## Abstract

In chromatic confocal microscopy, the signal characteristics influence the accuracy of the signal processing, which in turn determines measurement performance. Thus, a full understanding of the spectral characteristics is critical to enhance the measurement performance. Existing spectral models only describe the signal intensity-wavelength characteristics, without taking the displacement-wavelength relation into consideration. These models require prior knowledge of the optical design, which reduces the effectiveness in the optical design process. In this paper, we develop a two-dimensional spectral signal model to describe the signal intensity-wavelength-displacement characteristics in chromatic confocal microscopy without prior knowledge of the optical design layout. With this model, the influence of the dimensional characteristics of the confocal setup and the displacement-wavelength characteristics and monochromatic aberrations of the hyperchromatic objective are investigated. Experimental results are presented to illustrate the effectiveness of our signal model. Using our model, further evaluation of the spectral signal can be used to enhance the measurement performance of chromatic confocal microscopy.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Confocal microscopy (CM) is a powerful tool in biological research [1], materials science [2], and for measurement of surface topography [3,4]. Axial scanning based on a mechanical device is often needed in confocal imaging [5,6] or precision metrology applications [7,8]. As a variant of conventional CM, chromatic CM can achieve fast axial scanning with a hyperchromatic objective [9–11]. The hyperchromatic objective is designed to have longitudinal chromatic aberration, in which different spectral components are focused at different axial positions [12]. Combined with confocal pinholes, chromatic CM can code displacement with spectral signals of different peak wavelengths [13], then decode displacement from the received spectral signals through signal processing [14].

The quality of the spectral signals influences the signal processing procedure, which ultimately influences the measurement performance of chromatic CM. From the point of view of signal modeling, the spectral signal is related to the design parameters of the chromatic CM, such as the dimensional characteristics of the confocal setup (e.g. the dimensions of the illumination and detection pinhole, the numerical apertures of the illumination and detection, and the numerical aperture of the hyperchromatic objective), and the dispersion characteristics and monochromatic aberrations of the hyperchromatic objective. Thus, accurate modeling of the spectral signal is critical to understand its dependence on the design parameters. This information can be used to optimize the optical design process, which eventually can be used to enhance the measurement performance of chromatic CM.

Derived from basic confocal signal models [15,16], Ruprecht et al. [17] established a signal model assuming an infinitesimal illumination pinhole and applying the paraxial diffraction approximation. These assumptions are usually valid in conventional CM, where the size of the illumination pinhole is much smaller than that of the detection pinhole [1]. However, the finite-sized illumination pinhole is common practice to improve the light efficiency in chromatic CM instruments with low-brightness sources such as LEDs [10]. Additionally, the same pinhole configuration for both the illumination and detection is common in chromatic CM for ease of alignment [18–20]. Thus, the influence of the illumination pinhole size needs further consideration. Moreover, the defocus distance within the axial confocal signal is usually a few micrometres [15,16], while that for chromatic CM (when one wavelength is in focus, other wavelengths are out of focus) is often tens or even hundreds of micrometres [13].The paraxial approximation is not valid when the defocus distance is greater than a few tens of micrometres [21,22].

To address the invalid assumptions of the paraxial diffraction theory and infinitesimal illumination pinhole, Hillenbrand et al. [13] proposed a one-dimensional spectral signal model for chromatic CM, with a finite-sized illumination pinhole, based on nonparaxial diffraction theory. However, this model requires the optical design parameters (e.g. lens radii, lens thickness, airspace between lenses and material properties) of the optical system in advance. The model is not valid before the optical design process, which reduces its effectiveness to enhance the design performance the instrument. Furthermore, the spectral signal can differ at different dispersion positions [23], and the actual displacement-wavelength relationship, which is vital to the measurement performance [18], is not captured in this one-dimensional spectral signal model.

In this paper, we develop a two-dimensional (2-D) spectral signal model to describe the signal intensity-wavelength-displacement characteristics. The model is established based on nonparaxial diffraction theory with the size of the illumination pinhole and the displacement-wavelength relationship taken into consideration, without prior optical design knowledge of the chromatic CM. The rest of the paper is organized as follows. Section 2 describes the 2-D spectral signal model. Section 3 investigates the influence of the design parameters of the chromatic CM on the spectral signal. Section 4 presents the experimental verification of the proposed model and Section 5 presents concluding remarks.

## 2. 2-D spectral signal model

The 2-D spectral signal is composed of a series of spectral confocal signal frames at different dispersion positions. The spectral confocal signal is a collection of intensities representing different monochromatically illuminated axial confocal signals at different defocus distances, and the relationship between the defocus distance and the illuminating wavelength is determined by the dispersion characteristics of the hyperchromatic objective. Figure 1 illustrates the relationship between the axial confocal signal, the spectral confocal signal and the 2-D spectral signal. The establishment of a 2-D spectral signal model requires the development of a monochromatically illuminated axial confocal signal model and determination of the displacement-wavelength relationship.

#### 2.1 Nonparaxial axial confocal signal with a finite-sized illumination pinhole

The optical layout of the chromatic CM usually involves several lenses, which are simulated using a “black box” imaging model, as shown in Fig. 2. The light travelling between the entrance pupil and the exit pupil can be described using geometrical optics, but diffraction effects need to be considered as the light travels from the exit pupil to the image [24]. In a coherent imaging system, the image is calculated as a convolution of the image predicted by geometrical optics with the point spread function (PSF) of the optical system. For incoherent illumination, the image intensity is found as a convolution of the intensity PSF with the ideal image intensity [25].

Although the development of the axial confocal signal model in chromatic CM assumes monochromatic illumination, the spatial coherence of the illumination needs further consideration. The spatial coherence is determined by the statistical dependencies of the phasor amplitudes across the illumination object [25]. When the object illumination is spatially coherent, the phasor amplitudes in the field at all object points are correlated. When the object illumination is spatially incoherent, the phasor amplitudes at all points on the object are uncorrelated. In this paper, a spatially incoherent light source such as LED is utilized, thus the illumination patterns at different points on the object are incoherent. And the imaging process is partily incoherent [1].

Figure 3 is a schema of the chromatic CM, which is composed of a broadband light source, a beam-splitter, a hyperchromatic objective, an illumination pinhole and a detector pinhole. The space radiance of the extended illumination source is assumed to be homogeneous, and the illumination field can be expressed as

If the optical system satisfies the Abbe sine condition [26], the lateral magnification from the illumination space (as shown in Fig. 3) to the object space is expressed as

where $N{A_{ill}}$ and $N{A_{obj}}$ are the numerical apertures (NAs) of the illumination space and the object space. It is emphasized that the lateral magnification definition is only valid when the optical pupil is fully-filled. Similarly, the lateral magnification from the object space to the detection space is expressed as where $N{A_{\det }}$ is the NA of the detection space. The total magnification of the illumination-object-detection process is expressed as It is noted that all these NAs are correlated to wavelength in chromatic CM. Thus, the intensity distribution of the ideal image, predicted by geometrical optics, for the extended illumination source is given byThe intensity distribution of the actual image is calculated as the convolution of the effective intensity PSF of the optical system with the ideal image intensity [13], thus

where $\Delta z$ is the defocus distance and ${h_e}$ is the effective PSF of the combined imaging process of the illumination and detection. The effective PSF is calculated as the convolution of the individual PSFs of the illumination and detection processes [27], thus where ${h_1}$ and ${h_2}$ are the complex amplitude PSFs of the illumination and detection processes, respectively. For a circular aperture detector, the intensity sensitivity can be written as*k*is the wavenumber. The parameter

*R*is the distance between a point $({{x_0},{y_0},0} )$ in the exit pupil plane and a point $({x,y,z} )$ in the detection plane, and is given by It should be emphasized that the amplitude PSF of the illumination process is usually defined in object space, whereas we have defined it here in detection space. Thus, there is a lateral scaling for the PSF calculation [29].

By examining Eq. (10), it can be seen that the monochromatic wavefront at the exit pupil needs to be determined in advance. Hillenbrand et al. [13] adopted a ray-tracing analysis of the optical system to determine the field ${u_0}$ and demonstrated the accuracy of the method by experiment. However, ray-tracing analysis implies that the design parameters of the optical system are specified. In addition, the ray-tracing operation in commercial software is time-consuming. Such prerequisites may lower the effectiveness of Hillenbrand’s method for optimizing the dimensional characteristics of the confocal setup and determining the aberration tolerances of the hyperchromatic objective. On the contrary, in our model, the field at the exit pupil is assumed in advance, through which the influence of the design parameters of chromatic CM (including the dimensional characteristics of the confocal setup, the dispersion characteristics of the hyperchromatic objective, and optical aberrations) on the spectral signal can be investigated before the optical design process.

In the absence of monochromatic aberration, the output field at the exit pupil is an ideal convergent spherical wave. In the Kirchhoff approximation [21], the field of a truncated spherical wave is given by

#### 2.2 Chromatic dispersion model

The displacement-wavelength relationship can be described with a polynomial model of the chromatic coordinate $\omega $ [18]

The sensitivity of chromatic CM is defined as the ratio between the corresponding increment of the wavelength output and the increment of the displacement input,

where the dispersion position is in micrometres and the wavelength is in nanometres.#### 2.3 Spectral confocal signals at different positions

In the case of monochromatic illumination, the axial confocal signal is acquired based on Eq. (9) for CM with a finite-sized illumination pinhole. The spectral confocal signal is obtained by calculating the intensities of the axial confocal signal for different monochromatic wavelengths at different corresponding defocus distances. Specifically, when the wavelength ${\lambda _m}$ is exactly at focus, the defocus distance for other wavelengths is expressed as

and the spectral confocal signal can be expressed as Therefore, the 2-D spectral signal is obtained from with the expression ${S_{inv}}$ representing the inverse function of the displacement-wavelength relationship.According to Eq. (10), the PSF is correlated with the distance ${z_o}$ between the illumination plane and entrance pupil plane, and the distance ${z_i}$ between the exit pupil and the detection plane, as shown in Fig. 2. Thus, the calculation of the spectral signal needs information about these two distances. In Hillenbrand’s method [13], these two wavelength-related distances are acquired from the ray-tracing data about the optical system, meaning that the spectral confocal signal is not available unless the optical design parameters have been determined. In the following, the influence of these distances (e.g. the distance between the illumination plane and the entrance pupil plane, and distance between the exit pupil plane and the detection plane) on the spectral confocal signal is investigated.

In our simulation, the dimensional characteristics of the confocal setup and sensitivity are set to ${r_{ill}} = {r_{\det }} = 25\textrm{ }\mathrm{\mu }\textrm{m}$, $N{A_{obj}} = 0.3$, $\textrm{N}{\textrm{A}_{ill}} = N{A_{\det }} = 0.05,$ and $K = 0.075{{\textrm{ nm}} / {\mathrm{\mu }\textrm{m}}}$. These are the default values unless otherwise stated. In Fig. 4, the normalized spectral confocal signals are illustrated at different distances. It can be seen that the spectral signal model is almost independent of the distances ${z_o}$ and ${z_i}$. This conclusion is beneficial when investigating the influence of the design parameters on the 2-D spectral signal prior to the optical design process.

In Fig. 5, the 2-D spectral signal is illustrated, and the left-top demonstrates the spectral confocal signal at the dispersion position $l = 2000\textrm{ }\mathrm{\mu }\textrm{m}$. The spectral confocal signals are illustrated at different dispersion positions in Fig. 6. The correlation coefficient between two frames of spectral confocal signals at different dispersion positions is almost equal to unity, which indicates that the spectral confocal signal models at different dispersion positions are almost identical. This means that the spectral confocal signal model is almost independent of the dispersion position.

In our simulations, the radiance and the NA of the hyperchromatic objective are assumed the same for different wavelengths. In fact, these parameters vary with the wavelength or the dispersion position. Thus, the actual spectral confocal signal model also varies at different positions.

## 3. Characteristics of the 2-D spectral signal

In this section, the influences of the optical design parameters of the chromatic CM, including the dimensional characteristics of the confocal setup, the dispersion characteristics and optical aberrations on the 2-D spectral signal are investigated. A thorough demonstration of all cases with different design parameters can be complex. Here, we assume that the same size for both the illumination and the detection pinholes [14] for simplicity in the following examples, which is also the typical configuration in chromatic CM. Further investigations can be demonstrated in a similar way to those in this section, since the proposed 2-D spectral signal model is applied to optical configurations with different illumination and detection pinholes [23,30].

#### 3.1 Influence of the dimensional characteristics on the 2-D spectral signal

In Fig. 7, the 2-D spectral signals are illustrated for different dimensional characteristics of the confocal setup. When the radius of the illumination/detection pinhole or the NA of the illumination/detection space increases, the 2-D spectral signal is broadened, as shown in Fig. 7(a) and Fig. 7(b). When the NA of the hyperchromatic objective decreases, the 2-D spectral signal is also broadened, as shown in Fig. 7(c). At the dispersion position $l = 2000\textrm{ }\mathrm{\mu }\textrm{m}$, the full width at half maximum (FWHM) of the spectral confocal signal is illustrated for different dimensional characteristics of the confocal setup, as shown in Fig. 8. The results indicate that the FWHM is linearly proportional to the illumination/detection radius, which agrees with the results in Ref. [13]. Furthermore, the FWHM is linearly proportional to the illumination/detection NA, and is inversely proportional to the square of the objective NA.

#### 3.2 Influence of the dispersion characteristics on the 2-D spectral signal

Assuming that the chromatic CM has a constant sensitivity, an increase of the sensitivity implies a decrease of the dispersion range when there is a fixed spectral bandwidth. In Fig. 9, the 2-D spectral signal is shown for a higher sensitivity, and an increase of the sensitivity leads to a broader signal. At the half dispersion range, the FWHM of the spectral confocal signal is plotted at different levels of the sensitivity, as shown in Fig. 10, and the FWHM is found to be linearly proportional to the sensitivity.

The chromatic CM could have different levels of sensitivity at different dispersion positions. Therefore, two chromatic CMs of the same specification (meaning the same spectral bandwidth $\Delta \Lambda $ and dispersion range $\Delta l$) could have different dispersion characteristics. To describe the dispersion characteristics of the chromatic CM of the same specification, its mean sensitivity is defined as

#### 3.3 Influence of optical aberrations on the 2-D spectral signal

The hyperchromatic objective is designed to cause different spectral components to be “perfectly” focused at different axial positions. However, there will be monochromatic aberrations (occurring in the optical design, manufacturing and assembly processes) along with the dispersion range. According to the theory of Seidel aberrations, a monochromatic aberration is composed of five different aberration terms, including spherical aberration, coma, astigmatism, field curvature and distortion. The presence of such geometrical aberrations means that the normal direction of the wavefront deviates from an ideal path, which presents wavefront aberration at the exit pupil.

When the dimensional characteristics of the confocal setup and sensitivity are specified, a further understanding of the influence of different degrees of optical aberration on the 2-D spectral signal in advance is beneficial to guide the design of the hyperchromatic objective. The degree of optical aberrations in chromatic CM is correlated with the illumination field position, making a full characterization complex.

Here, isoplanatic imaging [31] is assumed across the pinhole illumination field, as the size of the illumination pinhole is much smaller than that of the exit pupil, which indicates that the degree of monochromatic aberration is the same across the pinhole illumination field. The assumption is applied to chromatic CMs for point [13], line [32] and full-field [20] measurement.

In this paper, the influence of the different degrees of spherical aberration on the spectral confocal signal is discussed for on-axis point measurement chromatic CM. Other aberration terms can also be taken into consideration similarly using our approach and will be the subject of a future publication.

There are several methods to characterize the wavefront aberration at the exit pupil, such as the Seidel coefficients and Zernike polynomials [31]. For on-axis point measurement chromatic CM, the wavefront aberration phase is expressed as

where $\rho $ is the normalized coordinate at the exit pupil, and ${W_{040}}$ and ${W_{060}}$ are the coefficients of the primary spherical aberration and the high-order spherical aberration, respectively. The defocus term is neglected since it only leads to a horizontal shift of the confocal signal [1]. The aberration phase is added into Eq. (10) to reformulate the complex field at the exit pupil, thus and the complex amplitude PSF is calculated using the aberrated wavefront expression according to Eq. (10).At the dispersion position $l = 2000\textrm{ }\mathrm{\mu }\textrm{m}$, the spectral confocal signal is plotted with different degrees of spherical aberration (see Fig. 13). When the hyperchromatic objective has a small amount of spherical aberration, the signal shape is almost unchanged. As the degree of spherical aberration increases, the spectral confocal signal becomes asymmetrical and broadened. If the degree of spherical aberration exceeds a limit, the spectral confocal signal has significant distortion. The RMS values of the spherical aberration for these two cases (${W_{040}} = 5, {W_{060}} = - 4$ and ${W_{040}}= 4.5, {W_{060}} = - 3.5$) are 0.48 and 0.43, however, the corresponding “degree of asymmetry” [33] is significantly different, which implies that even a small increase in the RMS value of spherical aberration can lead to significant asymmetry of the spectral confocal signal. Thus, the degree of spherical aberration needs to be carefully controlled in the optical design of the hyperchromatic objective.

As the hyperchromatic objective has an equivalent degree of spherical aberration at different dispersion positions, the 2-D spectral signal is plotted with different degrees of spherical aberration, as shown in Fig. 14. The ideal displacement-wavelength relationship is given by Eq. (17) in advance, and the actual relationship is obtained to extract peak wavelengths of the spectral confocal signals at different dispersion positions. The spectral confocal signal is simulated with a small wavelength sampling interval to avoid peak wavelength extraction errors [33]. The parallelism between the ideal curve and the actual curve indicates that the actual displacement-wavelength relationship will not be distorted for the hyperchromatic objective, with an equivalent degree of optical aberration at different dispersion positions.

As the hyperchromatic objective has different degrees of spherical aberration at different dispersion positions, the 2-D spectral signal is plotted with changes in the aberration coefficients, as shown in Fig. 15. The aberration coefficients are assumed to vary linearly with the dispersion position. It can be seen that the actual shape of the displacement-wavelength relationship deviates from the ideal shape, even though the mean values of the aberration coefficients ($\overline {{W_{040}}} = 3,\overline {{W_{060}}} ={-} 2$) are much smaller than those in Fig. 14(c) (${W_{040}} = 5,{W_{060}} ={-} 4$). Moreover, a larger change in the aberration coefficients, or larger mean coefficients with the same change in amplitude, causes a more significant deviation from the ideal displacement-wavelength relationship, as shown in Fig. 15(b) and Fig. 15(c). Thus, the consistency of the aberration coefficients, along with the dispersion position, needs attention in the optical design process.

## 4. Experimental verification

We developed a chromatic CM, as shown in Fig. 16, to test our signal models against experimental results. In our experimental setup, two self-designed chromatic confocal probes (A type and B type, as shown in Fig. 17) are utilised. The broadband source (MWWHF2, Thorlabs, USA) has an optical spectrum of 380 nm to 780 nm. The spectrometer (Ocean Maya Pro2000, USA) has a resolution of 0.46 nm/pixel. Multi-mode fibres of 50 μm diameter and a matched fibre coupler are used as the light transmitters for the chromatic confocal head, light source and the spectrometer.

#### 4.1 Comparisons between theoretical and actual spectral confocal signals

For the chromatic CM with the A type chromatic confocal head, the NA of the illumination/detection space is 0.065, and the NA of the hyperchromatic objective is approximately 0.34 at a wavelength approximately 550 nm. The sensitivity at 550 nm is approximately 0.09 nm/μm.

A comparison between the theoretical and experimental spectral confocal signals is shown in Fig. 18. When the imaging characteristics of the spectrometer are not taken into consideration, there is a large discrepancy between the experimental and the theoretical spectral confocal signals. The spectrometer has its own transfer function to broaden the ideally monochromatic light into the frame of the spectral signal with a finite bandwidth [34]. This bandwidth expansion can be determined using a process defined by the spectrometer manufacturer [35]. When the broadening effect of the spectrometer is taken into consideration, the theoretical spectral confocal signal coincides well with the experimental signal.

In Hillenbrand’s research [13], the experimental chromatic CM has a larger illumination/detection NA and a smaller hyperchromatic objective NA than those of our experimental setup. The spectral confocal signal in Hillenbrand’s setup is much broader than that of our chromatic CM, which results in the broadening effect of the spectrometer being negligible.

#### 4.2 Comparisons between different dimensional characteristics and sensitivities

In Fig. 17, the two self-designed chromatic confocal probes have the same hyperchromatic focusing layout and different collimating lenses with different illumination/ detection NAs. The A type chromatic confocal head has an illumination/detection NA of 0.065, while the B type has an illumination/detection NA of 0.115. The experimental spectral confocal signals of these two chromatic CMs are illustrated in Fig. 19. The FWHMs of the spectral confocal signals are 2.75 nm and 4.45 nm, respectively. The ratio between these two FWHMs is almost the same as the ratio between the two illumination/detection NAs. Demonstrations of other dimensional characteristics of the confocal setup, such as the illumination radius and the NA of hyperchromatic objective can be verified using similar comparisons, which is, however, omitted for brevity.

Experimental spectral confocal signals, from the B type chromatic CM at different dispersion positions with different sensitivities, are shown in Fig. 20. The FWHMs of these spectral signals are 2.91 nm, 3.61 nm, 4.24 nm, and 10.38 nm, with dispersion sensitivities of 0.038 nm/μm, 0.053 nm/μm, 0.083 nm/μm, and 0.155 nm/μm. The FWHM of the spectral confocal signal is verified to be linearly proportional to the dispersion sensitivity after eliminating the broadening effect of the spectrometer, which coincides well with the simulation results shown in Fig. 10.

#### 4.3 Comparisons for the displacement-wavelength relationship

The comparison between the theoretical and the experimental 2-D spectral signal is illustrated in Fig. 21. The theoretical 2-D spectral signal is simulated for the B type chromatic CM, with the displacement-wavelength relationship obtained from the ray-tracing information. The correlation coefficient between the two frames of 2-D signal is 0.96. It can be concluded that the theoretical result coincides well with the experimental result, which demonstrates the effectiveness of the 2-D spectral signal model.

The actual displacement-wavelength relationship of the experimental 2-D spectral signal is extracted with an accurate peak wavelength localization algorithm [6]. The deviation of the actual displacement-wavelength relationship from the ideal displacement-wavelength relationship that is obtained from the optical design, is illustrated in Fig. 22. The RMS radius of the spot diagram in the optical design is shown in Fig. 23, where the RMS of spot diagram reflects the degree of the optical aberration. Comparing Fig. 22 and Fig. 23, there is a small and constant deviation from the ideal displacement-wavelength relationship at dispersion displacements from 20 μm to 60 μm, with a small change in the RMS of the spot diagram, and a larger deviation from the ideal relationship at dispersion displacements from 80 μm to 100 μm, with a larger change in the RMS of the spot diagram. The discrepancy between the results in Fig. 22 and in Fig. 23 can be attributed to the additional aberrations caused in the manufacturing and assembly processes. Although, the control of the consistency of the optical aberrations along different dispersion positions has been taken into consideration in the optical design, there are unavoidable differences in optical aberrations at different dispersion positions. Thus, more accurate characterization methods (such as the neural network [36,37]) for the displacement response are needed in signal processing [18].

## 5. Conclusions

A two-dimensional nonparaxial spectral signal model is proposed for chromatic CM with the size of the illumination pinhole and displacement-wavelength relationship taken into consideration. The model requires little prior knowledge of the optical design layout. With this model, the intensity-wavelength-displacement characteristics are described, and the influences of the design parameters, including the dimensional characteristics of the confocal setup, dispersion characteristics and monochromatic aberrations, are illustrated. Experimental results have verified the validity of our model. In this research, some new information, such as the dependence on the dimensional characteristics, and the importance of the consistency of optical aberrations, is disclosed for the first time. On the basis of this research, further links between the two-dimensional spectral signals and the measurement performance will be built in our future research, through which, more direct guidelines will be offered for the optimized configuration of the dimensional characteristics, and the optical design of the hyperchromatic objective.

## Funding

National Natural Science Foundation of China (51875227, 51905200); National Science Foundation for Distinguished Young Scholars of Hubei Province of PRC (2019CFA038); Shenzhen Technical Project (JCYJ2017030717134710); Science Challenge Project (TZ2018006-0102-02).

## Disclosures

The authors declare no conflicts of interest.

## References

**1. **M. Gu, *Principles of three dimensional imaging in confocal microscopes* (World Scientific, 1996).

**2. **B. V. R. Tata and B. Raj, “Confocal laser scanning microscopy: Applications in material science and technology,” Bull. Mater. Sci. **21**(4), 263–278 (1998). [CrossRef]

**3. **C. Chen, J. Wang, X. J. Liu, W. L. Lu, H. Zhu, and X. Q. Jiang, “Influence of sample surface height for evaluation of peak extraction algorithms in confocal microscopy,” Appl. Opt. **57**(22), 6516–6526 (2018). [CrossRef]

**4. **L. Li, J. Liu, Y. Liu, C. Liu, H. Zhang, X. You, K. Gu, Y. Wang, and J. Tan, “A promising solution to the limits of microscopes for smooth surfaces: fluorophore-aided scattering microscopy,” Nanoscale **10**(20), 9484–9488 (2018). [CrossRef]

**5. **L. Qiu, D. Liu, W. Zhao, H. Cui, and Z. Sheng, “Real-time laser differential confocal microscopy without sample reflectivity effects,” Opt. Express **22**(18), 21626–21640 (2014). [CrossRef]

**6. **C. Chen, J. Wang, R. K. Leach, W. Lu, X. Liu, and X. Jiang, “Corrected parabolic fitting for height extraction in confocal microscopy,” Opt. Express **27**(3), 3682–3697 (2019). [CrossRef]

**7. **W. Zhao, Z. Li, L. Qiu, H. Ren, and R. Shao, “Large-aperture laser differential confocal ultra-long focal length measurement and its system,” Opt. Express **23**(13), 17379–17393 (2015). [CrossRef]

**8. **J. Liu, Y. Wang, K. Gu, X. You, M. Zhang, M. Li, and J. Tan, “Measuring profile of large hybrid aspherical diffractive infrared elements using confocal profilometer,” Meas. Sci. Technol. **27**(12), 125011 (2016). [CrossRef]

**9. **L. C. Chen, D. T. Nguyen, and Y. W. Chang, “Precise optical surface profilometry using innovative chromatic differential confocal microscopy,” Opt. Lett. **41**(24), 5660–5663 (2016). [CrossRef]

**10. **B. Jiao, X. Li, Q. Zhou, K. Ni, and X. Wang, “Improved chromatic confocal displacement-sensor based on a spatial-bandpass-filter and an X-shaped fiber-coupler,” Opt. Express **27**(8), 10961–10973 (2019). [CrossRef]

**11. **L. C. Chen, “Confocal Microscopy for Surface Profilometry,” Metrology, 1–34 (2019).

**12. **A. Ruprecht, K. Koerner, T. Wiesendanger, H. Tiziani, and W. Osten, “Chromatic confocal detection for high-speed microtopography measurements,” Proc. SPIE **5302**, 53–60 (2004). [CrossRef]

**13. **M. Hillenbrand, B. Mitschunas, F. Brill, A. Grewe, and S. Sinzinger, “Spectral characteristics of chromatic confocal imaging systems,” Appl. Opt. **53**(32), 7634–7642 (2014). [CrossRef]

**14. **C. Chen, W. Yang, J. Wang, W. Lu, X. Liu, and X. Jiang, “Accurate and efficient height extraction in chromatic confocal microscopy using corrected fitting of the differential signal,” Precis. Eng. **56**, 447–454 (2019). [CrossRef]

**15. **J. Liu and J. Tan, *Confocal Microscopy* (Morgan & Claypool, 2016).

**16. **R.K. Leach, *Optical Measurement of Surface Topography* (Springer, 2011).

**17. **A. K. Ruprecht, T. F. Wiesendanger, and H. J. Tiziani, “Chromatic confocal microscopy with a finite pinhole size,” Opt. Lett. **29**(18), 2130–2132 (2004). [CrossRef]

**18. **W. Lu, C. Chen, J. Wang, R. K. Leach, C. Zhang, X. Liu, Z. Lei, W. Yang, and X. Jiang, “Characterization of the displacement response in chromatic confocal microscopy with a hybrid radial basis function network,” Opt. Express **27**(16), 22737–22752 (2019). [CrossRef]

**19. **L. Chen, Y. Chang, and H. Li, “Full-field chromatic confocal surface profilometry employing digital micromirror device correspondence for minimizing lateral cross talks,” Opt. Eng. **51**(8), 081507 (2012). [CrossRef]

**20. **S. Cha, P. C. Lin, L. Zhu, P. Sun, and Y. Fainman, “Nontranslational three-dimensional profilometry by chromatic confocal microscopy with dynamically configurable micromirror scanning,” Appl. Opt. **39**(16), 2605–2613 (2000). [CrossRef]

**21. **J. J. Stamnes, *Waves in focal regions* (Hilger, Bristol, UK, 1986).

**22. **D. Claus and M. R. Nizami, “Influence of aberrations and roughness on the chromatic confocal signal based on experiments and wave-optical modeling,” Surf. Topogr.: Metrol. Prop. **8**(2), 025031 (2020). [CrossRef]

**23. **B. S. Chun, K. Kim, and D. Gweon, “Three-dimensional surface profile measurement using a beam scanning chromatic confocal microscope,” Rev. Sci. Instrum. **80**(7), 073706 (2009). [CrossRef]

**24. **L. Rayleigh, “On the theory of optical images, with special reference to the microscope,” in *Scientific papers by Lord Rayleigh, John William Strutt*, (Cambridge University Press, Cambridge, 1903), Vol. 4, pp. 235–260.

**25. **J. W. Goodman, *Introduction to Fourier optics* (McGraw-Hill, San Francisco, 1968).

**26. **W. J. Smith, *Modern Optical Engineering: The Design of Optical Systems* (McGraw-Hill, New York, 1990).

**27. **W. Gong, K. Si, and C. J. R. Sheppard, “Optimization of axial resolution in a confocal microscope with D-shaped apertures,” Appl. Opt. **48**(20), 3998–4002 (2009). [CrossRef]

**28. **M. Hillenbrand, A. Hoffmann, D. P. Kelly, and S. Sinzinger, “Fast nonparaxial scalar focal field calculations,” J. Opt. Soc. Am. A **31**(6), 1206–1214 (2014). [CrossRef]

**29. **D. Claus, G. Pedrini, T. Boettcher, M. Taphanel, W. Osten, and R. Hibst, “Development of a realistic wave propagation-based chromatic confocal microscopy model,” Proc. SPIE **10677**, 106770X (2018). [CrossRef]

**30. **Q. Yu, K. Zhang, C. Cui, R. Zhou, F. Cheng, R. Ye, and Y. Zhang, “Method of thickness measurement for transparent specimens with chromatic confocal microscopy,” Appl. Opt. **57**(33), 9722–9728 (2018). [CrossRef]

**31. **V. N. Mahajan, Optical Imaging and Aberrations, *Part II. Wave Diffraction Optics* (SPIE, Bellingham, Wash., 2001).

**32. **P. C. Lin, P. Sun, L. Zhu, and Y. Fainman, “Single-shot depth-section imaging through chromatic slit-scan confocal microscopy,” Appl. Opt. **37**(28), 6764–6770 (1998). [CrossRef]

**33. **C. Chen, J. Wang, C. Zhang, W. Lu, X. Liu, Z. Lei, and X. Jiang, “Influence of optical aberrations on the peak extraction in confocal microscopy,” Opt. Commun. **449**, 24–32 (2019). [CrossRef]

**34. **C. Olsovsky, R. Shelton, O. Carrasco-Zevallos, B. E. Applegate, and K. C. Maitland, “Chromatic confocal microscopy for multi-depth imaging of epithelial tissue,” Biomed,” Biomed. Opt. Express **4**(5), 732–740 (2013). [CrossRef]

**35. **https://www.oceaninsight.com/products/spectrometers/high-sensitivity/maya-series.

**36. **X. Chen, Y. Chen, K. Gupta, J. Zhou, and H. Najjaran, “Slicenet: A proficient model for real-time 3d shape-based recognition,” Neurocomputin **316**, 144–155 (2018). [CrossRef]

**37. **X. Chen, Y. Chen, and H. Najjaran, “End-to-end 3d object model retrieval by projecting the point cloud onto a unique discriminating 2d view,” Neurocomputing **402**, 336–345 (2020). [CrossRef]