Numerical studies of focal modulation microscopy in high-NA system

Bingzhao Zhu; Shuhao Shen; Yao Zheng; Wei Gong; Ke Si

doi:10.1364/OE.24.019138

1. Introduction

Confocal microscopy (CM) has been widely used in many fields, especially in biological research and medical diagnosis [1–3]. It has both submicron spatial resolution and optical sectioning capability by rejecting the out-of-focus background. This is achieved by introducing a small pinhole to the detection plane which acts as a spatial filter. However, it severely suffers from a limited imaging penetration depth in thick tissue as a result of multiple scattering [4], which significantly reduces the spatial resolution. In order to retain high resolution images in thick tissue, a number of gating methods have been implemented to sort out unscattered or less scattered photons to reconstruct a high-quality image of an object embedded in thick tissue. These gating methods include spatial gating methods [5], polarization gating methods [6], time gating methods [7], angle gating methods and coherence gating methods [8]. Focal modulation microscopy (FMM) is one of the spatial gating methods.

Specifically, FMM is an optical heterodyne technique based on the confocal framework, which is proposed to provide superior imaging contrast with sub-micron spatial resolutions at large penetration depth in highly scattering media such as biological tissue. To achieve this goal, FMM utilizes the coherence property of the light source. The illumination light at back focal plane of the objective lens is divided into several spatially non-overlapped sub-beams. Then a spatial-temporal modulation strategy is used to modulate these sub-beams with different frequencies. After focused by the objective lens, they generate an intensity modulated signal periodically moving at the focal region, which is similar with the structure illumination pattern [9]. After demodulation of the collected fluorescence signal at the designated modulation frequency, We can discriminate the in-focus fluorescence from the multiple-scattered background, thus the signal to background ratio (SBR) can be greatly enhanced compared with confocal microscopy [10]. Previous research has proved that FMM can be performed with various different geometries for the two non-overlapping illumination beams. Particularly, focal modulation microscopy with annular apertures (AFMM) can further enhance the background rejection compared with D-shaped apertures (DFMM), thus the penetration depth can be further extended [11].

The previous analysis of FMM was based on the scalar diffraction theory, which is applicable only in the low numerical aperture (NA) of the objective lens [12]. In case of high-NA objective lens, polarization effects cannot be neglected, thus vectorial diffraction theory must be applied [13]. In practice, a high-NA objective lens is preferred for better spatial resolution [14]. Moreover, by controlling the polarization state of the excitation beam, we can always generate a smaller focal spot, which can directly lead to an improvement of the spatial resolution of a laser scanning microscopy [15–19].

Here, we develop a theoretical model for FMM based on vectorial diffraction theory. Different polarization patterns are studied to optimize the spatial resolution. Other properties, such as integrated intensity and background rejection capability are also presented. The results show that the polarization of the input light beam could have an effect on the performance of the FMM system.

2. Theory

Figure 1 illustrates the geometry of the FMM system with high NA objective lens. Compared to confocal system, we insert a polarization filter and a custom-made spatial phase modulator into the illumination beam path. After polarized by a filter, the expanded illumination beam is modulated by a spatial phase modulator to get two spatially non-overlapped half-beams with different phase modulation frequency. When passing through the objective lens, the two half-beams interfere with each other to generate an illumination pattern periodically scanning in the focal region, similar as structure illumination pattern. Then the collected fluorescence light is collected by the collection lens into a detector with a circular pinhole mask and demodulated at the designated frequency by a lock-in amplifier to get the imaging signal.

Fig. 1 Geometry of the FMM system combined with annular aperture and polarized beams.

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In this paper, we use a high-NA oil-immersion (NA = 1.4) objective lens. Based on the vectorial diffraction theory, the electric field vector near the focal spot can be derived from the generalized Debye integral [20] as

\begin{matrix} \vec{E} (r_{2}, φ_{2}, z_{2}) = i C \iint_{Ω} \sin θ \cdot A_{1} (θ, φ) \cdot A_{2} (θ, φ) \cdot [\begin{matrix} p_{x} \\ p_{y} \\ p_{z} \end{matrix}] \cdot \exp {i k n [z_{2} \cos θ \\ + r_{2} \sin θ \cos (φ - φ_{2})]} d θ d φ . \end{matrix}

Here

\vec{E} (r_{2}, φ_{2}, z_{2})

is the electric field vector at the point

(r_{2}, φ_{2}, z_{2})

expressed in cylindrical coordinates with their origin at the focal point. C is the normalized constant.

A_{1} (θ, φ)

is the amplitude function of the input light, which often equals 1 or the deformation of the fundamental Gaussian function.

A_{2} (θ, φ)

is a 3 × 3 matrix related to the structure of the imaging lens. Further,

A_{2} (θ, φ)

could be expressed as

A_{2} (θ, φ) = f (θ) \cdot V (θ, φ)

, where

f (θ) = \sqrt{\cos θ}

is the apodization factor obtained from energy conservation and geometric considerations [21] and

V (θ, φ)

is the conversion matrix of the polarization from the object field to the image field. In particular, they can be derived as

\begin{matrix} V (θ, φ) = \\ [\begin{matrix} 1 + (\cos θ - 1) \cos^{2} φ & (\cos θ - 1) \cos φ \sin φ & - \sin θ \cos φ \\ (\cos θ - 1) \cos φ \sin φ & 1 + (\cos θ - 1) \sin^{2} φ & - \sin θ \sin φ \\ \sin θ \cos φ & \sin θ \sin φ & \cos θ \end{matrix}], \end{matrix}

where

[p_{x}; p_{y}; p_{z}]

is a matrix unit vector about the polarization of input light. Table 1 lists several polarization patterns and corresponding unit vector matrixes.

Table 1. Polarization Patterns and Matrixes

View Table

3. Effects of polarization on spatial resolution

Our previous study based on scale diffraction theory has proposed AFMM with inner and outer radii of Ɛa and a (0<Ɛ<1), respectively. We found that by increasing the inner radius, both the axial resolution and transverse resolution can be further enhanced, with the costs of signal level and modulation depth [11]. Here we further investigate high NA AFMM with different polarization patterns. Considering all the factors, such as spatial resolution, background rejection capability and modulation depth, we select Ɛ = 0.707, in which case the area of the two modulated segments is equal. Based on Richards and Wolf integrals [22], the energy density of transverse and longitudinal components of AFMM signal can be obtained.

Figure 2 shows the energy distribution along the x and y directions where both annular and circular incident beams are right-circular polarized. Here v_x and v_y are the optical coordinates related to the true distance from focal point by v_x = 2πx(sinα)/ λ and v_y = 2πy(sinα)/ λ, respectively, where sinα is the numerical aperture of the objective lens and λ is the wavelength. Noticed that only components polarized in the same direction can interact with each other. For circular polarization illumination, the transverse component is desirable for its tight focusing spot (Figs. 2(a) and 2(b), Figs. 2(d) and 2(e)), whereas the Z-polarized component is donut-shaped (Figs. 2(c) and 2(f)), thus undesired. This is opposite to the result of radial polarized illumination, in which case X, Y-polarized components are donut-shaped [16]. For the improvement of lateral resolution in FMM, intensity point-spread-function (IPSF) of modulated and unmodulated area should be polarized in the same direction in focus, and polarized in vertical direction out of focus.

Fig. 2 Energy density distribution of x (left), y (center) and z (right) components for (a-c) circular aperture with radius Ɛa (Ɛ = 0.707); (d-f) annular aperture with radius a and inner radius Ɛa (Ɛ = 0.707). Incident beam is right-circular polarized and axes are in wavelength units.

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FMM signal is used for imaging in focal modulation microscopy to obtain higher spatial resolution as well as deeper penetration depth. We can extract the FMM component of the signal by using a reference signal after passing through a lock-in amplifier. In high-NA cases, FMM signal is also made up of different polarized components. For the purpose of tighter point spread function (PSF), it is meaningful to study the contribution of different polarized components to the spot. Figure 3 shows the FMM signal of objective lens with annular aperture and right-circular polarized incident beam. From the figure, we conclude that the energy of FMM signal comes mainly from X, Y-polarized components. Z-polarized component is relatively weak, but donut-shape will harm the spatial resolution. Therefore, spatial resolution can be further optimized if Z-polarized component is blocked out.

Fig. 3 Energy density distribution of x (left), y (center) and z (right) components for FMM signal of annular aperture lens with equal modulated and unmodulated area. Incident beam is right-circular polarized and axes are in wavelength units.

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4. Image of a point object in high-NA system

For better understanding of performance of FMM in high-NA system, we study the three dimensional (3D) intensity image of a point object. Neglecting the Stokes’ shift, the corresponding 3D time-varying image of a point object with a finite size detector can be obtained from

I (r, φ, z, t) = | h_{1 A} (r, φ, z) + h_{1 B} (r, φ, z) e^{i 2 δ ω t} |^{2} \times (| h_{2} (r, φ, z) |^{2} \otimes_{2} D (r, φ)),

Where

\otimes_{2}

denotes the 2D convolution,

D (r, φ)

is the intensity sensitivity of the detector, whose size v_d is normalized by the illumination wavelength.

h_{1 B} (r, φ, z)

and

h_{1 B} (r, φ, z)

are the 3D amplitude point spread functions (APSF) of the annular and circular objective lens, and they can be obtained using Eq. (1).

h_{2} (r, φ, z)

is the APSF of the collection lens. With a long pass filter blocking out the excitation light, total signal detected can be expressed by FMM signal

I_{i l l u} = (h_{1 A x} h_{1 B x}^{*} + h_{1 A x}^{*} h_{1 B x}) + (h_{1 A y} h_{1 B y}^{*} + h_{1 A y}^{*} h_{1 B y}) + (h_{1 A z} h_{1 B z}^{*} + h_{1 A z}^{*} h_{1 B z}),

I_{FMM} = I_{i l l u} (| h_{2} |^{2} \otimes_{2} D),

where

h_{1 A x}

,

h_{1 A y}

and

h_{1 A z}

refer to X, Y, Z polarized components of

h_{1 A} (r, φ, z)

, and

h_{1 B x}

,

h_{1 B y}

,

h_{1 B z}

for

h_{1 B} (r, φ, z)

, respectively.

I_{i l l u}

is the illumination pattern of FMM in high-NA system. Total illumination pattern and X, Y, Z polarized components are shown in the Visualization 1, as a function of time in z = 0 plane. From this, we study behaviors of differently polarized components in time-varying illumination and thus the image of a point object in high-NA system can be obtained.

Spatial resolutions in both axial direction and transverse direction of FMM in high-NA system are presented in Fig. 4(a). In focal modulation microscopy, FMM signal is used to generate the images. Compared with CM, FMM with equal area apertures shows improvements in z direction in high-NA condition, while remains the same in x direction. Close analysis on half-width at half-maximum (HWHM) shows that FMM with right-circular polarized incident beam is improved up to 20.1% in axial direction, which is close to 17.8% based on scalar diffraction theory [11]. However, in the discussion above, we assume an ideal detector with v_d = 0. With finite detector sizes, IPSF in transverse direction is shown in Fig. 4(b). As v_d increases, the IPSF becomes boarder. Numerical study shows that HWHMs for v_d = 0, 0.2, 0.4, 0.6 are 0.120λ, 0.133λ, 0.175λ, 0.176λ, respectively. We then conclude the negative impact on spatial resolution of v_d = 0.2 (10.8%) is relatively smaller than that of v_d = 0.4 (45.8%). For cases requiring tight IPSF, the detector radius should be less than 0.2λ. Figure 4(c) shows variations of HWHM as a function of v_d for FMM, compared with a conventional confocal microscopy. From the plot, as detector size increases, the advantages of FMM over CM become more obvious. According to our knowledge, this has not been reported in previous work.

Fig. 4 (a) Cross sections of the IPSF for CM and FMM (in high-NA system) with equal area in z direction and x direction, respectively. (b) Transverse resolution for FMM in high-NA system with finite detector size. (c) Half-width at half-maximum (HWHM) in x direction as a function of detector size for FMM and CM in high-NA system. Both detector radius and axes are in wavelength units.

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To estimate the performance of FMM in high-NA imaging system with infinite small detector size, we use a spoke-like sample which is shown in Figs. 5(a). Figures. 5(b) and 5(c) show the image of FMM and CM in horizontal (XY) plane. Figures. 5(d) and 5(e) study the axial resolution with sample placed in cross-section (ZX) plane. The spatial resolution can be estimated by measuring the size of the unresolved area. For the comparison between FMM and CM on lateral resolution, their images of sample are similar. However, when it comes to images in cross-section plane, the edge of Fig. 5(d) is clearer than that of Fig. 5(e). Figure. 5(f) shows the intensity profiles along the lines in Fig. 5(d) and (e). We notice that there is 6%-10% intensity improvement near the edge of figures. That is to say FMM has better axial resolution than CM. Moreover, it is of practical significance that FMM is suitable for cases where high-NA AL and finite-sized detector are used.

Fig. 5 Simulation results of a sample with a spoke-like pattern. (a) Sample. (b, c) Image of sample in the horizontal (XY) plane for FMM and CM. (d, e) Image of sample in the cross-section (ZX) plane for FMM and CM, respectively. (f) Signal profile along the lines. The comparison is studied in high-NA system, using vectorial diffraction theory. The axes are in wavelength units.

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5. Background rejection of FMM in high-NA system

The imaging penetration depth is an important parameter for optical microscopy and can be measured by integrated intensity [23]. Here we only consider the single scattering and neglect multiple scattering [24]. The contribution to the background in the focal plane from a distribution of particles with a distance z is shown below, where $I (r, φ, z)$ represents the intensity image of a point:

I_{int} (z) = \iint I (r, φ, z) r d r d φ

The integrated intensity is used to measure the amount of background rejection from an infinitely thin autofluorescing sheet. Figure 6 illustrates the comparisons of the integrated intensity of CM between FMM with both scalar and vectorial theory. The previous work shows that the intensity falls off as $1 / u^{2}$ roughly for CM with a point detector [25]. Here the optical coordinate u is related to the true distance from the focal point z by u = 8πzsin²(α/2)/ λ with λ the excitation or emission wavelength, α the semi-angular aperture of the lens. From Fig. 6, the plot based on vectorial diffraction theory is slightly different from plot on scalar theory. It is apparent that background rejection can be effectively improved in both point and finite-sized detectors.

Fig. 6 Comparison of the integrated intensity of CM, FMM on both scalar and vectorial theory for v_d = 0 (solid lines) and v_d = 0.4 (dash lines). Both detector radius and axes are in wavelength units.

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For a thick object focused at the top with thickness $z_{0}$ , the total background detected can be expressed as:

I_{b g d} (z_{0}) = \int_{0}^{z_{0}} I_{int} (z) d z

The background detected of point (solid lines) and finite-sized (dash lines) detectors is compared in Fig. 7. For the cases of FMM, either on scalar theory or vectorial theory, the background is suppressed. We also conclude that with the increase of detector size, the advantage of FMM becomes more obvious.

Fig. 7 The total background as a function of defocus distance for CM, FMM on both scalar and vectorial theory for v_d = 0 (solid lines) and v_d = 0.4 (dash lines). Both detector radius and axes are in wavelength units.

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6. Polarization patterns for FMM in high-NA system

As described in [13], the intensity distribution around the focal spot is relative to the polarization pattern of the incident beam, especially in high-NA conditions. To further enhance spatial resolution in focal modulation microscopy, we study different types of polarization patterns in high-NA system. Both axial and transverse resolutions are presented in Fig. 8, with different polarization patterns as radial, circular and linear, all of which are based on vectorial diffraction theory. Azimuthal polarization is not included because it appears to be a donut-shape in the horizontal (XY) plane, which is undesirable in CM [21]. In Fig. 8, transverse resolutions are shown in solid lines, and axial resolution dash lines. From the plot, we conclude that Radial polarization is not suitable for FMM in high-NA system. While circular and linear polarization have similar results in z direction, Linear X is better than Circular in y direction but worse in x direction. However, all the differences are less than 10% in view of HWHM.

Fig. 8 The intensity distribution for different polarization patterns of radial, circular and linear, in the cases of AFMM with v_d = 0. The axis units are in wavelength.

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To further study the effects of polarization patterns on penetration depth, the integrated intensity for FMM signal is shown in Fig. 9, as a function of z. Higher integrated intensity values means deeper penetration depth. It can be seen that circular and linear polarized beams have better performance than radial polarized beams. The result is valid for both infinite small detector and finite-sized detector. From the discussion above, it can be concluded that either linear polarization or circular polarization could be used for FMM systems.

Fig. 9 The integrated intensity as a function of z for different polarization patterns, in the cases of AFMM with v_d = 0 (solid lines) and with v_d = 0.4 (dash lines). The axis units are in wavelength.

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7. Conclusion

In this paper, we have investigated the performance of focal modulation microscopy with annular apertures (AFMM) in high-NA systems. We build our theory of FMM using vectorial diffraction theory and show that different polarization patterns can result in varieties of spatial resolutions. Different polarization components of the focal spot is provided and analysis show improvements in spatial resolution and background rejection. Close analysis on half-width at half-maximum (HWHM) shows that FMM with right-circular polarized incident beam is improved up to 20.1% in axial direction, which is a more accurate estimate than 17.8% based on scalar diffraction theory. Also, numerical studies on different polarization patterns suggest that radial and azimuthal polarized beams are not suitable to obtain tighter spot and deeper penetration depth. To extend our results, we show that the contribution of transverse polarization component should be enhanced to shrink the focus size, since longitudinal component is undesired. Our work not only analyzes FMM’s performance in high-NA systems, but also gives guidance on the choice of polarization patterns.

Funding

National Basic Research Program of China (973 Program) (No. 2015CB352005); National Natural Science Foundation of China (NSFC) (No. 6142780065, 81527901, 31571110); Natural Science Foundation of Zhejiang Province of China (No. Y16F050002), and the Fundamental Research Funds for the Central Universities.

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Polarization	Linear X	Linear Y	Right circular	Left circular	Ellipse X	Radial	Azimuthal
Pattern
$[\begin{matrix} p_{x} \\ p_{y} \\ p_{z} \end{matrix}]$	$[\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}]$	$[\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}]$	$\frac{1}{\sqrt{2}} [\begin{matrix} 1 \\ i \\ 0 \end{matrix}]$	$\frac{1}{\sqrt{2}} [\begin{matrix} i \\ 1 \\ 0 \end{matrix}]$	$\frac{1}{\sqrt{5}} [\begin{matrix} 2 \\ i \\ 0 \end{matrix}]$	$[\begin{matrix} \cos θ \\ \sin θ \\ 0 \end{matrix}]$		$[\begin{matrix} - \sin θ \\ \cos θ \\ 0 \end{matrix}]$

Numerical studies of focal modulation microscopy in high-NA system

Abstract

1. Introduction

2. Theory

3. Effects of polarization on spatial resolution

4. Image of a point object in high-NA system

5. Background rejection of FMM in high-NA system

6. Polarization patterns for FMM in high-NA system

7. Conclusion

Funding

References and links

Supplementary Material (1)

Cited By

Figures (9)

Tables (1)

Equations (7)

Optics Express