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Abstract
Imaging depth of field is shallow in applications with high magnification and high numerical aperture, such as microscopy, resulting in images with in- and out-of-focus regions. Therefore, methods to extend depth of field are of particular interest. Researchers have previously shown the advantages of using freeform components to extend depth of field, with each optical system requiring a specially designed phase plate. In this paper we present a method to enable extended depth-of-field imaging for a range of numerical apertures using freeform phase plates to create variable cubic wavefronts. The concept is similar to an Alvarez lens which creates variable spherical wavefronts through the relative translation of two transmissive elements with XY polynomial surfaces. We discuss design and optimization methods to enable extended depth of field for lenses with different numerical aperture values by considering through-focus variation of the point spread function and compare on- and off-axis performance through multiple metrics.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Improving the quality of optical imaging systems is an ongoing goal of researchers. Performance criteria for image quality vary with the type of imaging systems and their applications. As one example, the lateral resolution in an imaging system can be quantified using the Rayleigh resolution criterion, defined as the minimum lateral distance, l, that can be resolved by the imaging system:
where λ is the source wavelength, and NA is the Numerical Aperture of the system [1]. Thus, imaging systems with higher NA values enable better lateral resolution. However, as the lateral resolution improves with higher NA, the system's depth of field degrades. Depth of field is the range of object distances from which the imaging system can create an acceptably in-focus image. This range of object distances map to a corresponding range of image-plane locations, with consequent defocus blur arising from longitudinal displacement of the observation plane with respect to the best-focus image plane. The range of image distances for which the defocus blur spot size is less than or equal to the diffraction blur is defined as the depth of focus (DoF) [1]:Strictly speaking, DoF is the image-space depth of focus, corresponding to depth of field in object space. Typical nomenclature in the community often refers to DoF as depth of field, and we follow that convention here.
In microscopy, different methods have been introduced to enable an Extended Depth of Field (EDoF) while maintaining a high NA for better lateral resolution. The confocal scanning microscope increases DoF by scanning the object with focused light along the optic axis, creating an in-focus image [2]. Wavefront coding is another technique to enable EDoF and provides the basis for this work [3]. In wavefront coding, a specially designed optical component is placed at the pupil of the imaging system to alter the Point-Spread Function (PSF) of the system and decrease its sensitivity to defocus. As a result, a blurry intermediate image of the object is created, and then deconvolution methods are used to retrieve an image with higher quality using the known PSF (Fig. 1). For example, logarithmic aspheres [4], axicons [5,6], deformable mirrors [7], annular tiered phase masks [8,9], and binary phase modulated pupil masks [10] have been used to enable EDoF in imaging systems.
Fig. 1. Schematic of immediate (blurry) and final (in-focus) images (using the “demo picture” provided for the image simulation in Zemax OpticStudio), created by an imaging system with EDoF phase plate.
With recent advances in design, fabrication, and metrology of freeform optics [11], rotationally asymmetric phase masks for wavefront coding have been used more widely in imaging applications. Examples include cubic [3,12–15] and logarithmic [16,17] phase plates. Figure 2 shows examples of through-focus spots for a 0.33 NA aspheric lens with and without a cubic phase plate at λ = 633 nm. The addition of the phase plate enables EDoF, by creating a PSF that is larger but less sensitive to defocus.
Fig. 2. Through-focus spot diagrams for 0.33 NA aspheric lens: (a) without and (b) with a cubic phase plate. The diffraction-limited spot size and location is indicated by the black spot in each diagram.
Imaging systems with specific DoF requirements and NA values require individual phase plates. Therefore, phase masks that enable EDoF for multiple NA values can be beneficial by reducing the number of optical components, potentially reducing weight and cost. It has been previously demonstrated that one pair of phase plates can extend DoF for multiple systems by relative rotation or translation of the phase components. For example, two asymmetric phase masks with cubic polynomial surfaces were shown to extend an imaging system’s DoF by creating a focus-invariant PSF through relative rotation of the phase masks [18]. Another example utilizes a pair of translated polynomial phase plates in a miniature multi-modal microscope to enable imaging different object distances mapping to the same image plane [19]. In these examples, the need to design and implement multiple fixed phase plates is eliminated by using a pair of transmissive phase plates that create a variable functionality through relative movement of the phase components. This concept is the basis of the Alvarez lens design [20], which is a variable focus lens composed of two identical, laterally shifted (along the x- or y-axes) freeform components with an XY polynomial surface equation. The focusing power of the composite surface generated by the freeform elements changes with the relative translation between the pairs. Such shifted optical components that create variable functionality have been used for multiple additional applications, including aberration correction [21,22], beam shaping [23,24], and head-mounted displays [25].
In this paper we propose a design method for a freeform phase plate pair that enhances DoF for multiple lenses with different NA values through lateral relative translation of the phase plates along the x-axis. We examine the DoF extension both qualitatively and quantitatively by considering through-focus variation of (1) ray-traced spot diagrams, and (2) diffraction-based PSF models, respectively. Researchers have previously reported focus-invariant systems through relative shifts of equal amounts along both the x- and y-axes between a pair of phase plates with surface profiles given by:
where k denotes the strength of the phase plate [26]. This approach is similar to the use of shifted freeform components along both axes for a variable-focus effect [27]. However, the basis for choosing this surface equation is not discussed, and the optimization method and criteria for determining the optimum k coefficient value are not reported [26].For our approach, we consider multiple lenses with different NA values. We first design fixed Cubic Phase Plates (CPP), one for each lens, that enable EDoF using an optimization method based on the Modulation Transfer Function (MTF). Each phase plate is a plano-freeform, with the equation of the freeform surface given by:
where the value of the cubic coefficient α depends on the corresponding lens NA and the optimization parameters. The resulting fixed phase plates are then replaced by one pair of phase plates with a 4th order (quartic) surface description that enable EDoF for all the lenses and NA values. Relative shifts between the quartic freeform pair create composite surfaces with similar functionality to the fixed cubic phase plates, as illustrated in Fig. 3.Fig. 3. Representative images of the EDoF systems considered in this manuscript. Two fixed Phase Plates (PP1 and PP2) may be replaced with one pair of freeform phase plates to enable EDoF through the relative translation of the phase plates along the x-axis.
2. Design approach
2.1 Variable phase plate design
Consider a pair of plano-freeform elements with and without relative shifts between the surfaces, as illustrated in Fig. 4:
Fig. 4. Schematic of a pair of plano-freeform components (a) with no lateral shift and (b) with shifts along the x-axis of equal amount and opposite direction.
There is no optical functionality at zero shift, but variable optical functionality is introduced to the system as the components are translated along the x-axis by amounts d in opposite directions, as shown. For this work, the goal is to obtain the same functionality of the cubic surface described in Eq. (4) where the value of the $\alpha $ coefficient is tuned by changing the shift d between the two freeform plates. For small values of d, we can apply a method introduced by Palusinski [21], to show that:
where zf(x,y) is the general equation for each freeform surface in the pair. Equations (4) and (5) are then used to obtain: whereIn Eq. (7), αmax is the cubic coefficient for the highest NA lens considered, dmax is the maximum desired shift between the phase plates, and n is the refractive index of the phase plate material at the design wavelength. This equation is derived considering the composite wavefront resulting from transmission through the shifted phase plate pair and its relation to the cubic phase plate coefficient for the highest NA lens. After calculating β, the relative shifts needed for lenses with lower NA values are calculated, using Eq. (8):
where α is the optimized cubic coefficient for each lens, and β is calculated using Eq. (7). We refer to the pair of components for variable EDoF with surfaces described by Eq. (6) as a Quartic Phase Plate Pair (QPPP) in the following discussion.2.2 Optical design and optimization
We consider three commercial aspheric singlets with 22.5 mm clear aperture and NA values of 0.25, 0.33, and 0.5 respectively as the basis for design of the example EDoF phase plates [28]. The EDoF phase plates are all assumed to be made of Polymethyl Methacrylate (PMMA) with a 22 mm clear aperture, 3 mm thickness, and design wavelength of 633 nm, giving a refractive index of 1.489. Simulation and optimization were performed in Zemax OpticStudioTM and imported into MATLAB and Excel for formatting and presentation where appropriate. Three CPPs, one for each lens, were designed to enable EDoF. For better performance, it is necessary to place the phase plates at or near the exit pupil of the imaging system. However, due to inaccessibility of the exit pupils for the selected lenses and optomechanical considerations for the test setup, the air gaps between the aspheric lenses and phase plates are set to 1 mm. The design goal is to create spots with less sensitivity to defocus over a targeted axial range from $- 6\Delta z$ to $+ 6\Delta z$, with $\Delta z$ given by:
where DoF is calculated using Eq. (2). The cubic coefficient α is optimized for each lens for image planes located at integer multiples of Δz across the target range. The optimization follows two main goals: (1) improving the system’s overall performance by increasing through-focus MTF values at specific spatial frequencies, and (2) achieving more consistent through-focus performance by minimizing differences between the through-focus MTF values for the selected frequencies.The optimization frequencies were selected by identifying the specific frequency ranges exhibiting low values in the through-focus MTF performance of each lens. This criterion can be directly implemented in the merit function within the design software and enables control of specific performance requirements. After selecting the spatial frequencies demonstrating this behavior, the phase plate parameters were optimized to increase the MTF and minimize the through-focus MTF variation for each lens. Table 1 summarizes the spatial frequencies selected for optimization for each lens.
Table 1. Optimization frequencies used to design EDoF phase plates for the selected aspheric lenses.
After optimizing the α coefficient for each lens, the β coefficient for the QPPP is calculated (β=9.205×10−6) using the EDoF requirement for the highest NA lens (αmax) and desired maximum shift amount (dmax = 2 mm) following Eq. (7). The air gap between the aspheric lens and first QPPP element is also set to 1 mm. The spacing between the first and second QPPP components is set to 400 µm to avoid collisions between the shifted parts and considers their overall sag over the full diameter, as well as manufacturing and optomechanical mounting constraints. Figure 5 shows the CPP surface for the 0.33 NA lens and one of the resulting QPPP surfaces.
Fig. 5. (a) Surface of designed CPP for 0.33 NA lens and (b) surface of one of the quartic phase plates.
3. Design results
3.1 Cubic phase plate (CPP) designs
Three CPPs were designed based on the criteria discussed in Sec. 2.2. Each of these phase plates enables EDoF for the selected lenses with specific NA values. Table 2 summarizes the optimization results for the designs. The cubic coefficients and the surface sags of the phase plates increase as the numerical aperture of the system increases.
Table 2. Cubic Phase Plate optimization results.
3.2 Quartic phase plate pair (QPPP) designs
One pair of variable phase plates with β=9.205×10−6 (as derived in Section 2.2) was used to replace the three CPPs through relative translation of the phase components along the x-axis. Table 3 summarizes the relative shift of each element in the QPPP systems needed for each lens NA.
Table 3. Quartic Phase Plate Pair design results.
4. Performance comparisons
4.1 On-axis system performance
4.1.1 Through-focus spot diagrams
Figure 6 qualitatively compares through-focus (−6Δz to +6Δz) spot diagrams for the selected lenses when used with the respective CPPs and QPPP. As expected, the addition of the EDoF phase plates results in an increase in the through-focus spot size but reduced variation in spot size through the defocus range. The spots for the QPPP are notably larger than those obtained for the CPP. We make more quantitative performance comparisons below.
Fig. 6. Through-focus spot diagrams for the three aspheric lenses with fixed CPPs and variable QPPP.
4.1.2 Through-focus MTF
As discussed previously, the addition of EDoF phase plates enables more consistent through-focus spots at the expense of lower system performance. The primary objectives for designing the cubic and quartic phase plates are to increase through-focus MTF at the selected frequencies and to achieve more consistent through-focus performance by minimizing the through-focus MTF differences at the selected spatial frequencies. We observe that adding the EDoF phase plates worsens the on-axis system MTF at the best image plane for both the CPP and QPPP, as shown in Fig. 7.
Fig. 7. Tangential MTF at the best focus for the (a) 0.25, (b) 0.33, and (c) 0.50 NA lenses with and without EDoF phase plates.
Figure 8 compares the on-axis through-focus MTF values for the 0.25, 0.33, and 0.5 NA lenses with and without EDoF phase plates at 80 cycles/mm as an example. The MTF varies less through focus ($- 6\Delta z$ to $\; + 6\Delta z$) with the addition of the phase plates, though the peak MTF values are lower, as expected.
Fig. 8. On-axis through-focus tangential MTF plots for the (a) 0.25, (b) 0.33, and (c) 0.50 NA lenses with and without EDoF phase plates at f = 80 cycles/mm.
4.1.3 Root-mean-square (RMS) deviation and slope of RMS deviation of PSF
The PSF of an optical system shows how the system images a point source. The PSF changes as the location of the observation plane varies from the best image plane. Figure 6 qualitatively shows how the through-focus spots vary for the selected lenses when used with the respective phase plates. For more quantitative comparisons, it is necessary to use diffraction-based models of the PSF. As an example, Fig. 9 shows the normalized PSF for the 0.33 NA lens with the designed CPP at best focus and ±4Δz image plane locations. Similar normalized PSF calculations were performed for the range of lenses, phase plates, and defocus distances. The results were then used to calculate both the RMS deviation of the PSF and the slope of the RMS deviation of the PSF as quantitative measures of the through-focus variability. The expectation is that with the addition of the EDoF phase plates, the PSF will remain consistent in shape and size through focus, resulting in lower RMS deviation values.
To this end, the RMS deviation of PSF is calculated between the normalized PSF at best focus and the defocus image planes (point by point):
Variations in the RMS deviation of the PSF for a system with no EDoF phase plate are expected as the through-focus location of the image plane changes. Similarly, systems with the EDoF phase plates should have lower through-focus RMS deviation with reduced sensitivity to defocus. Figure 10 compares the normalized PSF RMS deviation through focus ($- 6\Delta z$ to $\; + 6\Delta z$) for each of the three lenses with different NA values with and without EDoF phase plates. For each lens, the addition of the phase plates enables a lower RMS deviation of PSF, resulting in a more consistent through-focus performance, and the CPPs show a slightly lower PSF variance than the QPPP.
Fig. 10. RMS deviation of PSF through focus for the (a) 0.25, (b) 0.33, and (c) 0.50 NA lenses with and without EDoF phase plates.
We also consider the slope of the RMS deviation of the PSF as another quantitative metric to compare through-focus spot variation for systems with and without the EDoF phase plates. The slope for each plot shown in Fig. 10 represents the rate at which the RMS deviation of PSF changes and can be calculated as:
where si is the slope of the RMS deviation of PSF for each two consecutive defocus image planes, and RMSi+1 and RMSi are the RMS deviations of PSF values calculated for the image planes at di+1 and di locations respectively (from −6Δz to +6Δz). With the addition of the designed EDoF phase plates, we expect to see a lower slope, as the phase plates decrease the spot variation through focus. Figure 11 shows the slopes of the RMS deviation of PSF for the three aspheric lenses with different NA’s, with and without the phase plates. The rate at which the through-focus PSF changes is lower for systems with CPP and QPPP, as expected. In addition, the difference between CPP and QPPP for each system is small, with a lower rate for CPP.Fig. 11. Slope of RMS deviation of PSF through focus for the (a) 0.25, (b) 0.33, and (c) 0.50 NA lenses with and without EDoF phase plates.
4.2 Off-axis system performance
4.2.1 Through-focus spot diagrams vs field angle
For imaging applications, it is also essential to consider system performance for off-axis object locations. Figure 12 compares through-focus spot diagrams for the 0.33 NA asphere with the CPP and QPPP designs at 0, 1 and 3-degree field angles along the y-axis. We note that for each field angle, the spot shapes are relatively constant, while the through-focus spot sizes change more noticeably for higher field angles, particularly for the CPP case.
Fig. 12. Through-focus spot diagrams for 0.33 NA lens with CPP and QPPP over 3-degree field angle range along the y-axis.
4.2.2 MTF vs field angle
As a representative example, we compare the tangential MTF at best focus and through focus for the 0.33 NA aspheric lens with and without EDoF phase plates. Figure 13 shows the MTF at best focus at field angles of 0, 1 and 3-degrees along the y-axis. We observe that the MTF drops as the field angle increases for the aspheric lens by itself as well as in both EDoF systems, with a caveat that the MTF at 1-degree field angle is slightly higher than seen with on-axis light for lower spatial frequencies for the EDoF systems.
Fig. 13. Best focus tangential MTF plots at 0,1 and 3-degree field angles along the y-axis for (a) 0.33 NA aspheric lens, and 0.33 NA lens with (b) CPP and (c) QPPP.
To examine this unexpected result more closely, we also consider the tangential MTF for 1-degree field angles at different orientations around the optical axis, as shown in Fig. 14.
Fig. 14. Best focus tangential MTF plots at 1-degree field angles at different orientations around the optical axis for (a) 0.33 NA aspheric lens, and the same lens with (b) CPP and (c) QPPP.
From Fig. 14 we observe that the relative MTF of all the systems (with and without EDoF phase plates) are sensitive to both field angle and the orientation of the field point around the optical axis. The systems containing EDoF phase plates appear to be less sensitive to orientation than the lens alone, though at the cost of reduced image contrast. The results observed in both Fig. 13 and Fig. 14 for both fixed CPP and variable QPPP are consistent with prior research showing decreased field angle sensitivity with static cubic phase plates [13].
As an additional example, Fig. 15 shows through-focus modulation (tangential) values at the lowest spatial frequency used for CPP design (61 cycles/mm) for the 0.33 NA aspheric lens, with and without the designed CPP and QPPP at 0, 1 and 3-degree field angles (along the y-axis). While the absolute results would vary as a function of the chosen spatial frequency, the modulation values for the systems containing EDoF phase plates are more consistent through focus (particularly for smaller field angles) than the lens alone, but again at the cost of reduced image contrast that worsens with increased field angle.
Fig. 15. Through-focus tangential modulation plots for 0.33 NA aspheric lens with and without EDoF phase plates at (a) 0, (b) 1, and (c) 3-degree field angle at f = 61 cycles/mm.
5. Discussion and conclusion
We have proposed a design method to enable variable extended depth of field imaging using a pair of transmissive freeform phase plates. First, we designed three fixed CPPs for commercial aspheric singlets with different numerical apertures using an on-axis MTF-based optimization method. The optimization routine was implemented to improve each system’s MTF while decreasing the through-focus MTF variation. Multiple CPPs were then replaced by one QPPP to enable multiple focus-invariant systems. This approach can decrease the number of parts needed to enable EDoF imaging for multiple systems and potentially reduce system costs.
The RMS deviation of PSF, and its slope, were proposed and calculated for the CPP and QPPP systems as quantitative metrics for through-focus variation. The RMS deviation and slope deviation of PSF for systems with CPPs and QPPP are lower than for a corresponding lens with no phase plate, illustrating the reduced through-focus variation that is desirable for an EDoF system. While the overall performance of the QPPP system in terms of spot size and PSF variability is generally lower than for the corresponding CPP systems, the QPPP still can serve as a suitable replacement for multiple CPP elements. We believe the difference in CPP and QPPP performance can be attributed to the air gap needed between the two phase plates in the QPPP systems, and the fact that the CPP elements are specifically optimized while the QPPP design is derived from the CPP elements, rather than optimized.
We compared on- and off-axis performance of fixed CPPs and variable QPPP for a range of numerical apertures. The through-focus spot diagrams showed that the CPPs and QPPP enable spots with less through-focus variation compared to systems without phase plates, with an increase in the spot sizes and reduction in the MTF for QPPP systems relative to the CPP systems. Comparisons also showed that the performance of both CPPs and QPPP decrease at higher field angles. Results from on- and off-axis fields demonstrate the sensitivity of EDoF phase plates to increasing field angles and their directionality. These sensitivities may also be attributable to asymmetries in the phase plates surfaces and the directionality of the shifts between the QPPP elements. These factors suggest the need for additional study of (1) phase plates with more design freedoms, (2) optimization methods that consider off-axis field angles, and (3) analyses based on the 2D MTF and recent related performance metrics such as the Minimum Modulation Curve [29].
Work is currently underway on fabricating the designed phase plates (CPPs and QPPP) for experimental performance characterization and comparison to theoretical predictions.
Funding
Division of Industrial Innovation and Partnerships (1338877, 1338898, 1822026, 1822049).
Acknowledgments
Portions of this work were previously presented in [30]. The authors would like to acknowledge valuable discussions with Dr. Christoph Menke from Carl Zeiss AG, Dr. Sébastien Héron from THALES, Dr. Matthew Davies, Dr. Glenn Boreman, and Dustin Gurganus from the University of North Carolina at Charlotte, and Dr. Jannick Rolland from the University of Rochester.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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