Open Access
Add to BibSonomy
Abstract
We propose a heterodyne confocal microscopy technique that uses the symmetry properties of conjugate phase-pupil filters to shift the axial intensity-response curves of two confocal systems symmetrically with respect to the geometric focal plane. A monotonic position-measuring curve is obtained by subtracting the output signals of the two confocal systems collected synchronously, while eliminating the influence of environmental noise. The system’s three-dimensional tomographic ability can be improved by optimizing the pupil filter. Theoretical analyses and experiments indicate that the proposed method improves the axial and transverse resolutions of differential confocal microscopy and simplifies system installation and integration.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Confocal microscopy (CM) has important applications in many research fields such as biomedicine, microelectronics, industrial precision detection, and material sciences [1–5]. With the miniaturization and large-scale integration of microstructural elements, high-speed and high-resolution 3D microscopy is becoming more popular in many industrial fields [6]. However, the imaging efficiencies of existing CM techniques are not sufficient for many industrial applications that require layer-by-layer focused scanning [7]. Besides, their axial resolutions are low because the detection focus corresponds to the axial response curve’s peak with the lowest sensitivity.
To perform fast and high-axial-resolution tomography, differential confocal microscopy (DCM) was proposed in [8,9], which was based on the sharp slopes of the CM axial response curve. On the slopes, the sample height variations caused a change in intensity signal. The method could perform surface profiling without axial scanning, and it was successfully used for measuring the nanometer-sized surface profiles of optical gratings and the nanometer-sized shape features in human red blood cells. However, it was susceptible to disturbances in the light-source intensity, ambient lighting, and reflection characteristics of the sample surface, because of using the slopes of the intensity response curve. Besides, DCM worked in the defocused linear segment of the CM axial response curve, which led to reduced transverse resolutions.
In order to solve the above-mentioned problems, much work had been done to improve the performance of the DCM. Liu et al. proposed DCM with an annular pupil filter to extend the axial dynamic range and improve the transverse resolution of 3D profilometry at the cost of sacrificing the axial resolution [10]. Lee et al. proposed a dual-detection confocal reflectance microscopy technique to suppress the influence of the reflectance characteristics [11]. The ratio of the axial response curves measured by the two detectors provided the relationship between the axial position of the sample and the ratio of the intensity signals. In these approaches, the axial response curve was symmetrical with respect to the focus and had no absolute zero position. Thus, these works were not always suitable for the ultraprecise measurement of discontinuous surface contours and 3D microstructural workpieces with a large sudden skip in position. Besides, only a one-sided slope of the CM axial response curve was used in the measurement, which meant that the axial dynamic range of the DCM was small.
In [12–15], a laser heterodyne confocal probe (LHCP) was proposed based on two defocus pinholes, to measure the standard step contours; it achieved an axial resolution of 2 nm. The LHCP used two pinhole detectors, which were placed behind and in front of the focal point of a pair of collecting lenses focused with the same offset. By the differential subtraction of the two shifted axial response curves, a new monotone intensity response curve was obtained. The LHCP had a bipolar measurement range and an absolute zero, which could realize expansion of the axial dynamic range and highly stable real-time measurement. Specially, the monotone intensity response curve of LHCP had a greater curve slope than the curve of DCM, which could further improve the axial resolution. However, the LHCP suffered from high complexity and difficulty in system mounting and adjusting, owing to the strict requirements on the defocusing positions of the pinhole detectors [13].
In order to realize fast and high-resolution tomography in a simple manner, a heterodyne CM technique based on the shifted-focus characteristics of the phase pupil is proposed in this paper. In this method, a pair of conjugate phase pupils is used to shift the two light-intensity response curves of the system, symmetrically, relative to the geometric focal plane of the collecting lens, which has been carried out for the first time in this work. Then, heterodyne confocal measurement is realized by the differential subtraction of the two signals. This method maintains the technical advantages of LHCP, and it has the characteristics of accurate and easy pinhole installation and adjustment. In addition, because the phase-pupil filter designed in this paper has transverse super-resolution effects, this method can further improve the transverse resolution of the system, when compared to DCM and LHCP.
2. Measurement principle
LHCP is based on two point detectors with a symmetrical axial offset to measure discontinuous surface contours and 3D microstructures. As shown in Fig. 1, two pinholes are placed, one before and the other behind the focal planes of the collecting lenses in the light path arrangement.
Fig. 1 Schematic of LHCP. LS: laser source; PBS: polarization beam splitter; BS: beam splitter; 1/4WP: quarter wave plate; L1: objective lens; L2A, L2B: collecting lenses; DA, DB: intensity detectors for confocal systems A and B.
The system can achieve the maximum sensitivity and best linearity only when the offset distance ZFA of pinhole A, from the focal plane, is equal to the offset distance ZFB of pinhole B, and the offset distance is the optimum, i.e., uF = 5.21 [12,14] (the optical coordinates of ZFA and ZFB are uFA and uFB, respectively). Thus, the pinhole mounting and adjustments are highly complex and difficult.
In order to solve this problem, a new symmetrical shifted-focus heterodyne confocal microscopy (SSHCM) technique, shown in Fig. 2, is proposed based on the shifted-focus character of phase-pupil filters. In the SSHCM, two different phase filters are placed before the collecting lens L2A and L2B. The two different filters can cause symmetrical translation of the axial responses of the two confocal systems, with respect to their geometrical focus.
Fig. 2 (a) System setup of SSHCM: schematic diagram of the experimental setup. LS: laser source; PBS: polarization beam splitter; BS: beam splitter; L1: objective lens; L2A, L2B: collecting lenses; PF2A, PF2B: phase filters. (b) Structure of phase filter.
For a reflected confocal system, the 3D point spread functions (PSFs) for the illuminating part and collecting part can be written as [16,17]
Here, subscript i = 1 or 2 denotes the illuminating and collecting parts of the SSHCM. For the reflected CM, h1(v,u) will be the same as h2(v,u), as P1(ρ) = P2(ρ). v and u are the transverse and axial normalized coordinates, ρ is the radial normalized coordinate in the pupil plane, J0 is a zero-order Bessel function, P1 denotes the pupil function of the illuminating part, and P2 denotes the pupil function of the collecting part.
where r1 is the radial coordinate of the objective L1, z1 is the axial displacement of an object, sinα1 and sinα2 are the numerical apertures of the objective L1 and collect lens L2, respectively, ZF is the axially shifted focus distance from the geometrical focus plane in the detecting space, and uF is the corresponding axial normalized coordinate of ZF. For a point detector, the intensity responses of detectors A or B (i.e., confocal systems A or B) in Fig. 2 arewhere |hi(v,u)|2 denotes the intensity PSF of the illuminating part or collecting part. For simplifying the analysis, a variable ρ2 = t + 0.5 is introduced, the pinhole is assumed to be an ideal point pinhole, and P1(ρ) = 1. Thus, Eq. (1) can be rewritten aswhere Qi(t) is the phase pupil function Pi(ρ) written as a function of t, and Q1(t) = 1.Based on [19], in the case of a function Q2(t), which is not symmetrical around t = 0, i.e., Q2(t) ≠ Q2(−t), an axial focus shift occurs for the collecting part of the SSHCM. In addition, we assume that we have two phase filters whose transmission functions Q2A(t) and Q2B(t) satisfy Q2A(t) = Q2B(−t) or Q2A(t) = Q*2B(t), where * indicates the conjugate operation. For the two cases, a symmetrical axial focal shift of the confocal system A and B may be observed.
As shown in Fig. 2, the SSHCM follows a dual-measurement light-path arrangement. Two pinholes are placed at the geometrical focal plane positions of the collection lenses L2A and L2B. The intensity responses of the detectors A and B were the same before two pupil filters were added to the collection parts of the confocal systems A and B, i.e., IA (u,0) = IB(u,0). When the two designed pupil filters are placed separately before the collection lenses L2A and L2B, we hope that the maximum of the two light intensity responses will be shifted symmetrically, i.e., IA(0,u) = IB(0,−u) and uFA = −uFB. Consequently, when the object is moved along the optical axis (i.e., v = 0), we can easily obtain the heterodyne response function Iout(0,u) from Eq. (5)
The advantage of this method is that accurate symmetry shifts can be achieved without fine adjustment of the pinhole offset. In this method, two pinholes are placed on the geometric focal plane of the collecting lens. When the designed phase pupil is added, the focal planes of the two collecting lenses will be shifted symmetrically, realizing defocused detection of the pinholes. Because the focus position is easy to find and the intensity response characteristics of the two measuring optical paths are identical before the pupil is added, the two pinholes can be adjusted to the same position by mutual reference. Therefore, compared to LHCP, the pinhole installation of this method is simpler and easier to implement.
3. Phase filter design
In the design, we use a three-zone binary phase pupil filter, as shown in Fig. 2(b)
where a0 is the phase factor to be fitted, t0 is the center position of phase ring in coordinate t, the value range of t is [−0.5, 0.5], and w is the width of the phase ring.The functions of the filter are: 1) enhance the axial and transverse resolutions, 2) shift the focus of the collection lens symmetrically along the axis, and 3) maintain the contrast in light intensity. In the following text, we present a case design for a phase filter that fulfills all the requirements mentioned above.
In this research, the influence of the pupil structure parameters on the super-resolution performance is analyzed first. Then, the super-resolution evaluation function is proposed to optimize the pupil's structural parameters. The super-resolution gains are useful tools for filter design. The calculate equations for these gains are given in [18,20]. Because the method that we propose makes use of the shifted-focus character of phase pupils, the position umax, where the axial intensity is approximately the maximum, must be found first by evaluating |h(0, u)|2 numerically from Eq. (1). Then, we calculate the super-resolving gains by using expansions of the axial and transverse intensities around this point. The second-order expansion of h(v, u) around umax can be expressed as
Then, the focus shift distance u0 from umax is calculated using Ledesma et al.’s Eqs. (4) and (8) [18], and the axial and transverse super-resolution factors GA and GT, and the Strehl ratio S are calculated using Eqs. (9)–(11) in [18]. Note that the focus position uF from the geometrical focus can be calculated accurately using Eq. (8).
In order to analyze the influence of the pupil's structural parameters on the collection part, let us assume a0 = 0.65. The focal position uF, as a function of t0, is shown in Fig. 3(a), where we have selected w = 0.1, 0.2, 0.3.
Fig. 3 (a) Focal shift uF versus t0 for a = 0.65 and w = 0.1, 0.2, 0.3; (b) Strehl rate S; (c) Axial gain GA; (d) Transverse gain GT.
It can be seen from Fig. 3(a) that asymmetric focus shift can be attained when the two pupil functions are symmetrical around t = 0, i.e., Q2A(t) = Q2B(−t). Then, S, GA, and GT will be as shown in Figs. 3(b)–3(d). Both GA and GT are greater than 1, which means that 3D super resolution can be achieved using this filter. The maximum value of S is 1, and the larger the S is, the smaller the light intensity attenuation caused by the filter will be.
From Figs. 3(b)–3(d), we note that S and GA are even-symmetrical with respect to t0 = 0, but GT is odd-symmetrical with respect to t0 = 0, i.e., the axial and transverse resolutions cannot be improved simultaneously when Q2A(t) = Q2B(−t). However, we note that axial and transverse super-resolutions (GA>1 and GT>1) can be achieved when t0 is less than zero. Thus, we choose t0<0 in this paper.
Next, we investigate the effect of a0 on the focal shift and super-resolution characteristics of the system. Based on the analysis above, we choose a set of parameters, w = 0.2, t0 = −0.2. The calculated uF, S, GA, and GT, as functions of a0, are shown in Fig. 4. For SSHCM, when uF is approximately equal to half the width of a half peak of the axial intensity PSF, i.e., |h2(0,u)|2, the linearity of the monotonic measuring interval of the system will be the best. This is because the linearity near the peak position of the axial characteristic curve is the worst. It is found from Fig. 4(a) that the focal shift uF is the maximum when a0 = ± 0.65. Furthermore, from Fig. 4(b), GT and GA are found to be symmetrical with respect to a0 = 0, which indicates that axial and transverse super-resolution can be attained simultaneously. Based on these symmetry properties, conjugate pupil filters are used in the system, i.e., Q2A(t) = Q*2B(t). Thus, we choose a0 = ± 0.65 in this paper. In addition, we note from Fig. 4(b) that the variation trend of S is contrary to that of GA and GT. In order to obtain a high intensity contrast when GA and GT are greater than 1, S should be greater than 0.4 [17,18].
Fig. 4 (a) Focal shift uF. (b) Axial and transverse gains and Strehl ratio versus a0. (c) Normalized axial intensity distribution of illuminating and collecting parts in confocal system A; (d) Normalized axial intensity curve for point objects with t0 = −0.2 and w = 0.2.
For a0 = 0.65, w = 0.2 and t0 = −0.2, the intensity PSFs, i.e., |h1(0,u)|2 and |h2(0,u)|2, and axial intensity IA(0,u) of the confocal system A are as shown in Fig. 4(c). It can be seen that the focus of the collection light paths is moved to uF when the filter is added in the collecting part. Then, the measurement curves of SSHCM are calculated using Eq. (5), as shown in Fig. 4(d), wherein the b1b2 section of the curve Iout(0,u) is the measurement interval of the system. The simulation results show that the SSHCM has a bipolar measurement range and an absolute zero. However, the peak of Iout is only one-third the peak of IA or IB, which indicates that the contrast of the intensity signals is low. In addition, the slope of the linear interval of Iout has not been increased compared with those of IA and IB, which indicates that the axial resolution has not been improved.
In order to obtain high axial resolution for the SSHCM, the pupil filter needs to be optimized further. The target evaluation function J(w,t0) is first established as:
where K is the slope of the b1b2 line segment of Iout. K is established by the application of differential calculus on Eq. (5), with respect to u:Because the axial resolution is approximately the same in the linear segment b1b2 of the measurement curve, as shown in Fig. 4(d), and the slope at u = 0 is used to represent the slope of the linear segment, Eq. (10) can be simplified further. Using the second-order expansion of IA(0,u) around u = 0, the slope K(0, u = 0) can be deduced as follows:
where * denotes the complex conjugate and In is the nth moment of the pupil function, defined asThe Strehl ratio S of the collect part of the confocal system A is calculated using Eq. (13)
Consequently, by changing the values of t0 and w, the maximum of J(w,t0) can be obtained under the condition S > 0.4. The corresponding w and t0 values are the optimization results. By setting a0 = 0.65 and varying w and t0, J(w,t0) is calculated using Eq. (9) and is shown in Fig. 5(a). The calculated results show that the maximum of J(w,t0) is at w = 0.48 and t0 = −0.18. Thus, the optimum results of the two pupil filters are Q1 (a0 = 0.65, w = 0.48, t0 = −0.18) and Q2 (a0 = −0.65, w = 0.48, t0 = −0.18).
Fig. 5 (a) Variation of target evaluation function J with w and t0; when a0 = −0.65, w = 0.48, t0 = −0.18, (b) Axial intensity PSFs of illuminating part, collecting part, and confocal system A; (c) Axial intensity response curves for point objects; (d) Transverse intensity PSFs of collecting part ; (e) Axial intensity response curves for a reflector.
In this case, IA, |h2D(0,u)|2, and |h2S(0,u)|2 are plotted as in Fig. 5(b), where |h2D(0,u)|2 and |h2S(0,u)|2 are intensity PSFs of the collecting parts for DCM and SSHCM. In this paper, the slope of the half-peak position of the normalized intensity curve represents the slope of the linear segment of the curve. The slopes of the half-peak positions D and S, as marked in Fig. 5(b), are kD = 0.134 and kS = 0.142, respectively. It shows that the axial intensity slope of the DCM collecting part is improved slightly by adding the pupil filter. At this time, the slope of the confocal system A is kA = 0.208, while the slope curve of Iout is K(0,0) = 0.391, as shown in Fig. 5(c). It shows that the axial resolution of SSHCM is increased twice, compared to that of DCM.
The calculated results of uF are 5.81. When the peak position of Iout is u = uF/2, the monotonic measuring interval of SSHCM has the best linearity. As shown in Fig. 5(c), the peak position of Iout is close to uF/2, indicating that the linear range of SSHCM is near the maximum. In Fig. 5(d), the dashed and solid curves are the transverse intensities of the collecting parts for DCM and SSHCM, respectively. It is seen that the width of the main lobe of the transverse intensity decreases by adding the filter, while the transverse gain GT is 1.165. Therefore, the transverse resolution of SSHCM is also improved in comparison with that of DCM. In addition, the Strehl ratio S = 0.543, which meets the design requirements.
In order to compare with the results of LHCP, the intensity response of SSHCM is calculated via Eqs. (5) and (14) under the condition that the measured object is a reflector [21], as shown in Fig. 5(e).
In Eq. (14), the symbol denotes the convolution operation. Figure 5(e) shows that the intensity sensitivity K is 0.567 and the shifted focus distance uF is 5.4. In [15], the intensity sensitivity K is 0.54 and the shifted focus distance uF is 5.21. Thus, it can be seen from the analysis above that our system offers better results than the method of shifted pinholes.
As expected, the pupil filters designed in this paper can realize symmetrical shifted focus. Compared to DCM and LHCP, the SSHCM can improve 3D resolution with high Strehl value. The dynamic measurement range of the system is also expanded further. However, in this method, the resolution is improved at the expense of signal to noise ratio. As a rule, a phase filter improves resolution accompanied by a decrease of the Strehl ratio and an increase of the sidelobe intensity. Therefore, when optimizing pupil, the Strehl ratio should be kept as high as possible.
4. Experiments
Figure 6 shows the experimental setup corresponding to the measurement principle shown in Fig. 2. The laser source used was a semiconductor laser with wavelength of 632.8 nm (Thorlabs Co., USA, LPD-PM635-FC). In addition, a microscope objective L1 (OLYMPUS Co., Japan) of 20 × magnification, 0.4 NA and collecting lenses L2A and L2B of 10 × magnification, 0.25 NA were also used. The intensity response was recorded using two photomultiplier tubes (PMTs) produced by Thorlabs Co., and the pinholes produced by Newport Co. were 10 μm in diameter. In the experiment, the measured object was a plane mirror, and a P-753.1CD piezoelectric ceramic (PI GmbH & Co. KG, Germany) displacement platform was used in the axial and transverse characteristic experiments.
Figure 7(a) shows the axial response curves of the confocal system A, before and after the pupil filter was added to the optical path. The light intensity signal was weakened after adding the pupil filter. In order to observe the influence of the pupil on the linear interval slope k, the two light intensity curves were unified to 1.5 V, and the slope kC and kB were represented by the slope of the half-peak positions C and B, as marked in Fig. 7(a). The experiment results show that the slopes kC and kB, before and after the pupil filter is added to system A, are 0.625 and 0.659, respectively. It indicates that the designed filters can slightly improve the axial resolution of the system A. In addition, the focus position is moved by 1.6 μm and the corresponding optical coordinate is u = uF/2 = 2.65, which is consistent with the theoretical value.
Fig. 7 (a) Axial intensity response (IA) of confocal system A before and after the pupil is added; (b) Axial intensity responses of DCM, SSHCM, and LHCP.
In the actual measurement, because the system is inevitably affected by the beam splitter’s splitting ratio, dark current, etc., unequal amplitudes and DC biases between the intensity responses of the two confocal systems A and B are observed, which lead to the reduction of the axial resolution of the SSHCM. Therefore, before adding the pupil filter, the gains and DC bias voltages of the two signals need to be adjusted to make them the same.
In Fig. 7(b), IDCM, Iout, and ILHCP are axial intensity responses of DCM, SSHCM, and LHCP, respectively. The slope K(0,0) of Iout is 1.204. It can be observed that the slope of Iout is twice that of DCM. Therefore, the axial resolution is improved further. Moreover, compared to LHCP, the slope and range of the measured curve are improved slightly. In addition, the peak value of Iout is the same as that of IA and IB, and the peak value of the sidelobe decreases obviously compared to that of IA, which indicates that the SSHCM system can maintain a high signal contrast.
To evaluate the stability of the proposed method, against environmental disturbances and vibrations, sixty measurements were made with two conjugate phase pupils in 30 minutes at a time interval of 30 s, as shown in Fig. 8. The standard deviations of the voltage signals for the two confocal systems (IA and IB) were 19 mV and 21 mV, respectively, and the standard deviation of the differential output voltage (Iout) signal was 10 mV, which meant that a high stability could be achieved by subtraction.
Additionally, the axial resolution of the SSHCM system was also tested by moving the plane mirror using a piezoelectric ceramic actuator. Figure 9 shows the response of the system to 2 nm and 4 nm step distances of the reflector, with 200 measurements taken at each position. The measurement results indicate that a resolution of 4 nm can be achieved by the system.
To evaluate SSHCM’s transverse resolution, we employed the SSHCM and LHCP systems to scan the straight edge of a Standard step STEP-OX-0.2-M-Q sample coated with a Cr reflecting film (the base material was SiO2), as shown in Fig. 10. In the experiment, the standard step was mounted on the piezoelectric ceramic P753.1cd, and the piezoelectric ceramic moved along the transverse direction at a step distance of 0.1 μm. The response curve of the straight edge obtained using SSHCM was sharper than that obtained using LHCP; the slopes of ABS and ABL sections in Fig. 10 were 1.09 and 0.89, respectively. It indicated that the SSHCM could improve the transverse resolution further.
However, the method in this paper also has some limitations. First, due to using the phase filter, the system imaging contrast is decreased, and it results in the resolution of the system decreased. In practice, the effect of contrast reduction can be suppressed by increasing the intensity of the light source and amplifying the detection signal. Second, when the objective lens is changed, the phase filter also needs be changed. It will cause inconvenience to use the method. In our future studies, the method can be used more flexibly by introducing a spatial light modulator (SLM).
5. Conclusions
In this study, we proposed a symmetric shifted-focus heterodyne confocal measurement method. By adding a pair of conjugate-phase pupil filters in the same two confocal systems, symmetrical shift of the axial characteristic curve was realized and the monotonic measuring curve of the axial position could be obtained by the differential reduction of the two intensity signals. The experiments and simulations showed that the method could obtain high axial position resolution with strong anti-interference ability and high stability. In addition, the method could also improve the transverse resolution of the system, and the light energy loss was controllable. The conjugate phase pupil was easy to fabricate, and the system was simple and easy to adjust, compared to LHCP. The method can also be used for the coplanar detection of two light intensity signals and is suitable for miniaturization of the system. Furthermore, the method can expand the application scope of CM, such as in 3D rapid measurement of discontinuous surfaces of microstructural devices with large steps. If a super-resolution pupil filter is added to the illumination part of the SSHCM system, the three-dimensional resolution of the system can be further improved.
Funding
National Natural Science Foundation of China (51475111, 51775148); State Administration for Science, Technology and Industry for National Defense (JSJL2014603A002); Natural Science Foundation of Heilongjiang Province (QC2018079).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
References
1. C. Zhang, C. Min, L. Du, and X.-C. Yuan, “Perfect optical vortex enhanced surface plasmon excitation for plasmonic structured illumination microscopy imaging,” Appl. Phys. Lett. 108(20), 201601 (2016). [CrossRef]
2. D. Wang, D. Meza, Y. Wang, L. Gao, and J. T. C. Liu, “Sheet-scanned dual-axis confocal microscopy using Richardson-Lucy deconvolution,” Opt. Lett. 39(18), 5431–5434 (2014). [CrossRef] [PubMed]
3. H. Yu, T. Chen, and J. Qu, “Improving FRET efficiency measurement in confocal microscopy imaging,” Chin. Opt. Lett. 8(10), 947–949 (2010). [CrossRef]
4. E. D. Lemma, B. Spagnolo, S. Sergio, M. Pisanello, M. De Vittorio, F. Pisanello, E. D. Lemma, M. Pisanello, M. De Vittorio, and S. Sergio, “Confocal imaging characterization of two-photon lithography microstructures for cell cultures,” in Proceedings of 2017 IEEE 17th International Conference on Nanotechnology (IEEE-NANO, 2017), pp. 720–724.
5. Z. Li, K. Herrmann, and F. Pohlenz, “Lateral scanning confocal microscopy for the determination of in-plane displacements of microelectromechanical systems devices,” Opt. Lett. 32(12), 1743–1745 (2007). [CrossRef] [PubMed]
6. D.-R. Lee, Y.-D. Kim, D.-G. Gweon, and H. Yoo, “High speed 3D surface profile without axial scanning: dual-detection confocal reflectance microscopy,” Meas. Sci. Technol. 25(12), 125403 (2014). [CrossRef]
7. N. Koukourakis, M. Finkeldey, M. Stürmer, C. Leithold, N. C. Gerhardt, M. R. Hofmann, U. Wallrabe, J. W. Czarske, and A. Fischer, “Axial scanning in confocal microscopy employing adaptive lenses (CAL),” Opt. Express 22(5), 6025–6039 (2014). [CrossRef] [PubMed]
8. C.-H. Lee and J. Wang, “Noninterferometric differential confocal microscopy with 2-nm depth resolution,” Opt. Commun. 135(4–6), 233–237 (1997). [CrossRef]
9. C.-H. Lee, W.-C. Lin, and J.-Y. Wang, “Using differential confocal microscopy to detect the phase transition of lipid vesicle membranes,” Opt. Eng. 40(10), 2077–2083 (2001). [CrossRef]
10. L. Liu, X. Deng, L. Yang, G. Wang, and Z. Xu, “Effect of an annular pupil filter on differential confocal microscopy,” Opt. Lett. 25(23), 1711–1713 (2000). [CrossRef] [PubMed]
11. D.-R. Lee, Y. D. Kim, D. G. Gweon, and H. Yoo, “Dual-detection confocal fluorescence microscopy: fluorescence axial imaging without axial scanning,” Opt. Express 21(15), 17839–17848 (2013). [CrossRef] [PubMed]
12. W. Zhao, Y. Sun, Y. Wang, L. Qiu, R. Shao, and H. Cui, “Three-dimensional super-resolution correlation-differential confocal microscopy with nanometer axial focusing accuracy,” Opt. Express 26(12), 15759–15768 (2018). [CrossRef] [PubMed]
13. W. Zhao, C. Liu, and L. Qiu, “Laser divided-aperture differential confocal sensing technology with improved axial resolution,” Opt. Express 20(23), 25979–25989 (2012). [CrossRef] [PubMed]
14. 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] [PubMed]
15. W. Zhao, J. Tan, L. Qiu, and L. Zou, “A new laser heterodyne confocal probe for ultraprecision measurement of discontinuous contours,” Meas. Sci. Technol. 16(2), 497–504 (2005). [CrossRef]
16. Y. Ma, C. Kuang, W. Gong, L. Xue, Y. Zheng, Y. Wang, K. Si, and X. Liu, “Improvements of axial resolution in confocal microscopy with fan-shaped apertures,” Appl. Opt. 54(6), 1354–1362 (2015). [CrossRef] [PubMed]
17. D. M. de Juana, J. E. Oti, V. F. Canales, and M. P. Cagigal, “Design of superresolving continuous phase filters,” Opt. Lett. 28(8), 607–609 (2003). [CrossRef] [PubMed]
18. S. Ledesma, J. Campos, J. C. Escalera, and M. J. Yzuel, “Simple expressions for performance parameters of complex filters, with applications to super-Gaussian phase filters,” Opt. Lett. 29(9), 932–934 (2004). [CrossRef] [PubMed]
19. S. Ledesma, J. Campos, J. Escalera, and M. Yzuel, “Symmetry properties with pupil phase-filters,” Opt. Express 12(11), 2548–2559 (2004). [CrossRef] [PubMed]
20. C. J. R. Sheppard, S. Ledesma, J. Campos, and J. C. Escalera, “Improved expressions for performance parameters for complex filters,” Opt. Lett. 32(12), 1713–1715 (2007). [CrossRef] [PubMed]
21. M. Gu, Principles of Three-Dimensional Imaging in Confocal Microscopes (World Scientific, 1996), Chap. 3.
