Spoke wheel filtering strategy for on-axis flattop shaping

Tao Liu; Jiubin Tan; Jian Liu

doi:10.1364/OE.18.002822

1. Introduction

Uniformly on-axis intensity shaping with extended depth of focus (EDOF) of an optical imaging system fits a variety of applications in optical microlithography [1], high density optical data storage system [2], and microscopic imaging [3], etc. Several methods have been presented to achieve this goal in past researches, including using thin annular lens [3,4], axicon [5–8], angular modulation phase plate-light sword optical element (LSOE) [9,10], diffractive optical element (DOE) [11–14], and pupil filters [15–25], etc. Among all the methods above, pupil filtering approach has been most widely investigated. For this kind of method, radial-symmetric pupil filters have attracted much attention for the ease of analysis and fabrication since a seminal contribution by Toraldo [15] and an effective approximation by Sheppard [16]. This mainly includes power absorbing apodizers [17,18], annular multi-zone pupil filters [19–22], annular continuous lenses [23,24], and continuous pure phase mask [25]. Nonetheless, while required large focal depth extension is achieved, on-axis intensity profile either exhibits obvious oscillations within the focal depth or long tailing effect with large side lobes occurs on both sides. Besides, the developed pupil parameters are not explicitly related with the on-axis shaping performance, and usually the required transverse superresolution factor inherently restricts how large the focal depth should be extended with on-axis flattop intensity. The above problem should be a major challenge for radial-symmetric pupil filters when they are used to achieve on-axis flattop intensity focusing with required EDOF.

Following the pupil filtering principle, but partially removing the restriction of circular symmetry, we propose spoke wheel filtering strategy for on-axis flattop intensity shaping with low intensity oscillations, neat transition sides and very flexible design of focal depth extension. A partially radial-symmetric pupil filter, spoke wheel filter (SWF), is used as the incident wave front modulation element. The three-dimensional amplitude and intensity distribution in the focusing region is theoretically analyzed and calculated according to the scalar diffraction theory. Introducing a super-Gaussian function as the on-axis flattop profile with expected EDOF, a nonlinear optimization model is established, which is solved by simulated annealing (SA) algorithm. The effect of normalized center-shaded radius of SWF on transverse in-focus spot is also analyzed. The simplest SWF, SWF₂, has been chosen for detailed comparison with two kinds of radial-symmetric pupil filters, annular three-zone binary phase filter [20] and annular continuous phase filter [23], respectively.

2. Spoke wheel filtering strategy

2.1 Spoke wheel filter

Consider a generalized complex pupil filter (CPF) incident by a converging monochromatic spherical wave front passing through the center of the pupil plane (Fig. 1 ), the amplitude U in the three-dimensional focal region may be written in cylindrical coordinates as [26]

U (v, φ, u)= \int_{0}^{1} \int_{0}^{2 π} P (ρ, θ) \exp {- j [v ρ \cos (θ - φ) + u ρ^{2} / 2]} d θ ρ d ρ,

where

P (ρ, θ) = T (ρ, θ) \exp [j ϕ (ρ, θ)]

is the generalized CPF with ρ, θ being the polar coordinates over the pupil plane, and ρ is normalized by the pupil radius;

T (ρ, θ)

,

ϕ (ρ, θ)

are the transmission function and phase function of CPF, respectively. r, φ, and z are the usual cylindrical coordinates with origin at the geometrical focus; v, u are the dimensionless optical coordinates given by v = k·NA·r and u = k·NA²·z with NA being the numerical aperture of optical system. k = 2π/λ is the wave number with λ being the light wavelength. Note that the premultiplying constant and phase variation have been neglected in Eq. (1). When P(ρ, θ) is circularly symmetric, that is P(ρ, θ) = P(ρ), Eq. (1) can thus reduce to its usual form [26],

U (v, u)= \int_{0}^{1} P (ρ) \exp (- j u ρ^{2} / 2) J_{0} (v ρ) 2 π ρ d ρ

, where J ₀ denotes the zero-order Bessel function of the first kind. Nonetheless, by partially removing the restriction of circular symmetry, we can subdivide the pupil plane into equal-sized sectors with respect to θ rather than rings related to ρ. Each sector has a constant amplitude transmittance and bends into a parabolic shape either upwards or downwards along the radial direction. The detailed structure of proposed pupil filter is illustrated in Fig. 2 , which takes a shape of spoke wheel as a whole and thus it is called spoke wheel filter (SWF). SWF is a combination of center-shaded angular crossed-sectors (CSACS). Each CSACS (see the inset at the right bottom in Fig. 2) comprises four π/2-separated sectors with the same center-shaded radius, and these four curved sectors within one CSACS are the same. Let N represent the number of CSACSs included in SWF, denoted by SWF_N, and k = 0, 1, …, N-1 denotes the k-th CSACS. Thus SWF₄ is depicted in Fig. 2. Correspondingly, the pupil function for SWF should be expressed using a piecewise function in terms of the angular coordinate θ, for the k-th CSACS when πk/2N≤θ<π(k + 1)/2N, as

P_{k} (ρ, θ) = A_{k} \exp [j ϕ_{k} (ρ, θ)],

where ε≤ρ≤1, ε is the normalized radius of center-shaded circle and A_k is the constant amplitude transmittance for the k-th CSACS; the phase function is further given by

ϕ_{k} (ρ, θ) = α_{k} ρ^{2} / 2 + β_{k},

which is a parabolic function with respect to the radial coordinate ρ, and α_k, β_k are the corresponding coefficients. It should be noted that for the k-th CSACS, Eqs. (2) and (3) only describe its first sector within 0≤θ<π/2, and they are simply repeated by the other three sectors according to our design. Each CSACS modulates the incident wave front, and the light field in the focusing region should be coherently superimposed and thus redistributed.

Fig. 1 (Color online) Schematic diagram of an optical system using SWF as the incident wave front modulated element.

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Fig. 2 (Color online) Structure of SWF₄ consisting of four CSACSs, marked by k = 0, 1, 2, and 3, respectively (top view). ε represents the normalized radius of center-shaded circle. The inset at the right bottom shows one CSACS of k = 0, and the parabolic shape in the radial direction is represented by gradual changing color.

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Consider the three-dimensional amplitude distribution in the focusing region, substituting Eqs. (2) and (3) into Eq. (1), it gives:

\begin{array}{l} U_{swf} (v, φ, u) \\ = \sum_{k = 0}^{N - 1} \sum_{n = 0}^{3} \int_{ε}^{1} \int_{π k / 2 N + n π / 2}^{π (k + 1) / 2 N + n π / 2} A_{k} \exp {j [(α_{k} - u) ρ^{2} / 2 - v ρ \cos (θ - φ) + β_{k}]} d θ ρ d ρ . \end{array}

Letting v = 0 in Eq. (4), the on-axis amplitude distribution of an optical system with SWF can be derived as (see appendix A)

U_{swf} (u) = \frac{π (1 - ε^{2})}{N} \sum_{k = 0}^{N - 1} A_{k} sinc [(1 - ε^{2}) (u - α_{k}) / 4 π] \exp {j [(α_{k} - u) (1 + ε^{2}) / 4 + β_{k}]},

where, sinc denotes the function defined as

sinc(t) = \sin (π t) / (π t)

. Again, when u = 0 in Eq. (4), the transverse amplitude distribution with SWF in the focal plane (or in-focus amplitude) can be expressed as (see appendix B)

\begin{array}{l} U_{swf} (v, φ) = \\ 2 \sum_{k = 0}^{N - 1} \int_{ε}^{1} A_{k} \exp [j (α_{k} ρ^{2} / 2 + β_{k})] \int_{π k / 2 N}^{π (k + 1) / 2 N} {\cos [v ρ \cos (θ - φ)] + \cos [v ρ \sin (θ - φ)]} d θ ρ d ρ . \end{array}

It can be seen clearly from Eq. (6) that the transverse light field, not only depending on the radial coordinate v, but also the observation direction φ over the focal plane, is thus not circularly symmetric due to the asymmetrical shape of SWF.

2.2 Flattop focusing with extended depth of focus

It can be seen from Eq. (5), the on-axis amplitude distribution with SWF is expressed as a superposition of sinc functions multiplied by corresponding phase variations. The phase term in Eq. (5) is dependent on the coefficients α_k and β_k, which makes Eq. (5) complicated to analyze because a small deviation of α_k or β_k in the phase term may lead to a rapid, unexpected change of the amplitude profile. However, when α_k and β_k are purposely chosen to satisfy, α_k(1 + ε ²)/4 + β_k = 0, called decoupling condition, the phase term is thus independent of α_k and β_k, and Eq. (5) reduces to:

U_{swf} (u) = \frac{π (1 - ε^{2})}{N} \exp [- j u (1 + ε^{2}) / 4] \sum_{k = 0}^{N - 1} A_{k} sinc [(1 - ε^{2}) (u - α_{k}) / 4 π] .

It can be seen from Eq. (7) that α_k determines the axial displacement of sinc function, and therefore varying α_k should be selected in order to obtain EDOF from the superposition of amplitudes. Let

α_{k} = \frac{4 η π}{1 - ε^{2}} (k - \frac{N - 1}{2}),

and thus

β_{k} = - \frac{α_{k} (1 + ε^{2})}{4} = η π \frac{1 + ε^{2}}{1 - ε^{2}} (\frac{N - 1}{2} - k),

where η>0 represents the axial relaxation factor. η specifies the on-axis peak distance between two adjacent sinc pulses with respect to u in Eq. (7). A big η means the curved sectors in SWF should be bended largely. Further, if the amplitude transmittance A_k satisfies A_k = A_N- ₁ _-k, called symmetric condition, the on-axis intensity profile will distribute symmetrically about the focal plane. The ultimate pupil function for SWF, should thus be described, using Eqs. (2), (3), (8) and (9), as

P_{k} (ρ, θ) = A_{k} \exp {j [\frac{2 η π}{1 - ε^{2}} (k - \frac{N - 1}{2}) ρ^{2} + η π \frac{1 + ε^{2}}{1 - ε^{2}} (\frac{N - 1}{2} - k)]},

where ε≤ρ≤1, πk/2N≤θ<π(k + 1)/2N, A_k = A_N- ₁ _-k, k = 0, 1, …, N-1, and η>0. Note that only the first sector of each CSACS within 0≤θ<π/2 is described in Eq. (10).

When substituting Eqs. (8) and (9) into Eq. (6), the transverse in-focus amplitude distribution with SWF can be calculated numerically. The on-axis and in-focus intensities, I _swf(u) and I _swf(v, φ), are calculated by the modulus squared of the amplitudes, U _swf(u) and U _swf(v, φ), respectively.

When P(ρ, θ) = 1 in Eq. (1), which is the case of clear pupil, the on-axis and in-focus amplitudes reduce to the usual results, expressed analytically as U _cp(u) = πsinc[u/(4π)]exp(-ju/4) and U _cp(v) = 2πJ ₁(v)/v, respectively. J ₁ denotes the first-order Bessel function of the first kind. The intensities, I _cp(u) and I _cp(v), are calculated by the modulus squared of the corresponding amplitudes. I _cp(v) represents the Airy pattern, which is circularly symmetric, independent of the observation direction φ compared to Eq. (6).

From Eq. (6), a conclusion can be derived, using Eqs. (8), (9) and A_k = A_N- ₁ _-k, as (see appendix B)

I_{swf} (v, π / 2 \pm φ) = I_{swf} (v, φ) .

It is shown by Eq. (11) that, although the in-focus intensity distribution for SWF cannot be expressed in a form of circular symmetry, as predicted, I _swf(v, φ) still exhibits two partially symmetric forms over the focal plane. I _swf (v, π/2 + φ) = I _swf (v, φ) shows the periodicity of π/2 for the in-focus intensity in terms of φ, while I _swf (v, π/2-φ) = I _swf (v, φ) indicates the in-focus intensity is symmetric with respect to the observation direction φ = π/4. Clearly, another three symmetry axes correspond to the directions φ = 0, π/2, and 3π/4 indicated by Eq. (11). Therefore, Eq. (11) implies the transverse in-focus intensity will exhibit an eight-petal shape in the outer ring encircling the central main lobe.

2.3 Optimization results

The focal depth of an optical system with SWF and clear pupil is evaluated by FWHM (Full Width at Half Maximum) using I _swf (u) and I _cp(u), respectively. Further, let L _swf and L _cp represent the depth of focus (DOF) of an optical system with SWF and clear pupil, respectively. Define γ as L _swf /L _cp, which is used to represent the times DOF of an optical system with SWF as large as that with clear pupil, and therefore γ>1 means EDOF is obtained.

It can be seen from Eqs. (7) and (8) that, the amplitude transmittance A_k and the axial relaxation factor η determine the on-axis amplitude distribution when the structure parameters N and ε are given. If each the amplitude transmittance satisfies A_k = 1, which is phase-only pupil filter, and further when a value for η is selected to calculate the on-axis intensity distribution, a rough flattop focusing profile may be obtained according to Eq. (7). Nonetheless, in order to get a better on-axis flattop intensity distribution with EDOF, the amplitude transmittance A_k may not always be unity and a suitable η should also be chosen correspondingly.

The flattop focusing profile can be approximated with a super-Gaussian distribution, expressed as

I_{sg} (u) = I_{sgmax} \exp [- {(u^{2} / r_{sg}^{2})}^{N_{sg}}],

where I _sgmax is the maximum of I _sg(u), r _sg denotes the radius of the super-Gaussian profile at 1/e of I _sgmax, and 2N _sg is the super-Gaussian order (N _sg>1). For a big N _sg, I _sg(u) describes a more rectangular function but with short, smoothly decaying tails on both sides. If Eq. (12) is used to describe the desired on-axis flattop profile, a nonlinear optimization model is obtained, as

\begin{array}{l} Minimize δ (A, η) = {\frac{\int_{u} {[I_{swf} (u) - I_{sg} (u)]}^{2} d u}{\int_{u} I_{sg}^{2} (u) d u}}^{1 / 2} \\ Subject to: \\ A_{k} = A_{N - 1 - k} \\ 0.9 \leq A_{k} \leq 1 \\ η > 0 \end{array}

where, RMSE (Root Mean Square Error) δ(A, η) is used as the objective function of our optimization problem. The radius r _sg is chosen to be L _edof/2 approximately. We have restricted −2r _sg≤u≤2r _sg as the integration interval in Eq. (13) due to the quick attenuation property of super-Gaussian function. Decision variables should be C = [A, η]^T = [A ₀, A ₁, …, A_m, η]^T, m = N/2-1 for N is even, or (N-1)/2 for N is odd, the superscript T means C is a column vector. Therefore there are (m + 1) variables to be optimized. The constraint of amplitude transmittance, 0.9≤A_k≤1, allows high light transmission. Our goal is to minimize δ(A, η) in order to get a sufficiently uniform profile for a given focal depth extension factor, γ. Comparing with the ideal rectangular function, the super-Gaussian function can approximate the on-axis flattop profile without sharp edges on both sides. On the other hand, for a practical pupil filtering optical system, the on-axis intensity distribution can hardly exhibit extremely sharp edges due to complicated light diffraction. This is the main consideration why Eq. (12), rather than the ideal rectangular function, is used as the expected flattop profile. The above nonlinear optimization problem is solved by the global optimization algorithm, simulated annealing (SA) [27], which is one of a group of stochastic optimization algorithms, and well-suited to finding a global minimum (or maximum) for a large number of programming models. Besides, SA is insensitive to the initial values and easy to implement.

Comparison of on-axis intensity distributions before and after optimization design is plotted in Fig. 3(a) where SWF₂ and SWF₅ selected. For SWF₂, the optimized filter parameters are A ₀ = A ₁ = 1, η = 1.3459 (SWF₂ after), which is phase-only pupil filter. Before optimization, the parameters are A ₀ = A ₁ = 1 for both cases, η = 1.40 (SWF₂ before I), and η = 1.45 (SWF₂ before II). The calculated relative peak-valley intensity oscillation percentages are 12.4% (SWF₂ before I), 27.4% (SWF₂ before II) and 1.2% (SWF₂ after), respectively. The DOF is 2.69 (SWF₂ after) times of that for clear pupil. For SWF₅, the optimized filter parameters are A ₀ = A ₄ = 0.9, A ₁ = A ₃ = 0.9231, A ₂ = 1.0, η = 0.9746 (SWF₅ after), which is complex pupil filter. Before optimization, the parameters are A ₀ = A ₁ = A ₂ = A ₃ = A ₄ = 1 for both cases, η = 0.9746 (SWF₅ before I), and η = 1.1 (SWF₅ before II). Relative peak-valley intensity oscillation percentages are 21.6% (SWF₅ before I), 41.5% (SWF₅ before II), and 9% (SWF₅ after), respectively. The DOF is 5.19 (SWF₅ after) times of that for clear pupil. It is shown that DOF has been extended obviously and the on-axis intensity curves exhibit obvious oscillations before optimization; however, the peak-valley intensity oscillations reduce significantly to 1.2% and 9% after optimization design for SWF₂ and SWF₅, respectively. The three-dimensional phase profile of SWF₅ after optimization is shown in Fig. 3(b). The curved sectors with different colors represent different amplitude transmittances. The refractive index of the substrate used to fabricate SWF is assumed to be n. The maximum relief height of ideal SWF₅ is approximately 2λ/(n-1). For a practical achievable SWF, the phase profile and amplitude transmittance may deviate from the expected values, which will worsen flattop shaping property, as shown in Fig. 3(a). However, there are still other fabrication errors which might influence the performance of SWF even though η and A_k have been optimized. These errors mainly include burr-like defect of the phase relief within each sector and steepness of transition edges between adjacent sectors. For simplicity, the effect of burr-like relief defect on flattop shaping property is simulated by adding a normally distributed noise with mean zero and different standard deviation σ = std × 2π to the ideal phase profile purposely. σ corresponds to the RMS error of phase relief of SWF. The results of one simulation have been shown in Figs. 3(c) and 3(d) for SWF₂ and SWF₅, respectively. Again, we use RMSE to evaluate how large the actual on-axis intenstiy profile for SWF with fabrication error deviates from its expected profile for ideal SWF. It is found by averaging ten simulations for each std that, RMSE value increases from zero to 0.04 for both SWF₂ and SWF₅ with std increasing from zero to 0.03. The DOF extension factor is found to be insensitive to relief defects. When std is larger than 0.03, which corresponds to the relief RMS error of 0.03λ/(n-1), the on-axis intensity profiles will be worsened much, and hence unacceptable (see short dashed lines denoted by std = 0.035 in Figs. 3(c) and 3(d) for SWF₂ and SWF₅, respectively). For example, if n = 1.4, the RMS error of phase relief is suggested to be no more than 0.075λ in order to obtain good on-axis flattop shaping property.

Fig. 3 (Color online) Comparison of on-axis intensity profiles before and after optimization design of SWF₂ and SWF₅ with the same center-shaded radius ε = 0.2 (a). (b) shows the three-dimensional phase profile of SWF₅ after optimization, where different colors represent different amplitude transmittances. The effect of burr-like phase relief defect on actual on-axis intensity profile is simulated by adding a normally distributed noise with mean zero and standard deviation σ = std × 2π to the ideal phase profile for SWF₂ (c) and SWF₅ (d), respectively. Black solid lines represent the ideal intensity profiles for ideal SWF₂ and SWF₅ without fabrication errors.

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Figure 4 shows the optimization results for various EDOF requirements using SWF₂, SWF₃, SWF₅, and SWF₇, respectively. The on-axis intensity distribution for clear pupil has also been partially plotted for clear comparison (black dashed line). Relative peak-valley intensity oscillation percentages are 1.2%, 9.4%, 9%, and 15.7% for SWF₂, SWF₃, SWF₅, and SWF₇, respectively. It is shown by Fig. 4 that, EDOF is achieved with low intensity oscillations using SWF and a larger focal depth extension is obtained when the number of CSACSs included in SWF increases. Besides, from Eq. (7), it can be seen that the on-axis intensity is proportional to 1/N ², which implies the strength will significantly decrease for a large N, and thus makes it impossible to use in practice. As a result, SWF only with small N can be used in practice. The detailed pupil parameters for SWF after optimization by SA with N ranging from 2 to 7 are given in Table 1 . It is shown that SWF is pure phase pupil filter only when N = 2. In the above operations of SA, the initial temperature t_s and final temperature t_f for SA are set to be 20 and 0.001 degrees, respectively. The cooling coefficient ζ is chosen to be 0.95 for the design requirement of low precision and 0.98 for high precision, respecticely. Therefore, from the simple relation ζ^M = t_f /t_s, the iteration number M should thus be approximately 193 and 490 for ζ = 0.95 and 0.98, respectively, which should be independent of SWF_N to be designed.

Fig. 4 (Color online) Normalized on-axis intensity profiles after optimization design of SWF₂, SWF₃, SWF₅, and SWF₇ with the same center-shaded radius ε = 0.2. The on-axis intensity profile for clear pupil is partially plotted for comparison.

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Table 1. Optimized pupil parameters for SWF_N (ε = 0.2) calculated by SA

View Table

3. Comparisons and discussions

The simplest SWF is SWF₂, which is partially radial-symmetric phase-only pupil filter with a center-shaded circle. Because the on-axis light intensity is proportional to 1/N ² by Eq. (7), SWF₂ will be the most energy efficient filter.

Figure 5 shows the transverse in-focus intensity profiles for SWF₂. Introduce G, which is the transverse superresolution factor defined as the ratio of the minimum intensity position of transverse main lobe for SWF to that for clear pupil. Thus G<1 means superresolution ability. The effects of normalized center-shaded radius ε on the transverse intensity distribution for SWF₂ along the observation direction φ = 0 and π/8 over the focal plane are shown in Figs. 5(a) and 5(b), respectively. Note that the intensity curves for SWF and clear pupil have been equally normalized by the peak value of I _cp(u) for comparison. The transverse intensity curves in Figs. 5(a) and 5(b) are different, as predicted by Eq. (6). It can be clearly seen that, with ε increasing from 0 to 0.4, G reduces to 0.822 and 0.807 for φ = 0 and φ = π/8, respectively. In other words, the superresolving capability is obtained using SWF₂, and a larger center-shaded radius ε leads to a lower G but with the intensity decreasing. The first side lobe for SWF₂ becomes much larger than that for clear pupil. The results about the filter performance with respect to ε obtained above agree well with that using center-shaded filter or annular lens [4]. Figures 5(c) and 5(d) show three-dimensional transverse intensity distributions in the focal plane for clear pupil and SWF₂ (ε = 0.2), respectively. The main lobe has been compressed for SWF₂, while the first side lobe becomes larger than that for clear pupil. The outer ring encircling the main lobe exhibits obviously an eight-petal shape, as predicted by Eq. (11). Figure 5(e) further shows the contour of the minimum intensity positions of the transverse main lobe for SWF₂ (ε = 0.2) and clear pupil, respectively. It can be seen from Fig. 5 that the transverse intensity profile shows partially circular symmetry and the central main lobe of transverse spot compresses with ε increasing.

Fig. 5 (Color online) Normalized transverse intensity profiles for SWF₂ in the focal plane. (a) and (b) show the effects of normalized center-shaded radius ε on the transverse intensity distributions for SWF₂ along the observation direction φ = 0 and π/8, respectively. (c) and (d) show three-dimensional transverse intensity distributions for clear pupil and SWF₂, respectively. (e) Contour of the minimum intensity positions of the transverse main lobe, with v ranging from 3.235 to 3.321 for SWF₂ and remaining constant 3.832 for clear pupil.

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In order to further evaluate the performance of SWF₂ [Fig. 6(a) ] for on-axis flattop shaping with required EDOF, typically we choose two kinds of radial-symmetric pupil filter, annular three-zone binary phase filter [Fig. 6(b)] [20] and annular continuous phase filter [Fig. 6(c)] [23] for comparison.

Fig. 6 (Color online) Phase profiles of pupil filters: (a) SWF₂, (b) annular three-zone binary phase filter, and (c) annular continuous phase filter, respectively (top view).

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For annular three-zone binary phase filter [20] in Fig. 6(b), the parameters are annular radii (0.3, 0.58, 1), and binary phase variations (0, π, 0). The selected parameters for SWF₂ are ε = 0.67, and η = 1.335. Figures 7(a) , 7(b), and 7(c) show the normalized intensity at the u-v plane (φ = 0) for clear pupil, annular three-zone binary phase filter, and SWF₂, respectively. The on-axis intensity curves are clearly plotted in Fig. 7(d). The DOF is extended to be 2.7 (annular three-zone binary) and 4.6 (SWF₂) times of that for clear pupil. Figure 7(e) shows the transverse in-focus intensity curves. The calculated superresolving factor G is 0.71 (annular three-zone binary) and 0.73 (SWF₂), respectively. It can be seen clearly that extremely flattop profile with EDOF has been obtained using both SWF₂ and annular three-zone binary phase filter, while the superresolving factors remain approximately the same. However, it can be observed obviously that the transitional edges have very large side lobes on both sides for annular three-zone binary phase filter, but smoothly reduced edges with neat tails for SWF₂.

Fig. 7 (Color online) Normalized intensity distributions for (a) clear pupil, (b) annular three-zone binary phase filter, and (c) SWF₂ at the u-v plane (φ = 0). (d) Normalized on-axis intensity curves with no peak-valley intensity oscillations observed both for annular three-zone binary phase filter and SWF₂. (e) Transverse intensity curves for clear pupil, annular three-zone binary phase filter, SWF₂ observed at φ = 0 and π/8, respectively.

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For annular continuous phase filter [23] in Fig. 6(c), the corresponding pupil function is rewritten here as

P (ρ) = {\begin{cases} 0, 0 < ρ \leq \sqrt{0.5} \\ \exp [j 4.4 π (ρ^{2} - 0.5)], \sqrt{0.5} < ρ \leq \sqrt{0.75} . \\ \exp [- j 4.4 π (ρ^{2} - 1)], \sqrt{0.75} < ρ \leq 1 \end{cases}

The selected parameters for SWF₂ are ε = 0.795, and η = 1.38. Figures 8(a) and 8(b) show the normalized intensity at the u-v plane (φ = 0) for annular continuous phase filter and SWF₂, respectively, while Fig. 8(c) shows the on-axis intensity curves compared with that for clear pupil. Three focusing peaks are observed clearly in Fig. 8(a). The calculated focal depth has been extended to be 7.4 times for both cases. The transverse superresolving factors are approximately the same, G = 0.7. It can be seen that a very large focal depth extension has been obtained for both cases. Nonetheless, the relative peak-valley intensity oscillation for annular continuous phase filter is more than 5 times as large as that for SWF₂. It indicates that SWF₂ highly outperforms annular continuous phase filter in the uniformity of intensity within the wide range of focal depth.

Fig. 8 (Color online) Normalized intensity distributions for (a) annular continuous phase filter, and (b) SWF₂ at the u-v plane (φ = 0). (c) Normalized on-axis intensity curves with the relative peak-valley intensity oscillations, 43% for annular continuous phase filter and less than 8% for SWF₂. (d) The transverse intensity curves for clear pupil, annular continuous phase filter, SWF₂ observed at φ = 0 and π/8, respectively.

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It should be noted that the intensity profile has been normalizd by its own peak value in Figs. 7 and 8 for clear comparison. However, the actual maximal intensity value for SWF₂ should be much lower than that for annular three-zone binary phase filter in Fig. 7, which are 0.05 and 0.26 times of the peak intenisty value for clear pupil, respectively, due to the large obstruction of incident intensity for SWF₂ (ε = 0.67). The actual maximal intensity values for SWF₂ and annular continuous phase filter in Fig. 8 have been both reduced greatly, which are 0.02 and 0.08 times of the peak intensity value for clear pupil, respectively.

It can be seen that the design of SWF for on-axis flattop shaping with EDOF is flexible and straightforward. SWF₂ is phase-only pupil filter, the parameters ε and η can be easily determined for on-axis flattop shaping with required focal depth extension. The center-shaded radius ε is used to sharpen the focusing spot, while the axial relaxation factor η used to extend the focal depth. The latter comes from the fact that with the curved sectors bending in different degrees to a parabolic shape (satisfying the decoupling condition) in the radial direction over the pupil plane, the convex sectors focus the incident beam to the near side of the geometrical focus, while the concave ones push the focusing pattern to the far side. The combination (superposition of amplitudes) results in extending depth of focus. When the curved sectors are bended appropriately, a flattop intensity profile can be obtained accordingly. Because the structure of SWF is partially radial-symmetry, the transverse intensity profiles exhibit partially circular symmetry, which is an inherent advantage of SWF comparing with other angular modulation elements, such as LSOE [9] (only spiral spot observed).

4. Conclusions

Spoke wheel filtering strategy has been developed for on-axis flattop shaping with required extended depth of focus. A partially radial-symmetric pupil filter, spoke wheel filter, is proposed as the incident wave front modulation element. By introducing a super-Gaussian function as the expected flattop profile, a nonlinear optimization model is thus established, which has been successfully solved by simulated annealing algorithm. Theoretical calculations show that a larger focal depth can be obtained by increasing the number of curved sectors composing SWF and further bending them, while the peak-valley intensity oscillation is remarkably reduced by choosing a suitable axial relaxation factor and slightly changing the amplitude transmittances. The transverse in-focus intensity exhibits partially circular symmetry, the main lobe is compressed and the superresolving ability is enhanced when enlarging the central obstruction radius. Comparison between SWF₂ and circularly symmetric pupil filters shows that SWF₂ outperforms annular three-zone binary phase filter and annular continuous phase filter in its largely reduced flattop intensity oscillation and very flexible design of focal depth extension. The designed SWF₂ can be fabricated using popular techniques of microlithography or direct laser writing, while an alternative implementation of great flexibility is to use liquid crystal spatial light modulator.

Appendix A

The on-axis amplitude distribution (v = 0) with SWF, from Eq. (4), is

\begin{matrix} U_{s w f} (u) = 4 \sum_{k = 0}^{N - 1} \int_{ε}^{1} \int_{π k / 2 N}^{π (k + 1) / 2 N} A_{k} \exp {j [(α_{k} - u) ρ^{2} / 2 + β_{k}]} d θ ρ d ρ \\ = \frac{2 π}{N} \sum_{k = 0}^{N - 1} \int_{ε}^{1} A_{k} \exp {j [(α_{k} - u) ρ^{2} / 2 + β_{k}]} ρ d ρ \\ = \frac{2 π}{N} \sum_{k = 0}^{N - 1} A_{k} \exp (j β_{k}) \frac{1}{j (α_{k} - u)} {\exp [j (α_{k} - u) / 2] - \exp [j (α_{k} - u) ε^{2} / 2]} \\ = \frac{π (1 - ε^{2})}{N} \sum_{k = 0}^{N - 1} A_{k} sinc [(u - α_{k}) (1 - ε^{2}) / (4 π)] \exp {j [(α_{k} - u) (1 + ε^{2}) / 4 + β_{k}]} . \end{matrix}

Appendix B

The transverse in-focus amplitude distribution (u = 0) with SWF, from Eq. (4), is

\begin{matrix} U_{swf} (v, φ) \\ = \sum_{k = 0}^{N - 1} \sum_{n = 0}^{3} \int_{ε}^{1} A_{k} \exp [j (α_{k} ρ^{2} / 2 + β_{k})] \int_{π k / 2 N + n π / 2}^{π (k + 1) / 2 N + n π / 2} \exp [- j v ρ \cos (θ - φ)] d θ ρ d ρ \\ = \sum_{k = 0}^{N - 1} \int_{ε}^{1} A_{k} e^{j (α_{k} ρ^{2} / 2 + β_{k})} \int_{π k / 2 N}^{π (k + 1) / 2 N} [e^{- j v ρ \cos (θ - φ)} + e^{j v ρ \sin (θ - φ)} + e^{j v ρ \cos (θ - φ)} + e^{- j v ρ \sin (θ - φ)}] d θ ρ d ρ \\ = 2 \sum_{k = 0}^{N - 1} \int_{ε}^{1} A_{k} e^{j (α_{k} ρ^{2} / 2 + β_{k})} \int_{π k / 2 N}^{π (k + 1) / 2 N} {\cos [v ρ \cos (θ - φ)] + \cos [v ρ \sin (θ - φ)]} d θ ρ d ρ . \end{matrix}

Substituting (φ + π/2) for φ in Eq. (B1), it yields immediately:

U_{swf} (v, φ + π / 2) = U_{swf} (v, φ) .

Let

f_{k} (v ρ, φ) = \int_{π k / 2 N}^{π (k + 1) / 2 N} {\cos [v ρ \cos (θ - φ)] + \cos [v ρ \sin (θ - φ)]} d θ,

and suppose N is even, Eq. (B1) can thus be rewritten as

\begin{array}{l} U_{swf} (v, φ) \\ = 2 \sum_{k = 0}^{N - 1} \int_{ε}^{1} A_{k} e^{j (α_{k} ρ^{2} / 2 + β_{k})} f_{k} (v ρ, φ) ρ d ρ \\ = 2 \sum_{k = 0}^{N / 2 - 1} [\int_{ε}^{1} A_{k} e^{j (α_{k} ρ^{2} / 2 + β_{k})} f_{k} (v ρ, φ) ρ d ρ + \int_{ε}^{1} A_{N - k - 1} e^{j (α_{N - k - 1} ρ^{2} / 2 + β_{N - k - 1})} f_{N - k - 1} (v ρ, φ) ρ d ρ] \\ = 2 \sum_{k = 0}^{N / 2 - 1} [\int_{ε}^{1} A_{k} e^{j (α_{k} ρ^{2} / 2 + β_{k})} f_{k} (v ρ, φ) ρ d ρ + \int_{ε}^{1} A_{k} e^{- j (α_{k} ρ^{2} / 2 + β_{k})} f_{N - k - 1} (v ρ, φ) ρ d ρ] . \end{array}

Here the symmetric condition A_i = A_N _-1- _i, Eqs. (8) and (9) have been used.

Again, substituting (π/2-φ) for φ in Eq. (B1), it yields:

\begin{matrix} U_{swf} (v, π / 2 - φ) \\ = 2 \sum_{k = 0}^{N - 1} \int_{ε}^{1} A_{k} e^{j (α_{k} ρ^{2} / 2 + β_{k})} \int_{π k / 2 N}^{π (k + 1) / 2 N} {\cos [v ρ \cos (θ - π / 2 + φ)] + \cos [v ρ \sin (θ - π / 2 + φ)]} d θ ρ d ρ \\ = 2 \sum_{k = 0}^{N - 1} \int_{ε}^{1} A_{k} e^{j (α_{k} ρ^{2} / 2 + β_{k})} \int_{π / 2 - π (k + 1) / 2 N}^{π / 2 - π k / 2 N} {\cos [v ρ \cos (θ - φ)] + \cos [v ρ \sin (θ - φ)]} d θ ρ d ρ \\ = 2 \sum_{k = 0}^{N - 1} \int_{ε}^{1} A_{k} e^{j (α_{k} ρ^{2} / 2 + β_{k})} f_{N - k - 1} (v ρ, φ) ρ d ρ \\ = 2 \sum_{k = 0}^{N / 2 - 1} [\int_{ε}^{1} A_{k} e^{j (α_{k} ρ^{2} / 2 + β_{k})} f_{N - k - 1} (v ρ, φ) ρ d ρ + \int_{ε}^{1} A_{N - k - 1} e^{j (α_{N - k - 1} ρ^{2} / 2 + β_{N - k - 1})} f_{k} (v ρ, φ) ρ d ρ] \\ = 2 \sum_{k = 0}^{N / 2 - 1} [\int_{ε}^{1} A_{k} e^{j (α_{k} ρ^{2} / 2 + β_{k})} f_{N - k - 1} (v ρ, φ) ρ d ρ + \int_{ε}^{1} A_{k} e^{- j (α_{k} ρ^{2} / 2 + β_{k})} f_{k} (v ρ, φ) ρ d ρ] \\ = 2 \sum_{k = 0}^{N / 2 - 1} {{[\int_{ε}^{1} A_{k} e^{- j (α_{k} ρ^{2} / 2 + β_{k})} f_{N - k - 1} (v ρ, φ) ρ d ρ]}^{*} + {[\int_{ε}^{1} A_{k} e^{j (α_{k} ρ^{2} / 2 + β_{k})} f_{k} (v ρ, φ) ρ d ρ]}^{*}} \\ = U_{swf}^{*} (v, φ), \end{matrix}

where * denotes the complex conjugate. The symmetric condition A_i = A_N _-1- _i, Eqs. (8) and (9) have been used again.

Likewise, we can deduce U _swf(v, π/2-φ) = U*_swf(v, φ) when N is odd.

From Eqs. (B2) and (B5), we clearly deduce that |U _swf(v, π/2 ± φ)|² = |U _swf(v, φ)|², which is

I_{swf} (v, π / 2 \pm φ) = I_{swf} (v, φ) .

Acknowledgements

This research was supported by the National Natural Science Foundation of China (Grant no. 50905048). The authors are grateful to Dr. Jie Lin for helpful discussions and Mr. Xianfang Wen for reviewing the whole manuscript.

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Spoke wheel filtering strategy for on-axis flattop shaping

Abstract

1. Introduction

2. Spoke wheel filtering strategy

2.1 Spoke wheel filter

2.2 Flattop focusing with extended depth of focus

2.3 Optimization results

3. Comparisons and discussions

4. Conclusions

Appendix A

Appendix B

Acknowledgements

References and links

Cited By

Figures (8)

Tables (1)

Equations (21)

Optics Express

N	η	A ₀	A ₁	A ₂	A ₃	A ₄	A ₅	A ₆	γ
2	1.3459	1.0	1.0	— ^a	—	—	—	—	2.69
3	1.0316	0.9	1.0	0.9	—	—	—	—	3.07
4	0.8416	0.9	1.0	1.0	0.9	—	—	—	3.30
5	0.9746	0.9	0.9231	1.0	0.9231	0.9	—	—	5.19
6	1.1235	0.9	1.0	1.0	1.0	1.0	0.9	—	7.22
7	1.0072	0.9	0.9553	1.0	0.9670	1.0	0.9553	0.9	7.70