the side lobes induced by the back aperture truncation. As presented in Sect.1.4p.13, the axial resolution can be defined as the distance, along the longitudinal axis, between the maximum of intensity and the first minimum of intensity which corresponds to the equation rz “ NA2λn2 «3.8 µm.
The red continuous curve corresponds to the non-truncated Gaussian distribution illumi-nation case with waistw“0.5D, derived from the theoretical Gaussian beam equations (see Sect. 1.3). The FWQM is « 2ˆ0.6 µm « 1.2 µm. The distance where the 3D PSF2 is equal to 1{e4 is « 1.5 µm. The two measured transverse resolutions approxi-mately match.
The red dashed curve represents the case of a truncated Gaussian distribution illumina-tion with waist w“ 0.5D. Similarly to the case of the truncated uniform distribution illumination, the truncation creates secondary lobes. The first minimum can still be observed at z“ NA2λn2 as in the case of the truncated uniform distribution. The FWQM is «2ˆ1.7 µm«3.4 µm. This approximately matches with the definition of axial res-olution for truncated back apertures: rz «3.8 µm. The truncation also enlarges axially the 3D PSF2 confinement relatively to the non-truncated case. This enlargement is a lot more accentuated than in the transverse direction.
As we observed for the transverse cuts, the reduction of the waist also enlarges the 3D PSF2 in the axial direction (blue dashed curve) relatively to the case of waistw“0.5D. The increasing of the waist tends to converge the truncated Gaussian distribution to a truncated uniform distribution resulting on the overlap between the green and black curves.
To summarize, the theoretical equation for the propagation of Gaussian beams (Eq. (1.1) p.11) can not be used to correctly characterize the propagation of the excitation beam through the finite back aperture. As one could expect, the truncation by the back aperture widens the 3D PSF2 in both transverse and axial directions, with the axial direction being more impacted.
2.4 Characterization of the 3D PSF
2in the presence of aberrations
In this section I study the impact of aberrations on the 3D PSF2. I simulate two 3D PSF2’s with 2 rad of coma aberration and 2 rad of spherical aberration respectively.
Here, the 3D PSF2’s are simulated with a truncated Gaussian distribution illumination and a waist w “ 0.5D. To better observe the effects of the aberrations on each 3D PSF2, I consider the same sampling parameters as those used in Sect. 2.3 except the oversampling factor which was set to k“8 instead of k“24 (see Table 2.3).
Figure 2.13 illustrates three different 2D profiles of the 3D PSF2 for 2 rad of coma aberration. These cuts were extracted at the position px, y, zq where the 3D PSF2 is maximal.
We can observe that the coma aberration induces a transverse displacement on the main lobe of the 3D PSF2. It also induces an elongation of this lobe in both transverse and
46 Chapter 2. Study of the impact of aberrations
Figure 2.13: Illustration of the 3D PSF2 for 2 rad of coma aberration (Z7). The maximum of the 3D PSF2 is located at x“ ´1.44 µm, y “0 µm and z“0 µm. (left)Transverse scan at z“0 µm;(center)Axial profileyzat x“ ´1.44 µm;(right)Axial profilexz at y“0 µm.
axial directions, the elongation being more accentuated in the axial direction.
Figure2.14 illustrates the three cuts that cross the maximum of the 3D PSF2 for 2 rad of coma aberration.
As expected, one can verify the elongation of the 3D PSF2. The FWQM corresponds now to 8.4 µm instead of 3.4 µm, a degradation of the 3D PSF2 axial resolution by a factor 2.47 relatively to the diffraction-limited case. We can also observe a larger trans-verse resolution: «3.11 µm in thex-direction and«1.06 µm in the y-direction instead of 0.8 µm on both directions on the diffraction-limited case. This represents a degrada-tion of the 3D PSF2 transverse resolution by a factor 3.89 and 1.33 on each direction respectively.
For what concerns the spherical aberration, Fig.2.15illustrates three different 2D profiles of the 3D PSF2 for 2 rad of spherical aberration. These profiles were extracted at the point where the 3D PSF2 is maximal.
We can observe that the spherical aberration induces an axial displacement (and not transverse) of the main lobe of the 3D PSF2. It also induces an elongation of this lobe in both transverse and axial directions.
In Figure2.16 I illustrate the three cuts that cross the maximum of the 3D PSF2 for 2 rad of coma aberration.
We can verify that the FWQM has increased to«9.9 µm. When diffraction-limited the FWQM is«3.4 µm. This represents an degradation of the 3D PSF2 axial resolution by a factor 2.9.
We can also observe a larger FWQM: « 1.2 µm in both transverse directions instead
2.4. Characterization of the 3D PSF2 in the presence of aberrations 47
x µm
-5 0 5
0 0.2 0.4 0.6 0.8 1
(x=-2.44, 0.25) (x=-0.67, 0.25) y = 0 µm, z = 0µm
y µm
-5 0 5
0 0.2 0.4 0.6 0.8 1
(y=-0.53, 0.25) (y=0.53, 0.25) x = -1.4339 µm, z = 0µm
z [µm]
-20 -15 -10 -5 0 5 10 15 20
10-5 10-4 10-3 10-2 10-1 100
(z=-4.2, 0.25)
(z=4.2, 0.25) x = -1.4339 µm, y = 0 µm
Figure 2.14: 3D PSF2 cuts for 2 rad of coma aberration (Z7). The maximum of the 3D PSF2is located atx“ ´1.44 µm,y“0 µmandz“0 µm. (upper)Cut along the transversexdirection (y “ 0, z “ 0); (center) Cut along the transverse y direction (x “ ´1.44 µm, z “ 0 µm);
(lower)Cut along the axial zdirection (x“ ´1.44 µm, y“0 µm); The red dashed horizontal curve represents the quarter of the maximum. The transverse resolution is given by the full width of the 3D PSF2 at the quarter of its maximum
of 0.8 µm of the diffraction-limited case. This represents a degradation of the 3D
48 Chapter 2. Study of the impact of aberrations
PSF2 at z = 7.9 µm
x [µm]
-10 -5 0 5 10
y [µm]
-10
-5
0
5
10
3DPSF2 YZ cut at x = 0 µm
y [µm]
-10 -5 0 5 10
z [µm]
-20
-15
-10
-5
0
5
10
15
20
3DPSF2 XZ cut at y = 0 µm
x [µm]
-10 -5 0 5 10
z [µm]
-20
-15
-10
-5
0
5
10
15
20
Figure 2.15: Illustration of the 3D PSF2 for 2 rad of spherical aberration (Z11). The maximum of the 3D PSF2 is located atx“0 µm, y “0 µm andz “7.9 µm. (left)Transverse scan at z“7.9 µm;(center)Axial profileyz atx“0 µm;(right)Axial profilexz at y“0 µm.
PSF2 transverse resolution by a factor 1.5.
2.4. Characterization of the 3D PSF2 in the presence of aberrations 49
x µm
-5 0 5
0 0.2 0.4 0.6 0.8 1
(x=-0.6, 0.25) (x=0.6, 0.25)
y = 0 µm, z = 7.9µm
y µm
-5 0 5
0 0.2 0.4 0.6 0.8 1
(y=-0.6, 0.25) (y=0.6, 0.25)
x = 0 µm, z = 7.9µm
z [µm]
-20 -15 -10 -5 0 5 10 15 20
10-5 10-4 10-3 10-2 10-1 100
(z=-3.25, 0.25)
(z=13.15, 0.25) x = 0 µm, y = 0 µm
Figure 2.16: 3D PSF2 cuts for 2 rad of spherical aberration (Z11). The maximum of the 3D PSF2 is located at x “ 7.9 µm, y “ 0 µm and z “ 7.9 µm. (upper) Cut along the transverse xdirection (y “0 µm, z “7.9 µm); (center)Cut along the transversey direction (x“0 µm, z“7.9 µm); (lower)Cut along the axial z direction (x“0 µm, y “0 µm); The red dashed horizontal line represents the quarter of the maximum. The transverse resolution is given by the full width of the 3D PSF2 at the quarter of its maximum.
50 Chapter 2. Study of the impact of aberrations