Improved spatial resolution using focal modulation microscopy with a Tai Chi aperture

Ao Deng; Ao Deng; Yameng Zheng; Yameng Zheng; Jiajia Chen; Jiajia Chen; Yao Zheng; Yao Zheng; Wei Gong; Wei Gong; Ke Si; Ke Si; Ke Si; Ke Si

doi:10.1364/OE.426600

1. Introduction

The advent of confocal microscopy (CM) [1,2] has greatly changed optical imaging in biomedical research. Its sub-cellular spatial resolution and optical sectioning capability eliminate the background caused by defocusing and scattering [3]. In confocal microscopy, a small pinhole is introduced to the detection plane as a spatial filter to block out-of-focus light. However, this selective detection mechanism becomes less effective when a thick sample is used, or the focus point is moved deeper into the sample. Multiple scattered photons eventually leak through the pinhole [4–6], thereby eroding the image contrast and signal noise ratio [7,8]. Several technologies have emerged to provide new tools for biomedical imaging, such as two-photon nonlinear fluorescence microscopy (2PM) [9–11], coherent optical tomography (OCT) [12–14] and photoacoustic tomography (PAT) [15–17]. While many of these approaches have been used for biomedical imaging, none are yet ideal. For instance, highly optimized PAT could enable high-resolution visualization of fluorescent proteins a few millimeters deep within living organisms. However, PAT has diffraction-limited resolution and even worse axial resolution. It is fundamentally limited by the effective bandwidth of the ultrasonic detector (20 MHz) [18]. There is a considerable gap in optical microscopy development that has high penetration depth and subcellular spatial resolution. Several gating methods combined with CM have been proposed to achieve the problem, including spatial gating methods [4], polarization gating methods [19], time gating methods [20] and coherence gating methods [13]. As one of the gating methods, focal modulation microscopy (FMM) [21] is a promising candidate for light-scattering in vivo imaging. Currently, the imaging depth of FMM has reached up to 600 $\mathrm{\mu}$m experimentally [21–23] in biological samples. By extending focal modulation technology into 2PM, the imaging depth penetration of two-photon focal modulation microscopy (2PFMM) is increased by a factor of 3 [24]. The performance in thick samples experimentally shows that the signal-to-background ratio of 2PFMM can be improved up to five times of 2PM at the depth of 500 $\mathrm{\mu}$m [25].

In FMM, the illumination light is divided into several parallels (two beams in most cases) but non-overlapped lights. A phase modulator is used to modulate half of the illumination light periodically. The objective lens focuses the lights at the focal volume, and the interference of half beams result in a periodically modulated fluorescence signal from the focal volume. A lock-in technique is implemented to demodulate the fluorescent signal from the ballistic light and reject the background signal caused by scattering because the out of focus signal cannot be modulated. The most common FMM system uses two non-overlapping D-shaped apertures (DFMM), which destroys the system’s spatial resolution symmetry. The difference in the transverse resolution of the two orthogonal directions of the DFMM system [23] leads to confusion in image interpretation. FMM with uniform imaging resolution has less previously been reported [26].

Here, we propose a theoretical model for TCFMM based on scalar diffraction theory. The three-dimensional (3D) optical transfer function (OTF) is presented using the Tai Chi aperture and quasi Tai Chi aperture. The spatial resolution, including both axial and transverse resolution and background rejection capability, is also investigated. We discuss the relationship between the geometry shape of Tai Chi aperture and the transverse resolution in two directions.

2. Theory

The system diagram of focal modulation microscopy is shown in Fig. 1. The laser is expanded by the beam expander (LBM), and then the expanded beam is split into two spatially separated and modulated beams by the spatial phase modulator (SPM). As shown in Fig. 1, the two aperture designs of SPM, Tai Chi shape, and D shape are applied to FMM. After passing through the objective lens, the beams interfere and result in an intensity modulation at the focal volume. The modulated beam goes through a dichroic mirror and a filter, and then the fluorescence light is collected by the collection lens into the photomultiplier tubes (PMT) with a pinhole. Finally, the image signal is demodulated from a lock-in amplifier.

Fig. 1. System diagram of focal modulation microscopy and geometry of the aperture pattern. LBE, laser beam expander; L, lens; SPM, spatial phase modulator; A, aperture; DM, dichroic mirror; LF, long-pass filter; CL, collection lens; PH, pinhole; PMT, photomultiplier tube; OL, objective lens; S, sample.

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In our simulation, the imaging results of FMM are analyzed using the scalar diffraction theory with paraxial approximation. To simplify the simulation, we assume that the wavelength of incident light and the emitted fluorescence is the same. Therefore, the corresponding 3D time-varying image of a point object with a point detector can be described as [22,23]

(1)$$I=\left ( \left | h_{1A} \right |^{2} +\left | h_{1B} \right |^{2}\right )\left | h_{2} \right |^{2}+\mathrm{Re}\left ( h_{1A}h_{1B}^{*} \right )\left | h_{2} \right |^{2}\mathrm{cos}\left ( 2\delta \omega t \right )+\mathrm{Im}\left ( h_{1A}h_{1B}^{*} \right )\left | h_{2} \right |^{2}\mathrm{sin}\left ( 2\delta \omega t \right )$$

where $h_{1A}$ and $h_{1B}$ are the 3D amplitude point spread functions (APSF) of the illumination apertures, and $h_{2}$ is the APSF of the collection lens. $\mathrm {Re}$ and $\mathrm {Im}$ represent the real and imaginary parts respectively, and $*$ denotes the conjugation of the parameter. $2\delta \omega t$ is the time-varying relative phase shift of two beams. In the manuscript, we use $\omega +\delta \omega$ and $\omega -\delta \omega$ to describe the phase difference between the two parts of the aperture, as shown in Fig. 1. According to the scalar diffraction theory, the APSF is defined as

(2)$$\begin{aligned} h_{1A,1B,2}\left ( \nu_{x},\nu_{y},u \right )= & \iint_{}^{}P_{1A,1B,2}\left ( \rho_{x},\rho_{y} \right )\mathrm{exp}\left ( iu\left ( \rho _{x}^{2}+ \rho _{y}^{2} \right ) / 2 \right )\\ & \mathrm{exp}\left ({-}i\left ( \rho_{x}\nu _{x}+ \rho_{y}\nu _{y}\right ) \right )\mathrm{d}\rho_{x}\mathrm{d}\rho_{y} \end{aligned}$$

where $P_{1A}$ and $P_{1B}$ are the pupil function of the illumination lens, $P_{2}$ is the pupil function of the collection lens. Here $\nu _{x}$, $\nu _{y}$ and $u$ are functions of the distance $x$, $y$, $z$ from a point in space to the focal point, expressed as $\nu _{x}=\left ( 2\pi /\lambda \right )xn_r\mathrm {sin}\alpha$, $\nu _{y}=\left ( 2\pi /\lambda \right )yn_r\mathrm {sin}\alpha$, $u=\left ( 8\pi /\lambda \right )zn_r\mathrm {sin}^{2}\left ( \alpha /2 \right )$, respectively. $\lambda$ donates the excitation or emission wavelength, $\alpha$ is the semi-angular aperture of the lens, and $n_r$ refers to the refractive index of the immersion medium. The distances in the pupil plane $\rho _{x}$ and $\rho _{y}$ is normalized by the pupil radius of $r$. It is worth noting that these expressions are not only applicable to TCFMM and DFMM but also other FMMs with non-overlapping apertures $P_{1A}$ and $P_{1B}$. The in-phase signal demodulated from the lock-in amplifier can be expressed as [22]

(3)$$I_{in}=\mathrm{Re}\left ( h_{1A}h_{1B}^{*} \right )\left | h_{2} \right |^{2}$$

Equation (3) determines that the image formation is linear in intensity, i.e. Eq. (3), the image of a point object, can be regarded as an intensity point spread function (IPSF). The lock-in amplifier can demodulate two kinds of signals. The other signal comes from the imaginary part of Eq. (1) [27]. In this simulation, only the in-phase signal is used.

3. Spatial resolution

The 3D OTF for TCFMM is given by the 3D Fourier transform of IPSF, which can be expressed as [23]

(4)$$C\left ( n,m,s \right )=\mathfrak{F}_{3}\left [ \mathrm{Re}\left ( h_{1A}\left ( \nu_{x},\nu_{y},u \right )h_{1B}^{*}\left ( \nu_{x},\nu_{y},u \right ) \right )\left ( \left |h_{2}\left ( \nu_{x},\nu_{y},u \right ) \right |^{2}\otimes_{2}\mathrm{D}\left ( \nu_{x},\nu_{y} \right ) \right ) \right ]$$

where $n$,$m$ are the spatial frequency of the radial spatial frequency normalized by $n_{r}\mathrm {sin}\alpha /\lambda$, and $s$ is the spatial frequency of axial spatial frequency normalized by $n_{r}\mathrm {sin}^{2}\left ( \alpha /2 \right )/\lambda$. $\mathrm {D}\left ( \nu _{x},\nu _{y} \right )$ is the intensity sensitivity of the detector, and a circular pinhole mask with a normalized radius $\nu _{d}$ is placed in front of the detector. $\nu _{d}=0$ is a point detector used in the system and $n_{r}$ is the refractive index of the immersion medium. $\mathfrak {F}_{3}$ donates the 3D Fourier transform operation and $\otimes _{2}$ donates the 2D convolution operation.

Figure 2 compares the 3D OTFs of confocal microscopy (CM) with those of focal modulation microscopy with D-shaped apertures (DFMM) and focal modulation microscopy with Tai Chi apertures (TCFMM) for various values of detector radii $\nu _{d}$. Each 3D OTF is normalized to maximum value of $C(n,m,s)$. The cut-off spatial frequencies of CM and FMM in the transverse and axial directions are 4 and 1, respectively. However, the 3D OTF of DFMM and TCFMM is broader than that of CM in $m$ and $n$ directions, which means DFMM and TCFMM systems have a better response at high spatial frequencies than CM systems. This phenomenon becomes more evident when $\nu _{d}$ increases. Therefore, an enhanced spatial resolution can be achieved by TCFMM. Moreover, the OTF of DFMM is much broader in the vertical ($n$) direction than in the horizontal ($m$) direction because of the asymmetric geometry of D-shaped apertures, while the OTF of TCFMM behaves nearly the same in both directions. As $\nu _{d}$ increases, the difference between two transverse directions is still relatively small in TCFMM compared with DFMM, resulting in better uniform imaging in both directions.

Fig. 2. Three-dimensional optical transfer functions for (a)CM with a point detector in the horizontal direction;(b) CM with a point detector in vertical direction; (c)CM with $\nu _{d}=2$ in the horizontal direction; (d) CM with $\nu _{d}=2$ in the vertical direction; (e) DFMM with a point detector in horizontal direction; (f) DFMM with a point detector in vertical direction; (g) DFMM with $\nu _{d}=2$ in horizontal direction; (h) DFMM with $\nu _{d}=2$ in vertical direction; (i) TCFMM with a point detector in horizontal direction; (j) TCFMM with a point detector in vertical direction; (k) TCFMM with $\nu _{d}=2$ in the horizontal direction; (l) TCFMM with $\nu _{d}=2$ in the vertical direction.

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In order to show the 3D OTF more intuitively, we draw the transverse cross-sections of the OTF of three systems, as shown in Fig. 3. The cross-section in $m$ direction of TCFMM is greater than that of DFMM. It can be seen that the cross-section of DFMM in $n$ direction shows an upward trend for normalized spatial frequency smaller than 0.4. When the normalized spatial frequency is greater than 0.4, the cross-section shows a downward trend. This is because the peak of OTF in $n$ direction is not at the center point. From Fig. 2(f), we can see that the OTF of DFMM has two peaks in $n$ direction. The phenomenon of double peaks results from the introduction of the spatial phase modulator. The two illumination pupils do not overlap because of the geometry design of the spatial phase modulator. And the phenomenon becomes more distinct as $\nu _{d}$ increases. It is worth noted that the gap between the curves of TCFMM in the two directions is much smaller than those of DFMM, implying minimal differences in transverse resolution. For instance, the OTF difference is 41.0$\%$ for DFMM and 3.2$\%$ for TCFMM when normalized spatial frequency is 1, as $\nu _{d}=2$. The smaller the difference in transverse resolution, the less likely it is to cause imaging confusion on the focal plane.

Fig. 3. Transverse cross-section of OTF with a point detector in (a) horizontal direction and (b) vertical direction; Transverse cross-section of OTF with $\nu _{d}=2$ in (c) horizontal direction and (d) vertical direction.

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The axial resolution of CM is normally characterized by the axial response of the system to an infinitely-thin fluorescent layer. The axial response is proportional to the inverse Fourier transform of the axial section of the 3D OTF in the s direction. Therefore, the axial cross-section of 3D OTF can express the spatial resolution in the axial direction. The axial cross-sections of 3D OTF of CM, DFMM, TCFMM are shown in Figs. 4(a) and (c). The axial cross-sections, $C(0,0,s)$, are discussed for point detector and finite-size detector, respectively. The cross-section is normalized to the max value of the curve in the simulation. We can see that the axial cut-off frequencies of $C(0,0,s)$ of the three systems are all 1. When $\nu _{d}=0$ , $C(0,0,s)$ of TCFMM and DFMM overlap with each other, but both of them are wider than CM. When $\nu _{d}$ increases, $C(0,0,s)$ of TCFMM is slightly broader than DFMM, indicating that TCFMM responds better at high frequencies in the axial direction.

Fig. 4. Axial cross-section of OTF for (a) a point detector and (c) a finite size detector with $\nu _{d}=2$; Intensity of thick fluorescent layer for (b) a point detector and (d) a finite size detector with $\nu _{d}=2$.

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The thick uniform fluorescent layer image in the axial direction is also an optional model to characterize the axial resolution [28]. Besides, a thick fluorescent sample is easier to prepare and implement in practice. The axial response is the Fourier transform of 3D OTF. The narrower the axial response, the smaller the crosstalk between two adjacent sections of the image, and therefore the higher the axial resolution [29]. The intensity of the thick fluorescent layer can be expressed as [23]

(5)$$I\left ( u \right )=\frac{1}{2}+\frac{1}{\pi}\int_{1}^{s_{c}}C\left ( m=0,n=0,s \right )\frac{\mathrm{sin}\left ( us \right )}{s}\mathrm{d}s$$

where $s_{c}$ is the cut-off spatial frequency in the axial direction. Here $s_{c}=1$. Figures 4(b) and (d) illustrates the image intensity of the thick fluorescent layer in CM, DFMM, TCFMM with a point detector and finite-size detector, respectively. It can be seen that DFMM and TCFMM have a narrower axial response than CM. It means they have better axial resolution than CM, regardless of the size of the detector. Besides, for point detectors, the axial resolution of TCFMM is slightly better than that of DFMM, and this phenomenon becomes more evident as $\nu _{d}$ increases. For instance, when $\nu _{d}=2$, the mean gradient of axial intensity response is improved by 26.91$\%$ and 29.67$\%$ for DFMM and TCFMM, respectively, compared with CM.

The transverse resolution can be investigated not only by the full width at half maximum (FWHM) of the point spread function, but also in the similar way above. We consider the image of a straight, sharp edge in the transverse direction [30]. Since neither TCFMM nor DFMM has symmetry in both directions, we take the sharp edge oriented in two orthogonal directions, $x$ and $y$, into consideration. In this paper we consider the image intensity of a thick edge and then move to the image intensity of a thin one. The image intensity of a thick, straight, and sharp edge can be expressed as [23]

(6)$$I_{y}\left ( \nu \right )=\frac{1}{2}+\frac{1}{\pi}\int_{1}^{l_{c}}C\left ( m,n=0,s=0 \right )\frac{\mathrm{sin}\left ( \nu m \right )}{m}\mathrm{d}m$$

(7)$$I_{x}\left ( \nu \right )=\frac{1}{2}+\frac{1}{\pi}\int_{1}^{l_{c}}C\left ( m=0,n,s=0 \right )\frac{\mathrm{sin}\left ( \nu n \right )}{n}\mathrm{d}n$$

where $l_{c}$ is the transverse cut-off spatial frequency. The image intensity of a thick, straight, and sharp edge in CM, DFMM, and TCFMM are shown in Figs. 5(a) and (c). The value is normalized by the value far from the edge. The intensity at the edge is 1/2 of its value far from the edge, as is expected for incoherent imaging in confocal.

Fig. 5. Images of a thick, straight, and sharp edge in CM, DFMM, and TCFMM for (a) a point detector and (c) a finite size detector with $\nu _{d}=2$; Images of a thin, straight and sharp edge in CM, DFMM and TCFMM for (b) a point detector and (d) a finite size detector with $\nu _{d}=2$. PSFs in (e) vertical and (f) horizontal directions for CM, DFMM and TCFMM as $\nu _{d}=2$.

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Systems with point detectors and finite-size detectors are also discussed, respectively. For DFMM, the mean gradient of the image intensity is improved from 45.86$\%$ for a point detector to 72.64$\%$ for $\nu _{d}=2$ compared with CM, when the edge is oriented in x direction. However, When the edge is oriented in y direction, the improvement of the mean gradient is 10.37$\%$ and 12.42$\%$ for a point detector and $\nu _{d}=2$. Due to the geometric asymmetry of the aperture, DFMM has a higher resolution in y direction, but the resolution in x direction is less different from the CM. The asymmetry of the resolution in DFMM sometimes causes uneven imaging because it is clearer in one direction than the other. Unlike DFMM, the intensity response of TCFMM in two directions can be maintained at a comparable level. When the edge is oriented in x direction, the mean gradient of the image is improved from 39.31$\%$ for a point detector to 61.61$\%$ for $\nu _{d}=2$. Besides, improvement of the mean gradient is 26.51$\%$ and 41.37$\%$ for a point detector and $\nu _{d}=2$ when the edge is oriented in y direction. It is indicated that a high transverse resolution is achieved in TCFMM. Meanwhile, the transverse resolution of DFMM in both directions almost maintain at the same level. As $\nu _{d}$ increases, the phenomenon of asynchronous improvement of the resolution in DFMM becomes more evident, while the improvement in TCFMM still is maintained at a comparable level. For instance, the intensity difference between $I_{x}$ and $I_{y}$ for DFMM and TCFMM is 12.49$\%$ and 1.76$\%$ when the spatial frequency is 2, respectively as $\nu _{d}=2$. It can be seen that $I_{x}$ of DFMM becomes sharper as $\nu _{d}$ increases, but the intensity of the hump near the edge also increases. The hump causes a negative intensity due to the modulation and demodulation process. However, the intensity of the hump near the edge in TCFMM is significantly reduced compared with DFMM, which improves the image performance.

For a thin, straight, and sharp edge, the intensity response can be expressed as [23]

(8)$$I_{y}\left ( \nu \right )=\frac{1}{2}+\frac{1}{\pi}\int_{1}^{l_{c}}C_{2}\left ( m,n \right )\frac{\mathrm{sin}\left ( \nu m \right )}{m}\mathrm{d}m$$

(9)$$I_{x}\left ( \nu \right )=\frac{1}{2}+\frac{1}{\pi}\int_{1}^{l_{c}}C_{2}\left ( m,n \right )\frac{\mathrm{sin}\left ( \nu n \right )}{n}\mathrm{d}n$$

where $C_{2}\left ( m,n \right )$ is the projection of 3D OTF $C\left ( m,n,s \right )$ at the focal plane. Figures 5(b) and (d) shows the intensity response of a thin, straight, and sharp edge in TCFMM, DFMM, and CM, with a point detector and a finite size detector, respectively. Similarly, when the edge is oriented in y direction, the intensity response of TCFMM is sharper than that of DFMM, while the intensity response of DFMM overlaps that of CM. The mean gradient improvement is 0.86$\%$ for DFMM and 8.95$\%$ for TCFMM. Both TCFMM and DFMM has better intensity response for edge oriented in x direction compared with CM. And the mean gradient is 26.82$\%$ and 20.33$\%$, respectively. The difference in resolution and the hump near the edge still become evident in DFMM as $\nu _{d}$ increases. The resolution of TCFMM is maintained at a comparable level, and the hump near the edge is also reduced, indicating it has better image quality.

As a conventional evaluation criterion, we also give the comparison of PSF between DFMM, TFCMM, and CM. Figures 5(e) and (f) shows the PSF of the three systems in two lateral directions. In the vertical direction, the PSF of CM and DFMM overlap each other. Compared with CM and DFMM, the improvement of PSF for TCFMM is 9$\%$. In the horizontal direction, DFMM has the narrowest PSF. The PSF of TCFMM is broader than that of DFMM but narrower than that of CM. The improvement of PSF is 10$\%$ for TCFMM and 20$\%$ for DFMM. The resolution improvement of TCFMM and DFMM is consistent with the results discussed above.

In summary, TCFMM has a higher transverse resolution than CM and a smaller resolution difference between two different directions than DFMM, regardless of the size of the detector.

4. Background rejection capability

As a parameter related to the background suppression capability, the imaging penetration depth is also an essential factor affecting the application scenarios of the optical system. To characterize the imaging penetration depth of an optical system, we introduce a scattering model named integrated intensity here. In this model, only a single scattering is considered [22,30]. If the intensity image of a point is $I\left ( \nu ,u \right )$ and its distance from the focal plane is $u$, then its contribution to the background intensity is [22]

(10)$$I_{int}\left ( u \right )=\int_{0}^{\infty }I\left ( \nu ,u \right )\nu \mathrm{d}\nu .$$

Further, we also discuss the changes of background signal by using CM, DFMM, TCFMM along the distance $u$. For a thick object focused at the top with the thickness of $u_0$, the total background detected can be given by [22]:

(11)$$I_{bgd}\left ( u_0 \right )=\int_{0}^{u_0}I_{int}(u) \mathrm{d}u.$$

Fig. 6. Integrated intensity of CM, DFMM, and TCFMM with (a) a point detector and (c) a finite size detector with $\nu _{d}=2$; Integrated intensity after logarithm to base 10 of CM, DFMM, and TCFMM with: (b) a point detector and (d) a finite size detector with $\nu _{d}=2$. The total background normalized by intensity at focal point for CM, DFMM and TCFMM at different values of (e) a point detector ($\nu _{d}=0$); (f) a finite size detector with $\nu _{d}=2$.

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The integrated intensity of CM, DFMM, and TCFMM with various detector sizes is shown in Figs. 6(a) and (c). The sharper the curve, the stronger the background suppression ability and, therefore, the higher the imaging penetration depth. Note that TCFMM and DFMM have sharper integrated intensity than CM, which means their superior imaging penetration depth. Moreover, the integrated intensity of TCFMM is slightly narrower than that of DFMM. For instance, when $\nu _{d}=2$, the background signal is reduced to 81.11$\%$ and 77.26$\%$ for DFMM and TCFMM, respectively. In our simulation, the signal-to-background ratio (SBR) is defined as the ratio between the signal at the focal point and the back-ground scattering [31]. When $\nu _{d}=2$, the SBR is 41.66, 51.09, and 51.35 for the CM, DFMM, and TCFMM cases, respectively, indicating an improvement for TCFMM. And the improvement of SBR is 22.64$\%$ and 23.26$\%$ for DFMM and TCFMM respectively. To clearly show the difference between these three systems, a logarithm of 10 is taken on the horizontal axis and vertical axis of the integrated intensity curve, shown in Figs. 6(b) and (d). The total background detected by CM, DFMM and TCFMM for $\nu _{d}=0$ and $\nu _{d}=2$ is shown in Fig. 6, respectively. The intensity is normalized at the focal point. It can be noticed that compared with CM, the background signal can be suppressed by either DFMM or TCFMM. For instance, when $\nu _{d}=0$,given $u_0=30$, the background signal is reduced to 68.7% and 61.2% for DFMM and TCFMM compared with CM, respectively. When the detector size increase to $\nu _{d}=2$, the background signal is further reduced to 52.3% and 48.7%, respectively. As $\nu _{d}$ increase, the background rejection ability of the three systems is reduced, but the relationship between the strength of the background rejection ability remains unchanged.

5. Image performance

To examine the imaging performance of CM, DFMM, and TCFMM with a finite size detector ($\nu _{d}=2$), we use a sample of simulated fluorescent beads shown in Fig. 7(a). Figures 7(b)-(d) illustrate the images of CM, DFMM, and TCFMM in the transverse plane, respectively. Figure 7(e) and Fig. 7(f) are the intensity distribution along the vertical and horizontal white dashed line labeled in Figs. 7(b)-(d), respectively.

Fig. 7. Simulation results of a sample with (a) the fluorescent beads image. Image of the sample in the transverse direction for (b) CM, (c) DFMM and (d) TCFMM. Signal profile along the white dashed lines (b-d) in (e)vertical and (f)horizontal direction.

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It can be seen that TCFMM has better edge resolution than DFMM and CM in the vertical direction. TCFMM can significantly distinguish the edge located at 765 pixels in Fig. 7(e). Besides, the intensity of TCFMM is stronger than that of the other two microscopes. For instance, compared with CM, the intensity improvement is up to 8.39$\%$ and 24.01$\%$ for DFMM and TCFMM in Fig. 7(e). In terms of the horizontal direction, TCFMM can sometimes maintain a similar intensity level with DFMM. And both of them can achieve an improvement in resolution compared to CM. Moreover, It is practical that TCFMM is more suitable than DFMM for cases where the finite-size detector is used.

6. Discussion on the shape of the aperture and TCFMM imaging

The inconsistency of the transverse resolution of DFMM is related to the asymmetry of the aperture, so FMM using Tai Chi aperture can address the problem to a certain extent. Here, we change the height of the peaks of Tai Chi patterns and discuss the impact of these quasi-Tai Chi patterns on the resolution of TCFMM. There are three types of TCFMM used in the simulation, the conventional TCFMM, the TCFMM with a lower peak of Tai Chi aperture (TCFMM-L), and the TCFMM with a higher peak of Tai Chi aperture (TCFMM-H), as shown in Fig. 8(a).

Fig. 8. (a) Geometry of focal modulation microscopy with D-shape aperture (DFMM), focal modulation microscopy with Tai Chi Aperture (TCFMM), with Tai Chi aperture, which has lower peaks (TCFMM-L), and with Tai Chi aperture, which has higher peaks (TCFMM-H). (b, c) 3D OTF of CM in two transverse directions. (d, e) 3D OTF of DFMM in two transverse directions. (f, g) 3D OTF of TCFMM-L in two transverse directions. (h, i) 3D OTF of TCFMM in two transverse directions. (j, k) 3D OTF of TCFMM-H in two transverse directions. (l, m) Transverse cross-section of 3D OTF in horizontal and vertical directions.

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The 3D OTFs of CM, DFMM, and TCFMM with three types of Tai Chi apertures with different heights of peaks are shown in Figs. 8(b)-(k). It can be seen that the width of the OTF in the two transverse directions varies with the height of the quasi-Tai Chi pattern. As the height of the peak of the Tai Chi pattern increases, the OTF in the n direction will become narrower. On the contrary, the OTF in the m direction will become border. The transverse cross-sections of the OTF of the systems in the m direction and the n direction are shown in Fig. 8(l) and Fig. 8(m). It can be seen from the cross-section curve that TCFMM-H has the highest OTF in n direction, but the lowest OTF in m direction. Similar to OTF of DFMM in m direction, the peak of TCFMM-H is not at the center point in n direction. For finite-size detectors ($\nu _{d}=2$), the OTF difference is 40.97$\%$ , 37.29$\%$ , 3.28$\%$ , and 18.33$\%$ for DFMM, TCFMM-L, TCFMM, and TCFMM-H when normalized spatial frequency is 1. TCFMM still has the best performance among these systems with the minimum OTF difference. It is easy to understand why TCFMM-L has the largest OTF difference among the three TCFMM systems. If the peak of the Tai Chi pattern gradually decreases, then TCFMM becomes DFMM when it decreases to 0. Here, we only discuss three TCFMMs with different peak heights as examples, and we believe that there will be an ideal value of peak heights that can achieve the best resolution in both transverse directions using the same method as we proposed.

Besides, we investigate the OTFs of Tai Chi-shape, sine-shape, quadrant-shape, linear-shape and D-shape quantitatively, which are summarized in Table 1. It can be noticed that when the spatial frequency is 1, the OTFs of all shapes (Tai chi-shape, sine-shape, and quadrant-shape) in $m$ direction and are greater than that of DFMM, as $\nu _{d}=2$. And the OTF of Tai chi-shape in $m$ direction is greater than those of other shapes. The OTF of D-shape in $n$ direction is greater than those of Tai Chi-shape, quadrant-shape, and linear-shape. The OTFs of sine-shape in $n$ direction are greater than those of other shapes. However, the OTF difference is 3.28$\%$ for Tai Chi-shape, 20.5$\%$ for sine-shape, 13.3$\%$ for linear-shape and 69.4$\%$ for D-shape. In the axial direction, the OTF of the Tai Chi-shape is greater than those of other shapes when the axial spatial frequency is 0.3. Tai-Chi shape has a smaller OTF difference compared with sine-shape, linear-shape, and D-shape, while it has greater OTFs than quadrant-shape. Overall, the Tai Chi shape offers a smaller lateral resolution difference and better axial resolution than other shapes mentioned above, so it is an optimal solution to improve spatial resolution using focal modulation microscopy.

Table 1. OTFs of Tai Chi-shape, Sine-shape, quadrant-shape, linear-shape, and D-shape

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7. Conclusion

In this paper, we introduce Tai Chi Aperture to focal modulation microscope and investigate its imaging performance. The edge gradient, SNR, and resolution consistency of TCFMM, compared with CM, DFMM, are summarized in Table 2. The image intensity gradient expresses the transverse resolution for various edges. The large gradient is corresponding to high transverse resolution. We use the signal-to-background ratio to characterize the penetration depth of the system. The high signal-background-ratio is corresponding to strong background rejection capability and, therefore, deep penetration depth.

It is shown that TCFMM has better transverse and axial resolution than CM. Meanwhile, compared with DFMM, TCFMM has a smaller spatial resolution difference between horizontal and vertical directions. For instance, when imaging a thick, straight, sharp edge and $\nu _{d}=0$, the improvement of resolution is 26.51$\%$ and 39.31$\%$ for TCFMM compared with CM, while the improvement for DFMM is 10.37$\%$ and 45.86$\%$. Therefore, TCFMM can reduce imaging confusion caused by the asymmetry of the aperture pattern. When a finite size detector is used in TCFMM, the gradient of axial intensity is increased by 29.67$\%$. The simulation results of background rejection ability and imaging performance prove that TCFMM and DFMM have deeper penetration depth than CM. In TCFMM systems, the normalized background signal is reduced to 77.26$\%$ and the SBR is improved by 23.26$\%$ for finite-size detectors. We also discussed the influence of the geometric shape of the Tai Chi aperture pattern on the transverse resolution, which determines the optimal aperture pattern and obtains the most uniform lateral resolution. Our work analyzes the performance of TCFMM and offers important insights into future implementations to further extend the penetration depth of confocal microscopy in imaging biological tissues.

Table 2. Comparison of the parameters of three kinds of microscopy

View Table | View all tables in this article

Funding

National Natural Science Foundation of China (61735016, 61975178, 81771877); Natural Science Foundation of Zhejiang Province (LR20F050002); Key R&D Program of Zhejiang Province (2020C03009, 2021C03001); CAMS Innovation Fund for Medical Sciences (2019-I2M-5-057); Fundamental Research Funds for the Central Universities; Alibaba Cloud.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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$ν_{d} = 2$	$O T F_{m}$ $(m = 1)$	$O T F_{n} (n = 1)$	$O T F_{s} (s = 0.3)$	OTF difference in $m / n$ directions
Tai Chi-shape	0.7124	0.7358	0.5471	3.28%
Sine-shape	0.6701	0.8075	0.5452	20.5%
Quadrant-shape	0.5834	0.5834	0.4707	0
linear-shape	0.4811	0.5452	0.4668	13.3%
D-shape	0.4698	0.7959	0.5075	69.4%

	Thick Edge gradient		Thin Edge Gradient		Signal Background Ratio		Resolution Consistency
	( $ν_{d} = 0$ )	( $ν_{d} = 2$ )	( $ν_{d} = 0$ )	( $ν_{d} = 2$ )	( $ν_{d} = 0$ )	( $ν_{d} = 2$ )
CM	0.36 (y)	0.31 (y)	0.42 (y)	0.38 (y)	31.39	41.66	Consistent
CM	0.36 (x)	0.31 (x)	0.42 (x)	0.38 (x)	31.39	41.66	Consistent
DFMM	0.39 (y)	0.35 (y)	0.43 (y)	0.40 (y)	36.08	51.09	Inconsistent
DFMM	0.36 (x)	0.53 (x)	0.54 (x)	0.56 (x)	36.08	51.09	Inconsistent
TCFMM	0.45 (y)	0.43 (y)	0.46 (y)	0.45 (y)	36.53	51.34	Slightly inconsistent
TCFMM	0.50 (x)	0.49 (x)	0.51 (x)	0.52 (x)	36.53	51.34	Slightly inconsistent

$ν_{d} = 2$	$O T F_{m}$ $(m = 1)$	$O T F_{n} (n = 1)$	$O T F_{s} (s = 0.3)$	OTF difference in $m / n$ directions
Tai Chi-shape	0.7124	0.7358	0.5471	3.28%
Sine-shape	0.6701	0.8075	0.5452	20.5%
Quadrant-shape	0.5834	0.5834	0.4707	0
linear-shape	0.4811	0.5452	0.4668	13.3%
D-shape	0.4698	0.7959	0.5075	69.4%

	Thick Edge gradient		Thin Edge Gradient		Signal Background Ratio		Resolution Consistency
	( $ν_{d} = 0$ )	( $ν_{d} = 2$ )	( $ν_{d} = 0$ )	( $ν_{d} = 2$ )	( $ν_{d} = 0$ )	( $ν_{d} = 2$ )
CM	0.36 (y)	0.31 (y)	0.42 (y)	0.38 (y)	31.39	41.66	Consistent
CM	0.36 (x)	0.31 (x)	0.42 (x)	0.38 (x)	31.39	41.66	Consistent
DFMM	0.39 (y)	0.35 (y)	0.43 (y)	0.40 (y)	36.08	51.09	Inconsistent
DFMM	0.36 (x)	0.53 (x)	0.54 (x)	0.56 (x)	36.08	51.09	Inconsistent
TCFMM	0.45 (y)	0.43 (y)	0.46 (y)	0.45 (y)	36.53	51.34	Slightly inconsistent
TCFMM	0.50 (x)	0.49 (x)	0.51 (x)	0.52 (x)	36.53	51.34	Slightly inconsistent

Improved spatial resolution using focal modulation microscopy with a Tai Chi aperture

Abstract

1. Introduction

2. Theory

3. Spatial resolution

4. Background rejection capability

5. Image performance

6. Discussion on the shape of the aperture and TCFMM imaging

7. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (8)

Tables (2)

Equations (11)

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