The investigations on the effects of leaky aperture on the images of point objects formed by coherent optical systems apodized by the complex Hanning amplitude filter in the presence of defect-of-focus and primary spherical aberrations have been evaluated using the expression (4) by employing Matlab simulation. The intensity distribution B (Z) in the images of point objects has been obtained for different values of dimensionless diffraction variable Z varying from – 12 to + 12.The image quality assessment parameters such as full-width at half-maximum (FWHM) and first maxima and first minima have been studied for various values of apodization, aberrations and leaky aperture transmission parameter.
In general, the intensity level of the first-order sidelobes and the FWHM of the central peak in the resulting intensity distribution are used to calculate the effect of pupil apodization. Therefore, in a variety of circumstances, the suggested apodization across the pupil reduces the light energy related to the first and higher order sidelobe levels of the PSF. According to the Rayleigh criterion [23], the transverse resolution of PSF grows as the FWHM of its central peak shrinks. In the presence of high aberrations, apodization (β = 1) across the pupil to reduce the higher frequency components at the edges of the pupil function, which lowers the FWHM of the central peak to a lower value. Here, the pupil transmission profile strongly depends on the value of β, as well as the amount and nature of aberrations considered in the optical system.
Figure 3(a-e) depicts the effect of the apodization parameter β on the intensity distribution of PSF under various situations. With increase in the apodization, there is a decrease in the intensity of the central lobe. As the apodization parameter β is increased from β=0 to β=0.6, the optical side lobes are completely suppressed. However, for higher orders of apodization (β=1), the radius of the first dark ring in the diffraction pattern becomes less than that of Airy case.
Figure 3(e) represents the intensity distribution profiles for higher values of defect-of-focus and primary spherical aberration, i.e., for ϕd=2π and ϕs=2π. For β=0.8 and β=1, the first minima and the side-lobes on both sides of the main peak reaches to zero and the intensity of the main peak is considerably improved. The point spread function modifies into a super-resolved point spread function with increase in the intensity above that of the Airy case with reduction in the width of the central lobe. It is observed that for β=0 (Airy case), in the presence of high degree of spherical aberration and defocus intensity of the principal maximum is highly distorted. Here the intensity of the secondary maxima is high and its axial shape or resolution is found to be poor with non-zero first minima. In the presence of defocusing effect and aberration, as the degree of apodization increases from 0.4 to 1 (as shown in figure.2), there exists a consistent improvement in the lateral resolution of the main peak. It is clearly observed that for highest degree of apodization (β=1), the central light flux exhibit high intensity compared to that of Airy case (β=0) and along with zero intensity in the first minima is measured as radius of the first dark ring, resulting in super-resolved point spread function. Figure 3(e) represents the highly aberrated PSF, for β=0.8 and β=1.0, the first minima intensity is zero. And the central peak has high intensity compared to that of Airy case β=0. This is one of the desirable features in the super-resolved point spread function.
Table. 1 Maxima Minima values
|
β
|
C. Max
|
F. Min
|
F. Max
|
Position
|
Value
|
Position
|
Value
|
Position
|
Value
|
|
0
|
0
|
0.9781
|
3.7813
|
0
|
5.1345
|
0.0204
|
|
0.2
|
0
|
0.7965
|
3.8862
|
0
|
5.2206
|
0.0139
|
ε=0.1, T=0.1
|
0.4
|
0
|
0.3938
|
4.3923
|
0
|
5.6814
|
0.0032
|
φd=φs=0
|
0.6
|
0
|
0.0683
|
5.9655
|
0
|
7.619
|
0.0016
|
|
0.8
|
0
|
0.0134
|
1.6358
|
0
|
3.9809
|
0.0391
|
|
1
|
0
|
0.1731
|
2.5014
|
0
|
4.4142
|
0.0611
|
|
0
|
0
|
0.6045
|
6.929
|
0.0073
|
7.9688
|
0.0094
|
|
0.2
|
0
|
0.4959
|
7.0128
|
0.0057
|
7.9658
|
0.0068
|
ε=0.1, T=0.1
|
0.4
|
0
|
0.262
|
10.7228
|
0
|
….
|
….
|
φd=φs=π
|
0.6
|
0
|
0.0909
|
5.8495
|
0.0011
|
7.1088
|
0.0015
|
|
0.8
|
0
|
0.0913
|
2.9343
|
0.0131
|
4.0624
|
0.0169
|
|
1
|
0
|
0.2073
|
2.8289
|
0.0022
|
4.5414
|
0.0332
|
|
0
|
0
|
0.3983
|
6.7397
|
0.0126
|
7.687
|
0.0143
|
|
0.2
|
0
|
0.3317
|
6.821
|
0.0099
|
7.6648
|
0.0108
|
ε=0.1, T=0.1
|
0.4
|
0
|
0.1926
|
….
|
….
|
….
|
….
|
φd=π, φs=2π
|
0.6
|
0
|
0.1009
|
8.8856
|
0.0006
|
10.3029
|
0.0008
|
|
0.8
|
0
|
0.1183
|
3.5462
|
0.0071
|
4.2107
|
0.0076
|
|
1
|
0
|
0.1975
|
3.0112
|
0.0009
|
4.6692
|
0.02
|
|
0
|
0
|
0.2238
|
6.7273
|
0.0178
|
7.5879
|
0.0192
|
|
0.2
|
0
|
0.1869
|
6.8452
|
0.0143
|
7.5451
|
0.0148
|
ε=0.1, T=0.1
|
0.4
|
0
|
0.1177
|
….
|
….
|
….
|
….
|
φd=2π, φs=π
|
0.6
|
0
|
0.0939
|
8.6398
|
0.0007
|
10.2787
|
0.0011
|
|
0.8
|
0
|
0.1424
|
3.9885
|
0.0011
|
4.648
|
0.0013
|
|
1
|
0
|
0.2116
|
3.2614
|
0
|
4.7627
|
0.0096
|
|
0
|
0
|
0.0962
|
2.1508
|
0.0692
|
3.3748
|
0.0753
|
|
0.2
|
0
|
0.0857
|
2.3465
|
0.0647
|
3.0904
|
0.0656
|
ε=0.1, T=0.1
|
0.4
|
0
|
0.0729
|
….
|
….
|
….
|
….
|
φd=2π, φs=2π
|
0.6
|
0
|
0.0885
|
8.2537
|
0.0007
|
10.0378
|
0.0011
|
|
0.8
|
0
|
0.1319
|
4.8428
|
0
|
7.3998
|
0.0034
|
|
1
|
0
|
0.1665
|
3.5331
|
0.0011
|
7.4489
|
0.0116
|
It is also observed that, from Table 1, for higher values of β with induced phase π, first minima intensity value almost reaches zero, for β=1 and T=0.1 when ϕd =2π & ϕs=2π.
Figure 4(a-e) depicts the variation of FWHM with the apodization parameter β in the presence of defocus and primary spherical aberration with transmission factor T is 0.1 and partition parameter for the central zone ε is 0.1 with phase. The apodization parameter β is increased from 0 to 1 for the β=0 and 0.6 the optical side lobes are completely suppressed, for higher orders of apodization parameter β=1 the radius of the first dark ring should be less than that of airy case.
It is also observed that, FWHM decreased for higher values of β with induced phase π and it attains minimum for β=1 and T=0.1 when ϕd =2π & ϕs=2π. It is more clearly shown from the Table 2.
Table 2: Full Width Half Maxima values