sampled by three parallel lines separated by the time delay $\tau $. The linear sum of the three lines is the 1D Fourier transformation of the measured signal ${S_y}({z,\tau } )$. (c) The spectral-spatial intensity profile $I_y^{({\textrm{target}} )}({x,\omega } )$ is defined as the original object to be reconstructed in simulations. Its spectral range is from 2.24 fs−1 (840 nm) to 2.48 fs−1 (760 nm). (d) The $\tilde{I}_y^{({\textrm{target}} )}({{k_x},t} )$ of the object in the reciprocal $({{k_x},t} )$ domain. (e) The reconstructed spectral-spatial intensity profile ${I_y}({x,\omega } )$ with a fixed time delay step size $\Delta \tau = $11.33 fs. (f) The reconstructed spectral-spatial intensity profile ${I_y}({x,\omega } )$ with a 10% random variations of the time delay step size $\Delta \tau = $ 11.33 fs.
Here we propose an alternative perspective to solve Eq. (1) by applying Fourier transformation on ${S_y}({z,\tau } )$ from the $z$-domain to its reciprocal ${k_x}$-domain (detailed derivations are shown in the Appendix).
This perspective of Fourier-transform spectral imaging is an analogy to the central slice theorem (or Fourier slice theorem) for computerized tomography [21]. In the tomography case, the 1D Fourier transform of the measured projection data is equivalent to the slice of the object’s 2D Fourier transform at the specific view angle, and the 2D object is reconstructed by an inverse Radon transform procedure [42]. However, in the Fourier-transform spectral imaging case, the object’s 2D Fourier transformation is also sampled by a line pair translated oppositely along the $t$-direction according to the time delay $\tau $, and each of the coupled two lines can in principle be reconstructed under the slow-varying approximation. Though slices in the Fourier-transform spectral imaging are parallel and do not pass the origin, the information of the object in the reciprocal space is sufficiently sampled and a similar image reconstruction procedure can be applied.
Now we investigate the possibility of Fourier-transform spectral imaging at a sub-Nyquist sampling rate within the perspective of the generalized central slice theorem. ${\tilde{I}_y}({{k_x},t} )$ varies so slowly over a time range close to or even higher than one half of the inverse of the incident light spectral bandwidth $\mathrm{\Omega }$ which is typically much smaller than ${\omega _0}$, so our previous assumption is justified that ${\tilde{I}_y}({k,ak \pm \tau } )$ has little change when the time delay varies by $\mathrm{\Delta }\tau > \omega _0^{ - 1}$ beyond the Nyquist sampling rate. In this case, the maximum time delay sampling step size is comparable to $\pi /\mathrm{\Omega }$.
One may notice that the above analysis is also valid even when $a = dz/d\omega = 0$ that the problem becomes the conventional Fourier-transform spectral imaging. However, it does not suggest the conventional technique without the inserted angular dispersive optics can work at a sub-Nyquist sampling rate, because the slow-varying profile of ${\tilde{I}_y}({{k_x},t} )$ is the Fourier transformation of ${I_y}({x,\omega - {\omega_0}} )$ rather than ${I_y}({x,\omega } )$. The Fourier transformation of the latter has an extra phase term ${e^{i{\omega _0}t}}$ that is no longer slowly varying in the $({{k_x},t} )$ domain. Thus the seeming paradox comes from the way of measuring the central frequency. In conventional Fourier-transform spectral imaging, the central frequency and the spectral envelop function are coupled and obtained both from the time-delay dependent interferograms, and sub-Nyquist sampling introduces spectral aliasing. One can conduct the undersampling procedure by carefully choosing a time delay sampling rate to avoid aliasing [37–41], at a maximum time delay step size less than $\pi /\mathrm{\Omega }$. However, this procedure requires information about the incident light spectrum in advance. In contrast, the introduction of an angular dispersive optical element does not require such priori spectral information, and it decouples the central frequency measurement from that of the slow-varying spectral envelop, because the former is only determined by the calibrated lateral position of the signal on the detector plane whereas the latter can be reconstructed at a sub-Nyquist sampling rate.
We further discuss the spectral and spatial resolutions of this new sub-Nyquist imaging scheme. Similar to conventional Fourier-transform spectral imaging, the spectral resolution is determined by the time delay sampling range, because the support of ${\tilde{I}_y}({{k_x},t} )$ along the $t$-direction [Fig. 1(b)] should be covered. The spatial resolutions along $x$- and $y$-directions should be discussed separately. Both spatial resolutions are determined by the imaging system, however, the x-directional one is also limited by the sampling strategy. The support of ${\tilde{I}_y}({{k_x},t} )$ along the ${k_x}$-direction should also be sampled over its full range to improve the spatial resolution along the $x$-direction, thus both the time delay sampling range and the spatial shifting rate induced by the angular dispersive optics affect the $x$-directional spatial resolution. The sub-Nyquist sampling procedure involves only the sampling rate, rather than the sampling range. Thus both spectral and spatial resolutions are not degraded by the sub-Nyquist sampling procedure.
We finally discuss the numerical method to quantitatively reconstruct the spectral-spatial intensity profile ${I_y}({x,\omega } )$ from the measured ${S_y}({z,\tau } )$ at different time delays. Similar to tomographic reconstruction with limited projection information, we adopt the algebraic reconstruction technique (ART) which is especially robust for mathematically ill-posed inverse problems with partially measured information about ${I_y}({x,\omega - {\omega_0}} )$, because line pairs $t = ak - \tau $ and $t = ak + \tau $ in Fig. 1(b) are coupled [43]. We implement ART by discretizing the $({x,\omega } )$ coordinates into $M \times N$ grids. Then ${I_y}({{x_m},{\omega_n}} )$ is expressed in a column vector I with elements ${I_j}$ ($n =\lfloor j/M \rfloor$ and $m = j - nM$), and ${S_y}({{z_p},{\tau_q}} )$ is expressed in another vector S with elements ${S_i}$ [$q =\lfloor i/({M + N - 1} )\rfloor$ and $p = i - q({M + N - 1} )$]. Thus Eq. (1) is discretized by defining the matrix O satisfying S = OI, and each of its elements is
Assuming a reasonable initial guess of $I_j^{(0 )}$, its maximum-likelihood solution is obtained by following a standard ART iteration scheme that
Figure 1(c-f) shows simulation results of reconstructions of an assumed spectral-spatial intensity object. The object is defined in the spectral range from $\omega {\; } = $ 2.24 to 2.48 fs−1, corresponding to wavelength from 760 to 840 nm [Fig. 1(c)]. The object in the reciprocal $({{k_x},t} )$ domain is shown in Fig. 1(d), and its support along the $t$-direction is around 220 fs, much larger than $\pi /\mathrm{\Omega } = $13.33 fs, which is the maximum time delay step size in conventional undersampling strategies [41]. Figure 1(e) shows the reconstruction result with a time delay step size of 11.33fs, demonstrating the capability of undersampling. We have also introduced noises to the “raw data” in the simulation, and no significant degradation on the reconstructed image quality is observed. However, if 10% random variations of the time delay step size are introduced during the reconstruction procedure, in order to simulate uncertainties in determining time delay positions in experiments, artifacts along the diagonal direction are observed [Fig. 1(f)]. Thus in our proposed undersampling scheme, the uncertainties of the time delay positions introduce image reconstruction artifacts.
3. Results and discussions
To prove the concept of sub-Nyquist sampling Fourier-transform spectral imaging, we first place a resolution target (USAF 1951 from Thorlabs) at the object plane [Fig. 1(a)], which is illuminated by a femtosecond laser pulse pair (1kHz repetition rate, 760-840 nm spectral range, 40 fs pulse duration). The time delay between the two pulse copies is accurately scanned by a piezo linear stage (P631.XCD from Physik Instrumente) from $\tau = {\; }$ −150 to 365 fs. Though the scanning step size of the time delay is 0.33 fs, corresponding to a moving step size of 100 nm, we select measured images ${S_y}({z,\tau } )$ separated by a much coarser time delay step size for data processing, equivalent to the sub-Nyquist sampling scheme. While the object plane is imaged onto a CCD camera (MV-CA023-10UM from HIKROBT), an angularly dispersive grating (600 lines/mm, GR25-0608 from Thorlabs) is placed between the imaging lens and the camera, introducing the spatial shift for different frequency components with $a = $ 6.8 mm·fs. The spatial shift rate is calibrated by inserting a narrow bandpass filter before the grating, and the spatial profiles of the incident light at different wavelengths are measured by rotating the filter. We have noted that the imaging quality for each wavelength is not degraded by the grating. The time delay range and the spatial shifting rate determine a spectral resolution of 0.28 nm and a spatial resolution of 30 µm. The whole optical system is carefully aligned and can work in a stable way over a long period.
We first put a USAF 1951 resolution target at the object plane of the single imaging lens which is sufficient for low numerical aperture applications. If higher resolution is needed, a 4-F imaging system can be applied with the grating placed at its Fourier plane. Figure 2(a) shows the measured ${S_y}({z,\tau } )$ at selected time delays ($\tau = $ −150, −100, 0, 100 fs). One can observe intensity modulations along the $z$-direction because of the interference between the two copies of the incident light, and the spatial frequency increases with the absolute value of the time delay. In addition, the main features of the object such as the horizontally and vertically separated bars and the figure “3” disappear due to spatial shifts for different frequency components.
Fig. 2. Experimental results of sub-Nyquist sampling Fourier-transform spectral imaging of a spatially complicated object. (a) $S({z,y,\tau } )$ of the object (a USAF 1951 resolution target) with selected time delays $\tau = $ −150, −100, 0, 100 fs. (b) The reconstructed spectral-spatial intensity profile of the object ${I_y}({x,\omega } )$ at $y{\; } = $ 0.07 mm. The time delay step size of 7 fs. (c) The 3D spectral-spatial intensity profile $I({x,y,\omega } )$ (the regions for intensity 0.2 times higher than the peak one are shown), obtained by stacking ${I_y}({x,\omega } )$ at different $y$-positions. The spatial intensity profiles for three wavelength channels at 780 nm (left), 800 nm (middle), 820 nm (right) are shown in the right panels. (d) The spectral intensity profile (blue line) of $\; I({x,y,\omega } )$ at $({x,y} )= $ (0,0), consistent with the independent measurement with another fiber spectrometer (red line).
To reconstruct the original 3D spectral and spatial intensity profile of the object, we have selected a sequence of ${S_y}({z,\tau } )$ separated by a time delay step size of 7 fs, exceeding the Nyquist sampling rate limit by 5.25 times. Figure 2(b) shows the reconstructed spectral-spatial intensity profile of the object ${I_y}({x,\omega } )$ at $y =$ 0.07 mm using the ART algorithm [see Eq. (4), Eq. (5)]. We have found the converged solution after hundreds of iterations is insensitive to the initial guess. By stacking together the reconstructed profiles of ${I_y}({x,\omega } )$ at different $y$-positions, the 3D spectral-spatial intensity profile $I({x,y,\omega } )$ is reconstructed [Fig. 2(c)], and key features of the object are spatially resolved for each frequency component. The field of view here is limited by the magnification of the imaging system and the chip size of the camera, however, it is not necessarily small and can be enlarged by modifying the imaging system. We have further compared the reconstructed spectral-spatial intensity profile at any specific spatial position, e.g. $I(x = $ 0$,y = $ 0$,\omega )$ [blue line, Fig. 2(d)], with an independent measurement using another fiber spectrometer [FLAME-S from Ocean Insight, red line, Fig. 2(d)], and the consistency justifies the spectrally resolving capability.
The lower limit of the time delay sub-Nyquist sampling rate should be explored, by selecting the measured ${S_y}({z,\tau } )$ with different time delay step sizes $\Delta \tau $ to reconstruct the object ${I_y}({x,\omega } )$ at $y = $ 0.07 mm. When the time delay step size is $\Delta \tau = $ 0.33 fs, the Nyquist sampling condition is satisfied and the reconstructed spectral-spatial intensity profile ${I_y}({x,\omega } )$ is close to the original object with no significant artifact [Fig. 3(a)]. Comparing it with the reconstruction with $\Delta \tau = $ 7 fs [Fig. 2(b)], the imaging quality is not degraded, showing the feasibility of Fourier-transform spectral imaging at a sub-Nyquist sampling rate. The $\Delta \tau $ further increases to 11.33 fs [Fig. 3(b)], artifacts appear in the reconstructed ${I_y}({x,\omega } )$ profile because less information of ${\tilde{I}_y}({{k_x},t} )$ is sampled in the $({{k_x},t} )$–domain. In this case uncertainties of measured time steps start to reduce the image reconstruction quality, introducing artifacts along the diagonal direction which are also shown in simulation results [Fig. 1(f)]. The reconstructed spectral intensity profiles at different $\Delta \tau $ and $({x,y} )= $ (0,0) are also compared [Fig. 3(c)]. When $\Delta \tau = $ 11.33 fs, it is consistent with the reference ($\Delta \tau = $ 0.33 fs), showing that imaging quality is retained even when the undersampling rate is comparable to the limit of conventional undersampling techniques (∼13.33 fs). However, when $\Delta \tau = $ 16.33 fs, the reconstructed spectral intensity profile deviates obviously from the reference, because the time delay step size is too large.
Fig. 3. Comparison of reconstruction qualities of the object ${{\boldsymbol I}_{\boldsymbol y}}({{\boldsymbol x},{\boldsymbol \omega }} )$ with different time delay step sizes. (a) The reconstruction with a time delay step size of $\Delta \tau = $0.33 fs as a reference of Nyquist sampling. (b) The reconstruction at $y = $ 0.07 mm with $\Delta \tau = $11.33 fs. (c) The reconstructed spectral intensity profiles of $I({x,y,\omega } )$ at $({x,y} )= $ (0,0) for three time delay step sizes of $\Delta \tau = $ 0.33 fs (blue line, as a reference), $\Delta \tau = $ 11.33 fs (red line), $\Delta \tau = $ 16.33 fs (black line). (d) The quantity $\mathrm{\sigma }({\Delta \tau } )$ as a function of the time delay step size $\Delta \tau $, describing the sub-Nyquist sampling reconstruction quality.
We quantitatively grade the reconstruction quality at different sub-Nyquist sampling rates or time delay step sizes, by comparing them with that at the time delay step size of 0.33 fs that
Considering that the object in Fig. 2 is so simple that all frequency components share the same spatial features, we further challenge the technique’s capability of resolving complicated 3D spectral-spatial intensity profiles. To construct such an object, we have placed a mounted bandpass optical filter (800-810 nm bandpass) at the object plane in Fig. 1(a). The $x < 0$ half of the incident laser beam is partially blocked by a narrow bandpass filter but the $x > 0$ half is not, leaving the central region ($x \approx 0$) completely blocked by the mounting ring of the filter [ Fig. 4(a)]. We have measured this 3D spectral-spatial intensity profile at a time delay step size of 7 fs, and the reconstruction result is shown in Fig. 4(b). The semi-pie-shaped spatial intensity profiles in the $({x,y} )$–domain are reconstructed, and their sharp edges prove a high spatial resolution to resolve fine spatial structures. The $x > 0$ region [Point I in Fig. 4(a)] has a broad bandwidth [blue line, Fig. 4(c)] the same as the incident light [red line, Fig. 4(c)]. However, for the $x < 0$ region (Point II), the bandwidth [blue line, Fig. 4(d)] is as narrow as 10 nm, consistent with the specification of the bandpass filter and the independent fiber spectrometer measurement [red line, Fig. 4(d)].
Fig. 4. Experimental results of sub-Nyquist sampling Fourier-transform spectral imaging of a spectrally and spatially complicated object. (a) Schematic diagram of the partially blocked incident light by the mounted narrow bandpass filter. Points I and II are unblocked and blocked by the filter respectively. (b) The 3D spectral-spatial intensity profile $I({x,y,\omega } )$ (the regions for intensity 0.2 times higher than the peak one are shown), obtained by stacking ${I_y}({x,\omega } )$ at different $y$-positions. The time delay step size of 7 fs. (c) The spectral intensity profile of reconstructed $I({x,y,\omega } )$ at Point I, which is unblocked by the filter (blue line). It is consistent with the independently measured spectrum (red line). (d) The same as (c) but for Point II which is blocked by the filter.
Previous two experiments have proven the concept of Fourier-transform spectral imaging at a sub-Nyquist sampling rate, obtaining the spectral-spatial intensity profile ${I_y}({x,\omega } )$ of an object without phase information. Recently, Fourier-transform spectral imaging has been applied to measure the spatiotemporal optical field profile, including both intensity and phase of a femtosecond laser pulse in 3D (2D for transverse spatial and 1D for longitudinal temporal information), leading to techniques called total E-field reconstruction using a Michelson interferometer temporal scan (TERMITES) and INSIGHT [44–46]. These techniques are powerful in characterizing complicated spatiotemporal couplings of femtosecond laser pulses in large-scale terawatt or petawatt laser facilities. However such laser facilities are typically operated at an extremely low repetition rate [47,48], requiring improved measurement efficiency.
We take the technique INSIGHT as an example to demonstrate how sub-Nyquist sampling Fourier-transform spectral imaging improves the measurement efficiency [45]. In Fig. 5(a), the incident femtosecond laser pulse (800 nm wavelength, 35 fs pulse duration) is divided into two copies, and the optical field at the object plane (OP) is imaged by an imaging lens L and angularly dispersed by a prism (angular dispersion 14.3 nm/mrad @ 800 nm, PS850 from Thorlabs). Different from Fig. 1(a), the imaged laser beam is divided by a beam splitter BS4, and the two split beams are captured by two CCD cameras, including CCD1 at the focal plane of L for the far-field intensity profile $I({{k_x},{k_y},\omega } )$ and CCD2 at the imaging plane conjugate to the OP for the near-field intensity profile $I({x,y,\omega } )$. To measure the spectral phase of the incident laser pulse, another beam splitter BS3 together with a hard aperture samples only the central part ($x = $ 0, $y = $ 0) of the incident laser beam, and couples it into a standard frequency-resolved optical gating (FROG) device [49]. The inset plots of Fig. 5(a) show the FROG trace of the pulse (left panel) and the reconstructed spectral phase (red line, right panel, subtracting chirp induced by beam splitters BS1 and BS2), as well as the spectral intensity profile (blue line) which is consistent with independently measured spectrum (black line).
Fig. 5. Sub-Nyquist sampling Fourier-transform spectral imaging for efficient spatiotemporal characterizations of femtosecond laser pulses. (a) Experimental setup modified from Fig. 1(a). The angularly dispersed beam is split into two copies and imaged with two CCD cameras for both far- and near-field intensity profiles. A standard FROG device measures the spectral phase of the incident light at its spatially central part, obtaining the original FROG trace in the left inset panel. The right inset panel shows the reconstructed spectral phase (red line, the extra chirp induced by beam splitters BS1 and BS2 is subtracted), and the reconstructed spectral intensity profile (blue line) which is consistent with independent fiber spectrometer measurement (black line). (b) Frequency- or wavelength-resolved intensity profile reconstructions $\; I({x,y,\omega } )$ and $I({{k_x},{k_y},\omega } )$ at the near- (top row) and far-field (bottom row) respectively, for wavelength channels at 780 nm (left), 800 nm (middle), 820 nm (right). The time delay step size is 3.3 fs. (c) The near-field spatial phase profiles or wavefronts for 780, 800, and 820 nm. (d) The 3D spatiotemporal optical field profile. The carrier wave frequency is reduced by 1.5 times and regions for absolute amplitude 0.4 times higher than the peak one are shown, to avoid too fast oscillations of the field and get a clear visualization.
The sub-Nyquist sampling Fourier-transform spectral imaging measurements of $I({{k_x},{k_y},\omega } )$ and $I({x,y,\omega } )$ are conducted by scanning the time delay from −127 to 364 fs at a time delay step size of 3.3 fs which is 2.5 times of the Nyquist sampling rate limit. The top and bottom rows in Fig. 5(b) show representative intensity profiles at near- and far-fields respectively, for three different wavelengths (780, 800, 820 nm) or frequency components of the incident light. For each frequency component, we apply a standard 2D phase retrieval procedure for the frequency-resolved spatial phase profile or wavefront information using the Gerchberg-Saxton algorithm [50]. The reconstructed spatial phase profiles [Fig. 5(c)] linked by the spectral phase measured by FROG form a 3D phase map of the incident laser pulse at the near-field, supplementing 3D amplitude information in Fig. 5(b) (top row). Complete phase and amplitude information yield a 3D map of the femtosecond laser optical field shown in Fig. 5(d), demonstrating the same capability of femtosecond laser pulse characterization as the conventional INSIGHT technique [45]. However, thanks to the angularly dispersive optics assisted sub-Nyquist sampling, the time delay sampling rate is reduced and the measurement efficiency is improved.
So far our discussion of sub-Nyquist sampling Fourier-transform spectral imaging is based on the assumption that the incident light has a narrow bandwidth significantly smaller than the central frequency ($\mathrm{\Omega } \ll {\omega _0}$). It is interesting to investigate the possibility of sub-Nyquist sampling Fourier-transform spectral imaging of a complicated spectral-spatial intensity profile $I_y^{({\textrm{target}} )}({x,\omega } )$ over a broad spectral range. We have simulated the reconstructions of the 2D original object $I_y^{({\textrm{target}} )}({x,\omega } )$ ($y$-dependent information is irrelevant) with a spectral range from 500 to 1300 nm [ Fig. 6(a)] from the calculated ${S_y}({z,\tau } )$ at different time delays with $a = $4.16 mm·fs. Figure 6(b) shows the reconstructed spectral-spatial intensity profile ${I_y}({x,\omega } )$ with a fixed time delay step size $\Delta \tau = $ 6.67 fs. Severe artifacts are observed especially for frequency components at the wavelengths of 571, 667, 800, and 1000 nm, corresponding to $\omega \Delta \tau $ of $7\pi $, $6\pi $, $5\pi $, $4\pi $ respectively. This effect can be explained that for the specific frequency component which is not well reconstructed, the time delay scanning step size is an integer time of $\pi /\omega $ that Eq. (3) has non-unique solutions.
Fig. 6. Simulation results of reconstructions with heterogeneous time delay step sizes for broadband incident light. (a) The spectral-spatial intensity profile $I_y^{({\textrm{target}} )}({x,\omega } )$ is defined as the original object to be reconstructed in simulations. Its broad spectral range is from 1.45 fs−1 (1300 nm wavelength) to 3.77 fs−1 (500 nm). (b) The reconstructed spectral-spatial intensity profile ${I_y}({x,\omega } )$ with a fixed time delay step size $\Delta \tau = $ 6.67 fs. (c) The reconstructed spectral-spatial intensity profile ${I_y}({x,\omega } )$with a linearly increased time-delay step sizes from 6.67 to 13.33 fs within 170 steps, changing the time delay from −700 to 992 fs. (d) The $\tilde{I}_y^{({\textrm{target}} )}({{k_x},t} )$ of the object in the reciprocal $({{k_x},t} )$ domain.
To fix this problem for broadband incident light, we propose a time delay sampling strategy with heterogeneous time-delay step sizes. The time delay is scanned from −700 to 992 fs within 170 steps, and the step size changes linearly from 6.67 to 13.33 fs. The reconstructed spectral-spatial intensity profile ${I_y}({x,\omega } )$ from calculated ${S_y}({z,\tau } )$ is shown in Fig. 6(c), without observable artifact compared to the original object. We have also tried other time delay sampling strategies with heterogeneous sub-Nyquist time-delay step sizes, and reconstructions in all these cases are satisfying. Moreover, the maximum time delay step size limit for the conventional undersampling technique is $\pi /\mathrm{\Omega } = $ 1.35 fs, also smaller than any of our heterogeneous step sizes by times [Fig. 6(d)]. Thus heterogeneous time-delay step size sampling may allow sub-Nyquist sampling Fourier-transform spectral imaging over a broad spectral range comparable to the so-called central frequency, finding applications for broadband few-cycle laser pulses and attosecond optical pulses.
4. Conclusion and outlook
In summary, we have revisited the technique of Fourier-transform spectral imaging with a new perspective of a generalized central slice theorem. In this perspective, the measurements of the central frequency (if it can be defined for narrow-band light) and the smooth spectral envelope are decoupled by the angularly dispersive optics. The central frequency information can be simply determined by the direction of light diffracted from the angular dispersive optics, whereas the spectral-spatial envelope information is reconstructed using the ART algorithm based on the generalized central slice theorem.
A direct consequence of the new generalized central slice theorem perspective on Fourier-transform spectral imaging is a sub-Nyquist sampling of the time delay without losses of spectral and spatial resolutions. One can understand the sub-Nyquist sampling capability by comparing conventional imaging spectrometers and Fourier-transform spectrometers. The former ones measure spectral information based on the angularly chromatic dispersion and the latter ones rely on the spectral interference between two time-delayed incident light copies. We take advantage of both techniques. On the one hand, we adhere to the Fourier-transform spectrometer design and keep its superior spectral resolution, but only measure the “narrow-band” spectral-spatial envelope without any concern of aliasing or the Nyquist time-delay sampling rate in order to speed up the measurement. On the other hand, the angularly dispersive optics in an imaging spectrometer is applied to determine the central frequency while the entrance slit which limits the spectral resolution is discarded, so the high spectral resolution is retained. Moreover, simulations have shown that sub-Nyquist sampling may be feasible for broadband incident light if the time delay step size is heterogeneously set.
Compared to previous undersampling techniques [37–39], the new perspective on Fourier transform spectral imaging takes advantage of the inserted angular dispersive optics directly which determines the carrier frequency of the incident light, so it is no longer necessary to roughly measure the incident light bandwidth in advance which is critical to conventional undersampling strategies. This benefit is convenient for short-wavelength range such as extreme ultraviolet (XUV) high harmonic generation, whose bandwidth is broad and difficult to estimate. In addition, though the absolute time delay should be measured with nano-scale accuracy, the actuator can move with a step size larger than the wavelength, relaxing the precision requirement on time delay control devices [51]. Thus the sub-Nyquist time delay sampling with heterogeneous step sizes may enable high-efficiency hyperspectral measurement of high harmonic generation or attosecond light pulses [52], contributing to efficient and robust experimental studies within the attosecond-time and the nanometer-spatial scales [53–56].
Appendix
We propose an alternative perspective to solve Eq. (1) by applying Fourier transformation on ${S_y}({z,\tau } )$ from the $z$-domain to its reciprocal $k$-domain.
Then when the time delay is $\tau - \Delta \tau $ and $\tau + \Delta \tau$, the same calculation procedure yields
Assume ${\tilde{I}_y}({k,ak \pm \tau } )$ has negligible changes when the time delay $\tau $ varies by $\Delta \tau < {\mathrm{\Omega }^{ - 1}}$ where $\mathrm{\ \Omega }$ is the spectral bandwidth, so we get:
Funding
National Natural Science Foundation of China (11875140, 12275094); Science and Technology on Plasma Physics Laboratory (6142A04200212); National Key Research and Development Program of China (2022YFB4601300); Innovation Project of Optics Valley Laboratory (OVL2021ZD001); Fundamental Research Funds for the Central Universities (2019kfyXJJS013, 2021GCRC006).
Acknowledgments
We acknowledge financial support from the National Natural Science Foundation of China, Science and Technology on Plasma Physics Laboratory, National Key R&D Program of China, Innovation Project of Optics Valley Laboratory, and Innovation Fund of WNLO. This work of CT is partly supported by the Fundamental Research Funds for the Central Universities (HUST) and National Natural Science Foundation of China.
Disclosures
The authors declare no conflicts of interest related to this article.
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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