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Abstract
We propose a super-resolution technique for multichannel Fourier transform spectrometers, which is advantageous for feeble-light spectroscopy. The spectral resolution of an area sensor is limited by the number of lateral pixels. Our method fills the signals with vertical pixels, which is practically equivalent to increasing the number of lateral pixels. When applying our proposed technique, the resolution of an Ar-lamp spectrum, which is obtained by using an area sensor with 659 lateral pixels, becomes comparable to that of an area sensor with 1,626 lateral pixels. The spectral resolution is improved at least twice. Thus, using our method, a spectrometer with an area sensor can overcome the Nyquist frequency limitations.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The multichannel Fourier transform (McFT) spectrometer is one of the most promising equipment for feeble-light spectroscopy and the observation of rapid phenomena, as it produces high throughput and does not use a scanning mechanism [1–6].
However, the resolution of the McFT spectrometer is limited by the number of sampling points, which depends on the spatial resolution of the array sensor. Thus, in terms of spectral resolution, the McFT spectrometer is generally inferior to the Michelson interferometer or the dispersive spectrometer. The optical difference and diffraction angle can be adjusted finely by the moving mirror in the Michelson interferometer and the width of the grating in the dispersive spectrometer, respectively. In order to improve the widespread utilization of the McFT spectrometer as commercial and generalized equipment, its resolution must be improved [7,8]. McFT spectrometers of higher resolution need array sensors with higher resolutions, leading to a more expensive system.
We propose a simple and inexpensive method to improve the spectral resolution, which uses an area sensor instead of an array sensor. In self-correlating interferometers, such as Fourier transform interferometers, it is noteworthy that the wavelength resolution is directly decided by how large phase difference the detector can take, i.e. corresponding to the number of stripes obtained by the detector. Improving the wavelength resolution by x times needs x-times larger detector, as long as the sampling theorem is satisfied. However, the price of an area sensor is expensive, especially in EMCCD for feeble light and CCD of near infrared region.
In order to acquire the phase difference as large as possible, with exactly the same detector, there is a need to make the stripe pattern of the inferferogram narrower as possible, by using shorter focal-length lenses or thicker Savart plate. The proposed method is a super-resolution method that produces a spatial interferogram with finer accuracy than the original sensor pitch, without changing the optical setup or structure of the sensor. We applied this method to fine interferogram enough to cause aliasing, for the sake of acquisition of interferogram with large phase difference by an area sensor with small pixel numbers. To investigate the validity of our method, an interferogram at a frequency higher than the Nyquist frequency was formed with a triangular common path interferometer [1,9]. A signal, which was deformed because of aliasing, when using the conventional method, was detected as a proper signal by our method.
So far, the super-resolution technique has been developed with digital image processing. Many image reconstruction methods have been reported, which were used for imaging in medical [10,11], satellite [12], and video [13] systems. Obtaining higher-resolution images with low-resolution sensors has advantage on not only the cost but also on terms of the digital zoom. Several super-resolution techniques to reduce the practical pixel size of the area sensor have been reported [14–20]. In these techniques, several pictures were obtained using multiple cameras or by moving a single camera with a step finer than a pixel pitch. These pictures were then computationally combined to reconstruct a higher-resolution image. Fuji Film Co. sensor [21], whose individual pixels were octangular. Each sensor line was displaced from the adjacent line by half the length of the sensor pitch, as a substitute for mechanical sensor shifting.
Contrary to these elaborate methods, our method improves the spectral resolution by slightly tilting the area sensor, without moving nor rotating a camera [22]. There is no need to precise adjustment, which is user friendly. Such a simple scheme could be achieved because of the distinguishing feature of the McFT interferogram, whose signal is one dimensional stripe pattern. The proposed method in this paper improves resolution using a single image, without mechanical movement of detector, nor altering the hardware structure. Our method is handy and low cost, because it needs neither mechanical actuators, nor substrate manufacturing, nor complex calculations as in the case of conventional methods.
2. Methods
The resolution of the Fourier transform spectrometer is determined by the number of bright/dark fringes in the observed interferogram. To be accurate, the spectral resolution of the Fourier transform interferometer is determined by the maximum path difference between two split light beams. However, this section deals with the number of bright/dark fringes, which is easily understandable, in order to compare the spectral resolution with the pixel pitch of the area sensor. A finer stripe pattern of the interferogram requires a spatially higher-resolution sensor to observe the precise interferogram signal.
The signal pattern of the interferogram, generated by our experimental setup is presented in Fig. 1(a). When the pixel pitch of an area sensor is larger than half the length of the interval between two bright/dark fringes, the received signal on the area sensor becomes undersampled. This is why a proper interferogram would never be obtained in Fig. 1 (b). Figure 1 (c) and Fig. 1 (d) are schematic images of the obtained signal with the same sampling pitch as that in Fig. 1 (b), but the sampling position is slightly shifted in the lateral direction. Though each of Figs. 1 (b)-(d) contains aliasing, combining all of them offers a signal that meets the sampling theorem. That is, the various separately obtained low-resolution signals can be combined into a single super-resolution signal, as shown in Fig. 1 (e). The intrinsic interferogram pattern, which appears to be a distorted signal in the low-resolution array sensors, can be reconstructed by interpolating several dispersive signals whose positions are slightly offset with respect to each other, such as the ones in Figs. 1 (b)-(d).
Fig. 1 Schematic image of sampling theorem. (a) Intrinsic interferogram. Dashed lines are the center positions of lateral pixels, and their intervals are equal to the pixel pitch. (b), (c), and (d) Disperse sampling points of low-resolution area sensors. The sets of (b) gray points, (c) black points, and (d) white points are slightly shifted with respect to each other. All signals of (b), (c), and (d) are combined to reconstruct (e).
This study reconstructs the high-resolution interferogram and aims to overcome the Nyquist frequency limitation, by reading signals in the vertical direction of an inclined fixed-area sensor.
In order to acquire the resolution improvement comparable to that of the conventional mechanical sensor-shifting method, the area sensor is set slightly inclined to the stripe pattern of the interferogram (Fig. 2). On the tilted area sensor, the signals on the vertically adjacent lines are shifted sequentially in the lateral direction, as shown in Fig. 2 (b), which realizes the principle of Fig. 1. Moreover, the mechanical actuator to move the sensor can be avoided in this proposed method of Fig. 2.
Fig. 2 Concept of positional relationship between the interferogram of Fourier transform spectroscopy and each pixel of the area sensor. (a) Conventional positioning, where the vertical lines of the area sensor are parallel to the stripe of the interferogram. The fringe marked with circles represents an identical bright fringe of the interferogram. (b) Method proposed in this study, where the area sensor is tilted slightly toward the stripe of the interferogram. The fringe marked with stars represents an identical bright fringe of the interferogram. θ is the tilting angle of the area sensor.
An identical bright fringe is detected at the same lateral position in Fig. 2 (a), where the vertical lines of the area sensor are parallel to the stripe of the interferogram. On the other hand, an identical bright fringe is detected at a different lateral position in Fig. 2 (b), i.e., the detected positions of the upper and lower ends are shifted by a single pixel. Reading the signal in the vertical direction provides us the interferogram signals from laterally differentiated positions, as one-dimensional gray scale information. The amount of lateral shift between vertically adjacent pixels can be designed by tuning the tilting angle θ. This lateral displacement makes virtual subpixels, and the fineness of the subpixel can be precisely and freely chosen, as long as θ is larger than the minimum angle θ0 = tan−1 (1/n), where n is the total pixel number of the area sensor in the vertical direction..
In order to confirm whether the method proposed in this paper works properly, we demonstrated the improvement of spectral resolution by using a common path interferometer modified to the polarized interferometer. The experimental setup is shown in Fig. 3. The sample was chosen to be an argon (Ar) lamp (Ocean Optics, AR-1). The incident Ar lamp light was polarized by polarizer 1 (wire grid, WP-25L-UB) and passed through lens 1 (N-BK7, 350 nm–2.0 μm, LA1145, f = 75 mm, ϕ = 50 mm). Then, it was split into two orthogonal polarized lights by the cube-mounted polarizing beam splitter (CCMI-PBS252/M, 620-1000 nm, dimension: 30 × 30 × 30 mm). Each light was reflected by a mirror and passed through both the beam splitter and lens 2 (N-BK7, 350 nm–2.0 μm, LA1417, f = 150 mm, ϕ = 50 mm). Subsequently, the same polarization components were extracted from the two beams by polarizer 2 (wire grid, WP-25L-UB), and both of them interfered on the plane of the area sensor. The angles of polarizer 2 were set to be parallel or vertical to those of polarizer 1, which are hereafter known as in-phase and anti-phase, respectively [23]. Both in-phase and anti-phase data were obtained through the rotating polarizer 2, for obtaining a better signal to noise ratio. The interferogram data presented in this manuscript are obtained from the subtraction of the in-phase and anti-phase data, since the background noise due to the sensitivity unevenness of each pixel is nonnegligible, as long as an area sensor is utilized [24,25].
Although McFT with Savart plate is more common setup than the triangle one, we used the latter one to simplify the essence of this paper. Since the two dimensional stripe pattern is often distorted [26], McFT with Savart plate would be more complicated. However, our proposed method could be applicable to the setup with Savart plate, if the warping compensation method is combined.
In order to improve the spectral resolution of the tilting area sensor, a low-resolution CCD camera (Bastler acA640-120gm, 659 × 494 pixels, pixel pitch: 5.6 μm) was used for the demonstration. The interferogram obtained by the CCD camera was sent to a PC and Fourier transformed to produce a spectrum. The results of the Ar spectrum obtained with / without our method were compared. In addition, another Ar spectrum was obtained using a high-resolution camera (Bastler acA1600-20 um, 1626 × 1236 pixels, pixel pitch: 4.4 μm) without our method, to obtain the intrinsic Ar lamp spectrum. The spectrum with the high-resolution camera was used as a reference, when validating the spectral shape obtained with the low-resolution CCD camera and our method.
It is noteworthy that the interval of the stripe shapes of the interferogram can be varied by tuning a mirror position. In the experimental setup of the demonstration, the mirror position was set so that the interferogram signal was wrongly obtained in a low-resolution by aliasing and correctly obtained in the high-resolution CCD camera.
3. Results and discussion
The obtained two-dimensional interferogram images with our method of tilting the area sensor is shown in Fig. 4 (a). The interferograms of 184th line and 291st line are shown in Fig. 4 (b), whose laterally center parts are enlarged. None of these graphs meet the sampling theorem, and results in undersampled images. The tilting angle θ can be read from the obtained interferogram images. The amount of tilt was estimated as the number of vertical pixels, which is hereafter called “one cycle”, where the exactly same bright/dark fringe was obtained in the horizontally adjacent pixel. We hereafter define Ny as the number of total vertical lines composing one cycle. In the case where θ is larger than θ0, reading one vertical line from the top end to the bottom means reading for more than one cycle and obtaining the same information repeatedly. Here, the precise θ was derived so as to read only one cycle, avoiding redundant reading. In the experiment, there is no need to adjust the tilting angle precisely. Even if a user tilts the detector slightly without concern, the tilting angle can be estimated by searching one cycle. Once the spectrometric equipment is settled, the vertical reading pixels are determined semi permanently; this task of searching one cycle should be done just once for users.
Fig. 4 (a) Two-dimensional interferogram signal with our method tilting the area sensor slightly. (b) The center-part enlarged interferograms of 184th line (bold gray line) and 291st line (dashed line) from the top end. (c) Reconstructed interferogram by the proposed method; center part corresponding to Fig. 4 (b) was enlarged.
As shown in Fig. 4 (b), the interferogram signals of the 184th line and 291st line were keenly shifted by one pixel. That is, one cycle and tanθ were estimated to be 108 lines and 1/108, respectively. The one-dimensional interferogram pattern is reconstructed in Fig. 4(c). This reconstructed signal is obtained by reading the two-dimensional signal from the 184th to the 291st line, in the vertical direction. Here, the range of phase difference of Fig. 4 (c) is identical to that of Fig. 4 (b). Comparing the range of lateral axis of Fig. 4 (b) and Fig. 4 (c), the vertically read pixel number of Fig. 4 (c) is 108 ( = Ny) times. This is because the fringe pattern originally read by a single lateral pixel in Fig. 4 (b) is reconstructed by 108 ( = Ny) vertical pixels in Fig. 4 (c). This corresponds to sampling with 1/108-size subpixel, therefore Fig. 4 (c) is a much finer interferogram, compared to Fig. 4 (b), and it fulfills the sampling theorem.
Here we have to consider the possibility that the lateral position of interferogram pattern of 184th line and that of 291st line are slight different each other, that is, two stars positions are different slightly in Fig. 2. Supposing that the rate of horizontal shift of 291st line from 184th line is called s, where −1/Ny < s < 1/Ny, the reconstructed interferogram is extended (s > 0) or shrunk (s < 0) by the ratio of s, compared with the reconstructed interferogram of s = 0 (Fig. 5). This extension/shrink in reconstruction interferogram causes the wavenumber shift in the Fourier transformed spectrum by the ratio of (s + 1), compared with the Fourier transformed spectrum at s = 0. However, in the practical Fourier transform spectroscopy, the wavelength calibration is done after the experimental set up settlement, this shift of wave number will be corrected through the wavelength calibration. In addition, the small signals of discontinuous noise appear periodically, at the point of reading horizontally adjacent column; i.e. sequential reading from bottom star position to the top star position in Fig. 2(b). This periodical pulse noise may form periodical peaks in frequency domain. Regarding to the convolution theorem, since the interferogram is multiplied by the truncated periodical pulse in the real space, the spectrum is convolved by the periodical pulses whose intensities are decreasing. Therefore the spectrum can be observed periodically. However, the gaps between truncated periodical pulses are narrower than the interferogram, the periodically generated spectra never overlap each other.
Fig. 5 The schematic image of the case when two stars positions are different slightly in Fig. 2. The interferogram pattern becomes enlarged at s > 0.
The spectra of the Ar lamp derived from the Fourier transformation of the interferograms in Fig. 4. (b) and Fig. 4 (c), are shown in Fig. 6 (a) and Fig. 6 (b), respectively. Both were measured by a low-resolution area sensor, whose lateral pixel number was 659. The verification experiment was also performed with a high-resolution area sensor whose lateral pixel number was 1626, originally fulfilling the sampling theorem without our method. The resultant intrinsic Ar lamp is described in Fig. 6 (c), which is also appeared as dotted line in Fig. 6 (a) and Fig. 6 (b). It is clear that the spectral shape of Fig. 6 (a) is totally different from that of Fig. 6 (c), because light of wavelength smaller than ~800 nm accompanied by aliasing. The spectrum of Fig. 6 (b) succeeded in reconstructing a proper Ar lamp spectrum by using our method of tilting the area sensor and reading the signal in the vertical direction. Since the optical path difference was designed to be 254.4μm in our experimental setup in Fig. 6 (a) and Fig. 6 (b), the theoretical spectral resolution without our proposed method was 39.3 cm−1. Both of Fig. 6 (b) and Fig. 6 (c) were observed by McFT setup. Figure 6 (b) was obtained with low-resolution (horizontally 659 pixels) area sensor combined with our method. Figure 6 (c) was obtained with high-resolution (horizontally 1626 pixels) area sensor without our method, which was only a reference data for validation of our method. It was clearly shown that the spectrum of Fig. 6 (b) was homologous to Fig. 6 (c), which is clearly seen in over layered image of Fig. 6 (b). The peak-intensity ratio between Fig. 6 (b) and Fig. 6 (c) are not identical, because each individual CCD has unique wavelength sensitivity.
Fig. 6 (a) (Above bold line) and (b) (Above bold line) are Fourier transformed spectra, of Fig. 4 (b) and Fig. 4 (c), respectively. Both figures are over layered by dotted reference spectrum, which appears in Fig. 6 (c) (Above). (a) (Bottom) and (b) (Bottom) are entire interferograms of Fig. 4 (b) and Fig. 4 (c), respectively. (c) (Above) Accurate Ar lamp spectrum and (c) (Bottom) the acquired interferogram using high-resolution area sensor.
The theoretical optical path difference of Fig. 6 (c), obtained by high-resolution area sensor without our method, was 492.5 μm, about double than the setup with low resolution area sensor in Fig. 6 (a) and 6 (b). The theoretical spectral resolution of Fig. 6 (c) is 20.3 cm−1. High-resolution image acquisition was achieved with our super-resolution technique, which could not be obtained with a low-resolution area sensor. Note that the intensities of Fig. 6 (a)-(c) are not the essential issue to discuss, since the intensity difference only occurs in the Fourier space due to the number of sampling points.
At last we would like to refer to the amount of improvement in spectral resolution. There is a relationship between spectral resolution enhancements versus tilting angle θ. That is, (resolution improvement) = Ny = 1/tanθ, as long as Ny, with the ideal CCD, which is constituted of the pixels of infinite small points. On the other hand, if employed the ideal CCD which has 100% aperture ratio, the enhancement limit of our proposed method is double [27,28]. However, the practical aperture ratio is between 0% and 100%. It is noteworthy that the resolution enhancement is originated in interpolating the non-sensitive area between laterally adjacent pixels of the sensor. Supposing that the lateral length of practically sensitive area of a pixel is “a”, and that the lateral size of a pixel is “d” in Fig. 7, the lateral length of insensible field is (d - a). The fill factor is written as a2/ d2. The maximum of the resolution enhancement should be 2d/a, since vertical pixels interpolate the distance of (d - a) by sensitive area length of a. In order to interpolate the non-sensitive area, Ny should be larger than 2d/a. The exact a value of the CCD used in our experimental setup was not open to the public. The effective fill factor of a CCD is usually designed to be larger than the original one by utilizing lens arrays, therefore it is difficult to calculate precise fill factor. It is said that the general fill factor of a CCD is approximately 30%, i.e. a/ d = 0.55, the maximum of the resolution enhancement, 2d/a, becomes 3.7 times.
Fig. 7 The schematic image of the lateral size of a area sensor and the practically sensitive area of a pixel.
Although our original aim was only to improve spectral resolution with low cost, our method of reading CCD vertically also contributes to the improvement of the signal to noise ratio, in the matter of fact.
Summary
We proposed a super-resolution technique, which realized high spectral resolution for even low-resolution area sensors, by just tilting the sensor slightly. The image data of the undersampled interferogram signal was interpolated by tuning the tilting angle of the area sensor. The super-resolution image was obtained in a simple manner, as the signal of the interferogram was one-dimensional gray-scale information. We confirmed the efficiency of our technique through triangular common-path interferometric spectroscopy. Though aliasing appeared without our method, the spectral resolution improved more than 2.4 times and an intrinsic spectrum was obtained with our method. This method can be widely applied to interferometers for detecting interferograms with spatial distributions. It also enables us to use low-cost high-resolution highly dispersive telescopes for analyzing bolometric luminosities or temperatures of stars [29]. Spectrometers for characterizing properties of medicine and semiconductors, and auto collimators for light design would also be improved spectrally. It is noteworthy that our method is applicable not only to the Fourier-transform type, but also applicable to commercial grating imaging spectrographs. Since spectral distortion such as tilting was often observed on the detector of a grating imaging spectrograph [30], our super-resolution algorithm can be applied for the acquisition of higher-resolution images without distortion correction. This study especially contributes to spectrometers with quite expensive detectors, through improving the resolution while obtaining the interferogram images. It might be also useful for the image processing involved in reading one-dimensional signals such as bar codes in factories and atomic layer structures of thin films, using electron microscopes [31].
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