Spectroscopic nanoscopy (SN), alternatively referred to as spectroscopic single-molecule localization microscopy or spectrally resolved super resolution microscopy [1–5], stands prominently in the domains of chemistry and cell biology, owing to its advanced capabilities in capturing spatial locations of individual molecules, along with their distinctive emission spectra. This technique functions by capturing the stochastic single-molecule fluorescence from numerous fluorescent molecules emitting at different times, treating each as a spatial point-spread-function (PSF), and reconstructing an image based on their estimated center positions. Moreover, SN enables the retrieval of emission spectra from the individual molecules via a spectral calibration process that correlates the obtained spatial information.

The SN system mainly consists of a typical wide-field fluorescence microscope and an imaging spectrometer (IS) [5]. This IS can be implemented by either a prism or a grating [3–8]. Although the prism-based IS offers higher transmission efficiency, the grating-based IS has been widely used due to its simplicity: It contains only a grating and a pair of lenses, making it a more attractive tool for many researchers.

Basically, a grating-based IS divides the emitted photons into spatial and spectral beams, using the grating. These two beams are focused by a lens and captured by a camera to simultaneously provide spatial and spectral images. In this case, the aberrations caused by a series of optics are unavoidable, and may reduce image quality. However, to the best of our knowledge there has been no rigorous investigation regarding the aberrations of a grating-based IS in an SN system, and their impact on SN imaging. In this work we theoretically examine the aberrations of a grating-based IS using a ray-tracing model, and further elucidate how these aberrations are linked to calibrating spectral images, which is a fundamental procedure in correlating the spatial location of single molecules with their spectroscopic signatures. Specifically, we estimate the influence of the grating-based IS aberrations on calibrating spectral images, given the SN imaging parameters, such as spectral dispersion (SD) and spectral calibration error (SCE).

Figure 1 shows the schematic of an SN system using a grating-based IS. The green beam indicates the excitation light, while the red beam represents the fluorescence emission collected by an objective lens. After passing a dichroic mirror and a long-pass filter, the emission signal is focused by a tube lens at an intermediate image plane confined by a slit. Then, the emission signal is divided into two diffraction orders by a grating, passes through relay optics, and is captured by a camera [for example, a scientific complementary metal oxide semiconductor (sCMOS) camera], to form spatial and spectral images. The dotted-line box shows the grating-based IS, which offers variable SD values by adjusting the grating position with respect to the image plane [5]. It should be noted that we focus on a tunable SD configuration in this work, which is the commonly used configuration in the SN to achieve the desired SD, as shown in Fig. 1. Given the emission properties of commonly used visible fluorophores, we set the wavelength range to 400–700 nm in this work. We change the SD values from 1.5 to 9 nm/pixel in the following simulations, which is sufficient to cover commonly used SD values (typically 3 to 9 nm/pixel) in SN [4].

Figure 1. Layout of spectroscopic nanoscopy using a grating-based IS. OL, objective lens; DM, dichroic mirror; L, lens; BE, beam expander; BPF, bandpass filter; LPF, long-pass filter; TL, tube lens; M, mirror; IS, imaging spectrometer; G, grating.

We first evaluate the spatial and spectral beams of the grating-based IS using Zemax. Figure 2(a) shows the schematic of the spatial-beam path, which consists of a tube lens, a grating, a collimating lens, and a focusing lens, when the SD is set to 6 nm/pixel. In the SN system the beam size can be changed, depending on the detection scheme. In our simulations we set the beam size to 5 mm. The details of the simulation are described in Table 1 [9].

Figure 2. Performance evaluation. (a) Layout of the spatial-beam path of the grating-based imaging spectrometer (IS). (b)–(d) Simulated spot diagrams at different wavelengths. (e) Layout of the spectral-beam path, and its magnified view. (f)–(h) Simulated spot diagrams at different wavelengths. The spectral dispersion (SD) value used in the simulation is 6 nm/pixel. The blue, green, and red colors indicate 400, 550, and 700 nm, respectively. Scale bars: 100 mm in (a) and (e), 50 μm in (b)–(d) and (f)–(h).

TABLE 1. List of parameters used in simulations.

λ (nm)	Groove Density (grooves/mm)	Grating Thickness (mm)	Grating Material	Tube Lens	Relay Optic Lens	Spectral Dispersion (nm/pixel)	Camera Pixel Size (µm)	Beam Size (mm)
400, 550, 700	150	3	B270	Ref. [9]	AC508-100-Aa) (Focal Length: 100 mm)	1.5, 3, 6, 9	11	5

^a)Thorlabs, NJ, USA..

Figures 2(b)–2(d) show the simulated spot diagrams for the spatial beam at different wavelengths. The black-line circle of the spot diagram represents the diffraction-limited criterion. We confirm that the 0^th-order spatial beam at all wavelengths offers a diffraction-limited spot size: The geometrical (GEO) radii for 400, 550, and 700 nm are 0.0 (the computed number from Zemax is 0.001), 4.4, and 7.0 μm, respectively, which are significantly lower than the diffraction-limited criteria of 19.6, 26.9, and 34.2 μm.

Figure 2(e) shows the schematic of the spectral-beam path. Although the spectral beam shows elongated spot shape for 550 and 700 nm, it offers diffraction-limited spot size, as shown in Figs. 2(f)–2(h). The GEO values at 400, 550, and 700 nm are 0.9, 17.5, and 17.4 μm, respectively. In addition, as the wavelength increases we observe that the spot shape is stretched elliptically along the spectral axis corresponding to the dispersion direction. The groove density in the simulation is set to 150 grooves/mm. It should be noted that in the simulation we adjust the focal distance (from the focusing lens to the camera plane) for focusing the 400-nm beam.

The PSF is a key factor in determining localization precision in localization microscopy, while showing the characteristics of a given imaging system. Thus we visualize simulated PSFs for the grating-based IS. Figures 3(a)–3(c) show the cross sections of the PSFs for the spatial beam at different wavelengths. As wavelength increases, the shape of the PSFs becomes more collapsed, as expected from the simulated spot diagrams of Fig. 2.

Figure 3. Cross sections of simulated point-spread-functions (PSFs) for the (a)–(c) spatial- and (d)–(f) spectral-beam paths of the grating-based imaging spectrometer (IS), at different wavelengths. The blue, green, and red colors indicate 400, 550, and 700 nm, respectively.

We first simulate the PSFs of the grating-based IS without a tube lens, to see clearly the aberrations from the grating and relay optics only. In the case where the IS is combined with the tube lens, the PSFs are illustrated in Fig. 4. Especially for 700-nm PSFs, the shape recovers significantly, compared to the configuration without the tube lens.

Figure 4. Cross sections of simulated point-spread-functions (PSFs) for the (a)–(c) spatial- and (d)–(f) spectral beam paths of the grating-based imaging spectrometer (IS) including a tube lens, at different wavelengths. The blue, green, and red colors indicate 400, 550, and 700 nm, respectively.

To visibly investigate the influence of the wavelength difference on the grating-based IS, primarily related to chromatic aberration, we further simulate the beam focus of the grating-based IS for different depth values. Figures 5(a)–5(c) and 5(d)–5(f) illustrate simulated spot diagrams for the spatial- and spectral-beam paths, respectively, for different depth and wavelength values. We visualize the spot diagrams over a 5-mm depth variation at the image plane. As we focus the IS for the 400-nm beam, both the spatial and spectral beams are focused well on the focal plane. The spatial beams at 550 and 700 nm show a slight focal shift of approximately 300 μm, approximately corresponding to a few tens of nanometers at the ample plane, which is negligible given an axial resolution of about 50 nm and the working range of three-dimensional (3D) imaging in localization microscopy (usually ±500 μm for typical astigmatism and biplane-based 3D imaging [10–13]). Notably, we observe significant aberrations for the spectral beam, as illustrated in the spot diagrams of Figs. 5(d)–5(f).

Figure 5. Beam-focus visualization. Simulated spot diagrams of the (a)–(c) spatial- and (d)–(f) spectral-beam paths, for different depth and wavelength values. The blue, green, and red colors indicate 400, 550, and 700 nm, respectively. Scale bar, 150 μm in (a)–(f).

Using Zernike-coefficient analysis, we further reveal that several kinds of aberrations, such as coma, defocus, and astigmatism, are convolved to form complex spectral PSFs. Specifically, we confirm that the dominant aberrations are x tilt, defocus, and vertical forms of astigmatism, coma, and trefoil. Further details of the aberrations of the spectral beam from the analysis are described in Table 2.

TABLE 2. Zernike coefficients of the spectral beam in the grating-based imaging spectrometer (IS).

Term	Name	Zernike Polynomial	Wavelength (nm)
			400	550	700
2	y Tilt	2r cos θ	0.00000000	0.00000000	0.00000000
3	x Tilt	2r sin θ	−0.00037012	−0.00378570	−0.00429201
4	Defocus	3 (2r2 − 1)	0.00295632	0.04001133	0.02099066
5	Oblique Astigmatism	6r2 sin 2θ	0.00000000	0.00000000	0.00000000
6	Vertical Astigmatism	6r2 cos 2θ	−0.00152451	−0.01723508	−0.02603409
7	Vertical Coma	8 (3r3 − 2r) sin θ	−0.00012858	−0.00133666	−0.00151552
8	Horizontal Coma	8 (3r3 − 2r) cos θ	0.00000000	0.00000000	0.00000000
9	Vertical Trefoil	8r3 sin 3θ	−0.00012191	−0.00016124	−0.00024990
10	Oblique Trefoil	8r3 cos 3θ	0.00000000	0.00000000	0.00000000
11	Spherical, 3rd	5 (6r4 − 6r2 + 1)	−0.00002152	0.00004103	0.00003856

Although aberrations themselves are critical in determining imaging quality, the more important question in SN is how those aberrations impact the SN imaging performance. Thus the goal of this study is to identify practical issues caused by the aberrations in calibrating spectral images. Specifically, we examine the influence of the aberrations on a spectral-calibration procedure, using lateral color information from the ray-tracing analyses. Basically, to mimic wide-field spectroscopic imaging we generate several fields at different real-image dimensions in the simulation. Then, across various field distributions we analyze the effect of the aberrations on the lateral color components, which indicates the distance of the spectroscopic signature from the spatial location of specific molecules in the spectral image.

Figure 6(a) shows a map of field generation, a 3 × 3 array with an image height of 3 mm. As the x-axis has no correlation with the SD [14], we ignore fields along the x-axis and only consider the two fields (0, 0) and (3, 3) as examples, in the spectral axis. Figure 6(b) shows relative location as a function of real-image height, which is aligned to the spectral axis, for different wavelengths. The blue, green, and red lines represent 400, 550, and 700 nm, respectively. As wavelength increases, the relative location changes more noticeably with image height. This means that the lateral color shift between 400 and 700 nm changes with the image height. In other words, the distance between the spatial location of a single molecule and its spectral position can be altered. We first quantify the amount of variation using the SD, which is one of the key parameters to influence not only spectral precision but also system configuration [14]. We calculate the SD difference for the image height for fields (0, 0) and (3, 3) as below.

Figure 6. Influence of different field distributions on calibrating a spectral image. (a) Map of field generation. (b) Relative location as a function of image height for different wavelengths, obtained from a lateral color analysis. (c) Spectral dispersion (SD) difference and (d) spectral calibration error as a function of SD. The blue, green, and red colors indicate 400, 550, and 700 nm, respectively.

where SD is the SD value at the center or field position; W_λ is the given wavelength range (in nm), here 300 nm; N is the number of pixels to cover the wavelength range; L is the lateral color shift, i.e., the relative position between 400 nm and 700 nm; and S_pixel is the pixel size, 11 μm.

We obtain a 0.3-nm/pixel SD difference when the SD is 6 nm/pixel, as the relative location values are 550 and 525 μm. This means that an SD difference of up to 0.3-nm/pixel can occur, depending on the calibration position. We further calculate the SD difference for the cases of 1.5, 3, and 9 nm/pixel. The SD difference becomes more significant for lower SD values, as shown in Fig. 6(c). Notably, the difference is 0.54 nm/pixel when the SD is 9 nm/pixel, which is quite substantial and may cause a shift error of up to 1 pixel during the SD estimation of the spectral calibration procedure.

More importantly, the aberrations impact not only the SD value, but also the spectral calibration process during image processing. Due to the aberrations, the specific spectral component for single molecules may have different spectral distances depending on the spatial location, even though the same calibration curve is applied. Here we define this wavelength shift as the SCE. We calculated the SCE as below.

where N_dif is the pixel difference between N_center and N_field; N_center and N_field are the number of pixels to cover the given wavelength range (pixel) at the center and field position, respectively; L is the lateral color shift, i.e., relative position between 400 and 700 nm; And S_pixel is the pixel size.

Figure 6(d) shows the SCE as a function of SD. We observe that SCE increases with SD. Surprisingly, the SCE values are about 9.6 and 17.2 nm for SD of 1.5 and 9 nm/pixel, respectively. These results imply that the variety of aberrations may significantly impact the calibration of spectral images. In particular, SCE becomes more critical at lower SD.

In general, lower SD values are known to be advantageous for achieving high spectral precision, with more photons in a single pixel and an improved signal-to-noise ratio per pixel. However, the spectral shift error (SSE) becomes more pronounced as the SD value decreases, thereby reducing the spectral precision [14]. Consequently, optimal spectral precision can only be attained at some specific SD value. Therefore, the SCE and SSE need to be collectively considered when selecting the appropriate SD value, for an SN system to achieve the desired spectral precision.

In spectral image processing, the weighted-centroid method is commonly used for spectral calibration and SD estimation [14]. As it estimates the weighted value of the PSF in a given integration window, this method is notably tolerant of defocus, unless aberrations cause asymmetrical distortions in the PSF. We observe that with defocus the PSF becomes bigger while maintaining the symmetrical elongation along the x axis, which governs the spectral calibration procedure. Considering the symmetrical shape of the defocused PSF shown in Fig. 5, we assume that the variation in centroid wavelength values remains within a single pixel. Therefore, we neglect the influence of defocus on the spectral calibration procedure for the grating-based configuration in this work.

In this study, we comprehensively evaluate the performance of the grating-based IS in SN using a ray-tracing model, focusing on the impact of aberrations on the spectral calibration procedure. Our analysis reveals that the spectral beam of the grating-based IS exhibits various aberrations, including coma, astigmatism, and defocus, which can significantly degrade the quality of spectral images and prohibit detailed analyses of the spectroscopic signatures of molecules. Notably, we identify that the aberration-induced SCE of the spectral beam in the grating-based IS depends on the spatial location of the molecules, which may affect both spectral accuracy and precision. These findings underscore the necessity for researchers to consider not only photon utilization per pixel, but also the influence of aberrations on the IS, to achieve optimal spectral precision. Furthermore, our findings provide a valuable clue for understanding the intricate relationship between aberrations and SN imaging, facilitating the optimization of SN technologies.

This work was supported by the National Research Foundation of Korea grant, funded by the Korean government (MSIT) (Grant no. NRF-2022R1C1C1002850).

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

The data that support the findings of this study are available from the corresponding author upon reasonable request.