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An ultra long high resolution beam with extension of depth of focus (DoF) in the axial direction as well as high resolution in the transverse direction has been demonstrated by a seven-zone rotationally symmetrical complex pupil filter imposed at the aperture of a focusing lens. Both amplitude and phase of the transmitted light are modulated in different zones. The scalar diffraction theory is used to optimize the zone parameters. Simulation results show that extended DoF of the beam is increased by 16 times while the spot size at the beam waist is reduced to 0.7 times.
©2007 Optical Society of America
1. Introduction
A reduction of the spot size of the point spread function (PSF) leading to better transverse resolution is highly desired in some optical imaging applications and can be realized by modifying the pupil function of a focusing lens. Due to the early demonstration of its feasibility, several techniques have been proposed to generate super resolved diffraction patterns. The normalized field ψ measured at the image plane in general form is given by [1]
where r is the normalized coordinate at the exit pupil, η is the normalized coordinate at the image plane and J 0 is the zeroth order Bessel function of the first kind. Current methods are based either on the control of the amplitude transmittance A(r) or on the modification of the phase function ϕ(r). Typical technique reported by Toraldo di Francia [2] placed a filter at the pupil of the imaging system to modify the incident light properly. Sheppard et al. [3] analyzed the diffracted intensity distribution near the geometrical focus by using amplitude pupil functions within the realm of scalar diffraction theory. In comparison with pure amplitude type pupil filters usually inducing optical intensity loss [4–8], in recent years, attentions have centered on the designs of phase-only pupil filters [9,10] due to its better transmittance which make it more practical. However, higher side lobes are always observed in pure phase pupil filters because that only phase functions of optical beam passing through different zones are controlled that leads to part of energy moving to side lobes. High side lobes cause degradation of imaging quality. Therefore, hybrid design or complex pupil filter [11] was introduced to improve the superresolving power of an optical imaging system. In this kind of complex pupil filter, both amplitude transmittance and phase function of optical beam passing through different zones are modified. The goal of the research reported in this article is to develop a complex pupil filter capable of forming an ultra long high resolution beam. In this paper we present a seven-zone design, whose excellent performance is demonstrated. The DoF has been extended to about 16 times greater than that of a clear pupil system, and high transverse resolution is achieved simultaneously.
2. Principle and design
Similar to the widely used rotationally symmetrical pupil filter structures, a seven-zone complex pupil filter is designed as shown in Fig. 1. This complex pupil filter can be made of different kinds of materials, such as silicon for near infrared wavelength region or glass, with etched or deposited annular structures. The thickness of annular structures is given by
where λ is the light wavelength and n 1 and n 2 is the refractive index of the ridges and fillers, respectively. To reduce thickness errors, e.g. phase errors, induced during fabrication procedures, two kinds of materials which have similar refractive indices for larger thickness d are selected.
Fig. 1. Schematic of the seven-zone complex pupil filter. Only phase changes are shown in this figure.
The seven-zone complex pupil filter is placed at the aperture of a focusing lens. We suppose that a uniform optical wave is used to illuminate the lens and the general complex pupil function can be written as P(ρ) = T(ρ)exp[iϕ(ρ)], where ρ is the normalized radial coordinate over the circular pupil, T(ρ) is the transmittance function, and ϕ(ρ) is the phase function. In the case of N annuli complex pupil filter, the normalized complex field amplitude G in the focal region can be written as
where r is the radial coordinate of the objective lens’s pupil plane. Tj and ϕj. defines the amplitude transmittance and phase function of zone j on the pupil plane respectively. rj defines the radial position of each zone, where r 0 = 0. We define v and u as the dimensionless radial and axial coordinates on the image side.
where R and Z are the genuine radial and axial coordinates on the image side. We use Eq. (3) to calculate the optical intensity distribution in the focal region generated by the system using the seven-zone complex pupil filter.
Several methods for pupil filter design and parameters optimization are reported, such as GLUSA which combines genetic/annealing algorithms with the hill-climbing method for continuous phase-only pupil filter [12] and pupil filter synthesis using Bessel functions from imaging side [13]. Recently, an approach for the design of an imaging system that exhibits high resolution as well as extended depth of field has been reported by E. Ben-Eliezer et al. [14]. Widely accepted methods regarding tailoring DoF in axial direction [15–18] based on the integral transforms, such as Fourier transform or Laplacian transform, of desired amplitude distribution of PSF in axial direction, are used for parameter optimization in our design. As a fabrication-oriented design, both amplitude transmittance and phase coefficient of each zone have been discretized into 0/1 and 0/π, respectively. In comparison with other reported designs, such as continuous phase filter [19] and multi phase filter [20] design, our binary design is more simplified and easier to be implemented by using currently available microfabrication techniques based on silicon or glass. The parameters of the complex pupil filter are listed in Table 1. In our design, special phase values, 0 and π, that have been chosen are the only values that prevent a focus displacement [13].
Table 1. Parameters of seven-zone complex pupil filter
3. Results
Fig. 2. (a). 3D PSF and (b). intensity distribution in the focal region of original system and (c). 3D PSF and (d) intensity distribution of the optical system optimized with seven-zone complex pupil filter.
Three dimensional PSF and intensity distribution in the focal region of the original pupil system without complex pupil filter are shown in Figs. 2(a) and 2(b). Figure 2(b) shows that the focused optical wave changes from a converging wave front to a diverging wave front within a very short distance, e.g. very short DoF. Optical wave propagating in the form of plane wave front in significantly extended DoF of the system with seven-zone complex pupil filter is shown in Figs. 2(c) and 2(d) as a comparison. From curve in magenta color shown in Fig. 3, the ripple of intensity in DoF is less than 7%. The spot size remains constant within the extended DoF region where the corresponding axial intensity is almost flat as shown in curve in magenta color, Fig. 3. Compared with the original system, the spot size of the system with seven-zone complex pupil filter is also smaller than the free space diffraction limit, as shown in Fig. 4. In Fig.4, curve in blue color corresponds to the intensity distribution in transverse direction of the original system, while curve in magenta color corresponds to that of the system with the seven-zone complex pupil filter. As the length of DoF in axial direction significantly increases, the number of side lobes in transverse direction also increases. As a result, DoF is extended to about 16 times in FWHM greater than that of the original system, and 0.7 times of the spot size at the beam waist in FWHM is also achieved. In comparison with recently reported pure phase type pupil filter [10], our proposed complex pupil filter has achieved better transverse resolution of 0.406λ/NA and lower side lobe intensity ratio Γ, defined as the ratio of the maximal intensity of the first side lobe and central lobe intensity, of only 16%.
Fig 3. Axial intensity distribution in the focal region: (blue line) original system without pupil filter and (magenta line) optimized system with seven-zone pupil filter.
Fig 4. Transverse intensity distribution in the focal region: (blue line) original system without pupil filter and (magenta line) optimized system with seven-zone pupil filter.
4. Summary
In conclusion, an ultra long high resolution beam obtained by using a seven-zone complex pupil filter and numerically simulated optical intensity distribution in focal region have been demonstrated. The method of our calculations was based on scalar diffraction theory which is a good approximation to vector diffraction theory under low NA focusing conditions [20]. Such a long high resolution beam can have a variety of applications in optical imaging, such as endoscopic optical coherence tomography (OCT) that uses low NA focusing for long DoF. the National University of Singapore (FRC Grant) and the Institute of Microelectronics (IME), A*STAR, Singapore. The authors gratefully thank Jason Teo Hui Siang, Kotlanka Rama Krishna, Wong She Mein and Edmund Khon Su Hwei in Bioelectronics & BioMEMS (BEBM) laboratory of IME for their full supports.
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