3.1.

Second Harmonic Generation Demonstration of Lateral Image Inhomogeneity

To demonstrate the field inhomogeneity in multiphoton imaging, the SHG images of thin segments of chicken leg fascia were obtained at different imaging depths using the three objectives. Fascia consists of parallel and interlaced collagen fibers (mainly of type 1), which produce intensive second harmonic generation signal from pulsed, IR laser irradiation were imaged. For fascia SHG imaging with the oil-immersion objective 2 (Fluar $40\times ∕\mathrm{NA}$ 1.3), images at depths of 16 and $22\mu \mathrm{m}$ show that the collagen fibers at the center of the images are more intense than those at the peripheral region [Fig. 1a ]. At the imaging depth of $30\mu \mathrm{m}$ , comparable intensity across the entire frame was observed. However, at the greater imaging depths of 37 and $44\mu \mathrm{m}$ , the fibers at the image center become darker than those in the peripheral region. In addition to the existence of off-axis image inhomogeneity, these SHG images reveal that up to a depth of $30\mu \mathrm{m}$ , the fibers located at the central part of the imaging field tend to become increasingly intense with imaging depths. This phenomenon is demonstrated by the horizontally oriented fibers of the fasciae sample in Fig. 1b, series 2. In the case of the two Apochromat-type objectives (objective 1: C-Apochromat $40\times ∕\mathrm{NA}$ $1.2\mathrm{W}$ , and objective 3: Plan-Apochromat $63∕\mathrm{NA}$ 1.4 oil), this phenomenon is less prominent [Fig. 1b, series 1 and 3].

Fig. 1

SHG images of chicken leg fasciae at different imaging depths. 1: C-Apochromat $40\times ∕\mathrm{NA}$ $1.2\mathrm{W}$ ; 2: Fluar $40\times ∕\mathrm{NA}$ 1.3 oil; 3: Plan-Apochromat $63\times ∕\mathrm{NA}$ 1.4 oil. Frame size 1 and 2: $325\times 325{\mu \mathrm{m}}^2$ ; 3: $206\times 206{\mu \mathrm{m}}^2$ .

3.2.

Qualitative and Quantitative Characterization of Lateral Image Inhomogeneity

While the existence of off-axis image inhomogeneity was revealed by the SHG imaging of chicken leg fasciae using oil-immersion objective 2 (Fluar $40\times ∕\mathrm{NA}$ 1.3 oil), uniform fluorescent SRB samples were used to quantify this effect across the two-photon imaging field. Portions of the axial images of the aqueous SRB sample (thickness $\sim 70\mu \mathrm{m}$ , frame size $325\times 325{\mu \mathrm{m}}^2$ ), acquired with the same objective (Fluar $40\times ∕\mathrm{NA}$ 1.3 oil) are shown in Fig. 2 . The corresponding depth dependence of the axial intensity is shown in Fig. 3 . For calculating the intensity profile along the optical axis, a circular region $10\mu \mathrm{m}$ in diameter located at the frame center was selected. For peripheral region analysis, two circular regions of the same size were selected at the opposite corners of the image frame located $220\mu \mathrm{m}$ away from the frame center. From Figs. 2 and 3, it is evident that the focal points of the scanning laser form a convex surface with its top located approximately at the center of the scanning field.

Fig. 2

Depth-dependent SRB TPF images. Single frame size- $325\times 325{\mu \mathrm{m}}^2$ . Objective 2 (Fluar $40\times ∕\mathrm{NA}$ 1.3 oil).

Fig. 3

Axial profiles of SRB TPF intensity at the frame center and corners (see Fig. 2).

For additional quantitative analysis, the derivatives of the axial intensity profiles obtained by the oil-immersion objective 2 (Fluar $40\times ∕\mathrm{NA}$ 1.3 oil) were computed and plotted with peak width [full width at half maximum (FWHM)] characterizing the integrated TPF axial response (IAR). Figure 4a illustrates that the maxima and minima of the curves correspond respectively to the axial positions of the front and back sample surfaces. The peripheral focal points $220\mu \mathrm{m}$ away from the center fall behind the central focal points by $\Delta Z_{\text{front}}=Z_{\max ,\text{corner}}\text{-}{Z}_{\max ,\text{center}}=4.9\mu \mathrm{m}$ . At the back surface ( $\sim 70\mu \mathrm{m}$ below the front surface), this value increased to $\Delta Z_{\text{back}}=Z_{\min ,\text{corner}}\text{-}{Z}_{\min ,\text{center}}=7.9\mu \mathrm{m}$ . The IAR increased from $1.9\mu \mathrm{m}$ at the center of the image frame to $2.8\mu \mathrm{m}$ at the corner (increase of 147%). At a depth of $70\mu \mathrm{m}$ , the respective IARs at the center and corner are 7.0 and $7.8\mu \mathrm{m}$ . For comparison, the axial SRB TPF profiles were determined with the water-immersion objective 1 (C-Apochromat $40\times ∕\mathrm{NA}$ $1.2\mathrm{W}$ ). The derived curves [Fig. 4b] show a much smaller concavity of the scanning surface. At last for the well-corrected oil-immersion objective 3 (Plan-Apochromat $63\times ∕\mathrm{NA}$ 1.4 oil), a flat scanning surface was observed (Table 1 ).

Fig. 4

(a) Derivatives of aqueous SRB axial intensity profiles (see Fig. 3). 1 is the frame center, and 2 is the frame corner (smooth curves-Gaussian fits). Objective 2: Fluar $40\times ∕\mathrm{NA}$ 1.3 oil. (b) Axial profiles and the corresponding derivatives for aqueous SRB TPF intensity (smooth curves-Gaussian fits). Objective 1: C-Apochromat $40\times \mathrm{NA}$ $1.2\mathrm{W}$ corr.

Table 1

Integral TPF axial response (IAR) in the image frame centers and corners along with the relative axial shifts (in micrometers).

To further investigate the field inhomogeneity in multiphoton imaging, additional image analysis was performed for the three objectives using $10\text{-}\mu \mathrm{M}$ SRB dissolved in a solution with a higher refractive index immersion liquid (IL, $n$ around 1.465). The results show that the IARs of the lateral image inhomogeneity effect at the front surface depend on the objective type (Table 1). However, when the field inhomogeneity is present, it strongly depends on RIM and image depth [Fig. 4a].

3.3.

Investigation of Depth-Dependent Lateral Profiles

To compare the depth-dependent image inhomogeneity effects, at image surface and at depths of 75 and $120\mu \mathrm{m}$ , lateral profiles of the uniform SRB samples in water [Fig. 5a ] and IL [Fig. 5b] at the two depths have been plotted. In addition, comparison between the axial profiles in the central and peripheral regions (Fig. 6 ) was made for the two fluorescent SRB samples. For plotting the lateral profiles, a rectangular region $4\mu \mathrm{m}$ wide along the frame diagonal was chosen. For plotting the axial profiles, a circular region $4\mu \mathrm{m}$ in diameter was selected at the frame center and corner ( $\sim 227\mu \mathrm{m}$ away from the center for objectives 1 and 2 and $140\mu \mathrm{m}$ for objective 3). The results show that in using the water-immersion objective 1 (C-Apochromat $40\times ∕\mathrm{NA}$ $1.2\mathrm{W}$ ) for imaging the aqueous SRB specimen, lateral profiles at a depth of $120\mu \mathrm{m}$ (data not shown) are very similar to those at the surface [Fig. 5a]. The lateral intensity profile $140\mu \mathrm{m}$ away from the center decreases by less than 15% [Fig. 5a] and no intensity decay in the axial direction is observed [Fig. 6a]. These results suggest that objective 1 is well corrected for aqueous samples with $170\text{-}\mu \mathrm{m}$ cover glass and that the axial degradation is negligibly small for $0\text{to}100\mu \mathrm{m}$ depth.

Fig. 5

Lateral profiles of SRB TPF at the surface (1, 2, 3) and at the depths of $120\mu \mathrm{m}$ $(1^{\prime },2^{\prime })$ and $75\mu \mathrm{m}$ $\left(3^{\prime }\right)$ . (a) Aqueous solution. (b) Immersion liquid. $\mathrm{n}$ around 1.465. 1 and $1^{\prime }$ are objective 1: C-Apochromat $40\times$ NA $1.2\mathrm{W}$ corr; 2 and $2^{\prime }$ are objective 2: Fluar $40\times ∕\mathrm{NA}$ 1.3 oil; and 3 and $3^{\prime }$ are objective 3: Plan-Apochromat $63\times ∕\mathrm{NA}$ 1.4 oil DIC.

Fig. 6

Axial profiles of SRB TPF intensity in the frame center (1, 2, and 3, left $y$ axis) and the intensity ratio from the corner to the center ( $1^{\prime }$ , $2^{\prime }$ , and $3^{\prime }$ , right $y$ axis). (a) Aqueous solution and (b) immersion liquid. $\mathrm{n}$ around 1.465. 1 and $1^{\prime }$ are objective 1: ${\mathrm{C}}^{\prime }$ -Apochromat $40\times$ NA $1.2\mathrm{W}$ corr; 2 and $2^{\prime }$ are objective 2: Fluar $40\times ∕\mathrm{NA}$ 1.3 oil, and 3 and $3^{\prime }$ are objective 3: Plan-Apochromat $63\times ∕\mathrm{NA}$ 1.4 oil DIC.

However, for the oil-immersion objectives, when RIM is essential for the aqueous samples, our results reveal that lateral intensity degraded more than 40% from center to corner [Fig. 5a], and more than 35 and 50% of degradation from surface to the depth of $80\mu \mathrm{m}$ were respectively observed for objective 2 (Fluar $40\times ∕\mathrm{NA}$ 1.3 oil) and objective 3 (plan-Apochromat $63\times ∕\mathrm{NA}$ 1.4 oil) [Fig. 6a]. More importantly, for objective 2, the image with maximum intensity heterogeneity was observed at the sample surface [Figs. 5a and 5b]. This large nonuniformity can be explained by the axial shift of the focal point axial coordinate at the corner away from that of the center [Fig. 4a]

As Figs. 5 and 6 illustrate, the image lateral inhomogeneity effects exist for both oil- and water-immersion objectives in IL with refractive index of around 1.465 as well. The effects in the imaging field are similar to those for the aqueous sample whose refractive index is $\mathrm{n}=1.33$ (Fig. 5). However, the results for the axial profiles were very different. In Fig. 6b, for the two oil-immersion objectives, up to a depth of $60\mu \mathrm{m}$ , a small signal loss was observed in the immersion liquid (curves 2 and 3), compared with the aqueous medium [Fig. 6a]. Thus, the oil-immersion objectives can be used to obtain uniform TPF signals along depths throughout much of the working distance in samples with refractive indexes of around 1.465. For the water-immersion objective, a nonmonotonous intensity decay profile was observed [Fig. 6b, curve 1]. This observation may be caused by the excitation light passing through media with three different refractive indexes 1.33 (water), 1.515 (cover glass), and around 1.465 (immersion fluid) in reaching the focal point.

3.4.

Off-Axis Broadening of Lateral Point Spread Function

As a final characterization of the 3-D image inhomogeneity, the lateral PSF was measured for the three objectives at different depths. For measuring the lateral PSF in the central region, four to five fluorescent beads inside a $10\text{-}\mu \mathrm{m}$ diameter circular region at the frame center were chosen. For the off-axis measurement, the beads were selected at approximately $140\mu \mathrm{m}$ away from the center. The PSF characterization results are shown in Fig. 7 and Table 2 .

Fig. 7

Lateral TPF intensity profiles of $0.1\text{-}\mu \mathrm{m}$ fluorescent beads near the water-agarose sample surface and FWHM of PSF at center (1) and corner (2) of the frame (smooth curves—Gaussian fits). Objective 2: Fluar $40\times ∕\mathrm{NA}$ 1.3 oil.

Table 2

Lateral PSF FWHMs in the water-agarose sample at different locations (in micrometers).

The physical reason for causing the 3-D image heterogeneity in multiphoton scanning of uniformly luminescent specimens can be numerous. Factors associated with change of the laser beam intensity along any coordinate, such as reduction of entrance aperture for inclined beams, increase of Fresnel losses, and total internal reflection on borders of mediums with different refractive indexes at larger scanning angles, absorption, and scattering of excitation and emission radiation can all contribute. Furthermore, the broadening of the PSF and shifting of focal position involving RIM at interfaces of media with different indices of refraction (immersion liquid/cover-glass/sample) can lead to shifts of the axial focal spot positions), which results in the distortion of geometrical form of specimen, reduction of image resolution, and loss in signal intensity level.

Regarding high numerical aperture objectives, when a paraxial ray refracts at interfaces of two media with different refractive indexes, the shifted axial and lateral focal positions induced a defocused laser beam, as illustrated in Fig. 8 .

Fig. 8

Effects of refractive index mismatch on focal spots with (a) excitation parallel, and at (b) an inclination angle to the optical axis $(n_i>n_s)$ .

Suppose that the effect of spherical aberration is not present, that is, when the refractive index $\left(n_i\right)$ of the immersion liquid or cover glass is equivalent to the refractive index $\left(n_s\right)$ of the sample, $d\left(d^{\prime }\right)$ is the crossing point of approximately vertical rays $v\left(v^{\prime }\right)$ whose focal point is positioned at $f\left(f^{\prime }\right)$ with axial position at $H$ . In the case that $n_i>n_s$ , the crossing point of the rays $l\left(l^{\prime }\right)$ and $o\left(o^{\prime }\right)$ becomes $g\left(g^{\prime }\right)$ . The coordinate of such a crossing point is $Z\left(\theta \right)$ and depends on the inclination angle. Note that at $\theta$ , defocus can take place within the limits of $\Delta Z=Z\left(0\right)\text{-}Z\left(\theta \right)=H(\sin \theta \sin \phi \text{-}\tan \theta ∕\tan \phi )$ . As the laser beam enters the objective, an additional dithering of PSF focal volume shown in Fig. 8b can occur. In addition, due to the fact that ${\theta }_2>\theta ,g^{\prime }$ is located above $g$ and PSF is more extended in the axial direction [Fig. 8a]. These effects may contribute to the multiphoton image field inhomogeneity effects.