Abstract

An approach for designing purely refractive optical elements that generate engineered, multi-order-helix point spread functions (PSFs) with large peak separation for passive, optical depth measurement is presented. The influence of aberrations on the PSF’s rotation angle, which limits the depth retrieval accuracy, is studied numerically and analytically. It appears that only Zernike modes with an azimuthal index that is an integer multiple of the number of PSF peaks introduce PSF rotation, and hence depth estimation errors. This implies that high-order-helix designs have superior robustness with respect to aberrations. This is experimentally demonstrated by imaging an extended scene in the presence of severe system aberrations using novel, cost-efficient phase elements based on UV-replication on the wafer-scale.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

The depth dependence of the Point Spread Function (PSF) of an imaging system can be customized by engineering the pupil phase with a Computer Generated Hologram (CGH). This approach has been widely studied with the goal to extend the depth of focus of conventional imaging systems [1–3]. In addition, it has gained interest for monocular, three-dimensional imaging [4]. Phase element designs have been proposed that generate two separated PSF peaks with a defocus dependent, double-helix shaped rotation [5, 6]. If the relationship between the rotation angle α and the distance of an object point z is known, a measurement of α enables retrieving the axial location of an object point in addition to its lateral location. So far, this method has mainly been applied for three-dimensional particle localization and tracking [7–13]. Further studies have demonstrated that the entire depth map of an extended, three-dimensional scene can be obtained for a broadband illumination [14,15].

Previous studies have commonly considered idealized optical systems with an imaging performance close to the diffraction limit. The imaging performance of state-of-the-art optical systems in application fields such as machine vision, automotive or consumer electronics is, however, commonly far away from the diffraction limit. It is therefore essential to investigate how susceptible helical PSFs are to aberrations and how the rotation of the engineered PSF and ultimately the depth retrieval may be corrupted. In fact, the required calibration of the relationship between the rotation angle α and the object distance z only partially accounts for aberration introduced rotation. The calibration step is commonly performed close to the optical axis, which does not allow for compensating for the effect of off-axis system aberrations such as astigmatism. A calibration of the entire field of view of the pupil engineered system would be necessary in order to eliminate respective rotation angle uncertainties. Moreover, this characterization would have to be performed for each nominal object distance setting, which requires an immense calibration effort. In addition, aberration conditions may vary for different application scenarios, e.g. due to thermally induced effects or imaging through thick media. Thus, the question arises if the influence could be mitigated by altering the design of the rotating PSF.

The particular influence of primary spherical aberration, as well as coma and astigmatism has been investigated in the literature for the specific case of double-helix PSFs [16–18]. In this paper, we study the robustness for aberrations of helical PSFs in a generalized framework in which we consider designs with N peaks (N = 2, 3, 4, ...). The aim of this analysis is to provide the necessary foundation for balancing the impact of different aberrations on the rotation of the PSF in order to develop high accuracy, pupil engineered optical systems for 3D image acquisition. Furthermore, we demonstrate how high-order-helix designs that feature more than two PSF peaks provide improved robustness regarding the rotation of the PSF with defocus. The respective designs thus allow for less stringent requirements on the optical system, e.g. with respect to the residual wave front error, which reduces system complexity and costs. Especially astigmatism is of considerable importance, as it scales quadratically with the field coordinates, and therefore usually dominates over other aberrations that scale with higher powers of the field coordinates.

Initially, we present a new design method for higher order helical PSFs and evaluate its performance in section 2. In section 3, we analyze the sensitivity of the PSF rotation to different aberrations using numerical and theoretical means. Experimental results are presented in section 4 and the main results of the paper are concluded in section 5.

2. Multi-order-helix designs

2.1. Design approach

A design approach to generate PSF distributions that provide a depth dependent rotation was initially introduced based on a set Bessel beams [19]. Subsequently, these designs were described using a composition of Gauss-Laguerre (GL) modes [20] and extended towards phase only elements [5,21]. An alternative design method using a set of Fresnel zones to generate rotating, single-helix PSFs (N = 1) was described by Prasad in [6] and extended for double-helix shapes by Roider et al. in [11]. These two design approaches provide a basis for customized double-helix CGH designs that are, however, limited in different ways. On the one hand, we find that the GL mode based designs provide limited capability to generate largely separated PSF peaks with an extended rotation range. In particular, the iterative optimization results in undesired side lobes that limit the depth retrieval accuracy as well as the image quality. On the other hand, the Fresnel zone designs do not allow for direct optimization of the PSF peak confinement over a specific axial range of interest. Moreover, both methods lead to very complex CGH designs with a significant number of phase discontinuities and dislocations that degrade the PSF shape when imaging using a broad wavelength spectrum.

Here, we propose a novel phase element design routine to overcome these limitations and moreover to allow for generating generalized Nth order helix PSFs. To this end, we initially generalize the Fresnel zone design approach [6,11] by exploiting three main design parameters. The number of zones that lie within the phase element radius R is defined by the design parameter L. Each of the l = 1, ..., L zones is bounded by circles of radius Rl−1 and Rl, where Rl = R(l/L). The second design parameter can be used to tailor the trade-off between the helix peak confinement and the shape invariance during rotation. The phase of the lth ring of the CGH is given by

Φl(ϕ)=[(l1)N+1]ϕ
where ϕ is the azimuthal coordinate. Here, we consider a third design parameter N in the phase definition, which represents a generalization of existing designs [6, 11]. It directly defines the peak number of the generalized multi-order-helix PSF. Note that for N = 1, Eq. (1) reduces to the original single-helix phase element design proposed in [6]. In general, the peak separation, the rotation rate as well as the total depth range of the corresponding, engineered PSF are jointly influenced by the three design parameters [N, L, ]. On the one hand, the maximum depth range is limited by ambiguities in case of a rotation beyond [−π/N, π/N] since the depth retrieval is based on relating the rotation angle α to the object distance z. Consequently, a high number of peaks also reduces the available depth range. On the other hand, the PSF rotates at a rate 1/(N · L) rad per unit defocus change for the particular case of = 0.5, according to the analytical PSF approximation given in [6]. Hence, the respective maximum, unambiguous defocus range equals 2πL, which is independent of the peak number N. This range can be related to the maximum available depth range Δzmax of the pupil engineered optical system according to
Δzmax=2λd02LR2+2λld0L,
where the variables λ, d0 and R denote the wavelength, the object distance from the entrance pupil and the entrance pupil radius, respectively. The relationship is more complex for a general parameter , but it appears empirically that the depth range increases for decreasing .

The generalized Fresnel zone elements are considered as start designs for the following design process. In fact, the parameter set [N, L, ] directly determines the number of peaks, the peak separation, the rotation rate, as well as the maximum depth range of the final element. The individual steps of the overall design process are depicted in Fig. 1 for the design of a phase distribution Φp of exemplary double- and triple-helix CGHs. For illustration purposes, Fig. 1 additionaly provides the corresponding in-focus PSF distributions, which are calculated based on a Fourier transformation and considering a circular aperture. Figure 1(a) and 1(b) show the two Fresnel zone start design examples with design parameters [N, L, ] equal to [2, 16, 0.8] and [3, 11, 0.8], respectively. Note that the corresponding PSFs provide comparable peak distances with respect to the optical axis. We find that each of the N PSF peaks can be allocated to a dedicated circular sector of the Fresnel zone design with azimuthal extensions of 2π/N, respectively. Accordingly, the next design step only considers a single sector of the overall pupil area. In fact, restricting the design optimization to an individual subsection is essential in order to circumvent phase dislocations in the final phase element design. Figure 1(c) and 1(d) illustrate the respective sections and the corresponding PSF peak. Next, a modified version of the iterative optimization approach as proposed for the GL mode design approach [5] is applied. It is based on an Iterative Fourier Transform Algorithm (IFTA) and applies two main constraints. The first constraint maximizes the energy in the main lobes of the helical PSF for a high peak confinement. The second constraint enforces a phase only transfer function in the aperture plane. In contrast to the original method, the third constraint in the GL modal plane is voided and the pupil aperture is restricted to the circular sector of a single peak. The optimization results in a well-confined peak over a rotation range of 2π/N even for the large peak separations of the two examples, which are on the order of ten times the peak diameter. The obtained exemplary phase distributions are shown in Fig. 1(e) and 1(f). The remaining N − 1 circular sections of the CGH are subsequently filled with rotated copies of the optimized subarea as illustrated in Fig. 1(g) and 1(h). This leads to N equal peaks in the focal plane, which form a completely symmetric multi-order-helix PSF. Note that phase dislocations inside the CGH are only located at distinct lines, which separate the individual circular section. The phase distributions within each section can thus be easily transformed into continuous profiles by using a conventional one-dimensional phase unwrapping method [22]. Figure 1(i) and 1(j) show the respective profiles for the two design examples. We emphasize that these continuous pupil phase shifts can be realized by purely refractive (free-form) optical elements. These elements provide an improved performance when a broad spectral range is used for imaging. They offer minimized color artifacts, which is particularly critical for designs with a large peak separation. In fact, the double-helix element shown in Fig. 1(i) is similar to the double wedge type designs that have already been proposed in the literature [23,24]. However, our design approach enables an improved PSF confinement with minimized peak spreading over a customized defocus range as shown in the next subsection.

 figure: Fig. 1

Fig. 1 The individual steps of the proposed design process for multi-order-helix PSFs are exemplary illustrated for a double- (left column) and a triple- (right column) helix design. The phase distributions Φp as well as the corresponding PSFs are shown for the individual design steps. The Fresnel zone start designs (a,b) are initially truncated to an angular section, dedicated to a single PSF peak (c,d). The respective section is optimized for a high peak confinement (e,f) and subsequently used to fill the overall phase element with rotated copies (g,h). A purely refractive design is finally obtained by phase unwrapping in an optional step (i,j).

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2.2. Performance evaluation

The appropriateness of engineered PSFs for depth estimation is typically measured by the Cramér-Rao Lower Bound (CRLB) [5,8,14], which quantifies the theoretical limit of precision in axial ranging. The CRLB for an imaged, broadband point source is given by [14]

σCRLB2(z)=1σN2v=1Vw=1W(zH[v,w;z])2
where
H[v,w;z]=λ1λ2η(λ)H[v,w;z;λ]dλ
is the incoherent superposition of the discretely sampled spectral PSF components and σCRLB defines the standard deviation of the lower limit to measurement precision. A system operating in the detector-limited regime is assumed in Eq. (3), which is applicable for imaging extended scenes with a generally high background level using low-end cameras with high readout noise. Accordingly, the readout noise and the shot noise of the background irradiance level dominate the shot noise from the peak signal. The parameter σN2 denotes the variance of per-pixel noise, which is taken to be the sum of the two dominating noise contributions. Note that the three-dimensional PSF distribution H′[v, w; z] is normalized such that their maximum irradiance is equal to one. A simplified, constant relative illumination spectrum η(λ) = 1, which ranges from λ1 = 450 to λ2 = 650 nm, is considered in the following.

The goal of the subsequent analysis is to compare the performance of different helical PSF designs under broadband imaging conditions. The optical system parameters used for the experimental study in section 4 are exemplarily applied for this investigation. A signal-to-noise ratio of 20 is assumed, which results in σN = 0.05. The designs under investigation include the double-helix Fresnel zone design, the optimized double- and triple-helix designs as well as the purely refractive double-helix design shown in Fig. 1(a), 1(g), 1(h) and 1(i), respectively. In addition, a nominal, clear aperture system is analysed as a reference. The broadband, axial distributions H′[v, w; z] of the considered pupil functions are calculated at different object distances z and illustrated in the top part of Fig. 2. The selected axial range of 600 mm corresponds to a rotation of approximately 180° of the double-helix PSF designs. The rotation continues beyond this range but the double-helix peaks start experiencing significant spreading. In fact, the triple-helix design is only optimized for a range of approximately 300 mm but continues rotating beyond this range with accelerated peak spreading as can be seen in Fig. 2.

 figure: Fig. 2

Fig. 2 Top: Simulated, axial PSF dependency H′[v, w; z] of five considered pupil functions as indicated in the left part: A, nominal PSF for a clear aperture; B, Fresnel zone double-helix PSF; C, optimized double-helix PSF; D, optimized triple-helix PSF; E, refractive double-helix PSF. A broadband illumination spectrum ranging from 450 – 650 nm and a nominal object distance of z = 1000 mm are assumed. Note that the individual PSF plots are normalized to their respective maximum irradiance level for illustration purposes. Bottom: Corresponding axial dependencies of the standard deviation of the lower limit to measurement precision σCRLB.

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The simulated axial PSF distributions in the top part of Fig. 2 qualitatively illustrate the improved peak confinement of the novel, double-helix CGH design over the entire rotation range in comparison to the double-helix Fresnel zone design. The CRLB analysis moreover allows for a quantitative comparison, which shows that σCRLB is improved by a factor of 2 at the edges of the axial range of interest. In general, the proposed pupil engineered PSF designs feature superior values for σCRLB that are at least an order of magnitude decreased compared to the clear aperture system, which is consistent with previous results reported in [4,8,14]. In addition, the PSF comparison of the refractive double-helix design with the diffractive elements demonstrates their improved peak confinement under broadband imaging conditions. In particular, the PSFs of the diffractive elements in Fig. 2 comprise increased peak spreading in addition to the presence of a central peak at the nominal object distance. Both impairments of the PSF shape originate from a mismatch of the 2π phase jumps in the CGH design at wavelength that differ from the nominal wavelength. Finally, Fig. 2 reveals that the novel triple-helix design comprises a reduced axial resolution limit around the nominal object distance in comparison to the optimized double-helix design. In contrast, the axial resolution limit is increased for defocus values of |Δz| ≳ 100 mm, which is in accordance with the reduced axial range that is considered for the design optimization.

3. Influence of aberrations on PSF rotation

In the following investigation, the effect of individual optical aberrations on the rotation angle of multi-order-helix PSFs is analyzed. Initially, the influence is investigated by numerical simulations for the two design examples given in section 2 as well as a tetra-helix example with design parameters [N, L, ] = [4, 8, 0.8]. Note that no single peak design (N = 1) is considered, since it cannot be applied for 3D measurement using the approaches presented in [14, 15]. Finally, an analytic rotation assessment is provided for general, multi-helix CGH designs in order to obtain a theoretical understanding on which aberrations cause rotation. To this end, a general, quantitative measure for rotation of multi-order-helix PSFs is defined and evaluated in order to derive a general condition for identifying aberrations that result in a (false) PSF rotation.

3.1. Numerical investigation of PSF rotation

The pupil phase Φ(u, v) of an aberrated, pupil engineered optical system is given by a superposition of the phase element part Φp and the aberrated part according to

Φ(u,v)=Φp(u,v)+n,mAnmZnm(u,v),
where (u, v) denote the Cartesian coordinates of the pupil plane. The aberrated part is described by a set of Zernike polynomials Znm and their respective amplitudes Anm. The individual polynomials Znm are defined in polar coordinates by
Znm(ρ,ϕ)=Rnm(ρ)cosmϕ
for even and
Znm(ρ,ϕ)=Rnm(ρ)sinmϕ
for odd polynomials, respectively. The term Rnm(ρ) denotes the radial dependency according to [25] and (m, n) are non-negative integers that satisfy mn, nm = even. The effect of first order Zernike aberrations on the double- and triple-helix PSF distributions is qualitatively shown in Fig. 3. The amplitude Anm of each Zernike aberration is successively increased in Fig. 3, which introduces an rms phase error in a range of [0, π/2] rad. Note that the first three modes associated with piston, tip and tilt errors are omitted, since they do not affect the PSF shape. The actual depth retrieval using helical PSFs is based on the desired rotation with an increasing amplitude of defocus (Z20). However, Fig. 3 reveals that other Zernike aberrations lead to distortions of the original PSF shape that also give rise to an apparent rotation. In order to quantify the effect on the rotation of the multi-order-helix PSF distributions, an overall rotation angle α is defined as the average
α=1Nj=1NΔαj,
where Δαj is the difference between the individual peak angle with and without aberrations, respectively. Note that the individual angles Δαj can be positive or negative depending on the rotation direction. The change of the overall rotation angle α with increasing amplitudes of the first 21 Zernike aberrations is shown in Fig. 4 for the three multi-order-helix designs under investigation. The individual values are obtained by fitting N Gaussian peaks to the respective PSF distributions and extracting the fitted peak angle difference Δαj. The numerical fitting accuracy is on the order of 1° due to shape distortions and side lobes of the aberrated peaks. The overview in Fig. 4 demonstrates that the effect on the PSF rotation angle significantly depends on the particular Zernike mode as well as the order N of the engineered PSF. Aberrations that cause a significant rotation, i.e. |α(π/2)| > 1°, are highlighted by rectangular boxes in Fig. 4. An increasing defocus rms error leads to the desired linear change of α for all three design examples, as expected. It can be seen that primary spherical aberration (Z40) results in a significant increase of the rotation angle α as well. In contrast, the numerical simulations reveal that coma (Z31, Z31) does not lead to a change of the overall rotation angle α. The effect of astigmatism (Z22, Z22) depends on the particular CGH design. No rotation can be observed for the triple- and tetra-helix design, whereas the double-helix design is subject to significant rotation for astigmatism in x. In fact, the PSF distributions shown in Fig. 3(b) indicate that the individual peak angles of the triple-helix PSF vary indeed. However, the differences Δαi appear to compensate each other. The results are in agreement with previous analyses of double-helix designs [16–18], which demonstrated a high rotation susceptibility to spherical aberration and astigmatism, but a robustness with respect to coma. A general trend can be observed in Fig. 4 that suggests that a higher number of peaks N leads to less modes that cause significant PSF rotation. The simulations particularly suggest that the overall rotation angle is zero unless the mode order m is a multiple of the PSF peak number N. In the following, we will develop a theoretical framework that explains this result.

 figure: Fig. 3

Fig. 3 Influence of primary Zernike aberrations on the shape of an engineered (a) double- and (b) triple-helix PSF design including the effect of defocus (Z20) as well as first order astigmatism (Z22,Z22), coma (Z31,Z31) and spherical aberration (Z40).

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 figure: Fig. 4

Fig. 4 Comparison of the rotation angle dependency α of multi-helix-PSFs on the rms phase error for the first 21 Zernike aberrations (excluding piston, tip and tilt). Aberrations that cause a significant rotation, i.e. |α(π/2)| > 1°, on a particular multi-helix design are highlighted by rectangular boxes with the respective plot color.

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3.2. Theoretical rotation assessment

Considering geometrical optics, the two-dimensional position vector c⃗ where a single ray intersects the image plane is determined by the local slope of the wave front W(u, v) in the corresponding pupil location. The (coherent) superposition of all individual rays that emerge from an object point and pass through the pupil provides the centroid location of the PSF. As shown in [26], the centroid vector 〈c⃗〉 = (〈cx〉, 〈cy〉) of the PSF according to wave optics is identical with that according to geometrical optics. It can be obtained by integrating the gradient distribution of the wavefront W(u, v) over the exit pupil (EP) area according to:

c=DEEPW(u,v)dudv
where D is the exit pupil distance to the image plane and E denotes a normalization factor. In the following, only a single Zernike aberration is considered. The wave front W(u, v) of the aberrated, pupil engineered system is related to the pupil phase in Eq. (5) by the wavelength λ according to:
W(u,v)=λ2πΦ(u,v)=λ2πΦp(u,v)Wp+λ2πAnmZnm(u,v)Wnm
and can be decomposed into a sum of the aberrated (Zernike) part Wnm(u,v) and the phase element part Wp(u, v). The multi-order-helix design approach presented in section 2 demonstrates that the phase element effectively divides the wavefront W(u, v) into N angular segments Sj in the pupil plane that are defined by the azimuthal range:
2π(j1)N(ϕuϕ0)<2πjN|j=1,,N,
where ϕ0 denotes a rotational offset. Each segment Sj directly corresponds to a particular PSF peak j. Therefore, the centroids 〈c⃗j of each partial wavefront Wj(u, v) can be considered individually. We want to emphasize that this approximation is also applicable to the conventional design approaches based on GL-modes or Fresnel zones. The individual centroids are obtained by inserting Eq. (10) into Eq. (9), which leads to:
cj=DESjW(u,v)dudv=DESjWp(u,v)dudvDESjWnm(u,v)dudv=pj+znmj
Figure 5(a) illustrates the respective subareas S1 and S2 of the phase element part Wp(u, v) for the double-helix design. Furthermore, the corresponding gradient vector distribution, which needs to be integrated over the subareas to determine the centroid vectors according to Eq. (9) is indicated. An equivalent plot of defocus aberration W20(u,v) is shown in Fig. 5(b). The respective vectors 〈p⃗j and znmj according to Eq. (12) are indicated in Fig. 5(c) for the exemplary case of a defocus aberration on a double-helix design. Note that the centroid positions of the individual PSF peaks do not quantitatively match the peak locations detected by a Gaussian fit, which is due to the side lobes in combination with spot deformations and the background irradiance level. However, both methods to locate the PSF peaks agree on the presence or absence of rotation, which is the main goal of this theoretical investigation.

 figure: Fig. 5

Fig. 5 Wavefront distribution of (a) phase element part Wp and (b) defocus aberration part WZ20 (b) as well as respective gradient fields ∇⃗Wp and ∇⃗Wz4. The two highlighted subareas S1 and S2 directly correspond to the two peaks of the double-helix PSF. (c) PSF distribution associated with Wp. The direction of the centroid vectors 〈p⃗〉 and 〈z⃗k〉 are indicated for both subareas S1 and S2, respectively.

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In general, the deformation of a ring of N engineered PSF peaks can be described geometrically by an overall shift, a scaling, an anisotropic stretch and a rotation, provided that the deformation of the shape of the sub-spots themselves is neglected. Accordingly, we define the (scalar) measure:

Mnm=z^(j=1Npj×znmj),
where is the unit vector in the z-direction (along optical axis). The measure Mnm is only sensitive to rotation and not affected by the other shape distortions. The detailed evaluation of the PSF rotation measure Mnm is disclosed in the appendix (see section 6). The theoretical analysis proves that an aberrated wavefront Wnm only leads to rotation of a multi-order-helix PSF (i.e. Mnm0), if the azimuthal order m is an integer multiple κ of the peak number N according to the derived condition
m=κN|κ=0,1,2,
which provides the major outcome of the assessment. It particularly demonstrates that all types of multi-order-helix PSFs are sensitive to spherical aberration of all orders (Z40,Z60, ...). Moreover, it reveals that double-helix designs are sensitive to astigmatism of all orders (Z2±2,Z4±2, ...), whereas higher-order designs are not. In fact, triple-helix designs are sensitive to trefoil of all orders (Z3±3,Z5±3, ...), tetra-helix designs to tetrafoil etc. In addition, the rotation measure for odd Zernike aberrations in Eq. (34) vanishes for the particular case of ϕ0 = 0. These results are in agreement with the numerical simulations performed in section 3.1 and qualitatively match the rotation angle investigation shown in Fig. 4. In general, the analysis verifies that a higher number of peaks N leads to less Zernike modes that result in (unwanted) rotation and thus provides an increased aberration robustness with regard to monocular depth measurements. However, it should be recalled from section 2 that a larger parameter N results in a reduced depth range of [−π/N, π/N] at the same time.

4. Experimental results

4.1. Optical setup

A demonstration setup based on a hybrid optical imaging system is realized according to the system concept proposed in [15]. The three phase element designs studied in section 3 (double-, triple- and tetra-helix) are implemented using state-of-the-art micro-optics manufacturing processes on wafer-scale. A thin photoresist layer is initially spun on a borosilicate glass substrate and subsequently structured using the LED writing lithography approach presented by Eckstein et al. in [27]. The next steps provide a significant advancement in the manufacturing of phase elements for monocular depth estimation compared to previous elements [9, 15]. The surface profile of the photoresist layer is directly transferred onto the borosilicate glass substrate by applying reactive-ion-etching to generate a highly accurate and robust imprinting tool. Subsequently, a thin polymer layer is deposited on a second borosilicate glass substrate. The final wafer is obtained by transferring the surface profile of the tool to the polymer layer of the second substrate using UV-imprinting. We want to emphasize that the etched borosilicate glass tool potentially allows for manufacturing large numbers of polymer-on-glass wafers containing the respective phase elements. In addition to the high cost-efficiency, the utilized hybrid polymers, such as ORMOCERE [28], provide increased temperature and mechanical stability, as well as improved transparency over the entire visual range compared to previous elements that were directly realized in a photo resist layer. However, the main challenge relies in minimizing surfaces distortions during the replication process, which would directly compromise the performance of the CGH with respect to the introduced wave front error. The final optical elements are obtained by dicing the manufactured wafers and mounting the individual designs in customized, opaque frames, which define circular apertures of 10 mm diameter. The measured surface profiles h(x, y) of the tool and the replication are plotted in Fig. 6 for the double-helix CGH design. The plot shows that the profile of the tool element accurately complies with the phase element design shown in Fig. 1 (g). Moreover, the surface measurement of the replicated element demonstrates the successful transfer of the tool profile using UV replication. The apparent shape deviation is likely due to a bending of the 1 mm thin replication substrate, which does, however, not affect the PSF distribution significantly. A quantitative analysis of the residual wavefront error would additionally require the profile of the substrates back side, which is not directly accessible using a white light interferometer measurement.

 figure: Fig. 6

Fig. 6 Top: Measured surface profile height h(x, y) of (a) the tool and (b) the replication of the double-helix CGH design. The dashed white circle indicates the aperture size of 10 mm used in the optical setup. Bottom: horizontal cross-section of the profile height h(x) at y = 1 mm as indicated by dashed blue lines in the respective upper plots.

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An optical system based on two achromatic doublets (Thorlabs AC254-200-A and AC254-100-A) is set up to verify the functionality of the new phase elements and further to demonstrate the enhanced aberration robustness of high-order-helix designs. A frame with a particular CGH design can be placed in front of the two lenses, where it defines the entrance pupil of the optical demonstration system. The resulting setup, which is characterized by a focal length of 71 mm and an F-number of 7.4, is placed in front of a commercial camera system (AVT Pike F-505). The monochrome 2/3 inch CCD sensor (Sony ICX625) provides a pixel size of 3.45 μm and a total pixel count of 2452 × 2054, which leads to an overall field of view (FOV) of 114 mm × 96 mm at the nominal object distance of 1 m. The system provides diffraction limited performance on the optical axis for the design wavelength of 540 nm. However, it is intentionally subject to significant off-axis aberrations dominated by astigmatism and field curvature.

4.2. Demonstration of on-axis PSF rotation

Initially, the depth dependent rotation of the engineered PSF is investigated for the three CGH designs of interest. To this end, an on-axis LED point source with a peak irradiance at 540 nm is imaged for different object distances in a range between 860 mm and 1180 mm, which covers the optimized 90° rotation range of the tetra-helix PSF design. It is pointed out that the double- and triple helix PSFs continue rotating beyond this range. The top part of Fig. 7 illustrates the respective PSF distributions at multiple object distances. The displayed irradiance levels are normalized to the maximum peak value at z = 1000 mm for each design, respectively. The individual plots demonstrate a continuous PSF rotation, featuring clearly defined peaks. Solely one of the tetra helix PSF spots exhibits accelerated spreading with increasing object distance. This may be attributed to minor CGH distortions introduced during the manufacturing process, which affect the individual focus position as well as the depth of focus of the PSF spots. Nevertheless, Fig. 7 demonstrates that the helical PSF shape of all realized PSFs is maintained throughout an extended axial range of at least 300 mm. Accordingly, the axial range significantly extends over the diffraction limited depth of field of the nominal PSF, which is in the order of λ/NA2 ≈ 22 mm. The bottom part of Fig. 7 shows a close to linear dependency of the rotation angle α on the object distance z. The individual data points are obtained by fitting N Gaussian peaks to the respective PSF images and averaging over the fitted angular changes Δαi according to Eq. (8). The center of each helical PSF is determined by calculating the center of gravity of the individual distributions. Note that the respective localization accuracy, which is limited by noise and a finite sampling, is in the same order as the accuracy of the peak localization and adds a comparable error contribution to the overall rotation angle measurement. In fact, the angle determination for the tetra-helix design is ambiguous for angles |α| > 45°, which generally prohibits a depth retrieval beyond this angle. Prior knowledge on the rotation is used in Fig. 7 to circumvent these ambiguities for comparison purposes only. The PSFs feature comparable rotation rates Δαz on the order of 0.3 °/mm, which is in accordance with the design parameters of the three CGHs due to the similar values of N · L. Moreover, the peak distances p of approximately 12 pixels also agree for the three designs. Finally, we point out that the proper functionality of the replicated phase elements with respect to the PSF rotation confirms the successful manufacturing by UV replication. Thus, we verified a cost-efficient manufacturing approach of pupil engineering CGHs with enhanced optical, thermal and mechanical properties compared to previous photoresist-on-glass elements for monocular, three-dimensional imaging.

 figure: Fig. 7

Fig. 7 Top: Measured on-axis distributions of the engineered double-, triple- and tetra-helix PSFs at multiple object distances z (λ = 540 nm). Bottom: Corresponding dependency of the rotation angle α on the object distance z.

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4.3. PSF rotation across an extended field of view

4.3.1. Image acquisition

We exploit the severe off-axis aberrations of the demonstration setup in order to provide experimental verification of the superior aberration robustness of high-order-helix designs. To this end, the pupil engineered image of a planar screen located at the nominal object distance of 1 m and homogeneously illuminated by the previously applied LED source, is acquired. Figure 8(a) shows the nominal image distribution if no phase element is mounted inside the aperture stop. The white screen features a random pattern of black dots, which provide the required spatial features for the following analysis. It can already be observed that the image quality is significantly decreased at the edges of the field of view due to field curvature and astigmatism. The three CGHs with the designs under investigation are successively incorporated to image the same scene. For comparison purposes, only the central portion of the overall FOV is shown in Fig. 8(b)–8(e) for each of the three PSF designs in addition to the nominal case without a CGH, respectively. The shown subimages illustrate the effect of pupil engineering, which can be modeled as a convolution of the helical PSFs, shown in Fig. 7, with the nominal subimage shown in Fig. 8(b).

 figure: Fig. 8

Fig. 8 (a) Nominal distribution of the imaged test screen if no CGH is implemented in the demonstration system. The subimage shown in (b) correspond to the inset indicated in (a). The subimages (c)–(e) show the same part of the image after the double-, triple- and tetra-helix CGHs are implemented in the demonstration system, respectively.

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4.3.2. Cepstrum distribution

The cepstrum approach previously introduced by us [15] is used in order to extract the lateral rotation angle distribution α(x, y) of the engineered PSFs from the acquired images. This method is based on calculating the image cepstrum in a local environment around (x, y) and searching for peaks in the cepstrum domain. The location of the cepstrum peaks provides the information on the local PSF rotation. The approach was initially developed for double-helix designs but can be extended in order to accommodate for higher-order-helix PSFs. In fact, the number of first order peaks K in the cepstrum domain is equal to N(N − 1) for the rotating PSF designs introduced in section 2. However, peaks in the cepstrum may overlap due to symmetry in the respective PSF. This is the case for high-order-helix PSFs with even peak numbers and N > 2. The corresponding number of distinguishable peaks reduces to N2/2. In addition, aberrations can change the number of cepstrum peaks that can be distinguished by changing the PSF shape and thus eliminating or adding symmetry in the PSF peak locations.

Figure 9 illustrates the cepstrum of the 256 × 256 pixel subimages shown in Fig. 8(c)–8(e), respectively, after a Hann window is applied to mitigate edge effects. Note that these distributions are smoothed and the regions of interest are truncated as described in [15] in order to isolate first order cepstrum peaks. It can be observed that the double- and triple-helix designs exhibit clearly distinguishable cepstrum peaks with a constant radial distance and comparable magnitudes. The peaks of the tetra-helix design are, however, located at different radial distances and partially overlap, which results in 8 distinguishable peaks.

 figure: Fig. 9

Fig. 9 Processed Ccepstrum distributions of the subimages shown in Fig. 8(c)–8(e).

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4.3.3. Rotation angle analysis

The information on the location of the cepstrum peaks is used to determine to the rotation angle α and in that way enable a local depth estimation. Due to the complex tetra-helix cepstrum distribution with varying peak distances and overlapping peaks, we limit ourselves to the double- and triple-helix design in order to demonstrate the difference in aberration robustness. The local cepstrum distributions are calculated for each image position (x, y) in a 256 × 256 pixel environment. The first order cepstrum peaks are located by fitting Gaussian functions to the peaks. The overall rotation angle α directly corresponds to the average of the K/2 peak angle changes in the cepstrum domain. It is noted that the cepstrum approach reduces the unambiguous angular rotation range of the triple-helix designs by a factor of two, i.e. [−30°, 30°], due to the PSF symmetry. The calculated overall rotation angle distribution of the test images are shown in Fig. 10(a) and 10(c). It can be observed that α(x, y) is increasing towards the edges of the FOV despite the constant depth of the imaged screen. This observation can be related to a significant field curvature, a field dependent defocus aberration (Z20) that results in undesired rotation for all multi-order-helix designs. Note that a comparison between the two plots yet reveals a significant shape deviation.

 figure: Fig. 10

Fig. 10 Measured rotation angle distribution α of the imaged demonstration scene using a (a) double- and (b) triple-helix PSF. (c) and (d) show the residual rotation angle distributions αr(x, y) for the two cases after eliminating the effect of field curvature by fitting and subtracting a parabolic function from α(x, y).

Download Full Size | PPT Slide | PDF

In order to eliminate the effect of field curvature and enable the analysis of effects from other aberrations, a parabolic function is fitted and subtracted from the rotation angle distribution α(x, y). In practice, this could be achieved by incorporating a customized field flattening element in close proximity to the image sensor. Figure 10(c) and 10(d) show the respective angular distributions αr(x, y) with field curvature removed. The remaining error is mainly due to field dependent astigmatism, which results in false PSF rotation for the double-helix design, as predicted in section 3. In contrast, the triple-helix design does not suffer from this error, which allows for a significantly increased accuracy of the depth measurement. The RMS error in the angle estimation over the entire FOV is reduced from 1.7° to 0.3°, which demonstrates the superior aberration robustness of a high-order-helix PSF compared to the conventional double-helix PSF shape. We emphasize that the improved robustness is particularly crucial for low-NA, large FOV optical systems in machine vision or automotive applications, where astigmatism is one of the major aberrations.

5. Conclusion

A novel approach for the design of phase elements to generate engineered, multi-order-helix PSFs with a depth dependent rotation is presented. The approach is based on a combination of two previously introduced design methods and enables PSF distributions with a large peak separation and a high peak confinement over an extended depth range. In addition, it allows for the design of purely refractive optical elements, which potentially provide an enhanced imaging performance with respect to color artifacts. The performance of the novel designs is assessed based on Cramér-Rao Lower Bound analysis. The influence of Zernike aberrations on the rotation of engineered, multi-order-helix PSFs has been investigated. The numerical analysis revealed that high-order-helix designs (N > 2) facilitate a more robust depth estimation in the presence of aberrations. In particular, a theoretical assessment showed that only Zernike aberrations with an azimuthal order m = κ · N (κ = 0, 1, 2, ...) lead to PSF rotation.

Furthermore, the successful manufacturing of cost-efficient phase elements based on UV-replication on wafer-scale is verified. A demonstration system has been implemented to demonstrate the proper depth dependent rotation of novel triple- and tetra-helix PSF designs. In addition to a theoretical assessment, the superior aberration robustness of high-order-helix designs is demonstrated experimentally by analyzing the rotation angle distribution of an imaged scene.

The main goal of this paper is to provide a basis for balancing optical aberrations during the design process of pupil engineered, 3D imaging systems, which require minimized unwanted PSF rotation for an accurate monocular depth measurement. In contrast, future work will include investigating the possibility for a dedicated measurement of aberrated wavefronts using optical systems that feature a multi-order-helix PSF. In particular, we aim to exploit the effect of different aberrations on the shape of the PSF in order to extract information on present aberrations for potential applications such as adaptive optics.

It is mentioned that higher order helix PSFs can also find application in the field of single molecule imaging, where imaging at relatively large depth in cells and tissue gains considerable importance, implying that aberration robust PSFs could potentially be of use.

6. Appendix: evaluation of rotation measure Mnm

The evaluation of the measure for PSF rotation Mnm, as introduced in Eq. (13), is provided in the following. The centroid positions 〈p⃗j of each multi-order-helix PSF peak j = 1, 2, ..., N can be written as:

pj=p(cosϕjsinϕj),
where:
ϕj=ϕ0+π2πN+j2πN
according to the design approach described in section 2. After transition to polar coordinates, the position vectors can be expressed as:
pj=pcos(ϕϕj)ρ^psin(ϕϕj)ϕ^,
where ρ̂ and ϕ̂ are unit vectors in the radial and azimuthal direction, and p is the distance of the peak to the PSF center. The Zernike centroid vector znmj is determined by the gradient in polar coordinates according to:
znmj=Wnmj=Wnmρρ^1ρWnmϕϕ^j
Inserting Eq. (17) and Eq. (18) into the rotation measure defined in Eq. (13) leads to the following expression:
Mnm=pj=1N1ρcos(ϕϕj)Wnmϕsin(ϕϕj)Wnmρj
The integration variable ψ = 2πj/N + ϕ0ϕ is used for the average over the pupil in Eq. (19) so that for all segments j the integration range is:
0ψ2πN
The rotation measure then follows as:
Mnm=pDEρ=01ψ=02π/Nj=1N[1ρsin(πNψ)Wnm(ρ,2πjN+ϕ0ψ)ψ+cos(πNψ)Wnm(ρ,2πjN+ϕ0ψ)ρ]ρdρdψ
First, the rotation measure is considered for even Zernike aberration functions Znm defined in Eq. (6), which includes modes with m = 0. It can be written as:
Mnm=2πpDAnmλE[mFssρ=01Rnm(ρ)dρ+Fccρ=01ρRnm(ρ)ρdρ]
using the definition of Wnm in Eq. (10). The two respective integrals over the azimuthal angle ψ are defined by:
Fss=ψ=02π/N[sin(πNψ)j=1Nsin(2πjmN+mϕ0mψ)]dψ
Fcc=ψ=02π/N[cos(πNψ)j=1Ncos(2πjmN+mϕ0mψ)]dψ
In fact, both integrals equal zero unless m is a multiple of the PSF peak number N according to:
m=κN|κ=0,1,2,
For these specific cases, the integrals take the form:
Fss=Nκ2N21[2κNsin(πN)cos(κNϕ0)]
Fcc=Nκ2N21[2sin(πN)cos(κNϕ0)]
The latter integral over the radial coordinate in Eq. (22) can be simplified by partial integration according to:
ρ=01ρRnm(ρ)ρdρ=1ρ=01Rnm(ρ)dρ
The remaining integral over the the radial Zernike polynomial Rnm turns out to be:
ρ=01Rnm(ρ)dρ=(1)(nm)/2(n+1)
This result was found using Mathematica, but no formal proof could be found in the literature. Thus, the rotation measure in Eq. (22) takes the form:
Mnm=2πpDAnmλE[Fcc+(mFssFcc)((1)(nm)/21(n+1))]
Inserting Eq. (26) and Eq. (27) in Eq. (30) and considering only Zernike aberrations that fulfill condition (25) results in:
MnκN=4πpDAnκNNλE(κ2N21)sin(πN)cos(κNϕ0)[1+((1)(nκN)/2(κ2N21)(n+1))]
For odd Zernike aberration functions defined in Eq. (7) the two integrals over the azimuthal angle ψ in Eq. (21) also equal zero unless condition (25) is fulfilled. In this case, the respective integrals take the form
Fsc=Nκ2N21[2κN(1)κsin(πN)sin(κNϕ0)]
Fcs=Nκ2N21[2(1)κsin(πN)sin(κNϕ0)]
and the rotation measure results in
MnκN=(1)κ4πpDAnκNNλE(κ2N21)sin(πN)sin(κNϕ0)[1((1)(nκN)/2(κ2N2+1)(n+1))]

Funding

German Federal Ministry of Education and Research (BMBF) (13N13667).

Acknowledgments

The authors would like to thank Marko Stumpf, Andre Matthes and Peter Dannberg for manufacturing the CGHs, as well as Jens Dunkel for performing the surface profile measurements.

References and links

1. E. R. Dowski and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34, 1859–1866 (1995). [CrossRef]   [PubMed]  

2. A. Castro and J. Ojeda-Castañeda, “Asymmetric phase masks for extended depth of field,” Appl. Opt. 43, 3474 (2004). [CrossRef]   [PubMed]  

3. Q. Yang, L. Liu, and J. Sun, “Optimized phase pupil masks for extended depth of field,” Opt. Commun. 272, 56–66 (2007). [CrossRef]  

4. A. Greengard, Y. Y. Schechner, and R. Piestun, “Depth from diffracted rotation,” Opt. Lett. 31, 181–183 (2006). [CrossRef]   [PubMed]  

5. S. R. P. Pavani and R. Piestun, “High-efficiency rotating point spread functions”, Opt. Express 16, 3484–3489 (2008). [CrossRef]   [PubMed]  

6. S. Prasad, “Rotating point spread function via pupil-phase engineering,” Opt. Lett. 38, 585–587 (2013). [CrossRef]   [PubMed]  

7. S. R. P. Pavani and R. Piestun, “Three dimensional tracking of fluorescent microparticles using a photon-limited double-helix response system,” Opt. Express 16, 22048–22057 (2008). [CrossRef]   [PubMed]  

8. S. R. P. Pavani, A. Greengard, and R. Piestun, “Three-dimensional localization with nanometer accuracy using a detector-limited double-helix point spread function system,” Appl. Phys. Lett. 95, 021103 (2009). [CrossRef]  

9. G. Grover, S. Quirin, C. Fiedler, and R. Piestun, “Photon efficient double-helix PSF microscopy with application to 3D photo-activation localization imaging,” Biomed. Opt. Express 2, 3010–3020 (2011). [CrossRef]   [PubMed]  

10. A. Barsic, G. Grover, and R. Piestun, “Three-dimensional super-resolution and localization of dense clusters of single molecules,” Sci. Reports 4, 5388 (2014). [CrossRef]  

11. C. Roider, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Axial super-localisation using rotating point spread functions shaped by polarisation-dependent phase modulation,” Opt. Express 22, 4029–4037 (2014). [CrossRef]   [PubMed]  

12. S. Quirin, S. R. P. Pavani, and R. Piestun, “Optimal 3D single-molecule localization for superresolution microscopy with aberrations and engineered point spread functions,” Proc. Natl. Acad. Sci. 109, 675–679 (2012). [CrossRef]   [PubMed]  

13. C. Roider, R. Heintzmann, R. Piestun, and A. Jesacher, “Deconvolution approach for 3D scanning microscopy with helical phase engineering,” Opt. Express 24, 15456 (2016). [CrossRef]   [PubMed]  

14. S. Quirin and R. Piestun, “Depth estimation and image recovery using broadband, incoherent illumination with engineered point spread functions [Invited],” Appl. Opt. 52, A367–A376 (2013). [CrossRef]   [PubMed]  

15. R. Berlich, A. Bräuer, and S. Stallinga, “Single shot three-dimensional imaging using an engineered point spread function,” Opt. Express 24, 5946 (2016). [CrossRef]   [PubMed]  

16. S. Ghosh and C. Preza, “Characterization of a three-dimensional double-helix point-spread function for fluorescence microscopy in the presence of spherical aberration,” J. Biomed. Opt. 18, 036010 (2013). [CrossRef]   [PubMed]  

17. Z. Cao and K. Wang, “Effects of astigmatism and coma on rotating point spread function,” Appl. Opt. 53, 7325–7330 (2014). [CrossRef]   [PubMed]  

18. M. Baránek, P. Bouchal, M. Šiler, and Z. Bouchal, “Aberration resistant axial localization using a self-imaging of vortices,” Opt. Express 23, 15316 (2015). [CrossRef]   [PubMed]  

19. R. Piestun and J. Shamir, “Generalized propagation-invariant wave fields,” J. Opt. Soc. Am. A 15, 3039–3044 (1998). [CrossRef]  

20. R. Piestun, Y. Schechner, and J. Shamir, “Propagation-invariant wave fields with finite energy,” JOSA A 17, 294–303 (2000). [CrossRef]   [PubMed]  

21. G. Grover and R. Piestun, “New approach to double-helix point spread function design for 3D super-resolution microscopy,” Proc. SPIE 8590, 85900M (2013). [CrossRef]  

22. K. Itoh, “Analysis of the Phase Unwrapping Algorithm,” Appl. Opt. 21, 2470 (1982). [CrossRef]   [PubMed]  

23. D. Baddeley, M. B. Cannell, and C. Soeller, “Three-dimensional sub-100 nm super-resolution imaging of biological samples using a phase ramp in the objective pupil,” Nano Res. 4, 589–598 (2011). [CrossRef]  

24. A. N. Simonov and M. C. Rombach, “Passive ranging and three-dimensional imaging through chiral phase coding,” Opt. Lett. 36, 115–117 (2011). [CrossRef]   [PubMed]  

25. M. Born and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light (Elsevier, 2013).

26. V. N. Mahajan, “Line of sight of an aberrated optical system,” J. Opt. Soc. Am. A2 (1985). [CrossRef]  

27. H.-C. Eckstein, U. D. Zeitner, R. Leitel, M. Stumpf, P. Schleicher, A. Bräuer, and A. Tünnermann, “High dynamic grayscale lithography with an LED-based micro-image stepper,” Proc. SPIE 9780, 97800T (2016). [CrossRef]  

28. K.-H. Haas and H. Wolter, “Synthesis, properties and applications of inorganic–organic copolymers (ORMOCERs),” Curr. Opin. Solid State Mater. Sci. 4, 571–580 (1999). [CrossRef]  

References

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  • |

  1. E. R. Dowski and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34, 1859–1866 (1995).
    [Crossref] [PubMed]
  2. A. Castro and J. Ojeda-Castañeda, “Asymmetric phase masks for extended depth of field,” Appl. Opt. 43, 3474 (2004).
    [Crossref] [PubMed]
  3. Q. Yang, L. Liu, and J. Sun, “Optimized phase pupil masks for extended depth of field,” Opt. Commun. 272, 56–66 (2007).
    [Crossref]
  4. A. Greengard, Y. Y. Schechner, and R. Piestun, “Depth from diffracted rotation,” Opt. Lett. 31, 181–183 (2006).
    [Crossref] [PubMed]
  5. S. R. P. Pavani and R. Piestun, “High-efficiency rotating point spread functions”, Opt. Express 16, 3484–3489 (2008).
    [Crossref] [PubMed]
  6. S. Prasad, “Rotating point spread function via pupil-phase engineering,” Opt. Lett. 38, 585–587 (2013).
    [Crossref] [PubMed]
  7. S. R. P. Pavani and R. Piestun, “Three dimensional tracking of fluorescent microparticles using a photon-limited double-helix response system,” Opt. Express 16, 22048–22057 (2008).
    [Crossref] [PubMed]
  8. S. R. P. Pavani, A. Greengard, and R. Piestun, “Three-dimensional localization with nanometer accuracy using a detector-limited double-helix point spread function system,” Appl. Phys. Lett. 95, 021103 (2009).
    [Crossref]
  9. G. Grover, S. Quirin, C. Fiedler, and R. Piestun, “Photon efficient double-helix PSF microscopy with application to 3D photo-activation localization imaging,” Biomed. Opt. Express 2, 3010–3020 (2011).
    [Crossref] [PubMed]
  10. A. Barsic, G. Grover, and R. Piestun, “Three-dimensional super-resolution and localization of dense clusters of single molecules,” Sci. Reports 4, 5388 (2014).
    [Crossref]
  11. C. Roider, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Axial super-localisation using rotating point spread functions shaped by polarisation-dependent phase modulation,” Opt. Express 22, 4029–4037 (2014).
    [Crossref] [PubMed]
  12. S. Quirin, S. R. P. Pavani, and R. Piestun, “Optimal 3D single-molecule localization for superresolution microscopy with aberrations and engineered point spread functions,” Proc. Natl. Acad. Sci. 109, 675–679 (2012).
    [Crossref] [PubMed]
  13. C. Roider, R. Heintzmann, R. Piestun, and A. Jesacher, “Deconvolution approach for 3D scanning microscopy with helical phase engineering,” Opt. Express 24, 15456 (2016).
    [Crossref] [PubMed]
  14. S. Quirin and R. Piestun, “Depth estimation and image recovery using broadband, incoherent illumination with engineered point spread functions [Invited],” Appl. Opt. 52, A367–A376 (2013).
    [Crossref] [PubMed]
  15. R. Berlich, A. Bräuer, and S. Stallinga, “Single shot three-dimensional imaging using an engineered point spread function,” Opt. Express 24, 5946 (2016).
    [Crossref] [PubMed]
  16. S. Ghosh and C. Preza, “Characterization of a three-dimensional double-helix point-spread function for fluorescence microscopy in the presence of spherical aberration,” J. Biomed. Opt. 18, 036010 (2013).
    [Crossref] [PubMed]
  17. Z. Cao and K. Wang, “Effects of astigmatism and coma on rotating point spread function,” Appl. Opt. 53, 7325–7330 (2014).
    [Crossref] [PubMed]
  18. M. Baránek, P. Bouchal, M. Šiler, and Z. Bouchal, “Aberration resistant axial localization using a self-imaging of vortices,” Opt. Express 23, 15316 (2015).
    [Crossref] [PubMed]
  19. R. Piestun and J. Shamir, “Generalized propagation-invariant wave fields,” J. Opt. Soc. Am. A 15, 3039–3044 (1998).
    [Crossref]
  20. R. Piestun, Y. Schechner, and J. Shamir, “Propagation-invariant wave fields with finite energy,” JOSA A 17, 294–303 (2000).
    [Crossref] [PubMed]
  21. G. Grover and R. Piestun, “New approach to double-helix point spread function design for 3D super-resolution microscopy,” Proc. SPIE 8590, 85900M (2013).
    [Crossref]
  22. K. Itoh, “Analysis of the Phase Unwrapping Algorithm,” Appl. Opt. 21, 2470 (1982).
    [Crossref] [PubMed]
  23. D. Baddeley, M. B. Cannell, and C. Soeller, “Three-dimensional sub-100 nm super-resolution imaging of biological samples using a phase ramp in the objective pupil,” Nano Res. 4, 589–598 (2011).
    [Crossref]
  24. A. N. Simonov and M. C. Rombach, “Passive ranging and three-dimensional imaging through chiral phase coding,” Opt. Lett. 36, 115–117 (2011).
    [Crossref] [PubMed]
  25. M. Born and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light (Elsevier, 2013).
  26. V. N. Mahajan, “Line of sight of an aberrated optical system,” J. Opt. Soc. Am. A2 (1985).
    [Crossref]
  27. H.-C. Eckstein, U. D. Zeitner, R. Leitel, M. Stumpf, P. Schleicher, A. Bräuer, and A. Tünnermann, “High dynamic grayscale lithography with an LED-based micro-image stepper,” Proc. SPIE 9780, 97800T (2016).
    [Crossref]
  28. K.-H. Haas and H. Wolter, “Synthesis, properties and applications of inorganic–organic copolymers (ORMOCERs),” Curr. Opin. Solid State Mater. Sci. 4, 571–580 (1999).
    [Crossref]

2016 (3)

2015 (1)

2014 (3)

2013 (4)

S. Quirin and R. Piestun, “Depth estimation and image recovery using broadband, incoherent illumination with engineered point spread functions [Invited],” Appl. Opt. 52, A367–A376 (2013).
[Crossref] [PubMed]

S. Ghosh and C. Preza, “Characterization of a three-dimensional double-helix point-spread function for fluorescence microscopy in the presence of spherical aberration,” J. Biomed. Opt. 18, 036010 (2013).
[Crossref] [PubMed]

S. Prasad, “Rotating point spread function via pupil-phase engineering,” Opt. Lett. 38, 585–587 (2013).
[Crossref] [PubMed]

G. Grover and R. Piestun, “New approach to double-helix point spread function design for 3D super-resolution microscopy,” Proc. SPIE 8590, 85900M (2013).
[Crossref]

2012 (1)

S. Quirin, S. R. P. Pavani, and R. Piestun, “Optimal 3D single-molecule localization for superresolution microscopy with aberrations and engineered point spread functions,” Proc. Natl. Acad. Sci. 109, 675–679 (2012).
[Crossref] [PubMed]

2011 (3)

2009 (1)

S. R. P. Pavani, A. Greengard, and R. Piestun, “Three-dimensional localization with nanometer accuracy using a detector-limited double-helix point spread function system,” Appl. Phys. Lett. 95, 021103 (2009).
[Crossref]

2008 (2)

2007 (1)

Q. Yang, L. Liu, and J. Sun, “Optimized phase pupil masks for extended depth of field,” Opt. Commun. 272, 56–66 (2007).
[Crossref]

2006 (1)

2004 (1)

2000 (1)

R. Piestun, Y. Schechner, and J. Shamir, “Propagation-invariant wave fields with finite energy,” JOSA A 17, 294–303 (2000).
[Crossref] [PubMed]

1999 (1)

K.-H. Haas and H. Wolter, “Synthesis, properties and applications of inorganic–organic copolymers (ORMOCERs),” Curr. Opin. Solid State Mater. Sci. 4, 571–580 (1999).
[Crossref]

1998 (1)

1995 (1)

1982 (1)

Baddeley, D.

D. Baddeley, M. B. Cannell, and C. Soeller, “Three-dimensional sub-100 nm super-resolution imaging of biological samples using a phase ramp in the objective pupil,” Nano Res. 4, 589–598 (2011).
[Crossref]

Baránek, M.

Barsic, A.

A. Barsic, G. Grover, and R. Piestun, “Three-dimensional super-resolution and localization of dense clusters of single molecules,” Sci. Reports 4, 5388 (2014).
[Crossref]

Berlich, R.

Bernet, S.

Born, M.

M. Born and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light (Elsevier, 2013).

Bouchal, P.

Bouchal, Z.

Bräuer, A.

R. Berlich, A. Bräuer, and S. Stallinga, “Single shot three-dimensional imaging using an engineered point spread function,” Opt. Express 24, 5946 (2016).
[Crossref] [PubMed]

H.-C. Eckstein, U. D. Zeitner, R. Leitel, M. Stumpf, P. Schleicher, A. Bräuer, and A. Tünnermann, “High dynamic grayscale lithography with an LED-based micro-image stepper,” Proc. SPIE 9780, 97800T (2016).
[Crossref]

Cannell, M. B.

D. Baddeley, M. B. Cannell, and C. Soeller, “Three-dimensional sub-100 nm super-resolution imaging of biological samples using a phase ramp in the objective pupil,” Nano Res. 4, 589–598 (2011).
[Crossref]

Cao, Z.

Castro, A.

Cathey, W. T.

Dowski, E. R.

Eckstein, H.-C.

H.-C. Eckstein, U. D. Zeitner, R. Leitel, M. Stumpf, P. Schleicher, A. Bräuer, and A. Tünnermann, “High dynamic grayscale lithography with an LED-based micro-image stepper,” Proc. SPIE 9780, 97800T (2016).
[Crossref]

Fiedler, C.

Ghosh, S.

S. Ghosh and C. Preza, “Characterization of a three-dimensional double-helix point-spread function for fluorescence microscopy in the presence of spherical aberration,” J. Biomed. Opt. 18, 036010 (2013).
[Crossref] [PubMed]

Greengard, A.

S. R. P. Pavani, A. Greengard, and R. Piestun, “Three-dimensional localization with nanometer accuracy using a detector-limited double-helix point spread function system,” Appl. Phys. Lett. 95, 021103 (2009).
[Crossref]

A. Greengard, Y. Y. Schechner, and R. Piestun, “Depth from diffracted rotation,” Opt. Lett. 31, 181–183 (2006).
[Crossref] [PubMed]

Grover, G.

A. Barsic, G. Grover, and R. Piestun, “Three-dimensional super-resolution and localization of dense clusters of single molecules,” Sci. Reports 4, 5388 (2014).
[Crossref]

G. Grover and R. Piestun, “New approach to double-helix point spread function design for 3D super-resolution microscopy,” Proc. SPIE 8590, 85900M (2013).
[Crossref]

G. Grover, S. Quirin, C. Fiedler, and R. Piestun, “Photon efficient double-helix PSF microscopy with application to 3D photo-activation localization imaging,” Biomed. Opt. Express 2, 3010–3020 (2011).
[Crossref] [PubMed]

Haas, K.-H.

K.-H. Haas and H. Wolter, “Synthesis, properties and applications of inorganic–organic copolymers (ORMOCERs),” Curr. Opin. Solid State Mater. Sci. 4, 571–580 (1999).
[Crossref]

Heintzmann, R.

Itoh, K.

Jesacher, A.

Leitel, R.

H.-C. Eckstein, U. D. Zeitner, R. Leitel, M. Stumpf, P. Schleicher, A. Bräuer, and A. Tünnermann, “High dynamic grayscale lithography with an LED-based micro-image stepper,” Proc. SPIE 9780, 97800T (2016).
[Crossref]

Liu, L.

Q. Yang, L. Liu, and J. Sun, “Optimized phase pupil masks for extended depth of field,” Opt. Commun. 272, 56–66 (2007).
[Crossref]

Mahajan, V. N.

V. N. Mahajan, “Line of sight of an aberrated optical system,” J. Opt. Soc. Am. A2 (1985).
[Crossref]

Ojeda-Castañeda, J.

Pavani, S. R. P.

S. Quirin, S. R. P. Pavani, and R. Piestun, “Optimal 3D single-molecule localization for superresolution microscopy with aberrations and engineered point spread functions,” Proc. Natl. Acad. Sci. 109, 675–679 (2012).
[Crossref] [PubMed]

S. R. P. Pavani, A. Greengard, and R. Piestun, “Three-dimensional localization with nanometer accuracy using a detector-limited double-helix point spread function system,” Appl. Phys. Lett. 95, 021103 (2009).
[Crossref]

S. R. P. Pavani and R. Piestun, “High-efficiency rotating point spread functions”, Opt. Express 16, 3484–3489 (2008).
[Crossref] [PubMed]

S. R. P. Pavani and R. Piestun, “Three dimensional tracking of fluorescent microparticles using a photon-limited double-helix response system,” Opt. Express 16, 22048–22057 (2008).
[Crossref] [PubMed]

Piestun, R.

C. Roider, R. Heintzmann, R. Piestun, and A. Jesacher, “Deconvolution approach for 3D scanning microscopy with helical phase engineering,” Opt. Express 24, 15456 (2016).
[Crossref] [PubMed]

A. Barsic, G. Grover, and R. Piestun, “Three-dimensional super-resolution and localization of dense clusters of single molecules,” Sci. Reports 4, 5388 (2014).
[Crossref]

S. Quirin and R. Piestun, “Depth estimation and image recovery using broadband, incoherent illumination with engineered point spread functions [Invited],” Appl. Opt. 52, A367–A376 (2013).
[Crossref] [PubMed]

G. Grover and R. Piestun, “New approach to double-helix point spread function design for 3D super-resolution microscopy,” Proc. SPIE 8590, 85900M (2013).
[Crossref]

S. Quirin, S. R. P. Pavani, and R. Piestun, “Optimal 3D single-molecule localization for superresolution microscopy with aberrations and engineered point spread functions,” Proc. Natl. Acad. Sci. 109, 675–679 (2012).
[Crossref] [PubMed]

G. Grover, S. Quirin, C. Fiedler, and R. Piestun, “Photon efficient double-helix PSF microscopy with application to 3D photo-activation localization imaging,” Biomed. Opt. Express 2, 3010–3020 (2011).
[Crossref] [PubMed]

S. R. P. Pavani, A. Greengard, and R. Piestun, “Three-dimensional localization with nanometer accuracy using a detector-limited double-helix point spread function system,” Appl. Phys. Lett. 95, 021103 (2009).
[Crossref]

S. R. P. Pavani and R. Piestun, “Three dimensional tracking of fluorescent microparticles using a photon-limited double-helix response system,” Opt. Express 16, 22048–22057 (2008).
[Crossref] [PubMed]

S. R. P. Pavani and R. Piestun, “High-efficiency rotating point spread functions”, Opt. Express 16, 3484–3489 (2008).
[Crossref] [PubMed]

A. Greengard, Y. Y. Schechner, and R. Piestun, “Depth from diffracted rotation,” Opt. Lett. 31, 181–183 (2006).
[Crossref] [PubMed]

R. Piestun, Y. Schechner, and J. Shamir, “Propagation-invariant wave fields with finite energy,” JOSA A 17, 294–303 (2000).
[Crossref] [PubMed]

R. Piestun and J. Shamir, “Generalized propagation-invariant wave fields,” J. Opt. Soc. Am. A 15, 3039–3044 (1998).
[Crossref]

Prasad, S.

Preza, C.

S. Ghosh and C. Preza, “Characterization of a three-dimensional double-helix point-spread function for fluorescence microscopy in the presence of spherical aberration,” J. Biomed. Opt. 18, 036010 (2013).
[Crossref] [PubMed]

Quirin, S.

Ritsch-Marte, M.

Roider, C.

Rombach, M. C.

Schechner, Y.

R. Piestun, Y. Schechner, and J. Shamir, “Propagation-invariant wave fields with finite energy,” JOSA A 17, 294–303 (2000).
[Crossref] [PubMed]

Schechner, Y. Y.

Schleicher, P.

H.-C. Eckstein, U. D. Zeitner, R. Leitel, M. Stumpf, P. Schleicher, A. Bräuer, and A. Tünnermann, “High dynamic grayscale lithography with an LED-based micro-image stepper,” Proc. SPIE 9780, 97800T (2016).
[Crossref]

Shamir, J.

R. Piestun, Y. Schechner, and J. Shamir, “Propagation-invariant wave fields with finite energy,” JOSA A 17, 294–303 (2000).
[Crossref] [PubMed]

R. Piestun and J. Shamir, “Generalized propagation-invariant wave fields,” J. Opt. Soc. Am. A 15, 3039–3044 (1998).
[Crossref]

Šiler, M.

Simonov, A. N.

Soeller, C.

D. Baddeley, M. B. Cannell, and C. Soeller, “Three-dimensional sub-100 nm super-resolution imaging of biological samples using a phase ramp in the objective pupil,” Nano Res. 4, 589–598 (2011).
[Crossref]

Stallinga, S.

Stumpf, M.

H.-C. Eckstein, U. D. Zeitner, R. Leitel, M. Stumpf, P. Schleicher, A. Bräuer, and A. Tünnermann, “High dynamic grayscale lithography with an LED-based micro-image stepper,” Proc. SPIE 9780, 97800T (2016).
[Crossref]

Sun, J.

Q. Yang, L. Liu, and J. Sun, “Optimized phase pupil masks for extended depth of field,” Opt. Commun. 272, 56–66 (2007).
[Crossref]

Tünnermann, A.

H.-C. Eckstein, U. D. Zeitner, R. Leitel, M. Stumpf, P. Schleicher, A. Bräuer, and A. Tünnermann, “High dynamic grayscale lithography with an LED-based micro-image stepper,” Proc. SPIE 9780, 97800T (2016).
[Crossref]

Wang, K.

Wolf, E.

M. Born and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light (Elsevier, 2013).

Wolter, H.

K.-H. Haas and H. Wolter, “Synthesis, properties and applications of inorganic–organic copolymers (ORMOCERs),” Curr. Opin. Solid State Mater. Sci. 4, 571–580 (1999).
[Crossref]

Yang, Q.

Q. Yang, L. Liu, and J. Sun, “Optimized phase pupil masks for extended depth of field,” Opt. Commun. 272, 56–66 (2007).
[Crossref]

Zeitner, U. D.

H.-C. Eckstein, U. D. Zeitner, R. Leitel, M. Stumpf, P. Schleicher, A. Bräuer, and A. Tünnermann, “High dynamic grayscale lithography with an LED-based micro-image stepper,” Proc. SPIE 9780, 97800T (2016).
[Crossref]

Appl. Opt. (5)

Appl. Phys. Lett. (1)

S. R. P. Pavani, A. Greengard, and R. Piestun, “Three-dimensional localization with nanometer accuracy using a detector-limited double-helix point spread function system,” Appl. Phys. Lett. 95, 021103 (2009).
[Crossref]

Biomed. Opt. Express (1)

Curr. Opin. Solid State Mater. Sci. (1)

K.-H. Haas and H. Wolter, “Synthesis, properties and applications of inorganic–organic copolymers (ORMOCERs),” Curr. Opin. Solid State Mater. Sci. 4, 571–580 (1999).
[Crossref]

J. Biomed. Opt. (1)

S. Ghosh and C. Preza, “Characterization of a three-dimensional double-helix point-spread function for fluorescence microscopy in the presence of spherical aberration,” J. Biomed. Opt. 18, 036010 (2013).
[Crossref] [PubMed]

J. Opt. Soc. Am. A (1)

JOSA A (1)

R. Piestun, Y. Schechner, and J. Shamir, “Propagation-invariant wave fields with finite energy,” JOSA A 17, 294–303 (2000).
[Crossref] [PubMed]

Nano Res. (1)

D. Baddeley, M. B. Cannell, and C. Soeller, “Three-dimensional sub-100 nm super-resolution imaging of biological samples using a phase ramp in the objective pupil,” Nano Res. 4, 589–598 (2011).
[Crossref]

Opt. Commun. (1)

Q. Yang, L. Liu, and J. Sun, “Optimized phase pupil masks for extended depth of field,” Opt. Commun. 272, 56–66 (2007).
[Crossref]

Opt. Express (6)

Opt. Lett. (3)

Proc. Natl. Acad. Sci. (1)

S. Quirin, S. R. P. Pavani, and R. Piestun, “Optimal 3D single-molecule localization for superresolution microscopy with aberrations and engineered point spread functions,” Proc. Natl. Acad. Sci. 109, 675–679 (2012).
[Crossref] [PubMed]

Proc. SPIE (2)

G. Grover and R. Piestun, “New approach to double-helix point spread function design for 3D super-resolution microscopy,” Proc. SPIE 8590, 85900M (2013).
[Crossref]

H.-C. Eckstein, U. D. Zeitner, R. Leitel, M. Stumpf, P. Schleicher, A. Bräuer, and A. Tünnermann, “High dynamic grayscale lithography with an LED-based micro-image stepper,” Proc. SPIE 9780, 97800T (2016).
[Crossref]

Sci. Reports (1)

A. Barsic, G. Grover, and R. Piestun, “Three-dimensional super-resolution and localization of dense clusters of single molecules,” Sci. Reports 4, 5388 (2014).
[Crossref]

Other (2)

M. Born and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light (Elsevier, 2013).

V. N. Mahajan, “Line of sight of an aberrated optical system,” J. Opt. Soc. Am. A2 (1985).
[Crossref]

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Figures (10)

Fig. 1
Fig. 1 The individual steps of the proposed design process for multi-order-helix PSFs are exemplary illustrated for a double- (left column) and a triple- (right column) helix design. The phase distributions Φp as well as the corresponding PSFs are shown for the individual design steps. The Fresnel zone start designs (a,b) are initially truncated to an angular section, dedicated to a single PSF peak (c,d). The respective section is optimized for a high peak confinement (e,f) and subsequently used to fill the overall phase element with rotated copies (g,h). A purely refractive design is finally obtained by phase unwrapping in an optional step (i,j).
Fig. 2
Fig. 2 Top: Simulated, axial PSF dependency H′[v, w; z] of five considered pupil functions as indicated in the left part: A, nominal PSF for a clear aperture; B, Fresnel zone double-helix PSF; C, optimized double-helix PSF; D, optimized triple-helix PSF; E, refractive double-helix PSF. A broadband illumination spectrum ranging from 450 – 650 nm and a nominal object distance of z = 1000 mm are assumed. Note that the individual PSF plots are normalized to their respective maximum irradiance level for illustration purposes. Bottom: Corresponding axial dependencies of the standard deviation of the lower limit to measurement precision σCRLB.
Fig. 3
Fig. 3 Influence of primary Zernike aberrations on the shape of an engineered (a) double- and (b) triple-helix PSF design including the effect of defocus ( Z 2 0) as well as first order astigmatism ( Z 2 2, Z 2 2), coma ( Z 3 1, Z 3 1) and spherical aberration ( Z 4 0).
Fig. 4
Fig. 4 Comparison of the rotation angle dependency α of multi-helix-PSFs on the rms phase error for the first 21 Zernike aberrations (excluding piston, tip and tilt). Aberrations that cause a significant rotation, i.e. |α(π/2)| > 1°, on a particular multi-helix design are highlighted by rectangular boxes with the respective plot color.
Fig. 5
Fig. 5 Wavefront distribution of (a) phase element part Wp and (b) defocus aberration part W Z 2 0 (b) as well as respective gradient fields ∇⃗Wp and ∇⃗Wz4. The two highlighted subareas S1 and S2 directly correspond to the two peaks of the double-helix PSF. (c) PSF distribution associated with Wp. The direction of the centroid vectors 〈p⃗〉 and 〈z⃗k〉 are indicated for both subareas S1 and S2, respectively.
Fig. 6
Fig. 6 Top: Measured surface profile height h(x, y) of (a) the tool and (b) the replication of the double-helix CGH design. The dashed white circle indicates the aperture size of 10 mm used in the optical setup. Bottom: horizontal cross-section of the profile height h(x) at y = 1 mm as indicated by dashed blue lines in the respective upper plots.
Fig. 7
Fig. 7 Top: Measured on-axis distributions of the engineered double-, triple- and tetra-helix PSFs at multiple object distances z (λ = 540 nm). Bottom: Corresponding dependency of the rotation angle α on the object distance z.
Fig. 8
Fig. 8 (a) Nominal distribution of the imaged test screen if no CGH is implemented in the demonstration system. The subimage shown in (b) correspond to the inset indicated in (a). The subimages (c)–(e) show the same part of the image after the double-, triple- and tetra-helix CGHs are implemented in the demonstration system, respectively.
Fig. 9
Fig. 9 Processed Ccepstrum distributions of the subimages shown in Fig. 8(c)–8(e).
Fig. 10
Fig. 10 Measured rotation angle distribution α of the imaged demonstration scene using a (a) double- and (b) triple-helix PSF. (c) and (d) show the residual rotation angle distributions αr(x, y) for the two cases after eliminating the effect of field curvature by fitting and subtracting a parabolic function from α(x, y).

Equations (34)

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Φ l ( ϕ ) = [ ( l 1 ) N + 1 ] ϕ
Δ z max = 2 λ d 0 2 L R 2 + 2 λ l d 0 L ,
σ CRLB 2 ( z ) = 1 σ N 2 v = 1 V w = 1 W ( z H [ v , w ; z ] ) 2
H [ v , w ; z ] = λ 1 λ 2 η ( λ ) H [ v , w ; z ; λ ] d λ
Φ ( u , v ) = Φ p ( u , v ) + n , m A n m Z n m ( u , v ) ,
Z n m ( ρ , ϕ ) = R n m ( ρ ) cos m ϕ
Z n m ( ρ , ϕ ) = R n m ( ρ ) sin m ϕ
α = 1 N j = 1 N Δ α j ,
c = D E EP W ( u , v ) d u d v
W ( u , v ) = λ 2 π Φ ( u , v ) = λ 2 π Φ p ( u , v ) W p + λ 2 π A n m Z n m ( u , v ) W n m
2 π ( j 1 ) N ( ϕ u ϕ 0 ) < 2 π j N | j = 1 , , N ,
c j = D E S j W ( u , v ) d u d v = D E S j W p ( u , v ) d u d v D E S j W n m ( u , v ) d u d v = p j + z n m j
M n m = z ^ ( j = 1 N p j × z n m j ) ,
m = κ N | κ = 0 , 1 , 2 ,
p j = p ( cos ϕ j sin ϕ j ) ,
ϕ j = ϕ 0 + π 2 π N + j 2 π N
p j = p cos ( ϕ ϕ j ) ρ ^ p sin ( ϕ ϕ j ) ϕ ^ ,
z n m j = W n m j = W n m ρ ρ ^ 1 ρ W n m ϕ ϕ ^ j
M n m = p j = 1 N 1 ρ cos ( ϕ ϕ j ) W n m ϕ sin ( ϕ ϕ j ) W n m ρ j
0 ψ 2 π N
M n m = p D E ρ = 0 1 ψ = 0 2 π / N j = 1 N [ 1 ρ sin ( π N ψ ) W n m ( ρ , 2 π j N + ϕ 0 ψ ) ψ + cos ( π N ψ ) W n m ( ρ , 2 π j N + ϕ 0 ψ ) ρ ] ρ d ρ d ψ
M n m = 2 π p D A n m λ E [ m F s s ρ = 0 1 R n m ( ρ ) d ρ + F c c ρ = 0 1 ρ R n m ( ρ ) ρ d ρ ]
F s s = ψ = 0 2 π / N [ sin ( π N ψ ) j = 1 N sin ( 2 π j m N + m ϕ 0 m ψ ) ] d ψ
F c c = ψ = 0 2 π / N [ cos ( π N ψ ) j = 1 N cos ( 2 π j m N + m ϕ 0 m ψ ) ] d ψ
m = κ N | κ = 0 , 1 , 2 ,
F s s = N κ 2 N 2 1 [ 2 κ N sin ( π N ) cos ( κ N ϕ 0 ) ]
F c c = N κ 2 N 2 1 [ 2 sin ( π N ) cos ( κ N ϕ 0 ) ]
ρ = 0 1 ρ R n m ( ρ ) ρ d ρ = 1 ρ = 0 1 R n m ( ρ ) d ρ
ρ = 0 1 R n m ( ρ ) d ρ = ( 1 ) ( n m ) / 2 ( n + 1 )
M n m = 2 π p D A n m λ E [ F c c + ( m F s s F c c ) ( ( 1 ) ( n m ) / 2 1 ( n + 1 ) ) ]
M n κ N = 4 π p D A n κ N N λ E ( κ 2 N 2 1 ) sin ( π N ) cos ( κ N ϕ 0 ) [ 1 + ( ( 1 ) ( n κ N ) / 2 ( κ 2 N 2 1 ) ( n + 1 ) ) ]
F s c = N κ 2 N 2 1 [ 2 κ N ( 1 ) κ sin ( π N ) sin ( κ N ϕ 0 ) ]
F c s = N κ 2 N 2 1 [ 2 ( 1 ) κ sin ( π N ) sin ( κ N ϕ 0 ) ]
M n κ N = ( 1 ) κ 4 π p D A n κ N N λ E ( κ 2 N 2 1 ) sin ( π N ) sin ( κ N ϕ 0 ) [ 1 ( ( 1 ) ( n κ N ) / 2 ( κ 2 N 2 + 1 ) ( n + 1 ) ) ]

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