Abstract
1. Introduction
Spectral imaging acquires a three-dimensional (3D) spectral data cube, in which additional spectral information contains a significant amount of object information. It plays an important role in many applications, such as astronomical imaging, remote sensing, and biomedical imaging[
Meanwhile, research has shown that optimizing the speckle light field can significantly improve the sampling efficiency and the reconstruction quality of ghost imaging, especially at low signal-to-noise ratios (SNRs) and low sampling rates[
In this study, by introducing a hybrid refraction/diffraction structure, we propose a method for generating super-Rayleigh speckles over a broad range of wavelengths. The design theory of dispersion control for broadband super-Rayleigh speckles was derived and verified through simulations and experiments. The experimental imaging results showed that the reconstruction quality of snapshot spectral ghost imaging with broadband super-Rayleigh speckles was significantly improved, especially in the case of a low SNR.
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2. Theory and Simulation
2.1. Theory
As shown in Fig. 1, an object is imaged on the first imaging plane by a lens, and the SLM modulates light from the first imaging plane ‘b,’ resulting in a light intensity distribution on the speckle plane ‘d’ to be detected by a CCD. The amplitude and phase (AP) distribution of the SLM is designed to generate super-Rayleigh speckles on the speckle plane ‘d’ by Liu et al.’s method[
Figure 1.Schematic of snapshot spectral ghost imaging with broadband super-Rayleigh speckles. (a) is the object plane; (b) is the first imaging plane; (c) is the virtual speckle plane; (d) is the speckle plane.
When the lens of focal length is achromatic, objects with different wavelengths are imaged to the same first imaging plane ‘b,’ and the light field in the speckle plane ‘d’ generated by the point source on the first imaging plane ‘b’ is calculated as
From Eqs. (4) and (5), the speckle contrast reaches a maximum at different planes depending on the wavelength , which is consistent with the dispersion effect in traditional diffractive lens imaging systems[
A series of achromatic methods has been developed to compensate for dispersion in traditional diffractive imaging[
The spatial resolution is determined by the correlation function of the speckles generated by two points at the same wavelength and different positions with a distance [
2.2. Simulation
To verify the results of the theoretical derivation, we carried out a numerical simulation based on the Fresnel diffraction formula. First, we generated a random phase matrix uniformly and randomly distributed between 0 and and used it as the phase applied to SLM. Then, the Rayleigh speckles in the virtual plane were obtained by the free propagation of the light field. The field of the super-Rayleigh speckle in the virtual speckle plane was obtained by numerically raising the value of the Rayleigh speckles to the power . Then, by inversely propagating the field of the super-Rayleigh speckle in the virtual speckle plane with a distance at a wavelength , the AP distribution of the obtained light field was extracted to the SLM. According to the Fresnel diffraction formula, we simulated the light field on the speckle plane generated by a point source at different , , and on the first imaging plane. Figure 2 shows that the simulated speckle contrast varies with the system parameters , , and , and the corresponding system parameters for maximum contrast satisfy Eq. (4).
Figure 2.Numerical simulation results for the speckle contrast varied with the system parameters z1, z2′, and λ. (a) The simulation results of the speckle contrast varied with the system parameters z1 and λ. Here, z2′ = 10.6 mm. (b) The simulation results of the speckle contrast varied with the system parameters z2′ and λ. Here, z1 = 60 mm. The solid black line is the theoretical curve based on Eq. (
Then, we set the simulation parameters in Fig. 1 as follows: , , and . The simulated dispersion characteristic of the lens with focal length satisfied Eq. (7). The AP and P distributions of were extracted to the SLM. By placing a source on the object plane, the light field on the speckle plane was simulated by the diffractive propagation of the light field under two cases of SLM with AP and P distributions, as shown in Fig. 3(a). This indicated that in the case of SLM with AP distribution, the generated speckles maintain the same super-Rayleigh distribution, as shown in Fig. 3(b), and the contrast of speckles with different wavelengths satisfied the theoretical results in Eq. (5). As shown in Fig. 3(c), in the case of SLM with P distribution, the contrast of the generated speckles is lower than in the case of SLM with AP distribution. However, it remained much higher than that of the Rayleigh speckles. We also simulate the case of the lens with focal length as achromatic and SLM with AP and P modulation, as shown in Fig. 3(c). The simulated result showed that the contrast of the speckles was equal to the above-mentioned cases at the center wavelength and decreased as the wavelength deviated from the center wavelength [the brown and green lines in Fig. 3(c)]. The correlation functions of the speckles generated by two points at the same wavelength and different positions with a distance , and of the speckles generated by two points at the same position and different wavelengths with a gap , are shown in Figs. 3(d) and 3(e), respectively. The simulation results (blue line) were consistent with the theoretical results (dotted black line) in Eqs. (9) and (10).
Figure 3.(a) Simulation speckles of different wavelengths on the detection plane. AP: the amplitude and phase of Uslm (r0, λ0) were extracted to the SLM. P: the phase-only of Uslm (r0, λ0) was extracted to the SLM. (b) Probability distribution of the normalized intensity of the speckles. (c) Simulation results of the speckle contrast when the dispersion of the lens satisfied Eq. (
In addition, the contrast of the generated speckles was also affected by other non-ideal factors [such as the bandwidth (), size () of the calibration source, modulation size (), fill factor, pixel size () of the SLM, the phase quantization level of the SLM, the aperture of the imaging lens, the pixel size () of the CCD, speckle magnification of lens 3, and stray light]. When these factors in the actual experiment were considered in the simulation (such as the calibration source’s bandwidth , the calibration source’s size , the SLM’s fill factor 0.95, the SLM’s pixel size , the SLM’s phase quantization level of 256, lens 1’s aperture , lens 2’s aperture , lens 3’s aperture , the speckles’ magnification by lens 3 , the pixel size of the CCD , and 1% stray light), the simulated speckles’ contrast of reduced from 19 to 1.28, and the simulated speckles’ contrast of reduced from 12,869 to 4.23. We noted that different values of affected the speckle contrast and the phase loaded on the SLM, but they did not affect dispersion compensation [shown in Eq. (7)]. To obtain better imaging quality, we increased the value of in the actual experiment to increase the contrast of the experimental speckle.
3. Experimental Results
Figure 4(a) shows the experimental setup for snapshot spectral ghost imaging with broadband super-Rayleigh speckles. According to Eq. (7), the parameters for generating the phase matrix were set as , , and . The parameters of the experimental system were set as , , and . The dispersion curves of the lens used in the experiment were calculated by importing the corresponding lens structure data into the optical design software, as shown in Fig. 4(b). The dispersion of Lens 2 [the red line in Fig. 4(b)] almost coincided with the theoretical curve according to Eq. (7) [black line in Fig. 4(b)]. The achromatic doublet lens (Lens 1) with a focal length of 500 mm (GCL-010611) and the K9 doubly convex lens (Lens 2) with a focal length of 150 mm (GCL-010212) formed a lens group to image objects of different wavelengths to different positions in front of the reflective SLM ( meadowlark optics, pixel size was , and fill factor was 95.7%). To achieve P modulation of the SLM, the light from the object passes through a polarizer (GCL-050003), whose polarization direction was consistent with that of the SLM with P modulation. A broadband bandpass filter of 500–700 nm (#84-743, #86-103, Edmund) was placed behind Lens 2. After being reflected by the SLM and the beam splitter with a beam splitting ratio of R:T=1:1, the light was magnified by an achromatic doublet lens (Lens 3) with a focal length of 30 mm (GCL-010650), the magnification is , and it was finally recorded by the CCD (Basler ace acA4112-30 µm).
Figure 4.(a) Experimental setup of the snapshot spectral ghost imaging with broadband super-Rayleigh speckles. The calibration setup shown in the bottom box was adopted instead of the object in the black box when calibrating. The SCL was a supercontinuum laser. (b) Dispersion curves of lenses used in the experiment.
Before the imaging process, a calibration process was required to obtain the intensity impulse response functions by scanning along the spatial and spectral dimensions using a monochromatic point source within the field of view (FOV)[
In the imaging process, the intensity distribution in the detection plane was regarded as an incoherent superposition of the intensity distributions generated by all points in the object plane. The image of the object can be obtained by calculating the second-order intensity correlation between the calibrated speckle patterns and the imaging intensity distribution in a single shot. The second-order correlation function is expressed as[
Meanwhile, the imaging process can be expressed as[
Figure 5.(a) Curve of the correlation function of the experimental speckles generated by two points at the same wavelength and different positions with a distance Δ
Figure 5(c) shows the experimental speckles of snapshot spectral ghost imaging with broadband super-Rayleigh speckles at different wavelengths. The experimental results demonstrate a scaling relationship between the speckles of snapshot spectral ghost imaging with broadband super-Rayleigh speckles at different wavelengths. Unlike non-dispersion compensated snapshot spectral ghost imaging with super-Rayleigh speckles, where the speckle contrast decreases as the wavelength deviates from the central wavelength, the broadband super-Rayleigh speckles modulation was realized, and the speckle contrast maintained a high level across the entire spectrum.
A transmissive butterfly target (shown in the first column of Fig. 6) was illuminated by a xenon lamp. Different SNRs, obtained by exposing 50 ms and 10 ms at a sampling rate of 40%, were demonstrated. To quantitatively analyze the quality of the reconstructed images, we calculated the peak SNR (PSNR) and the structural similarity index (SSIM)[
Figure 6.Experimental imaging results with different exposure times, while the sampling rate remained at 40%. The mPSNR and mSSIM are also shown. (a) Exposure time of 50 ms. (b) Exposure time of 10 ms.
4. Conclusion
In conclusion, we theoretically derived the dispersion condition for realizing broadband super-Rayleigh speckle modulation. Moreover, we verified this by implementing broadband super-Rayleigh speckles in simulations and experiments. The experimental imaging results indicated the noise immunity of snapshot spectral ghost imaging with super-Rayleigh speckles, and its imaging quality was significantly improved. In this paper, we experimentally achieved dispersion control at 500–700 nm. This wavelength range is mainly determined by the system parameters and the selected component dispersion. Since this paper mainly verifies the feasibility of the scheme in principle, the dispersive lens was first selected, and then the parameters of the modulation system were set according to its dispersion. In practical applications, the parameters of the modulation system are generally determined according to the spatial resolution, spectral resolution, and other system indices. The dispersive lens is then customized according to the Eq. (7), and its fit will limit the final dispersion compensation range. We expect this to be applied in low SNR spectral imaging scenarios such as microscopy[
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