Tailoring of Defocused Point Spread Function of Asymmetrically Apodized Four Zone Aperture Optical Imaging System in the presence of Primary Spherical Aberration

doi:10.21203/rs.3.rs-2471766/v1

In this paper we present the attempts made for the improvement of the resolution of an optical system in the presence of higher orders of defocus and primary spherical aberration function by introducing four level asymmetric filters of Triangular, Connes, Polynomial and Hanning. Four zone aperture shading amplitude in each zone have been tested for better results. Using the above four filters placing Triangular filter in the inner zone, Connes filter in the second zone, Polynomial filter in the third zone and Hanning filter in the outer zone at higher degrees of both defocusing and primary spherical aberration situation provides the optical system to achieve super-resolver state. The highest degree of the amplitude apodization improves the lateral resolution of the central maxima and suppression of side lobes which helps to study the two point resolution.

Point spread function

Hanning function

Connes function

Polynomial function

Triangular function

defocus

primary spherical aberration

super resolution

Among the mechanisms available nowadays to study an optical imaging system, the point spread function (PSF) plays an important role. The Point Spread Function of image forming optical systems is determined by the parameters of the optical system and the distance or depth of the object being imaged. In real time imaging, defocused point spread functions are being used for the optical sectioning as part of microscope image visualization system. The response of a defocused optical system to line frequencies in the object studied analytically [1]. Pera A. Stokseth [2], has analyzed the optical properties of an aberration-free defocused optical system used to image incoherently illuminated objects. Li et al. [3] have measured and analyzed defocused point spread functions and optical transfer functions of a microscope. The main purpose of their analysis was to verify the accuracy of analytical approximations of these functions, which were presented but not extensively studied in the literature. Alan R. Fitz Gerrell et.al [4] presented a two-dimensional function that graphically illustrates the effects of defocus on the optical transfer function (OTF) associated with a circularly symmetric pupil function. All these studies were relative to defocusing analysis of Point Spread Function of optical systems with symmetric apodized pupils. In this approach, the object is decomposed into infinitesimal point sources and the image changes as the superposition of the field distribution corresponding to each point-source object changes. In order to improve the resolution of the optical system of four zone aperture, asymmetric apodization was introduced, the results had sharp central peak and with the help of different filters and suitable combinations of filters, reduced the side-lobes. Suitable combinations of different filters have been examined in terms of the reduction of secondary side-lobes by modifying the four zone aperture with different degrees of amplitude apodization β with defocus and primary spherical aberration. The present research provides a significant contribution to study the resolution of PSF. Depending on the investigations done in the four zone apodization process, it can be inferred that the Triangular amplitude filter in the first zone within the limits 0-0.3, Connes amplitude filter in the second zone in the limits 0.3–0.5, Polynomial filter in the limits 0.5–0.7 and Hanning amplitude filter in the outer zone of 0.7-1 could be the solution for modifying the point spread function of the optical system under the influence of defect-of-focus and primary spherical aberration.

The aim of this study was to examine the effect of four zone filter with different combinations on the optical system. The Hanning amplitude mask is placed in the outer zone and the other three filters are interchanged, the best outcome was for the combination of Triangular amplitude filter in the first zone and Connes, Polynomial filter in the second and third zone respectively and Hanning filter in the outer zone with defocus and primary spherical aberration. The general expression for diffraction field of four amplitude filters is given by:

Where f₁(x) is triangular amplitude pupil function, f₂(x) is Connes amplitude pupil function f₃(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.

For the defocussed planes (\({{\varnothing }}_{{d}}\)), the amplitude distribution of the four zone PSF is given by

Upon introducing the primary spherical aberration (\({{\varnothing }}_{{s}}),\)eq. (2) becomes

The coefficient \({{\varnothing }}_{{s}}\)is the amount of primary spherical aberration. The coefficient \({{\varnothing }}_{{d}}\) is the amount of defocus aberration. For the calculation of intensity response, the amount of aberration is expressed in terms of dimensionless quantities, π and 2π. In current study, the pupil functions we have considered are Triangular, Connes, Polynomial and Hanning filter of second order respectively which can be represented by:

\({ f}_{1}\left(x\right)=(1-\beta x)\) (Triangular filter) (4)

\({ f}_{2}\left(x\right)={(1-{\beta }^{2}{x}^{2})}^{2}\) (Connes Filter) (5)

\({ f}_{3}\left(x\right)=1-{4\beta x}^{2}+4\beta {x}^{4}\) (Polynomial Filter) (6)

\({ f}_{4}\left(x\right)=cos\left(\pi \beta x\right)\) (Hanning Filter) (7)

where β is the apodization coefficient, which controls the degree of apodization.

The intensity PSF I(Z) which is the measurable quantity can be obtained by taking the squared modulus of S(Z). Thus,

The amplitude apodizer controls the transmission of amplitude of the optical system. Figure 1 shows the pupil transmittance as a function of the radial coordinate x with varying degrees of apodization coefficient β. For higher values of β, 90% attenuation of light transmission from the pupil edges has been shown to have a substantial effect on the response of the aberrated optical system. The reason for selecting this apodization function is that it corrects the severe effects of aberrations such as increased side lobes, nonzero first minima, displacement of internal energy from the central peak etc.

Eqns. (3), (4), (5), (6), (7) and (8) have been used to draw the point spread functions of the apodised apertures with variable apodisation using four amplitude filters using Matlab. Figure 2–6 depicts the intensity distribution curves for various degrees of apodisation parameter for given four-zone aperture combination in the absence and presence of various degrees of defocus and primary spherical aberration (a = 0.3, b = 0.5, c = 0.7) with Triangular filter in the inner zone, Connes filter in the second zone, polynomial filter in the third zone and Hanning filter in the outer zone. From the pattern of the intensity distribution curves it is evident that for β = 0.50, i.e., for partial apodisation there appears to be a total elimination of the optical side-lobes thus shaping the point spread function to the desired profile. However, for extreme apodisation β = 1, the intensity in the central lobe tailors to the desired profile there by an increase in the intensity of the central maximum and reduction in the value of full width at half maxima (FWHM) is the outcome by employing the variable apodisation.

Figure 2 and Table 1, depicts the effect of the apodization parameter β on the intensity distribution of PSF in the absence of defocus and spherical aberration. 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 = 0.5, 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.

Table 1

Intensities and positions of maxima and minima of the PSF in the absence of defocus and primary spherical aberration
		c. max		f. min		f. max		s. min		s. max
	β	Pos	Value	Pos	Value	Pos	Value	Pos	Value	Pos	Value
a = 0.3, b = 0.5, c = 0.7 φd = φs = 0	0	0	1	3.8317	0	5.1356	0.0175	7.0156	0	8.4172	0.0042
	0.25	0	0.6802	4.0195	0	5.2449	0.0076	7.0143	0	8.4054	0.0022
	0.5	0	0.2202	5.3447	0	6.6308	0.0006	9.2508	0	9.8401	0
	0.75	0	0.0042	3.4205	0.0467	7.9587	0.0056	10.0755	0	11.4906	0.001
	1	0	0.0579	1.9442	0	4.1001	0.073	6.5887	0	8.1685	0.0124

Figure 3 and Table 2, depicts the effect of the apodization parameter β on the intensity distribution of PSF defocus and primary spherical aberration i.e., for ϕ_d = π and ϕ_s = π. 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 = 0.75, the intensity is decreased. However, at apodization (β = 1), the intensity is increased than that of β = 0.5 but there is no suppression of side lobes.

Table 2

**Intensities and positions of maxima and minima of the PSF at defocus ϕ**_d **= π and primary spherical aberration ϕ**_s **= π**
		c. max		f. min		f. max		s. min		s. max
	β	Pos	Value	Pos	Value	Pos	Value	Pos	Value	Pos	Value
a = 0.3, b = 0.5, c = 0.7 φd = π φs(r⁴/4)=π	0	0	0.6133	6.9436	0.0057	8.0547	0.0081	10.1256	0.0011	11.4418	0.0025
	0.25	0	0.4179	6.9599	0.003	8.126	0.0047	10.2088	0.0004	11.6187	0.0014
	0.5	0	0.1705	11.5163	0	--	--	--	--	--	--
	0.75	0	0.1057	6.1048	0.0014	7.736	0.0051	9.8268	0.0011	11.1185	0.0018
	1	0	0.1873	2.8217	0.0177	4.2314	0.0345	6.4193	0.0036	7.9949	0.0122

From Fig. 4 and Table 3, it is evident that the effect of the apodization parameter β on the intensity distribution of PSF defocus and primary spherical aberration, i.e., for ϕ_d = π and ϕ_s = 2π. 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 = 0.5, the intensity is decreased. However, at apodization (β = 0.75 and β = 1), the intensity is increased than that of β = 0.5 but still side lobes are present.

Table 3

Intensities and positions of maxima and minima of the PSF for different degrees of defocus ϕ_d = π and primary spherical aberration ϕ_d = 2π
		c. max		f. min		f. max		s. min		s. max
	β	Pos	Value	Pos	Value	Pos	Value	Pos	Value	Pos	Value
a = 0.3, b = 0.5, c = 0.7 φd = π φs(r⁴/4) = 2π	0	0	0.4026	6.8443	0.0107	7.7248	0.012	10.0541	0.0026	11.2337	0.0037
	0.25	0	0.2783	6.8822	0.0059	7.8631	0.007	10.2521	0.0011	11.4379	0.0017
	0.5	0	0.1452	11.7061	0	--	--	--	--	--	--
	0.75	0	0.1489	5.8934	0.0014	7.6077	0.0039	9.5461	0.0015	10.8973	0.0022
	1	0	0.2297	3.2327	0.011	4.3977	0.0178	6.2876	0.0044	7.889	0.0106

From Fig. 5 and Table 4, it is clear that the effect of the apodization parameter β on the intensity distribution of PSF defocus and primary spherical aberration, i.e., for ϕ_d = 2π and ϕ_s = π. 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 = 0.5, the intensity is decreased. However, at apodization β = 0.75, the intensity is increased than that of when β = 0.5 and at β = 1, the intensity is considerably increased than that in the case of unapodized, and the first minima is also decreased when compared to airy case.

Table 4

**Intensities and positions of maxima and minima of the PSF for different degrees of defocus ϕ**_d **= 2π and primary spherical aberration ϕ**_s **= π**
		c. max		f. min		f. max		s. min		s. max
	β	Pos	Value	Pos	Value	Pos	Value	Pos	Value	Pos	Value
a = 0.3, b = 0.5, c = 0.7 φd = 2π φs(r⁴/4)=π	0	0	0.2217	3.1695	0.082	6.9543	0.0167	7.4643	0.017	10.1021	0.0036
	0.25	0	0.1506	7.0567	0.0099	7.609	0.0101	10.402	0.0017	11.2856	0.002
	0.5	0	0.1039	--	--	--	--	--	--	--	--
	0.75	0	0.174	5.1854	0.0006	7.4251	0.0037	9.3018	0.0016	10.8359	0.0026
	1	0	0.2769	3.5924	0.0023	4.64	0.0055	5.9046	0.0024	7.8135	0.0102

Table 5

Intensities and positions of maxima and minima of the PSF for different degrees of defocus ϕ_d = 2π and primary spherical aberration ϕ_s = 2π
		c. max		f. min		f. max		s. min		s. max
	β	Pos	Value	Pos	Value	Pos	Value	Pos	Value	Pos	Value
a = 0.3, b = 0.5, c = 0.7 φd = 2π φs(r⁴/4) = 2π	0	0	0.094	2.0176	0.0713	3.4264	0.0806	10.0076	0.0062	10.8588	0.0067
	0.25	0	0.068	1.9043	0.0578	3.2598	0.0626	--	--	--	--
	0.5	0	0.0825	--	--	--	--	--	--	--	--
	0.75	0	0.1724	5.424	0.001	7.2785	0.0019	8.8356	0.0011	10.718	0.0025
	1	0	0.2568	3.9076	0	4.8615	0.0012	5.5622	0.0009	7.7575	0.0071

Figure 6 and Table 5, represents the intensity distribution profiles for higher values of defect-of-focus and primary spherical aberration, i.e., for ϕ_d = 2π and ϕ_s = 2π. For 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 first 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 0.5 to 1 (as shown in figure.5), 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 7 depicts the intensity distribution when apodization is maximum and defocus at ϕ_d = 2π, for different degrees of primary spherical aberration. We can say that for ϕ_s = 2π, the intensity is almost equal to that of airy case, as well as optical side lobes are completely suppressed.

Figure 8 depicts the intensity distribution when apodization is maximum and primary spherical aberration at ϕ_s = 2π, for different degrees of defocus. We can say that for ϕ_d = 2π, the intensity is almost equal to that of airy case, as well as optical side lobes are completely suppressed.

The characteristics of an aberrated optical system under asymmetric apodization have been demonstrated theoretically.

The present work is done to improve the resolution of an optical system with four zone aperture by using Hanning amplitude filter, Triangular amplitude filter, Connes amplitude filter and Polynomial amplitude filter with all possible combinations. From present studies we can conclude that in the presence of combined effect of higher orders of defocusing and primary spherical aberration there is a significant improvement in the resolution of asymmetrically apodized optical systems in the case of four dimensional complex pupil functions. On the whole it can be emphasized that higher orders of defocusing and primary spherical aberration is found to be more effective in enhancing the resolution in terms of the intensity in the central lobe at higher degrees of apodization for the above said four zone optical system. These characteristics would make the system more effective in resolving the composite image of two line objects which are widely varying in their intensities.

Acknowledgements

We are grateful to Optics research group, Department of Physics, University College of Science, Osmania University, Hyderabad and their contributed publications.

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No competing interests reported.

Tailoring of Defocused Point Spread Function of Asymmetrically Apodized Four Zone Aperture Optical Imaging System in the presence of Primary Spherical Aberration

Status:

Version 1

Abstract

Figures

1. Introduction

2. Theory

3. Experiment And Discussions

4. Conclusions

Declarations

Acknowledgements

References

Additional Declarations

Status:

Version 1