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Enhancement of the resolution of well-distributed multi-channel spectrometers beyond the pixel Nyquist limits while maintaining their inherent advantages such as high-speed measurement, compactness, and robustness will represent a critical step toward real-time monitoring of dynamic events. Here, we report the first super spectral resolution measurement beyond pixel Nyquist limits which was made possible by employing the Moiré effect in a commercially available Czerny-Turner type spectroscope of 4.63nm-spectral resolution. The experimental results show that the spectral resolution can be enhanced up to 0.31 nm by a factor of more than 10 with exceeding the 50-μm pixel Nyquist limits.
© 2016 Optical Society of America
1. Introduction
Spectroscopy presently encompasses our scientific and industrial scenes as an indispensable optical measurement tool [1–3], and there is an increasing need for much finer spectral treatment in various fields [4–7]. The spectroscopic technique can be roughly classified into interferometric and dispersive methods [8]. They have both pros and cons in terms of measurement speed, spectral resolution, observable spectral range, compactness, and maintainability. Among the dispersive methods, the use of a multi-channel spectrometer has been widespread because of its important advantages such as high-speed measurement, compactness, and robustness without the need for any mechanical scanning architecture. Although multi-channel spectrometer has been widely used because of these advantages, its spectral resolution is inferior to that of other methods. The spectral resolution of multi-channel spectrometers may be improved by using a larger dispersion value, smaller pixel size of each photodetector, or narrower entrance slit width. The dispersion value is basically limited by both the grating groove density and the diffraction order. Echelle grating spectrometers with increased high order dispersion would indisputably be a good solution for specialized high-end purposes but they can be very sizable and expensive when compared to conventional well-distributed spectrometers [9,10]. The pixel size of each photodetector has a certain size limitation (several μm to a few tens of μm) in terms of the sensitivity, fabrication process rule, and so on [11], whereas a line spread function (LSF) of an optical system can be much smaller than a pixel size. Therefore, although narrowing the entrance slit width is the only option to upgrade the spectral resolution regardless of increase in size and cost, the entrance slit width cannot be made smaller than the pixel size, and this “pixel Nyquist limits issue” potentially restricts the spectral resolution of current conventional multi-channel spectrometers [12].
As is well known, the Moiré effect can magnify very small mutual displacements of two similar patterns, and it has therefore been widely used in the field of metrology [13]. In fact, the super resolution method using the Moiré effect has recently been successful in exceeding the diffraction limit, which restricts spatial resolution in microscopy [14–16]. The first introduction of the Moiré effect for enhancement of spectral resolution in a multi-channel spectrometer is the externally dispersed interferometer (EDI), which is a hybrid spectrograph of purely dispersive and interferometric systems [12,17,18]. The EDI can prevent LSF influence through heterodyne detection by using interferometric sinusoidal fringes, and it therefore enables much simpler measurement compared to Echelle grating spectrometers [18]. Nevertheless, it cannot resolve a spectral difference smaller than a pixel size because it uses an interferometric sinusoidal fringe that can work only after detection. Several attractive image processing methods have been proposed for overcoming the pixel-limited resolution of digital imagers, but they require additional assistance of relatively complex post calculations and additional operations [19,20].
In this work, we propose Moiré effect-based super spectral resolution of multi-channel spectrometer beyond the pixel Nyquist limits using multiplication-type Moiré pattern [21]. To the best of our knowledge, this is the first report on spectral resolution enhancement beyond the pixel Nyquist limits in a multi-channel spectrometer. Further, the experimental results successfully verify the effectiveness of spectral resolution enhancement beyond the pixel Nyquist limits, so that the spectral resolution can be enhanced up to ∼0.31 nm by a factor of more than 10 compared to the original resolution (4.63 nm).
2. Technical background and proposed method
2.1. Previous works and pixel Nyquist limits
The EDI is representative of previous efforts to improve the spectral resolution of a multi-channel spectrometer, and it aims at overcoming the degradation by LSF in an optical system, as shown in Fig. 1.
Fig. 1 Pixel Nyquist limits issue in an externally dispersed interferometer (EDI): Fine details beyond LSF of an optical system in the Fourier transform of an EDI-detected signal can be shifted by an interferometer delay of τ but an EDI apparatus cannot relieve the blur caused by pixel Nyquist limits (determined by the Nyquist frequency).
Interferometric sinusoidal fringes in the EDI enable resolution of fine spectral details in a spatial distribution along the dispersion axis by heterodyning with spectral features to provide a Moiré pattern. However, since the interferometric sinusoidal fringe itself can appear only after detection by a photodetector array, the EDI-detected signal Sedi (x) should be pixelized before heterodyning.
The actual complex disturbance Uedi (x) just before a photodetector array in the EDI apparatus is unpixelized and given by,
where So (x) and τ are the spatial versions of the input spectrum and the relative delay time between two arms of an interferometer in the EDI apparatus, respectively. Since Uedi (x) is first pixelized by each photodetector window before detection at a photodetector array, Uedi (x) is rewritten in the pixelized version , which is given by, where Wpixel is the pixel width. Here, note that we simplified pixelization process to focus on pixel Nyquist limits caused by the sampling theorem. Assuming the LSF(x) distribution is much smaller than the pixel size, we can approximate it to δ(x), and the EDI-detected signal Sedi (x) is rewritten to the pixelized version , which is given by, is expressed in Fourier space as, where S̃edi and fx are the Fourier transform of Sedi and the spatial frequency along the dispersion axis, respectively. Consequently, the Nyquist sampling theorem requires the following condition for the bandwidth BWedi in the EDI,This result suggests that the bandwidth BWedi in the EDI is still limited by pixel Nyquist limits (determined by the Nyquist frequency) even if we use an ideal optical system and a high spatial frequency interferometric sinusoidal fringe.
2.2. Proposed method for the Moiré effect-based super spectral resolution beyond pixel Nyquist limits
2.2.1. Moiré effect for exceeding pixel Nyquist limits
We propose a Moiré effect-based super spectral resolution method beyond pixel Nyquist limits for a multi-channel spectrometer. Figures 2 and 3 show the typical schematic configuration and the conceptual procedure of the proposed method.
Fig. 2 Schematic configuration of the proposed method for the Moiré effect-based super spectral resolution beyond pixel Nyquist limits: Two repetitive structures RS1 and RS2 are set at the entrance and exit planes in a multi-channel spectrometer. An image of RS1 is superposed on RS2 for the Moiré effect generation under an imaging conditions between the entrance and exit planes.
Fig. 3 Procedure of the proposed method for the Moiré effect-based super spectral resolution beyond pixel Nyquist limits: A measured input light Ein is multiplied by RS1. After spectral decomposition, colored images of RS1 are imaged on RS2, and they form colored Moiré fringes.
Here, we set two repetitive structures at the entrance and exit planes in a multi-channel spectrometer, as shown in Fig. 2. Note that the entrance and exit planes are set to an imaging relation. In general, the Moiré effect can be observed when we superpose repetitive structures such as sinusoidal ones with slightly different periods over each other. A measured input beam Ein is guided to an entrance repetitive structure RS1(x) with the period of p1(Ⓐ) and multiplied by RS1(Ⓑ). Here, we simply consider a slit array with a period of p1 as a typical example of RS1. Assuming Ein(x) is a normalized spatially uniform plane wave, Ein(x) after multiplication by RS1 is corresponding to RS1, which are periodically aligned beam lines with the period p1 (Ⓒ). After spectral decomposition at a dispersive grating, if the measured light Ein is polychromatic, we could have colored multiple images of RS1, which is spatially shifted depending on each color (Ⓓ). Here, focusing on a blue spectral component, we could have a single RS1 image, which is equivalent to periodically aligned blue lines with p1 on the observational plane. A blue RS1 image is superposed by an exit slit array RS2 with a different period p2 (Ⓔ) and forms a blue Moiré fringe on the left side (Ⓕ). This is the magnified version of the original blue spectral line. Unlike a general magnifying image formation optical system, a formed Moiré fringe has a latent potential for super spectral resolution because it can magnify fine spatial details smaller than the spatial resolution, as described below. Returning to the case of polychromatic spectral components, each colored RS1 image is likewise superposed by an exit slit array RS2 and forms each Moiré fringe for super spectral resolution at different positions (Ⓔ and Ⓕ).
To help understanding the details of the proposed concept, we consider sinusoidal structures G1 and G2 with periods of p1 and p2 as a typical example of a repetitive structure. The intensity transmittance of the sinusoidal structures along with the x axis, G1(x) and G2(x) are given by,
where fp1 and fp2 are the spatial frequency of G1(x) and G2(x), and they correspond to 1/p1 and 1/p2, respectively. When a normalized monochromatic plane wave is inputted into a multi-channel spectrometer, G1(x) and G2(x) are mutually multiplied at the exit plane just before detection by a photodetector array, as shown in Fig. 4.Fig. 4 Moiré effect generation between sinusoidal repetitive structures: A multiplication-type Moiré pattern is generated from multiplication of G1(x) and G2(x). A generated Moiré pattern has a magnified form of G1(x).
The product of G1 and G2 is described by,
where Δfp12 is the difference between fp1 and fp2, and we used the condition of p1≃p2 for approximation. The fifth term in Eq. (8) essentially works as the so-called multiplication-type Moiré fringe, and it has a f1/Δfp12 times magnified form of G1. When the input wavelength changes, a corresponding spatial displacement of the imaged G1(x) occurs on G2(x). By introducing a displacement ΔG1 of G1(x), the rewritten fifth term is given by,Here, we express G1(x) in the notation of the convolution between a comb function combp1(x) and an elemental wavelet WLcos(x) for one cycle to express the structural detail of G1(x), as shown in Fig. 5. This result suggests that the Moiré effect can periodically magnify both the structure and the displacement of WLcos(x) in fine spectral detail by fp1/Δfp12 times. In exchange for the magnification capability, the observable spectral range is narrowed to one cycle of WLcos(x) because of the periodicity of the magnified WLcos(x).
Fig. 5 Sinusoidal repetitive structure G1(x) in the notation of the convolution between a comb function combp1(x) and an elemental wavelet WLcos(x) for one cycle.
When we assume that p1 and p2 are sufficiently smaller than Wpixel, the product of G1 and G2 is also pixelized after detection. Consequently, high frequency terms in Eq. (8) are blurred, and the detected signal is rewritten as,
Interestingly, Eq. (10) suggests that the magnified G1(x) (the second term in Eq. (10)) can individually survive regardless of the spatial resolution of an imaging optical system in front of a linear image sensor as well as the pixel Nyquist limits because of the inherent feature of the multiplication-type Moiré pattern [21]. As shown in Fig. 6, the frequency of the original G1 (a blue line) can be down-scaled by Δfp12/fp1 to that of the magnified G1 (a red line) so that Δfp12 of the magnified G1 is sufficiently lower than the Nyquist frequency. The magnified displacement ΔG1 can be likewise observed even if ΔG1 is much smaller than a pixel size.
Fig. 6 Fourier space description of the Moiré effect-based super spectral resolution using sinusoidal repetitive structures: The Fourier transform of the original G1 (a blue line) can be down-scaled by Δfp12/fp1 to that of the magnified G1 (a red line) so that the Fourier transform of the magnified G1 is sufficiently lower than the pixel Nyquist limits (a pale blue dotted line; Nyquist frequency).
In fact, expressed in Fourier space as,
where G̃1 and fx are the Fourier transform of G1 and the spatial frequency along the dispersion axis, respectively. Here, we omitted the displacement term because it is transformed to a phase term after Fourier transform. From Eq. (11), G̃1 is down-scaled by Δfp12/fp1 times in Fourier space although the upper limit of the allowable frequency remains unchanged from 1/2Wpixel of the pixel Nyquist limits. Consequently, the upper limit of the effective frequency of the original G1 is up-scaled to fp1/Δfp12 times as high as 1/2Wpixel, as described by Eq. (12). It can be said from the above results that it is possible to resolve a fine displacement of a spectral line with a slight spectral change beyond pixel Nyquist limits using the Moiré effect in a multi-channel spectrometer.2.2.2. Sinusoidal repetitive structure limitation
Down-scaling using sinusoidal structures can successfully relieve the spatial frequency component of fp1 from the pixel Nyquist limits but the spectral resolution is still limited by the spread of the magnified WLcos(x), as shown in Fig 7.
Fig. 7 Spectral resolution limitation due to sinusoidal repetitive structures: The resolvable size on a linear image sensor array is limited to 1/2Δfp12.
The well-known Rayleigh criterion determines the minimum resolvable size Δres (=1/2Δfp12), which is the distance between the first minimum and the maximum of the magnified WLcos(x). As a result, the number of resolvable points is limited to 2 within the observable spectral range 1/Δfp12. This result causes a different bandwidth limitation due to the sinusoidal structure.
The above sinusoidal structure limitation would be solved by replacing a sinusoidal structure G1(x) with a comb structure combp1(x) with a δ-like WLδ (x) because Δres is limited by the spread of the magnified WLcos(x). When a normalized polychromatic plane wave with a spectrum WLsp (λ) is inputted into a multi-channel spectrometer, each δ-like WLδ (x) in a combp1(x) forms a spectral structure WLsp (x) on G2(x). Consequently, a repetitive spectral structure of WLsp (x) is imaged on G2(x) at the exit plane, as shown in Fig. 8. Here, we assume that p1 is sufficiently larger than the spread of the spectrum.
Fig. 8 Repetitive structure of a spectral structure WLsp (x) imaged on G2(x) in the notation of the convolution between a comb function combp1(x) and an elemental wavelet WLsp (x) for one cycle.
As shown in Fig. 9, WLsp (x) has a certain bandwidth BWWL and its Fourier transform W̃Lsp (fx) occupies an area painted with the pale blue in the Fourier space, and it is sampled by c̃ombp1(fx) because of its repetitive structure with the period of p1.
Fig. 9 Fourier space description of the Moiré effect-based super spectral resolution using a repetitive structure with a more general spectral structure WLsp (x) and a sinusoidal one: W̃Lsp (fx) (pale blue area) cannot be completely down-scaled by Δfp12/fp1 and aliasing occurs.
In the Fourier space, the product of WLsp (x) and G2(x) corresponds to the convolution between W̃Lsp (fx) (blue lines) and G̃2(fx) (purple triple lines). The purple line component at fp2 shifts the first, second, and third blue line components at fp1, 2 fp1, and 3 fp1 to components at Δfp12, fp1, and 2 fp1, respectively. W̃Lsp (fx) after convolution by the purple line component at fp2, still occupies a broad area shown in pale red. Consequently, W̃Lsp (fx) is not down-scaled by Δfp12/fp1 times and cannot completely exceed the pixel Nyquist limits. Furthermore, W̃Lsp (fx) convolved by the DC component of G̃2(fx) (the purple line at zero frequency) completely overlaps with the original W̃Lsp (fx), and aliasing occurs. The cause of these problems is the absence of the Moiré effect in a high frequency region because of the lack of frequency components in G̃2(fx), which is to be paired with those in combp1(x).
2.2.3. Ideal repetitive structure for super spectral resolution
The above problems would be simply solved by replacing a sinusoidal structure G2(x) with a comb structure combp2(x) so as to provide frequency components that can be paired with those in combp1(x). As shown in Fig. 10, c̃ombp2(fx) (purple lines) provides frequency components neighboring to those of W̃Lsp (fx) (blue lines).
Fig. 10 Fourier space description of the Moiré effect-based super spectral resolution using a repetitive structure with a more general spectral structure WLsp (x) and a comb structure combp2(x): W̃Lsp (fx) (pale blue area) can be completely down-scaled by Δ fp12/fp1 and aliasing does not occur.
As a result of the convolution between W̃Lsp (fx) (blue lines) and c̃ombp2(fx) (purple lines), the purple line components at fp2, 2 fp2, and 3 fp2 down-scales the first, second, and third blue line components at fp1, 2 fp1, and 3 fp1 to components at Δfp12, 2Δfp12, and 3Δfp12, respectively. Consequently, the W̃Lsp (fx) after convolution by the purple line components occupies an area shown in pale red, and it is successfully down-scaled by Δfp12/fp1 times. The above sinusoidal structure limitation with aliasing has been solved and the allowable bandwidth is given by,.
Figure 11 shows a schematic of the Moiré effect generation between a repetitive function of the measured spectrum So (x) with the period p1 and combp2(x). From Fig. 11, each δ function in a combp2(x) samples each different point of each spectrum function and forms a magnified spectrum function as a Moiré fringe. This suggests that the Moiré effect can be interpreted as the spatial version of an equivalent sampling method. In fact, equivalent sampling enables similar reconstruction of broadband signals from slower sampling rates in the time domain. By referring to equivalent sampling, the achievable spectral resolution and observable spectral range are equivalent to the shift amount of a sampling point Δp12(=p2 − p1) and the period of combp1(x). It can be said that a pair of ideal repetitive structures is a pair of square wave shape structures with duty ratios D1 and D2 of Δp12/p1 and Δp12/p2, respectively. The number of resolvable points in an observable spectral range and a spectral resolution are 1/D1. For example, since D1 of a sinusoidal repetitive structure can be regarded as 0.5, the number of resolvable points is estimated to 2 so as to be consistent with the preceding result.
Fig. 11 Moiré effect generation between a repetitive function of the measured spectrum So (x) with the period p1 and a comb structure combp2(x): A generated Moiré pattern is discretely sampled with intervals p2, and it can be interpreted as the spatial version of an equivalent sampling.
3. Experimental verification and discussion
We experimentally verify our proposed Moiré effect-based super spectral resolution using a commercially available Czerny–Turner type spectroscope. The spectroscope is originally composed of 300-μm entrance and exit slits, a 74-mm focal length concave mirror, and a 830-lpmm grating, and the catalog value of its spectral resolution is 4.63 nm. An entrance plane is imaged on an exit plane through a 1.5-times magnification imaging system in the spectroscope.
We replace the initial entrance and exit slits with a pair of slit arrays for verification. Figures 12 and 13 show the experimental setup for verification. The spectral distribution on the exit plane is observed using an InGaAs linear image sensor (Hamamatsu Photonics K.K., G9203-256S) with a 50-μm pixel size. The measured light is guided to the entrance plane as a 2.5-mm diameter Gaussian beam through an optical fiber and a collimator.
Fig. 12 Experimental setup: Two slit arrays SLA1 and SLA2 are set at the entrance and exit planes in a commercially available Czerny–Turner type spectroscope. The entrance plane is imaged on the exit plane through a 1.5-times magnification imaging system.
3.1. Spectral line shift measurement beyond pixel Nyquist limits
First, we verify the ability of spectral line shift measurement beyond pixel Nyquist limits. To examine beforehand the correspondence relation between the wavelength difference and the pixel size of an InGaAs linear image sensor, we set only the entrance slit with a 300-μm width in the spectroscope configuration. We observed that the spectral distribution shifts on the exit plane with tuning the center wavelength of the input measured light radiated from a communication-band wavelength tunable laser (Yokogawa, AQ2201). The center wavelength was tuned from 1551.00 nm to 1553.00 nm with steps of 0.10 nm. Figure 14 shows a series of experimentally observed distributions.
Fig. 14 Experimental results of spectral line shift measurement using a normal spectrometer configuration with a 300-μm width entrance single slit: The observed LSF of the 300-μm width entrance single slit is magnified by about 1.5 times (∼450μm width) on the linear image sensor. The center of the LSF moves 4 pixels between two white dotted lines for a 2.0-nm wavelength change.
The observed spectral distribution corresponds to the LSF of the entrance slit, and the central maximums at 1551.00 nm and 1553.00 nm are indicated by the two white dotted lines in Fig. 14. From Fig. 14, the position of the center of the LSF moves 4 pixels between the two white dotted lines for a 2.0-nm wavelength change, and the pixel size can be converted into a 0.50-nm wavelength difference according to the dispersive conversion rate 0.5 nm/50 μm. To confirm the ability of the spectral line shift measurement beyond the pixel Nyquist limits, we only have to demonstrate sensing of a fine wavelength change of less than 0.50 nm by using our proposed method.
To generate the Moiré effect, we set custom slit arrays SLA1 and SLA2 with periods of 100 μm and 180 μm at the entrance and exit planes, respectively. Both slit widths are set to be 10 μm less than the 50-μm pixel size. In addition, the duty ratios of SLA1 and SLA2 are also set to be less than 1/10. Since SLA1 is magnified by about 1.5 times on SLA2 at the exit plane, the effective period and slit width of SLA1 are expected to be 150 μm and 15 μm, respectively. In fact, the effective periods SLA1 and SLA2 on a linear image sensor are experimentally evaluated as 156 μm and 187 μm, respectively. Since an actual Δp12 is one-fifth of p1, the Moiré fringe is expected to resolve a spectral line shift as small as 31 μm in this configuration. The 31μm spectral line shift corresponds to 0.31 nm wavelength change, which is less than the pixel Nyquist limits of 0.50 nm. In addition, this shows that the spectral resolution can be enhanced by a factor of more than 10 (∼0.31 nm) compared to the original one (4.63 nm). The observable spectral range is estimated to be about 1.50 nm (=156μm×0.50nm/50μm) according to the dispersive conversion rate 0.50 nm/50 μm.
Figure 15 shows a series of experimentally observed Moiré fringe distributions with tuning of the center wavelength from 1549.20 nm to 1553.00 nm with steps of 0.05 nm. The periodicity of the Moiré fringe with a 1.50-nm wavelength interval was observed, and the intensity of both end parts is relatively weak due to the Gaussian input beam shape. When focusing on the region between 1549.50 nm and 1551.00 nm (inside a rectangular frame of white dotted lines for an interval), a strong peak of the Moiré pattern moves as the center wavelength shifts and five peaks can be observed within a 1.50-nm spectral range. This is consistent with the above expected spectral resolution (∼0.31 nm) because a 0.30-nm spectral resolution was experimentally obtained. From Fig 15, the ability of spectral line shift measurement beyond pixel Nyquist limits can be confirmed because the Moiré fringes apparently change in response to wavelength changes much smaller than 0.50 nm.
Fig. 15 Experimental results of a series of observed Moiré fringes with tuning of the center wavelength from 1549.20 nm to 1553.00 nm with steps of 0.05 nm. The actual periods of the entrance and exit slit arrays on the linear image sensor are evaluated as 156 μm and 187 μm, respectively.
3.2. Polychromatic light spectrum measurement beyond pixel Nyquist limits
Next, we verify the ability of polychromatic light spectrum measurement beyond pixel Nyquist limits. We used two spectral lines from two CW lasers as a polychrimatic light spectrum for verification. One center wavelength is fixed at 1550.00 nm and the other one is varied from 1549.20 nm to 1553.00 nm with steps of 0.05 nm to assess the minimum resolvable difference between two spectral lines. Figure 16 shows a series of experimentally observed Moiré fringe distributions with tuning of the center wavelength from 1549.20 nm to 1553.00 nm with steps of 0.05 nm.
Fig. 16 Experimental results of a series of observed Moiré fringes formed by two spectral lines from two CW lasers as a polychrimatic light spectrum. One center wavelength is fixed at 1550.00 nm and the other one is varied with 0.05 nm steps from 1549.20 nm to 1553.00 nm.
When focusing on the region between 1549.50 nm and 1551.00 nm (inside a rectangular frame of white dotted lines for an interval), two peaks of the Moiré fringe (at pixel positions 5 and 9) can be apparently resolved after one tunable center wavelength reaches 1550.30 nm. This is consistent with the above expected spectral resolution (∼0.31 nm) because a 0.30-nm spectral resolution was experimentally obtained for a polychromatic light spectrum measurement. From Fig 16, the ability of a polychromatic light spectrum measurement beyond pixel Nyquist limits can be successfully confirmed because the Moiré fringes apparently resolve two spectral lines with a wavelength difference of much less than 0.50 nm.
3.3. Discussion
Here, we discuss the achievable spectral resolution performance of the various repetitive structures to provide a design guideline of appropriate repetitive structures to overcome the pixel Nyquist limits. Figure 17 shows Δres as a function of p1 for different values of Wbeam.
Fig. 17 Achievable spectral resolution as a function of the entrance slit period p1 for different values of beam diameter Wbeam. Appropriate p1 and Wbeam are determined so as to satisfy the demand of spatial resolution Δres from the pixel size limit Wpixel.
When the beam diameter and the period of a repetitive structure at an entrance plane are Wbeam and p1, the number of sampled points Nsample is given by Wbeam/p1 according to equivalent sampling. As a result, the achievable spatial resolution Δres and the appropriate period p2 of a repetitive structure at an exit plane are given by and p1 + p1/Nsample, respectively. Appropriate p1 and Wbeam are determined so as to satisfy the demand of spatial resolution Δres from the pixel size limit Wpixel. The duty ratio D1 should be also minimized in order to improve the spectral resolution for the fixed p1 because Δres is represented by p1D1. Hereafter, we focus on Δres for comparison with pixel size because Δres can be promptly converted into the corresponding spectral resolution using the dispersive conversion rate of each spectroscope. The model results and the experimental values are given in Fig. 17. The pale blue dotted line shows the pixel Nyquist limits of 50 μm in this experiment. To overcome the pixel Nyquist limits using the p1 of 156 μm, we can reduce Wbeam to less than 1 mm. The red line shows the case where Wbeam of 2.5 mm. From Fig. 17, the achievable Δres for p1 of 156 μm is ideally estimated to about 10 μm using an appropriate repetitive structure with the period p2 of about 166 μm at the exit plane. Although we set the period p2 to about 187 μm, the actual Δres of 31 μm in this experiment was also able to overcome the pixel Nyquist limits.
4. Conclusion
In this work, we have shown that the Moiré effect can boost the effective resolving power of a conventional multi-channel spectrometer, and it can exceed the pixel Nyquist limits. The spectral resolution is dramatically enhanced by employing a downscaling function based on the Moiré effect. In the current work, we used a pair of slit arrays with slightly different periods, but the concept can be extended for the use of a pair of different repetitive structures provided they can form Moiré fringes and can be paired to each other with sufficient spatial frequency components to cover the measured fine spectral details. This method completely differs from previous techniques regarding super resolution beyond pixel Nyquist limits, but it should be still used in combination with the other techniques so as to be selectively used in accordance with measurement speed, spectral resolution, observable spectral range, compactness, and maintainability. For the future of this technology, we envision a variable spectral resolution multi-channel spectrometer to focus on the target with varying the slit array’s magnification power like a microscope’s revolving nosepiece. We also note the potential of this approach for astronomy especially, in which the use of our proposed method as the last assistance of the EDI for solar Doppler shift measurements can resolve a spectrograph by a factor of more than 10 in a limited observation spectral range. For example, more fine details less than a wavelength difference of 0.2 cm−1 between two peaks (e.g., at 5139.3 cm−1 and 5139.5 cm−1) could be also resolved because the ordinary solar spectrum has been able to resolve the corresponding single peak (blurred two peaks) with FWHM of 0.7 cm−1. The method has the potential to enhance the capability of the well-distributed multi-channel spectrometer for real-time monitoring of dynamic events in a wide range of fields from life science to astronomy.
Funding
This work was supported by the Osaka University LLP (Lean LaunchPad Program) Gap Fund.
Acknowledgments
This study was supported by the Japan Science and Technology Agency (JST), particularly in terms of patent cooperation treaty (PCT) application and JSPS KAKENHI Grant Number JP16H04365. The authors thank Dr. R. Nakamura for the technical support.
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