Abstract

Design aspects of multiple-order diffraction engineered surface (MODE) lenses are discussed that result in significant improvement of geometrical off-axis performance. A new type of aberration that is characteristic of this type of segmented lens, which is called zonal field shift, is minimized by curving front intercepts of zone transitions. Three MODE designs are compared, based on a 240 mm aperture, 1 m focal length system with a 0.125° half field angle over the astronomical R wavelength band (589 nm to 727 nm). Optimized curved-front designs indicate diffraction-limited monochromatic geometrical performance over the full field of view. A technique is implemented with a combination of a non-sequential ray-trace model and a diffraction code to model physical optical effects, which indicates that the modulation transfer function (MTF) of MODE lenses are significantly improved compared to a first-order equivalent refractive achromat.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

The multi-order diffractive engineered surface (MODE) lens is a new type of high-performance ultralightweight optical element that could enable efficient large-aperture space telescopes [1], among other applications. The initial motivation for MODE lenses is to design a new type of space telescope that provides a cost-effective solution suitable for exoplanet research through transit studies. In comparison with commonly used reflective space telescopes, MODE primary lenses have (1) unobscured apertures, (2) lightweight structures, (3) less sensitivity to manufacturing and alignment errors, (4) construction from stable optical materials compared to thin membranes, and (5) potential for easy replication in a space telescope array. This paper discusses important geometrical optics design aspects of MODE lenses that can be used to improve off-axis performance, and implications of the design aspects is analyzed with physical optics simulation.

The application for large-aperture space telescopes presents certain optical and mechanical constraints on practical implementation of new technology. First, the optical performance must meet science requirements of the mission. These missions often, but not always, require diffraction-limited imaging. An example of a non-diffraction limited mission requirement is the spectroscopic measurement of a transiting exoplanet, where, while the star and planet system remain spatially unresolved, temporal modulations in the combined spectroscopic signal are used to characterize the planetary atmosphere’s transmission spectrum [2]. Second, the system must be as simple as possible and robust in the difficult environment of space. MODE lenses are a new concept in space telescope design, where a large-diameter, ultrathin glass lens is used as the primary focusing element in the telescope. The single lens has both diffractive and refractive properties, which are different than single-harmonic diffractive Fresnel lenses (DFLs) that have been discussed for this purpose [35]. One advantage of a MODE lens is that it maintains nearly the ultralight nature of a DFL, but with a greatly reduced range of focal dispersion versus wavelength. Scaling the aperture diameter of MODE lenses and potential cost advantages for space science missions are discussed in [1].

A recent publication describes principles of first-generation MODE lenses, as shown conceptually in Fig. 1, which have a high-harmonic multiple-order diffractive (MOD) lens on the front surface and a DFL on the back surface [6]. The MOD front surface is comprised of circularly symmetric zones that act like separate lenses directed to a common focal point ${f_0}$ at the design wavelength λ0. The zone transitions are designed such that integer M waves of optical path difference in transmission is between the two sides of each transition at λ0, as shown in Fig. 1 for the first zone transition with a thickness change of Mh at the boundary, where h is the surface height change for one wavelength of optical path difference (OPD) in air. In common optical glasses and plastics with refractive index n ∼ 1.5, h given by

$$h = {\lambda _0}/(n - 1) \approx 1\,\mathrm{\mu }\textrm{m}.$$
These MODE lenses exhibit both refractive and diffractive components of the longitudinal chromatic aberration (LCA). A high harmonic MOD surface as defined here satisfies M >250. For example, M = 1000 requires a transition height of $M{\lambda _0}/({n - 1} )$ ∼ 1 mm for visible wavelengths. The back surface is a single-harmonic (M = 1) DFL, which has the effect of making each MOD zone achromatic to limit the refractive portion of LCA shown in Fig. 1.

 figure: Fig. 1.

Fig. 1. Multiple-order-diffraction engineered surface (MODE) lens. The front surface is a multiple-order-diffraction (MOD) lens, and the back surface is a diffractive Fresnel lens (DFL). h is the glass thickness of index n that produces one wave of optical path difference in transmission. A MOD surface with high M number produces a small value of LCA. The DFL reduces the refractive part of LCA, making each MOD zone achromatic.

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Radial positions of the zone boundaries are determined using a simple OPD model with an infinitely thin lens. Incident light rays from an on-axis source located at infinity are parallel to the optical axis at the vertex plane of the lens. At the lens, light rays are bent in a straight line toward the focal point f0. OPD is defined as the difference between the length of the line and f0. Zone transitions occur along the radius whenever the OPD is a multiple of 0. For this case, zone transition radii $\rho _i^{}$ are defined by

$$\rho _i^{} = \,\,\,{f_0}\sqrt {\frac{{2({i - 1} )M\lambda }}{{{f_0}}} + {{\left[ {\frac{{({i - 1} )M\lambda }}{{{f_0}}}} \right]}^2}} \,\,, $$
where i is an integer that indicates the zone transition number. For example, i = 2 identifies the transition between central zone 1 and the next radial zone, which is zone 2.

Space telescope imaging instruments (cameras) are typically designed to sample a relative narrow field of view. For example, Hubble Space Telescope’s WFC3 and ACS “wide field” and “survey” cameras [7] are designed for ∼0.05° full field. Missions with telescopes and instruments specifically designed for wide-field imaging (such as WISE and the Roman Space Telescope) have fields more than an order of magnitude larger on the largest side of about 0.78° (with Roman Space Telescope ∼ 200x in area compared to Hubble Space Telescope instruments). In contrast, a common cell phone telephoto lens has a 16° or larger full field.

Even though the field of view is relatively small for space telescopes, it must be well corrected to meet science requirements. A primary contribution of this paper is the analysis and correction of off-axis aberrations for MODE lenses. Although work has been presented concerning off-axis aberrations of DFLs on flat and curved surfaces [8,9], these analyses assume that the diffracting surface is infinitely thin. While this assumption can lead to accurate results for a single-harmonic DFL, it does not accurately describe the behavior of a MODE lens, due to its high-harmonic structure. Our results indicate that the off-axis aberration behavior of MODE lenses is well described by geometrically considering each MOD zone as a separate lens, and off-axis performance is improved by changing positions of the transition points at zone boundaries.

MOD lens structures with transition positions aligned in a plane, like the planar-front surface shown in Fig. 1, have been discussed in the literature for relatively low harmonic M numbers [1012]. Application of the theory presented in those references to high-harmonic systems (M > 250) indicate that the range of axial focal dispersion is approximately f0/M. For example, a high-harmonic system with M = 1000 and f0 = 1 m results in an axial focal range of Δf ∼ 1 mm. Higher M results in smaller Δf and smaller corresponding blur circle B. As demonstrated in [6], a high-harmonic M = 1000 system with a planar front displays the diffractive LCA characteristics similar the theory of [1012]. However, it is not known if a curved-front design modified for improved off-axis performance will produce the same diffractive behavior.

Although work has been reported on more complicated multi-diffractive-element harmonic diffractive lenses [13,14], these systems involve using two or more diffractive lens structures in close proximity. This arrangement is clearly impractical for a large-diameter primary lens on a space telescope, which would dramatically increase fabrication and alignment difficulties. In addition, the emphasis in these references is on improving diffraction efficiency over a wide wavelength range, not in reducing the diffractive component of LCA, which is of primary importance in imaging experiments.

The front MOD surface produces a cyclic range of focus given by approximately f0/M. For the design presented in this paper, the rear surface has a single-order (M = 1) DFL, whose purpose is to provide refractive color correction of each zone. Due to the relatively low focusing power of this surface, it does not significantly affect the cyclic range of focus of the MOD surface. Although it may be possible to use a higher-harmonic lens on the back surface, the harmonic-order number should not be greater than the design wavelength divided by the correction bandwidth. Otherwise, compounding cyclic variations of the focal position with wavelength would be produced. For the R-band astronomical wavelength band design example, 658 nm/(727 nm -589 nm) = 4.8, so the maximum harmonic order for the rear surface is approximately 4.

In section 2 of this paper, geometrical aspects of MODE lenses are discussed, including the introduction of zonal field shift, which is a new type of aberration characteristic of these segmented lens systems. By curving the front of the MODE lens transition positions, zonal field shift can be minimized or eliminated. Section 3 discusses diffractive behavior of MODE lenses, and section 4 provides a summary and conclusions.

2. Geometrical design considerations of MODE lenses

In this section, geometrical aspects of designing a 240 mm aperture MODE lens with M = 1000 are discussed in detail. By understanding that each MOD zone acts geometrically as a separate lens, it is found that shaping the MOD surface with a non-planar front significantly improves off-axis performance. Several to-scale lenses are shown in Fig. 2 for comparison. The nearest-refractive-equivalent achromatic doublet lens of Fig. 2(a) is more than 80-mm thick. An achromatic lens with similar performance is achieved with a MODE lens only 5-mm thick. As a result, the volume of a MODE lens is 10 to 15 times lower than a traditional achromatic doublet. Figure 2(b) shows a planar-front MODE design. Figures 2(c) and 2(d) show curved-front MODE designs. These designs are compared in the following paragraphs, along with theory that describes zonal field shift (ZFS).

 figure: Fig. 2.

Fig. 2. Achromatic lenses with same effective focal length: (a) traditional achromatic doublet made with BK7 and SF5; (b) Planar-front MODE design; (c) ZFS-free MODE design; and (d) Spherical-front MODE design. MODEs demonstrate significantly lower volume than classical achromats.

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Lenses shown in Fig. 2 have aperture diameters of 240 mm and design focal lengths f0 = 1 m. They are optimized for astronomical R-band wavelengths from 589 nm to 727 nm with a center wavelength of 658 nm. The maximum half-field angle is 0.125°, which results in an image height of 2.182 mm. MODE lenses considered here are designed with L-BSL7 glass, which is a low-temperature glass suitable for molding. In the design, the refractive index of L-BSL7 is slightly modified to account for mold cooling at a rate of 0.5° C/sec [15]. Detailed Zemax lens design files are available in Code 1 (Ref. [16]), Code 2 (Ref. [17]) and Code 3 (Ref. [18]). Base curvatures and aspheric coefficients of each zone are designed to focus the object at f0 geometrically over the R-band by optimization of each zone separately in the ray-trace program. Spot diagrams at 658 nm of the MODE lens designs are shown in Fig. 3. Figure 3(a) shows an on-axis spot diagram for the planar-front design. The black circle is the Airy disk diameter of 6.7 μm at λ0 = 658 nm, and Fig. 3(a) displays ray intercepts as colored symbols that are well within the Airy disk boundary. Symbol colors indicate ray intercepts from different zones. Since on-axis performance of MODE designs in Figs. 2(b)-(d) are similar, only the on-axis result for the planar-front design is shown. As field angle increases, Fig. 3(b) indicates that the planar-front design exhibits significant displacements of ray intercepts that are a function of the zone number, where the central MOD zone has no displacement and the outer MOD zone has the greatest displacement from the central-zone chief ray. This behavior is the ZFS of the lens.

 figure: Fig. 3.

Fig. 3. Monochromatic spot diagram at central wavelength (658 nm) of MODE designs. (a) On-axis spot diagram of planar-front design; (b) Maximum field angle spot diagram of planar-front design; (c) Maximum field angle spot diagram of the ZFS-free design; and (d) Maximum field angle spot diagram of spherical-front design. Only the on-axis spot diagram of the flat-front design is shown, because the on-axis spots are very similar for all designs.

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Note that the off-axis behavior of the planar front in Fig. 3(b) is considerably different than the off-axis behavior predicted by assuming an infinitely thin and planar diffractive surface, as described by [8,9], which results in dominant comatic aberration with a root-mean-square diameter of 43 μm. Instead, individual zonal spot radii, which are displayed as different colors in Fig. 3(b), are much smaller than the Airy spot diameter of 6.7μm at λ0 = 658 nm, and the maximum extent of the ZFS displacement due to the outer zone is less than 20 μm. In effect, ZFS is a geometrical effect due to the refractive structure of each zone.

To understand effects of ZFS in Fig. 3(b), a first-order geometrical analysis is performed by considering each MOD zone as a separate optical system. Figure 4(a) illustrates this concept for the planar-front design. The first zone is a standard lens with thickness t, where the base curvatures are chosen to image the object at f0 for λ0. The second zone (i = 2) is a different lens where the diameter of the first zone is effectively cored out from it and replaced with the zone 1 lens (i = 1). As with the first zone, the base curvature of the first surface is calculated to image the object at f0 for λ0. The step height at the transition of the 1st and 2nd zones corresponds to the MOD number M times the height h that corresponds to one wave of OPD at the design wavelength λ0. The effective axial thickness of the second zone lens is Mh greater than the first zone lens. Although no geometrical rays pass through the cored-out section of the lens describing the second zone, it is important to keep in mind the complete zonal lens description to understand its first-order properties. Higher-number zones follow this same trend, with correspondingly larger axial distances. The vertex of each zone lens is set forward of the first zone vertex by (i-1)Mh, where i is the zone number.

 figure: Fig. 4.

Fig. 4. Illustration of two configurations of MOD surfaces: a) Planar front; and b) Spherical front. Cored-out sections of effective lenses from MOD zones are shown as dotted lines.

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The spherical-front configuration displayed in Fig. 4(b) is like the planar front, except each zone is setback by an amount s from the vertex of the first zone. The thickness of each zone at its first transition point is t, which is the axial thickness of the first zone. Although the effective axial thicknesses of each zone lens is the same as in the planar-front design, the distance that each lens is set forward is (i-1)Mhs. Effectively, each zone is simply shifted toward the image, so that the transition points follow a specified function s.

A model for the lens in each zone is shown in Fig. 5, in which the cored-out section is replaced with the full refractive geometry. Planes 1 through 4 sequentially are the system stop, first refractive surface with power, a planar back surface and the image plane. A paraxial analysis of the lens is enough to understand its first-order optical characteristics, and the restriction of surface 3 as planar does not significantly affect the analysis. Distances z12, z23 and z34 are standard thicknesses between surfaces. z2a is the distance between the front vertex of the zone lens and the front vertex of the i = 1 lens (plane a), zab is the axial thickness of the i = 1 lens, and zb3 is the distance between the back vertex of the i = 1 lens (plane b) and the back vertex of the zone lens. For the i = 1 lens, both z2a and zb3 are zero. BFL is the back focal length from the planar lens surface to the image plane. BFLb is the distance from plane b to the image plane. The system stop is placed at an arbitrary distance z12 for this analysis, but in practice it is typically at the front vertex of the i = 1 lens.

 figure: Fig. 5.

Fig. 5. Simplified model of a lens representing a single MOD zone.

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A paraxial raytrace with a marginal ray from infinity (ω1=0) and ray height y2 at surface 2 produces the following relationships (ϕ is optical power, n is refractive index, u is paraxial marginal angle, y is the marginal ray height, $\bar{u}$ is paraxial chief ray angle, $\bar{y}$ is chief ray height, and ω is marginal optical angle given by nu. Prime indicates value in the subsequent space.):

$$\begin{array}{l} {{\omega ^{\prime}}_2} ={-} {y_2}{\phi _2}\\ {y_3} = {y_2} + {{\omega ^{\prime}}_2}{z_{23}}/n\\ {{\omega ^{\prime}}_3} = {{\omega ^{\prime}}_2} \end{array}$$
Effective focal length EFL is given by
$$EFL ={-} \frac{{{y_2}}}{{{{u^{\prime}}_3}}} = \frac{1}{{{\phi _2}}}\,\,\,\,\,\,\,\,\,.$$
Back focal length BFL = z34 for each zone is
$$BFL ={-} \frac{{{y_3}}}{{{{u^{\prime}}_3}}} = 1/{\phi _2} - {z_{23}}/n\,\,\,\,\,\,.$$
The first requirement for the focusing system is that back focal length BFLb defined by BFL + zb3 is the same for all zones. BFLb with zb3 = 0 is the BFL for zone 1, and zb3 = 0 for all zones of the planar geometry. In terms of first-order parameters,
$$BF{L_b} = BFL + {z_{b3}} = \,1/{\phi _2}\, - {z_{ab}}/n\,\, - \,\,({i - 1} )Mh/n\,\, + s\,\,,$$
where z23 is given by ${z_{ab}} + ({i - 1} )Mh$.

With the planar geometry, s = 0, and BFLb is a function of i unless ϕ2 is also a function of i. If

$$1/{\phi _2} = \,BF{L_b}\, + {z_{ab}}/n\,\, + \,\,({i - 1} )Mh/n\,\,\,,$$
where BFLb is a constant, the EFL of Eq. (4) is a function of i, and image-plane field height of the chief ray is now also a function of i, which is given by
$$\bar{y} = 1/{\phi _2}\,\bar{u} = \,[{BF{L_b}\, + {z_{ab}}/n\,\, + \,\,({i - 1} )Mh/n} ]\,\,\bar{u}\,\,\,\,.$$
That is, the image height depends on the zone number. The composite image from an off-axis object point is distributed radially outward from the zone 1 intercept into multiple image points, one for each zone. This condition is the zonal field shift (ZFS). Deviation of the zonal chief ray height from the zone 1 intercept is
$$\Delta \bar{y} = \,\,({i - 1} )Mh\bar{u}/n\,\,\,\,\,\,.$$
Substitution of the maximum zone number derived from Eq. (2) with ρ equal to the radius of the MODE lens into Eq. (8) produces the surprising result that maximum ZFS is not a function of M for a fixed MODE lens diameter. With equal BFLb on all zones, an M = 1000 planar MODE lens at a design wavelength of 658 nm and 0.125° field angle with n = 1.5 and 11 zones produces a maximum paraxial $\Delta \bar{y}$ of ∼19.1 µm, which is clearly displayed in Fig. 3(b). If BFLb is adjusted to compensate for the i- dependent term in Eq. (6), $\Delta \bar{y} = \,\,0$, but the resulting geometrical blur circle diameter B for each zone is
$$B = \,\,\frac{{({i - 1} )Mh}}{{n\,f\# }}\,\,\,\,\,\,.$$
For the system mentioned above and f/# = 4.17, the maximum blur circle diameter is 2.1 mm. Clearly, ZFS is a limiting issue with planar-front designs.

Curved-front designs have an additional degree of freedom in the choice of the function s, where Eq. (6) is used to define s with constant ϕ2 and BFLb such that

$$s = \,\,\,({i - 1} )Mh/n\,\,$$
defines a transition-point function that eliminates ZFS in a first-order sense, and this condition is called a ZFS-free design. The spot diagram of Fig. 3(c) indicates dramatically improved off-axis performance compared to Fig. 3(b), where the optimized real-ray ZFS-free design results in only a small amount of distortion in terms of shift of the centroid of the ray intercepts compared to the chief-ray intercept at the center of the Airy circle. Additional ray deviation is from residual coma and astigmatism in the system. However, that small shift has an insignificant effect on device performance, because it is much less than the Airy spot diameter.

A second choice is to set s on a spherical front that is concentric with the image point, which is called a spherical-front design. The functional form of s at each transition in this case is

$${s_i} = \,\,\,{f_0} - \sqrt {f_0^2 - \rho _i^2} \,\,, $$
where f0 is the design EFL and ρi is the radius of the zone transition defined by
$$\rho _i^{} = \,\,\,{f_0}\sqrt {\frac{{2({i - 1} )M\lambda }}{{{f_0}}} - {{\left[ {\frac{{({i - 1} )M\lambda }}{{{f_0}}}} \right]}^2}} \,\,. $$
The spot diagram of Fig. 3(d) shows that the optimized spherical-front design produces a symmetric geometric spot profile that falls well within the Airy diameter. More generally for any function s, zone transition radii are found from
$$\rho _i^{} = \,\,\,{f_0}\sqrt {\frac{{2({i - 1} )M\lambda }}{{{f_0}}} + {{\left[ {\frac{{({i - 1} )M\lambda }}{{{f_0}}}} \right]}^2} - \frac{{2s}}{{f_0^2}}} \,\,, $$
and the general paraxial expression for ZFS is
$$\Delta \bar{y} = \,\,({i - 1} )Mh\bar{u}/n\,\, - s\bar{u}\,\,\,.$$
The ratio of Eq. (14) to the Airy spot diameter is a useful metric for defining an improved optical system using a MODE lens. This ratio is the ZFS ratio and is defined by
$${r_{ZFS}} = \,\frac{{\,[{({i - 1} )Mh\bar{u}/n\,\, - s\bar{u}} ]{\,_{\max }}}}{{1.22\lambda /NA}}\,\,, $$
where the bracketed expression is evaluated at its maximum value and NA is numerical aperture of the MODE lens. Values of ${r_{ZFS}} \le 1$ define a system that is diffraction limited in the sense of ZFS, and values of rZFS greater than one are often acceptable, up to a maximum value that depends on the application. For the spherical-front and ZFS-free front designs presented here, rZFS <1.

In terms of residual ZFS, application of Eq. (14) to an M = 1000, f0 = 1 m MODE lens at λ0 = 658 nm and 0.125° field angle with n = 1.5 and 11 zones produces $\Delta \bar{y} = \,\,4.8\textrm{ }\mu \textrm{m}$, although the optimized optical design real-ray spot diagram of Fig. 3(d) indicates a slightly smaller $\Delta \bar{y}$ value for the spherical front. In either case, $\Delta \bar{y}$ for the spherical-front design is significantly smaller than for the planar design with equal BFLb.

3. Diffractive behavior of MODE lenses

In this section, the diffractive behavior MODE lenses is analyzed with respect to the planar, spherical-front and ZFS-free designs. First, the simulation model and technique are described, and then both point-spread function (PSF) and modulation transfer function (MTF) analyses are presented.

MODE front and back surfaces have multiple zones, as shown in Fig. 4, where each zone is a lens with its own geometrical optical characteristics and is shifted forward as described above. The first step for design of the complete lens is to optimize each zone segment separately to achieve a minimum spot rms radius in the image plane over the specified field angle. This zonal optimization can be accomplished in most commercial sequential lens design programs. All MODE zone lens segments are combined to construct the composite surface for calculation of OPD data. In a Code V non-sequential ray-trace model, the vertex of each zone lens segment is shifted along the optical axis according to each design type. It is efficient to first design each zone segment in a sequential ray-trace program and then perform detailed optimization in a non-sequential model.

Unlike simple refractive optical systems, the complete defocus function versus wavelength of a MODE lens cannot be simulated by raytracing. The MODE focus shift is a combination of refractive dispersion and MOD/DFL diffraction effects. To perform the calculation, rays are traced in Code V through the system, and OPD is calculated at the exit pupil. In order to calculate the effect of the DFL, a Sweatt surface is used for ray tracing that effectively gives the correct OPD in transmission in a scalar sense through the DFL as a function of wavelength [19]. More complex finite-difference-time-domain simulations or other electromagnetic solvers are not appropriate for this system, due to the physical size of the MODE lens. Field amplitudes of the point-spread function (PSF) near the focal plane of wavelengths across the R-band are calculated through well-known Fresnel propagation of field amplitudes using Hankel transformations in Matlab with appropriate phase factors, and the best focal positions are taken where axial irradiance values maximize. The Hankel transform used instead of a two-dimensional Fourier transform, because the on-axis geometry is symmetric around the optical axis [20]. Figure 6 shows transverse-axial cross-section irradiance images of the three designs [1618] at λ0 = 658 nm. The axial behavior is essentially identical for the planar-front, spherical-front and ZFS-free designs. Visualization 1 displays a dynamic sequence of transverse-axial cross sections over λ = λ0 +/- 10 nm.

 figure: Fig. 6.

Fig. 6. Transverse-axial cross section of the focal irradiance distribution of the three designs [1618] at λ0 = 658 nm. On-axis behavior of the three systems is essentially identical. (See Visualization 1 for a dynamic simulation over a +/- 10 nm wavelength range around λ0 = 658 nm.)

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Two different calculations are used to generate modulation transfer function (MTF) data. Monochromatic MTFs of MODE designs at center wavelength λ0 = 658 nm are generated using non-sequential Code V lens models, and these results are displayed as dashed lines in Fig. 7. However, Code V limits the number of wavelengths in a polychromatic MTF calculation to 21. Therefore, a hybrid model is implemented, where Code V is used to generate up to 1400 OPD data sets across the R-Band sampled at 0.1 nm, and then Matlab is used to generate the corresponding PSF image profiles, like those shown in Fig. 6.

 figure: Fig. 7.

Fig. 7. Diffraction-based modulation transfer function (MTF) calculations. The designs are illustrated in Fig. 2. SF = spherical front design, ZF – ZFS-free design. Two groups of calculation results are displayed relative to the diffraction-limited MTF. The first is monochromatic illumination at the center design wavelength λ0 = 658 nm, and the second group is a polychromatic average over the astronomical R-Band (589 nm to 727 nm).

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For the MTF calculations shown here, the non-sequential Code V lens models are used to generate OPD versus pupil coordinate in the exit pupil for an on-axis polychromatic source with a distribution of wavelengths over the astronomical R-Band (589 nm to 727 nm). The wavelengths are chosen over five separate evenly-spaced center points over the R-Band at 589 nm, 623 nm, 658 nm, 692 nm, and 727 nm. Around each center point, calculations are made with seven discrete wavelengths at 0.1 nm spacing. The seven closely-spaced wavelengths adequately sample over a diffractive cycle at each of the center points. The total number of wavelength sample points is 35. These data are then used in a Matlab scalar diffraction program with a forward Hankel transform to calculate electric field distributions in the focal plane. Point-spread-functions (PSFs) are then calculated by taking the magnitude squared of the electric field distribution at each wavelength. PSF distributions are summed, and the inverse Hankel transform of the aggregate PSF is normalized at zero spatial frequency to generate the on-axis R-Band plots in Fig. 7, which are displayed with solid lines. It was necessary to calculate the PSF with 10,000 evenly-spaced sample points over a range of 0 mm to 2 mm from the optical axis in order to calculate accurate MTF data at low spatial frequencies.

Figure 7 displays MTF calculation results from the achromat, spherical-front (SF) and ZFS-free (ZF) designs shown in Fig. 2. The maximum spatial frequency of 400 mm−1 is the axial diffraction limit [1/λmin(f/#)]. On-axis and off-axis radial (RAD) and tangential (TAN) dashed-line curves are displayed in Fig. 7 for the monochromatic calculation at λ0. On-axis performance is excellent with Strehl ratio (SR) = 0.96, and off-axis performance is slightly degraded due to astigmatism with SR = 0.90 for RAD and 0.84 for TAN directions. Polychromatic results shown as solid lines indicate that MODE designs outperform the achromat on axis over most spatial frequencies, with slightly worse MTF values at spatial frequencies below 25 mm−1. Corresponding SR values are 0.15 for the achromat, 0.19 for the spherical front (SF) and 0.21 for the ZFS-free (ZF) designs. Although not shown, both the SF and ZF designs also outperform the achromat off axis. MODE designs show an insignificant change of polychromatic MTF performance with field angle over the range out to 0.125°.

Differences between the MODE monochromatic and polychromatic results are mostly due to simple chromatic focal shift over the diffractive cycle and the residual longitudinal chromatic aberration (LCA). The residual LCA can be compensated by classical design techniques with an additional optical lens group near the focus to produce an apochromat. Compensation for LCA over the diffractive cycle is an active area of research and will be reported in a future publication.

4. Summary and conclusions

Several important design aspects of MODE lenses are presented. The segmented-zone MODE lens is efficiently designed by first optimizing each zone with a sequential ray-trace program. By considering each zone segment as a separate optical system, the offset of the focus spot from the chief-ray image-plane intercept of the central zone characterizes a new type of aberration called zonal field shift (ZFS), which can be minimized or eliminated by choosing curved-front values of the axial offset for each zone segment. Several designs are compared, based on a 240 mm aperture, 1 m focal length lightweight system with a maximum field angle of 0.125°. Both the spherical-front and ZFS-free designs show significantly better geometrical off-axis performance than the planar-front design. Since MODE lenses exhibit characteristics of both a geometrical lens and a diffractive lens, special consideration is given in the design for both effects. In terms of diffraction, the M number of the front surface is designed so that the diffractive focal range is approximately equal to the achromat secondary spectrum formed with the DFL back surface of the MODE lens. To simulate physical optics effects and optimize detailed design aspects of the MODE lens, the surface shape is transferred to a non-sequential model in Code V. This non-sequential model calculates OPD data that are used with a Matlab diffraction code to simulate polychromatic on-axis PSF and MTF. MTF performance of the MODE lens is significantly better than a refractive achromat that is optimized over the same wavelength range. Although imaging performance with SR ∼0.21 is adequate for temporal intensity and spectroscopic analysis, system performance should be improved for imaging applications. The concept and demonstration of a color corrector, which is a secondary lens system close to the focal plane of the MODE primary, has been discussed for larger-aperture DFL systems that improve the system to nearly diffraction-limited performance [3,4]. Construction of a 240 mm aperture MODE lens and color corrector is now an active project of the authors and others. The high-precision glass molding process for MODE lenses discussed in [21] will be used for fabrication.

Funding

Gordon and Betty Moore Foundation (7728).

Acknowledgements

We thank D. Apai of Steward Observatory for motivation to do this work and Glenn Schneider of Steward Observatory for useful comments.

Disclosures

The authors declare no conflicts of interest.

References

1. D. Apai, T. D. Milster, D. W. Kim, A. Bixel, G. Schneider, R. Liang, and J. Arenberg, “A Thousand Earths: A Very Large Aperture, Ultralight Space Telescope Array for Atmospheric Biosignature Surveys,” Astron. J. 158(2), 83 (2019). [CrossRef]  .

2. S. Seager and D. Deming, “Exoplanet atmospheres,” Annu. Rev. Astron. Astrophys. 48(1), 631–672 (2010). [CrossRef]  .

3. R. A. Hyde, “Eyeglass. 1. Very large aperture diffractive telescopes,” Appl. Opt. 38(19), 4198–4212 (1999). [CrossRef]  .

4. P. Atcheson, J. Domber, K. Whiteaker, J. A. Britten, S. N. Dixit, and B. Farmer, (2014, August). “MOIRE: ground demonstration of a large aperture diffractive transmissive telescope,” In Space Telescopes and Instrumentation 2014: Optical, Infrared, and Millimeter Wave (Vol. 9143, p. 91431W). International Society for Optics and Photonics.

5. D. Sham, R. Hyde, A. Weisberg, J. Early, M. Rushford, and J. Britten, Development of Large-Aperture, Light-Weight Fresnel Lenses for Gossamer Space Telescopes (No. UCRL-JC-148223). (Lawrence Livermore National Lab., 2002).

6. T. D. Milster, Y. S. Kim, Z. Wang, and K. Purvin, “Multiple-order diffractive engineered surface lenses,” Appl. Opt. 59(26), 7900–7906 (2020). [CrossRef]  .

7. https://www.nasa.gov/content/hubble-space-telescope-advanced-camera-for-surveys

8. E. Delano, “Primary aberrations of meniscus Fresnel lenses,” J. Opt. Soc. Am. 66(12), 1317–1320 (1976). [CrossRef]  .

9. L. N. Hazra, Y. Han, and C. Delisle, “Curved kinoform lenses for stigmatic imaging of axial objects,” Appl. Opt. 32(25), 4775–4784 (1993). [CrossRef]  .

10. D. Faklis and G. M. Morris, “Spectral properties of multiorder diffractive lenses,” Appl. Opt. 34(14), 2462–2468 (1995). [CrossRef]  .

11. D. W. Sweeney and G. E. Sommargren, “Harmonic diffractive lenses,” Appl. Opt. 34(14), 2469–2475 (1995). [CrossRef]  .

12. T. Stone and N. George, “Hybrid diffractive-refractive lenses and achromats,” Appl. Opt. 27(14), 2960–2971 (1988). [CrossRef]  .

13. S. Noach, Y. Arieli, and N. Eisenberg, “Wave-front control and aberration correction with a diffractive optical element,” Opt. Lett. 24(5), 333–335 (1999). [CrossRef]  .

14. C. Xue and Q. Cui, “Design of multilayer diffractive optical elements with polychromatic integral diffraction efficiency,” Opt. Lett. 35(7), 986–988 (2010). [CrossRef]  .

15. Technical Data: Pressing and forming for low Tg optical glasses, Version 8.2, Ohara Corporation, September 9, 2017.

16. Z. Wang, “Zemax design for flat-front MODE lens,” figshare (2020), https://doi.org/10.6084/m9.figshare.12954641

17. Z. Wang, “Zemax design for spherical-front MODE lens,” figshare (2020), https://doi.org/10.6084/m9.figshare.12954659

18. Z. Wang, “Zemax design for ZFSfree-front MODE lens,” figshare (2020), https://doi.org/10.6084/m9.figshare.12954662

19. W. C. Sweatt, “Describing holographic optical elements as lenses,” J. Opt. Soc. Am. 67(6), 803–808 (1977). [CrossRef]  .

20. J. D. Gaskill, Linear systems, Fourier transforms, and optics, Vol. 576, (Wiley, 1978), p. 317.

21. Y. Zhang, R. Liang, O. J. Spires, S. Yin, A. Yi, and T. D. Milster, “Precision glass molding of diffractive optical elements with high surface quality,” Opt. Lett. 45(23), 6438–6441 (2020). [CrossRef]  .

References

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  1. D. Apai, T. D. Milster, D. W. Kim, A. Bixel, G. Schneider, R. Liang, and J. Arenberg, “A Thousand Earths: A Very Large Aperture, Ultralight Space Telescope Array for Atmospheric Biosignature Surveys,” Astron. J. 158(2), 83 (2019)..
    [Crossref]
  2. S. Seager and D. Deming, “Exoplanet atmospheres,” Annu. Rev. Astron. Astrophys. 48(1), 631–672 (2010)..
    [Crossref]
  3. R. A. Hyde, “Eyeglass. 1. Very large aperture diffractive telescopes,” Appl. Opt. 38(19), 4198–4212 (1999)..
    [Crossref]
  4. P. Atcheson, J. Domber, K. Whiteaker, J. A. Britten, S. N. Dixit, and B. Farmer, (2014, August). “MOIRE: ground demonstration of a large aperture diffractive transmissive telescope,” In Space Telescopes and Instrumentation 2014: Optical, Infrared, and Millimeter Wave (Vol. 9143, p. 91431W). International Society for Optics and Photonics.
  5. D. Sham, R. Hyde, A. Weisberg, J. Early, M. Rushford, and J. Britten, Development of Large-Aperture, Light-Weight Fresnel Lenses for Gossamer Space Telescopes (No. UCRL-JC-148223). (Lawrence Livermore National Lab., 2002).
  6. T. D. Milster, Y. S. Kim, Z. Wang, and K. Purvin, “Multiple-order diffractive engineered surface lenses,” Appl. Opt. 59(26), 7900–7906 (2020)..
    [Crossref]
  7. https://www.nasa.gov/content/hubble-space-telescope-advanced-camera-for-surveys
  8. E. Delano, “Primary aberrations of meniscus Fresnel lenses,” J. Opt. Soc. Am. 66(12), 1317–1320 (1976)..
    [Crossref]
  9. L. N. Hazra, Y. Han, and C. Delisle, “Curved kinoform lenses for stigmatic imaging of axial objects,” Appl. Opt. 32(25), 4775–4784 (1993)..
    [Crossref]
  10. D. Faklis and G. M. Morris, “Spectral properties of multiorder diffractive lenses,” Appl. Opt. 34(14), 2462–2468 (1995)..
    [Crossref]
  11. D. W. Sweeney and G. E. Sommargren, “Harmonic diffractive lenses,” Appl. Opt. 34(14), 2469–2475 (1995)..
    [Crossref]
  12. T. Stone and N. George, “Hybrid diffractive-refractive lenses and achromats,” Appl. Opt. 27(14), 2960–2971 (1988)..
    [Crossref]
  13. S. Noach, Y. Arieli, and N. Eisenberg, “Wave-front control and aberration correction with a diffractive optical element,” Opt. Lett. 24(5), 333–335 (1999)..
    [Crossref]
  14. C. Xue and Q. Cui, “Design of multilayer diffractive optical elements with polychromatic integral diffraction efficiency,” Opt. Lett. 35(7), 986–988 (2010)..
    [Crossref]
  15. Technical Data: Pressing and forming for low Tg optical glasses, Version 8.2, Ohara Corporation, September 9, 2017.
  16. Z. Wang, “Zemax design for flat-front MODE lens,” figshare (2020), https://doi.org/10.6084/m9.figshare.12954641
  17. Z. Wang, “Zemax design for spherical-front MODE lens,” figshare (2020), https://doi.org/10.6084/m9.figshare.12954659
  18. Z. Wang, “Zemax design for ZFSfree-front MODE lens,” figshare (2020), https://doi.org/10.6084/m9.figshare.12954662
  19. W. C. Sweatt, “Describing holographic optical elements as lenses,” J. Opt. Soc. Am. 67(6), 803–808 (1977)..
    [Crossref]
  20. J. D. Gaskill, Linear systems, Fourier transforms, and optics, Vol. 576, (Wiley, 1978), p. 317.
  21. Y. Zhang, R. Liang, O. J. Spires, S. Yin, A. Yi, and T. D. Milster, “Precision glass molding of diffractive optical elements with high surface quality,” Opt. Lett. 45(23), 6438–6441 (2020)..
    [Crossref]

2020 (2)

2019 (1)

D. Apai, T. D. Milster, D. W. Kim, A. Bixel, G. Schneider, R. Liang, and J. Arenberg, “A Thousand Earths: A Very Large Aperture, Ultralight Space Telescope Array for Atmospheric Biosignature Surveys,” Astron. J. 158(2), 83 (2019)..
[Crossref]

2010 (2)

1999 (2)

1995 (2)

1993 (1)

1988 (1)

1977 (1)

1976 (1)

Apai, D.

D. Apai, T. D. Milster, D. W. Kim, A. Bixel, G. Schneider, R. Liang, and J. Arenberg, “A Thousand Earths: A Very Large Aperture, Ultralight Space Telescope Array for Atmospheric Biosignature Surveys,” Astron. J. 158(2), 83 (2019)..
[Crossref]

Arenberg, J.

D. Apai, T. D. Milster, D. W. Kim, A. Bixel, G. Schneider, R. Liang, and J. Arenberg, “A Thousand Earths: A Very Large Aperture, Ultralight Space Telescope Array for Atmospheric Biosignature Surveys,” Astron. J. 158(2), 83 (2019)..
[Crossref]

Arieli, Y.

Atcheson, P.

P. Atcheson, J. Domber, K. Whiteaker, J. A. Britten, S. N. Dixit, and B. Farmer, (2014, August). “MOIRE: ground demonstration of a large aperture diffractive transmissive telescope,” In Space Telescopes and Instrumentation 2014: Optical, Infrared, and Millimeter Wave (Vol. 9143, p. 91431W). International Society for Optics and Photonics.

Bixel, A.

D. Apai, T. D. Milster, D. W. Kim, A. Bixel, G. Schneider, R. Liang, and J. Arenberg, “A Thousand Earths: A Very Large Aperture, Ultralight Space Telescope Array for Atmospheric Biosignature Surveys,” Astron. J. 158(2), 83 (2019)..
[Crossref]

Britten, J.

D. Sham, R. Hyde, A. Weisberg, J. Early, M. Rushford, and J. Britten, Development of Large-Aperture, Light-Weight Fresnel Lenses for Gossamer Space Telescopes (No. UCRL-JC-148223). (Lawrence Livermore National Lab., 2002).

Britten, J. A.

P. Atcheson, J. Domber, K. Whiteaker, J. A. Britten, S. N. Dixit, and B. Farmer, (2014, August). “MOIRE: ground demonstration of a large aperture diffractive transmissive telescope,” In Space Telescopes and Instrumentation 2014: Optical, Infrared, and Millimeter Wave (Vol. 9143, p. 91431W). International Society for Optics and Photonics.

Cui, Q.

Delano, E.

Delisle, C.

Deming, D.

S. Seager and D. Deming, “Exoplanet atmospheres,” Annu. Rev. Astron. Astrophys. 48(1), 631–672 (2010)..
[Crossref]

Dixit, S. N.

P. Atcheson, J. Domber, K. Whiteaker, J. A. Britten, S. N. Dixit, and B. Farmer, (2014, August). “MOIRE: ground demonstration of a large aperture diffractive transmissive telescope,” In Space Telescopes and Instrumentation 2014: Optical, Infrared, and Millimeter Wave (Vol. 9143, p. 91431W). International Society for Optics and Photonics.

Domber, J.

P. Atcheson, J. Domber, K. Whiteaker, J. A. Britten, S. N. Dixit, and B. Farmer, (2014, August). “MOIRE: ground demonstration of a large aperture diffractive transmissive telescope,” In Space Telescopes and Instrumentation 2014: Optical, Infrared, and Millimeter Wave (Vol. 9143, p. 91431W). International Society for Optics and Photonics.

Early, J.

D. Sham, R. Hyde, A. Weisberg, J. Early, M. Rushford, and J. Britten, Development of Large-Aperture, Light-Weight Fresnel Lenses for Gossamer Space Telescopes (No. UCRL-JC-148223). (Lawrence Livermore National Lab., 2002).

Eisenberg, N.

Faklis, D.

Farmer, B.

P. Atcheson, J. Domber, K. Whiteaker, J. A. Britten, S. N. Dixit, and B. Farmer, (2014, August). “MOIRE: ground demonstration of a large aperture diffractive transmissive telescope,” In Space Telescopes and Instrumentation 2014: Optical, Infrared, and Millimeter Wave (Vol. 9143, p. 91431W). International Society for Optics and Photonics.

Gaskill, J. D.

J. D. Gaskill, Linear systems, Fourier transforms, and optics, Vol. 576, (Wiley, 1978), p. 317.

George, N.

Han, Y.

Hazra, L. N.

Hyde, R.

D. Sham, R. Hyde, A. Weisberg, J. Early, M. Rushford, and J. Britten, Development of Large-Aperture, Light-Weight Fresnel Lenses for Gossamer Space Telescopes (No. UCRL-JC-148223). (Lawrence Livermore National Lab., 2002).

Hyde, R. A.

Kim, D. W.

D. Apai, T. D. Milster, D. W. Kim, A. Bixel, G. Schneider, R. Liang, and J. Arenberg, “A Thousand Earths: A Very Large Aperture, Ultralight Space Telescope Array for Atmospheric Biosignature Surveys,” Astron. J. 158(2), 83 (2019)..
[Crossref]

Kim, Y. S.

Liang, R.

Y. Zhang, R. Liang, O. J. Spires, S. Yin, A. Yi, and T. D. Milster, “Precision glass molding of diffractive optical elements with high surface quality,” Opt. Lett. 45(23), 6438–6441 (2020)..
[Crossref]

D. Apai, T. D. Milster, D. W. Kim, A. Bixel, G. Schneider, R. Liang, and J. Arenberg, “A Thousand Earths: A Very Large Aperture, Ultralight Space Telescope Array for Atmospheric Biosignature Surveys,” Astron. J. 158(2), 83 (2019)..
[Crossref]

Milster, T. D.

Morris, G. M.

Noach, S.

Purvin, K.

Rushford, M.

D. Sham, R. Hyde, A. Weisberg, J. Early, M. Rushford, and J. Britten, Development of Large-Aperture, Light-Weight Fresnel Lenses for Gossamer Space Telescopes (No. UCRL-JC-148223). (Lawrence Livermore National Lab., 2002).

Schneider, G.

D. Apai, T. D. Milster, D. W. Kim, A. Bixel, G. Schneider, R. Liang, and J. Arenberg, “A Thousand Earths: A Very Large Aperture, Ultralight Space Telescope Array for Atmospheric Biosignature Surveys,” Astron. J. 158(2), 83 (2019)..
[Crossref]

Seager, S.

S. Seager and D. Deming, “Exoplanet atmospheres,” Annu. Rev. Astron. Astrophys. 48(1), 631–672 (2010)..
[Crossref]

Sham, D.

D. Sham, R. Hyde, A. Weisberg, J. Early, M. Rushford, and J. Britten, Development of Large-Aperture, Light-Weight Fresnel Lenses for Gossamer Space Telescopes (No. UCRL-JC-148223). (Lawrence Livermore National Lab., 2002).

Sommargren, G. E.

Spires, O. J.

Stone, T.

Sweatt, W. C.

Sweeney, D. W.

Wang, Z.

Weisberg, A.

D. Sham, R. Hyde, A. Weisberg, J. Early, M. Rushford, and J. Britten, Development of Large-Aperture, Light-Weight Fresnel Lenses for Gossamer Space Telescopes (No. UCRL-JC-148223). (Lawrence Livermore National Lab., 2002).

Whiteaker, K.

P. Atcheson, J. Domber, K. Whiteaker, J. A. Britten, S. N. Dixit, and B. Farmer, (2014, August). “MOIRE: ground demonstration of a large aperture diffractive transmissive telescope,” In Space Telescopes and Instrumentation 2014: Optical, Infrared, and Millimeter Wave (Vol. 9143, p. 91431W). International Society for Optics and Photonics.

Xue, C.

Yi, A.

Yin, S.

Zhang, Y.

Annu. Rev. Astron. Astrophys. (1)

S. Seager and D. Deming, “Exoplanet atmospheres,” Annu. Rev. Astron. Astrophys. 48(1), 631–672 (2010)..
[Crossref]

Appl. Opt. (6)

Astron. J. (1)

D. Apai, T. D. Milster, D. W. Kim, A. Bixel, G. Schneider, R. Liang, and J. Arenberg, “A Thousand Earths: A Very Large Aperture, Ultralight Space Telescope Array for Atmospheric Biosignature Surveys,” Astron. J. 158(2), 83 (2019)..
[Crossref]

J. Opt. Soc. Am. (2)

Opt. Lett. (3)

Other (8)

Technical Data: Pressing and forming for low Tg optical glasses, Version 8.2, Ohara Corporation, September 9, 2017.

Z. Wang, “Zemax design for flat-front MODE lens,” figshare (2020), https://doi.org/10.6084/m9.figshare.12954641

Z. Wang, “Zemax design for spherical-front MODE lens,” figshare (2020), https://doi.org/10.6084/m9.figshare.12954659

Z. Wang, “Zemax design for ZFSfree-front MODE lens,” figshare (2020), https://doi.org/10.6084/m9.figshare.12954662

https://www.nasa.gov/content/hubble-space-telescope-advanced-camera-for-surveys

P. Atcheson, J. Domber, K. Whiteaker, J. A. Britten, S. N. Dixit, and B. Farmer, (2014, August). “MOIRE: ground demonstration of a large aperture diffractive transmissive telescope,” In Space Telescopes and Instrumentation 2014: Optical, Infrared, and Millimeter Wave (Vol. 9143, p. 91431W). International Society for Optics and Photonics.

D. Sham, R. Hyde, A. Weisberg, J. Early, M. Rushford, and J. Britten, Development of Large-Aperture, Light-Weight Fresnel Lenses for Gossamer Space Telescopes (No. UCRL-JC-148223). (Lawrence Livermore National Lab., 2002).

J. D. Gaskill, Linear systems, Fourier transforms, and optics, Vol. 576, (Wiley, 1978), p. 317.

Supplementary Material (4)

NameDescription
» Code 1       Zemax flat-front
» Code 2       Zemax spherical front
» Code 3       Zemax ZFSfree front
» Visualization 1       Point-spread functions for axial objects with three different versions of a multiple-order diffractive lens that have different leading-edge shape functions.

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Figures (7)

Fig. 1.
Fig. 1. Multiple-order-diffraction engineered surface (MODE) lens. The front surface is a multiple-order-diffraction (MOD) lens, and the back surface is a diffractive Fresnel lens (DFL). h is the glass thickness of index n that produces one wave of optical path difference in transmission. A MOD surface with high M number produces a small value of LCA. The DFL reduces the refractive part of LCA, making each MOD zone achromatic.
Fig. 2.
Fig. 2. Achromatic lenses with same effective focal length: (a) traditional achromatic doublet made with BK7 and SF5; (b) Planar-front MODE design; (c) ZFS-free MODE design; and (d) Spherical-front MODE design. MODEs demonstrate significantly lower volume than classical achromats.
Fig. 3.
Fig. 3. Monochromatic spot diagram at central wavelength (658 nm) of MODE designs. (a) On-axis spot diagram of planar-front design; (b) Maximum field angle spot diagram of planar-front design; (c) Maximum field angle spot diagram of the ZFS-free design; and (d) Maximum field angle spot diagram of spherical-front design. Only the on-axis spot diagram of the flat-front design is shown, because the on-axis spots are very similar for all designs.
Fig. 4.
Fig. 4. Illustration of two configurations of MOD surfaces: a) Planar front; and b) Spherical front. Cored-out sections of effective lenses from MOD zones are shown as dotted lines.
Fig. 5.
Fig. 5. Simplified model of a lens representing a single MOD zone.
Fig. 6.
Fig. 6. Transverse-axial cross section of the focal irradiance distribution of the three designs [1618] at λ0 = 658 nm. On-axis behavior of the three systems is essentially identical. (See Visualization 1 for a dynamic simulation over a +/- 10 nm wavelength range around λ0 = 658 nm.)
Fig. 7.
Fig. 7. Diffraction-based modulation transfer function (MTF) calculations. The designs are illustrated in Fig. 2. SF = spherical front design, ZF – ZFS-free design. Two groups of calculation results are displayed relative to the diffraction-limited MTF. The first is monochromatic illumination at the center design wavelength λ0 = 658 nm, and the second group is a polychromatic average over the astronomical R-Band (589 nm to 727 nm).

Equations (16)

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h = λ 0 / ( n 1 ) 1 μ m .
ρ i = f 0 2 ( i 1 ) M λ f 0 + [ ( i 1 ) M λ f 0 ] 2 ,
ω 2 = y 2 ϕ 2 y 3 = y 2 + ω 2 z 23 / n ω 3 = ω 2
E F L = y 2 u 3 = 1 ϕ 2 .
B F L = y 3 u 3 = 1 / ϕ 2 z 23 / n .
B F L b = B F L + z b 3 = 1 / ϕ 2 z a b / n ( i 1 ) M h / n + s ,
1 / ϕ 2 = B F L b + z a b / n + ( i 1 ) M h / n ,
y ¯ = 1 / ϕ 2 u ¯ = [ B F L b + z a b / n + ( i 1 ) M h / n ] u ¯ .
Δ y ¯ = ( i 1 ) M h u ¯ / n .
B = ( i 1 ) M h n f # .
s = ( i 1 ) M h / n
s i = f 0 f 0 2 ρ i 2 ,
ρ i = f 0 2 ( i 1 ) M λ f 0 [ ( i 1 ) M λ f 0 ] 2 .
ρ i = f 0 2 ( i 1 ) M λ f 0 + [ ( i 1 ) M λ f 0 ] 2 2 s f 0 2 ,
Δ y ¯ = ( i 1 ) M h u ¯ / n s u ¯ .
r Z F S = [ ( i 1 ) M h u ¯ / n s u ¯ ] max 1.22 λ / N A ,

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