The point spread function of the optical system in the presence of defocus and primary, secondary and tertiary spherical aberration with Hanning and Connes amplitude filter is studied. A noted increase in the profile of the point spread function has been achieved. Employment of the Hanning and Connes amplitude pupil function under the higher degree of spherical aberration and defocusing effect helps the optical systems increases the resolution. The lateral resolution of the central peak is improved by the highest degree of the amplitude apodization parameter β. The presence of first minima with zero intensity necessary for Rayleigh criterion can be used to study two-point resolution.
Research Article
Study of the intensity of point spread function of two zone aperture optical imaging system under the combined influence of defocus and various spherical aberrations
https://doi.org/10.21203/rs.3.rs-2455991/v1
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point spread function
amplitude apodization
primary
secondary and tertiary spherical aberration
defocus
two-zone aperture
Hanning function
Connes function
super-resolution
Apodization is the process for removal of secondary side-lobes or side-bands in the diffraction field also known as the point spread function (PSF) which is required for any optical system to act as super resolver. And this can be acquired by properly choosing the transmittance of the pupil function of the optical system, the intensity around the focused fields can be totally suppressed or at least considerably reduced without increasing the dimensions of the pupil. In the current study, the imaging characteristics of the diffracted field of rotationally symmetric optical systems with Hanning and Connes amplitude filters have been investigated in terms of the reduction of secondary side-lobes by modifying the two zone aperture with different degrees of amplitude apodization β using defocus and primary, secondary and tertiary spherical aberrations. To study the imaging properties of the optical systems, the knowledge of the PSF is an important parameter in designing optical imaging systems. The present study provides a significant contribution to the resolution studies. We know that by employing suitable apodization function, the point spread function in the maximum out-of-focus image plane can be modified according to the axial shape requirements. A suitable aperture of shading is very helpful to correct the Seidel aberration effect in the image plane of the optical system. Based on the investigations done in the two zone apodization process, it can be inferred that the Hanning amplitude filter in the outer zone and Connes amplitude filter in the inner zone could be the solution for modifying the point spread function of the optical system under the strong combined influence of defect-of-focus and primary spherical aberration. In the present study, we studied the two zone aperture with the second order Hanning and Connes amplitude mask, to modify the distribution of light radiation in the focal plane of an aberration made optical systems.
The current study is done to study the effect of the Hanning and Connes amplitude filter on the optical system which is under the combined influence of high Seidel aberration and maximum defect of focus. The Connes filter is placed in the inner zone and Hanning filter is placed in the outer zone of the two zone pupil of the apodised optical system. The point spread function is subjected to a higher degree of defocusing and primary, secondary and tertiary wave aberration effect. The general expression for diffraction field of two amplitude filters is given by:
$${S}\left({Z}\right)=2{\int }_{0}^{{a}}{{f}}_{1}\left({x}\right){{J}}_{0}\left({Z}{x}\right){x}{d}{x}+2{\int }_{{a}}^{1}{{f}}_{2}\left({x}\right){{J}}_{0}\left({Z}{x}\right){x}{d}{x}$$
1
Where f1(x) is Connes amplitude pupil function, f2(x) is Hanning amplitude pupil function of the optical system; Z is the dimension less variable which forms the distance of the point of investigation from the centre of the diffraction field; and J0 (Zx) is the zero order Bessel function of the first kind; ‘x’ is the reduced radial coordinate on the exit-pupil of aberrations influenced optical system.
The general expression for Hanning and Connes amplitude mask of the two zone pupil function is written as:
The generalized expression for the amplitude impulse response of the pupil function in the presence of higher degree of primary spherical aberration and defocusing can be written as:
$${S}\left({{\varnothing }}_{{d}},{{\varnothing }}_{{s}}, {Z}\right)=2{\int }_{0}^{{a}}{{f}}_{1}\left({x}\right)\mathbf{exp}\left[-{i}\left({{\varnothing }}_{{d}}{\frac{{x}}{2}}^{2}+\frac{1}{4}{{\varnothing }}_{{s}}{{x}}^{4}\right)\right]{{J}}_{0}\left({Z}{x}\right){x}{d}{x}+2{\int }_{{a}}^{1}{{f}}_{2}\left({x}\right)\mathbf{exp}\left[-{i}\left({{\varnothing }}_{{d}}{\frac{{x}}{2}}^{2}+\frac{1}{4}{{\varnothing }}_{{s}}{{x}}^{4}\right)\right]{{J}}_{0}\left({Z}{x}\right){x}{d}{x}$$
(2)
in the presence of higher degree of secondary spherical aberration and defocusing can be written as:
$${S}\left({{\varnothing }}_{{d}},{{\varnothing }}_{{s}}, {Z}\right)=2{\int }_{0}^{{a}}{{f}}_{1}\left({x}\right)\mathbf{exp}\left[-{i}\left({{\varnothing }}_{{d}}{\frac{{x}}{2}}^{2}+\frac{1}{4}{{\varnothing }}_{{s}}{{x}}^{6}\right)\right]{{J}}_{0}\left({Z}{x}\right){x}{d}{x}+2{\int }_{{a}}^{1}{{f}}_{2}\left({x}\right)\mathbf{exp}\left[-{i}\left({{\varnothing }}_{{d}}{\frac{{x}}{2}}^{2}+\frac{1}{4}{{\varnothing }}_{{s}}{{x}}^{6}\right)\right]{{J}}_{0}\left({Z}{x}\right){x}{d}{x}$$
(3)
And in the presence of higher degree of tertiary spherical aberration and defocusing can be written as:
$${S}\left({{\varnothing }}_{{d}},{{\varnothing }}_{{s}}, {Z}\right)=2{\int }_{0}^{{a}}{{f}}_{1}\left({x}\right)\mathbf{exp}\left[-{i}\left({{\varnothing }}_{{d}}{\frac{{x}}{2}}^{2}+\frac{1}{4}{{\varnothing }}_{{s}}{{x}}^{8}\right)\right]{{J}}_{0}\left({Z}{x}\right){x}{d}{x}+2{\int }_{{a}}^{1}{{f}}_{2}\left({x}\right)\mathbf{exp}\left[-{i}\left({{\varnothing }}_{{d}}{\frac{{x}}{2}}^{2}+\frac{1}{4}{{\varnothing }}_{{s}}{{x}}^{8}\right)\right]{{J}}_{0}\left({Z}{x}\right){x}{d}{x}$$
(4)
Here Φd, Φs, are the defect-of-focus and the primary, secondary, tertiary spherical aberration parameters respectively. In current study, the pupil functions we have considered are Connes amplitude filter and Hanning filter of second order respectively which can be represented by:
$${{f}}_{1}\left({x}\right)={(1-{{\beta }}^{2}{{x}}^{2})}^{2}$$
5
$${{f}}_{2}\left({x}\right)={c}{o}{s}\left({\pi }{\beta }{x}\right)$$
6
Where ‘β’ is the amplitude apodization parameter controlling the non-uniform transmission of the pupil function.
The intensity PSF I(Z) which is the measurable quantity can be obtained by taking the squared modulus of S(Z). Thus,
$${I}\left({Z}\right)={\left|{S}\left({Z}\right)\right|}^{2}$$
7
|
|
β
|
c. max |
f. min |
f. max |
s. min |
s. max |
|||||
|
pos |
val |
pos |
val |
pos |
val |
pos |
val |
pos |
val |
||
|
a= 0.3 φd =2π φs(r4/4)=2π |
0 |
0 |
0.094 |
2.0175 |
0.0713 |
3.4266 |
0.0806 |
10.0084 |
0.0062 |
10.8589 |
0.0067 |
|
0.2 |
0 |
0.0845 |
2.1325 |
0.0675 |
3.2037 |
0.0704 |
10.1108 |
0.0047 |
10.8268 |
0.0049 |
|
|
0.4 |
0 |
0.0747 |
13.6255 |
0.0004 |
14.4355 |
0.0004 |
۔۔۔ |
۔۔۔ |
۔۔۔ |
۔۔۔ |
|
|
0.6 |
0 |
0.0952 |
8.8769 |
0.0005 |
9.9344 |
0.0006 |
12.4625 |
0.0001 |
13.6695 |
0.0002 |
|
|
0.8 |
0 |
0.1447 |
4.8609 |
0.0002 |
7.4024 |
0.005 |
9.3493 |
0.002 |
10.8076 |
0.0031 |
|
|
1 |
0 |
0.1848 |
3.7888 |
0.0006 |
7.4835 |
0.0143 |
9.673 |
0.0056 |
10.8482 |
0.0068 |
|
Table: 1
|
a= 0.5 φd =2π φs(r4/4)=2π |
0 |
0 |
0.094 |
2.0175 |
0.0713 |
3.4264 |
0.0806 |
10.0075 |
0.0062 |
10.8593 |
0.0067 |
|
0.2 |
0 |
0.0853 |
2.1287 |
0.0687 |
3.1789 |
0.0714 |
10.1548 |
0.0047 |
10.8128 |
0.0048 |
|
|
0.4 |
0 |
0.0799 |
13.7475 |
0.0004 |
14.1625 |
0.0004 |
۔۔۔ |
۔۔۔ |
۔۔۔ |
۔۔۔ |
|
|
0.6 |
0 |
0.112 |
8.4851 |
0.0009 |
9.7213 |
0.0011 |
12.59 |
0.0002 |
14.0027 |
0.0003 |
|
|
0.8 |
0 |
0.1796 |
4.9906 |
0.0005 |
7.2854 |
0.0035 |
8.8747 |
0.002 |
10.7298 |
0.0044 |
|
|
1 |
0 |
0.2363 |
3.9907 |
0 |
7.3451 |
0.0105 |
9.1378 |
0.0056 |
10.8163 |
0.0095 |
Table: 2
Table 1-2: Maxima minima position and values of the point spread function of two zone aperture for different degrees of Connes filter (first zone) (0-0.0.3,0.5) and Hanning filter (second zone) (0.3,0.5-1) apodization for primary, secondary and tertiary spherical aberration.
Table: 3
|
a= 0.7 φd =2π φs(r6/6)=2π |
0 |
0 |
0.1952 |
10.0296 |
0.0044 |
11.0049 |
0.0051 |
13.1969 |
0.0013 |
14.4206 |
0.002 |
|
0.2 |
0 |
0.1768 |
10.088 |
0.0034 |
10.9888 |
0.0038 |
13.2533 |
0.0009 |
14.466 |
0.0013 |
|
|
0.4 |
0 |
0.1521 |
10.4754 |
0.0014 |
13.588 |
0.0002 |
14.7504 |
0.0003 |
۔۔۔ |
۔۔۔ |
|
|
0.6 |
0 |
0.1737 |
9.8563 |
0.0007 |
10.6611 |
0.0007 |
۔۔۔ |
۔۔۔ |
۔۔۔ |
۔۔۔ |
|
|
0.8 |
0 |
0.2445 |
5.0627 |
0.0002 |
7.6195 |
0.0047 |
9.7021 |
0.0017 |
11.1527 |
0.0024 |
|
|
1 |
0 |
0.3016 |
3.9249 |
0.0022 |
4.5971 |
0.0028 |
5.8139 |
0.0007 |
7.9119 |
0.0093 |
Table: 4
|
a= 0.7 φd =2π φs(r8/8)=2π |
0 |
0 |
0.2592 |
10.0696 |
0.0035 |
11.0927 |
0.0042 |
13.2084 |
0.001 |
14.4709 |
0.0017 |
|
0.2 |
0 |
0.2296 |
10.1268 |
0.0027 |
11.0759 |
0.0031 |
13.2553 |
0.0007 |
14.5134 |
0.0011 |
|
|
0.4 |
0 |
0.1788 |
10.5179 |
0.0011 |
13.5559 |
0.0001 |
14.7857 |
0.0002 |
۔۔۔ |
۔۔۔ |
|
|
0.6 |
0 |
0.1753 |
5.5575 |
0.0053 |
5.9827 |
0.0053 |
9.7201 |
0.0007 |
10.8922 |
0.0008 |
|
|
0.8 |
0 |
0.2338 |
5.2985 |
0.0003 |
7.6224 |
0.0059 |
9.7822 |
0.0017 |
11.2539 |
0.0026 |
|
|
1 |
0 |
0.2915 |
3.8517 |
0.0051 |
4.3905 |
0.0056 |
5.8897 |
0.0012 |
7.9015 |
0.0108 |
Table: 5
Table 3-5: Maxima minima position and values of the point spread function of two zone aperture for different degrees of Connes filter (first zone) (0-0.7) and Hanning filter (second zone) (0.7-1) apodization for primary, secondary and tertiary spherical aberration.
Using any one equation from (2), (3), (4) and combining with (5), (6) and (7), the observations on the intensity distribution in the Point Spread Function of the apodised optical system has been obtained, for the two zone aperture by using matlab program to study the effects of the Hanning and Connes amplitude apodization, defect-of-focus and the primary, secondary, tertiary spherical aberration on the imaging efficiency of optical imaging systems. The apodization parameter β varies from 0 to 1 in steps of 0.2. With β = 0 the optical system is said to be unapodized optical system. β = 0, corresponds to the Airy case (perfect lens) and for the values of β ≠ 0 represents the apodised system. The influence of defect-of-focus (ϕd) on the optical system aberrated with the primary, secondary and tertiary spherical aberration (ϕs) is investigated analytically for various degrees of the apodization parameter β. Fig. 1-8 explains the intensity distribution profile for two zone aperture using various combinations of Hanning and Connes amplitude filters when the optical system is under the combined influence of high degrees of the primary, secondary and tertiary spherical aberration as well as defect-of-focus.
From fig. 4-8 and Table 1-5: It is observed that for β = 0 (Airy), in the presence of high degree spherical aberration (ϕs = 2π) the peak intensity of the central maximum is decreased for the maximum out-of-focus plane (ϕd = 2π). The Airy PSF lost its axial shape or resolution and non-zero first minima. Similar pattern results are noticed in the case of β = 0.2 and β = 0.4. For β = 0.6, the main peak intensity starts to increase. Whereas for β = 0.6, the first minima and the side-lobes on the both sides of the main peak reaches to zero intensity and the intensity of the main peak is considerably improved. It helps in detection of the direct image of the faint companion in every direction around the bright companion, known as two-point resolution studies. In the presence of defocusing effect and primary spherical aberration, as the degree of apodization increases from 0.6 to 1 (as shown in the Fig: 4-8), there exists a consistent improvement in the lateral resolution of the main peak.
From fig: 6 and table 3, It is evident that for highest degree of apodization (β = 1), the central light flux exhibit maximum intensity compared to that of Airy case (β = 0) and along with zero intensity in the first minima is measured, resulting in super resolved point spread function. For the highest degree of amplitude apodization (β = 1), The FWHM of the main peak obtains lower value than any other case. Fig.6 concludes that the lateral resolution of the aberrations made PSF is technically improved by placing Connes filter in the first zone (0-0.7) and Hanning filter (0.7-1) in the second zone amplitude apodization under maximum defocus and primary spherical aberration.
Hence, from Figs. 9 & 10, it is found that employing Connes filter (0-0.7) and Hanning amplitude filter (0.7-1) is effective in achieving a super-resolved PSF for higher values of amplitude apodization (β = 1), defocusing (ϕd = 2π) and primary spherical aberration (ϕs = 2π). The process of apodising the optical system, suppresses fully or partially the optical side-lobes. For β = 0.6, these side-lobes are eliminated. For β = 0.6, the axial shape and the lateral resolution of the PSF is modified into the required component. On the whole it is emphasized that the two zone aperture with Connes and Hanning amplitude apodization filters has got good result in terms of the intensity for the two zone optical system under the combined influence of defocusing effect and the primary spherical aberration.
1. No funding was received to assist with the preparation of this manuscript.
2. The authors have no financial or proprietary interests in any material discussed in this article.
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No competing interests reported.