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Abstract
Large diameter, high-harmonic diffractive lenses could find applications in future space telescopes. Residual chromatic aberrations from these lenses can cause significant blurring. Solutions to reduce chromatic dispersion and other aberrations to diffraction-limited performance are discussed. A design example based on a 240-mm-diameter, 1-m focal length multi-order diffractive engineered lens operating over the astronomical R-Band (589–727 nm) is presented. The design example uses a relay subsystem with four times smaller diameter than the primary. This color corrector includes both refractive and diffractive optical elements and reduces the longitudinal chromatic aberrations by more than a factor of 30 compared to the primary lens alone, while maintaining the effective focal length and numerical aperture of the system.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. INTRODUCTION
Multiple-harmonic optical elements have interesting and useful optical properties, and they may be used for lightweight optical components in future space telescopes [1], terahertz imaging [2], beam shaping [3], remote sensing [4], and other applications. The optical principle behind all these systems is to design the focal spot of a diffractive lens into a high diffractive order, rather than the first diffractive order common to single-order diffractive Fresnel lenses (DFLs). Like DFLs, multiple-harmonic systems display a dramatic change in focal position with wavelength. However, the axial range $\Delta f$ of this change is limited to approximately ${f_0}/M$, where $M$ is the focal diffraction order at the design wavelength, and ${f_0}$ is the design focal length [5–7]. This paper considers color correction of the residual $\Delta f$ for high-harmonic diffractive lenses ($M\; \gt \;{250}$).
Correction of the focal change with wavelength, otherwise known as focal dispersion or longitudinal chromatic aberration (LCA), for diffractive optical systems is an important topic, given that the focal length as a function of wavelength $f(\lambda)$ is inversely proportional to wavelength, with
where ${\lambda _0}$ is the design wavelength, and $\lambda$ is the evaluation wavelength. For a system that images objects in the visible spectrum (400–700 nm) with a center wavelength of 550 nm, a DFL with $M = {1}$ exhibits ${\gt}\;{50}\%$ $\Delta f$ change in focus. The resulting blur creates unacceptable images without correction. High-harmonic lens systems have considerably less focal dispersion. For example, an $M = {250}$ system used for the visible spectrum exhibits only a 0.4% change in focus and is apochromatic in the sense that there are multiple wavelengths ${\lambda _p}$ that focus to ${f_0}$ given by [6] where $p$ is an integer that specifies the diffracted order.Since high-harmonic diffractive lenses and DFLs are ultralightweight, they are attractive for use as large-aperture primary lenses in space telescopes [1,8,9]. However, the focal dispersion $\Delta f$ should be corrected for high-quality imaging. A secondary optical system placed near the focus of the primary lens is acceptable for spacecraft, if its volume and mass are not large. Although Schupmann-type color correctors (CCs) have been discussed for large-aperture space-telescope DFL systems [8], they are undesirable in telescopes with low $f$-number primary objective lenses, due to the requirement of refocusing a virtual image with large marginal ray angles, as discussed below. In addition, the Schupmann configuration by itself is not enough to compensate for the unique diffractive effects observed in $M\; \gt \;{1}$ systems. In this paper, we introduce a new type of color correction that corrects the focal dispersion for high-harmonic lenses and overcomes this limitation.
Our design example is based on a multiple-order diffraction engineered (MODE) telescope primary lens that combines a multi-order diffractive (MOD) surface [5,6] and single-order DFL, as shown in Fig. 1 [7]. The MOD surface is composed of annular zones with transition heights of $Mh$, where $h$ is the glass thickness of refractive index $n$ that produces one wave of optical path difference (OPD) in transmission. The LCA has two components: Type 1 LCA due to dispersion of the glass and the DFL, and type 2 LCA due to diffractive characteristics of the MOD surface. Over a wide spectral range, type 1 LCA exceeds type 2 LCA. As suggested in [5], the DFL on the back surface is designed to partially compensate for refractive dispersion. However, the zonal MODE DFLs are optimized individually. Each MODE zone is achromatic in the classical sense, where two wavelengths in the spectrum focus at $f_{0}$. Previous results with an $M = {1000}$ MODE lens indicate that the residual type 2 $\Delta f$ is approximately equal to the type 1 LCA component [7]. A second design example with an $M = {1000}$ MODE lens [10] shows a similar characteristic. Even with this achromatic compensation, polychromatic imaging performance leads to a Strehl ratio of only 0.12 in the best design over the astronomical R-band of wavelengths (589–727 nm). The goal of this paper is to increase the polychromatic Strehl ratio to diffraction-limited performance with a Strehl ratio ${\gt}\;{0.9}$ over the astronomical R-band.
Fig. 1. Multiple-order-diffraction engineered (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 refractive index $n$ that produces one wave of optical path difference in transmission. A MOD surface with high $M$ number ($M\; \gt \;{250}$) produces a small value for type 2 longitudinal chromatic aberration (LCA). The DFL reduces type 1 focal dispersion, making each MOD zone achromatic.
Fig. 2. Overview of components and structure of the color-corrected MODE telescope system in a to-scale raytrace drawing. A three-dimensional rendering of the MODE primary lens is shown below the raytrace. The color corrector (CC) is magnified in a detailed view.
The layout of the system described here is diagramed in Fig. 2, which shows a 240-mm-diameter aperture MODE primary lens with effective focal length (EFL) of 1 m and a CC system near the focus of the primary. Design of the primary lens is similar to those discussed in [10], and detailed design files are provided as supplementary material in Code 1, Ref. [11]. The magnified view of the CC shows that it has several parts, including a field lens, four doublets (A, B, C, and D), a DFL, and a new type of component that we call an Arizona total energy CC (AZTECC) lens. The primary MODE lens has five circular zones with $M = {2196}$ and is designed over the astronomical R-band of wavelengths. The field lens is an essential part of the system to reduce the height of off-axis rays and enables effective color correction over the entire 0.25° full field of view. The main part of the CC is essentially a four-doublet relay lens that refocuses the light and significantly reduces type 1 LCA. The DFL adds dispersive power to the four-doublet relay to form classical apochromatic type 1 LCA correction, and the AZTECC lens corrects type 2 LCA. In this example, the CC is designed with unity magnification.
This paper is divided into several sections. In Section 2, background is presented with discussion of existing solutions to reduce chromatic aberrations of transmissive diffractive telescopes based on DFLs, which are based on Schupmann chromatic correction. Section 3 discusses design theory that includes principles of type 1 and type 2 LCA in MODE primary lenses and a method to correct them. Section 4 presents performance results from a design example that exhibits diffraction-limited performance with a Strehl ratio ${\sim}{0.99}$. Sections 5 and 6 present a short discussion about lightweight and larger bandwidth color-corrected MODE systems and the conclusions, respectively.
2. BACKGROUND
Previously, large-aperture DFL Eyeglass [8] and MOIRE [9] telescope designs addressed the issue of color correction for very-large-aperture space telescopes using Schupmann-type CCs. LCA of the primary lens is proportional to the EFL of the primary in these designs. Without correction, LCA due to the Eyeglass 2500 m EFL primary lens is ${\sim}{200}\;{\rm m}$ over the spectral range of 0.48–0.52 µm. The corresponding MOIRE LCA due to a 32.5 m EFL primary is ${\sim}{2.3}\;{\rm m}$ over the spectral range of 625–670 nm. Principles of color correction for these systems are discussed below.
The Eyeglass conceptual telescope design uses a CC based on a design from Faklis and Morris [12], which is shown in Fig. 3(a) with an object at infinity. The principle of color correction is shown conceptually in Fig. 3(b). A long-wavelength point image and a short-wavelength point image formed by the primary holographic element (HOE) due to a distant object are shown as ${i_l}$ and ${i_s}$, respectively, where the primary HOE focal dispersion displaces the longer wavelength image closer to the primary. The negative-power second holographic optical element 2 (HOE 2) exhibits long-wavelength and short-wavelength focal lengths ${f_l}$ and ${f_s}$, respectively. The focal length and axial position of HOE 2 are chosen so that the virtual images of both wavelengths, $i^{\prime}_{l}$ and $i^{\prime}_{s}$, are coincident and arranged with equal transverse magnification for both wavelengths. The highly divergent light from HOE 2 is then focused to the image plane by the second achromat, which is not shown in Fig. 3(b). By including realistic refractive index values, Faklis and Morris show an example system that is apochromatic with three wavelengths that focus at the image plane. Residual LCA is ${\sim}{40}\;\unicode{x00B5}{\rm m}$ over a spectral band from 400 to 700 nm. The MOIRE CC is a similar concept, except the relay lens is replaced by a collimator [9].
Fig. 3. Simplified illustrations of modified Schupmann configurations used to correct single-harmonic DFL dispersion of a primary lens in a telescope. (a) Color correction structure demonstrated in [12] and adapted by Eyeglass; (b) schematic diagram that shows the Schupmann-type color correction condition, where a portion of the optical system is illustrated with axial intercepts ${i_l}$ and ${i_s}$ foci from the primary lens, ${f_l}$ and ${f_s}$ front focal lane positions of HOE 2, and the virtual image locations of light transmitted through HOE 2. Subscripts $l$ and $s$ refer to long and short wavelengths of the design spectrum, respectively.
Fig. 4. (a) Raytrace of a DFL telescope system with a Schupmann-type color corrector. Foci from the primary DFL lens are shown for focal lengths of 727 nm, 658 nm, and 589 nm. The virtual focus created by HOE 2, diverges strongly after this negative element. The primary-lens DFL is not shown. (b) OPD aberration plot of the output of (a), which shows that when the aperture of the primary DFL is large, spherochromatism cannot be corrected and results in unacceptable performance.
Single-harmonic DFL telescope primary lenses corrected by Schupmann-like structures share several common problems that make them unsuitable for large-aperture, wide-field-of-view applications. A 240-mm-diameter, 1- m focal length, ${\rm F}/{4}$ DFL primary lens, the first-order parameters of which are the same as the high-harmonic design shown in later sections, is used as an example to demonstrate these issues over the astronomical R-band (589–727 nm with center wavelength 658 nm). The primary-lens DFL exhibits 212 mm of LCA over the astronomical R-band without color correction. An attempt is made to correct this system with a geometry similar to the one shown in Fig. 3(a), where the first achromat is replaced by an ideal field lens for simplicity. Since the second achromat serves only to reimage the virtual focus and does not contribute significantly to color correction, it is not included in the example. As shown in Fig. 4(a), the color-corrected output has an $f$-number approximately 0.5, which makes design of the second achromat impractical when constrained by implementing a CC system that is lightweight compared to the primary lens. In addition, when the aperture of the primary DFL is moderately large, spherochromatism, which is variation in the fourth-order shape of the OPD versus wavelength as measured in the exit pupil, becomes dominant over LCA, as shown in the OPD plot in Fig. 4(b). Spherochromatism affects rays diverging from the virtual focus by slightly changing ray angles for different wavelengths as a function of radius in the divergent cone, as shown with the inset of Fig. 4(a). Adding aspheric terms to either the primary or HOE 2 does not significantly improve performance. Also, fabrication of HOE 2 is problematic, due to its low $f$-number. For example, the outer-most period of the diffractive pattern for a DFL with $f/\# = {0.5}$ at 658 nm is ${\sim}{0.7}\;\unicode{x00B5}{\rm m}$.
Fig. 5. Schematic diagram of type 1 LCA color correcting structure used in the MODE system. One goal of the color corrector is to correct ${{\rm SS}_1}$ and make the output apochromatic, as shown in insets, which show LCAs before and after correction. In all insets, the horizontal axis is LCA in micrometers, and the vertical axis is wavelength in nanometers.
To realize larger-aperture and lower $f$-number ultralightweight astronomical telescopes, a different solution is necessary. As shown in the following sections, designing a MODE lens with a new type of CC could satisfy the goal of providing diffraction-limited imaging quality with an ultralight transmissive optical system. Although computer processing techniques show good promise for enhancing high-harmonic and wide-bandwidth images [4], they are not discussed here.
3. DESIGN THEORY
This section presents the basic theory of color correction for high-harmonic MODE lenses, and it may also be useful for other high-harmonic designs. Each MODE zone is optimized to be achromatic, where two wavelengths come to focus at the same axial location, by combination of the zonal refractive lens and the zonal DFL. Correction of this type 1 LCA is similar to classical color correction, with the added constraint of a reduced diameter for the correction optics. Unlike the slowly varying type 1 LCA, the MODE lens also exhibits a cyclic variation in focal length versus wavelength [7], where the range of focal values in each cycle is $\Delta f\;\sim f/M$, and the cycle period is $\Delta \lambda \;\sim{\lambda _0}/M$. This cyclic variation is type 2 LCA, which is due to the MOD front surface [5,6]. The design example illustrates that type 1 and type 2 LCA are additive, as has been our experience with several other similar designs. The design theory below is divided into first solving the type 1 LCA problem and then solving the type 2 LCA problem. A discussion about a new effect called zonal confusion is included that is a consequence of type 2 LCA correction.
A. Type 1 LCA
To understand the principle of correcting type 1 LCA, it is useful to examine a simplified optical system by considering only paraxial properties and axial color. The task is to correct LCA of an achromatic lens to make it apochromatic, where three wavelengths exhibit the same axial focus. It is not possible add lenses at the primary because of weight/mass considerations for space telescopes. The Schupmann configuration is also not desirable, because it results in spherochromatism and a virtual focal point with a low $f$-number that must be refocused onto the image plane. The selected method to correct the axial color is to use a relay lens at the focal point of the primary lens and then re-image the focal point onto the image plane, as suggested in [13]. As shown in Fig. 5, type 1 LCA at the primary-lens focal point ${f_1}$ exhibits an achromatic form, and the focal point difference between the central wavelength focus and shortest and longest wavelength foci is the secondary spectrum of the primary lens (${\rm SS_1}$), which is also the residual type 1 LCA of the primary lens. The CC is a finite-conjugate relay lens, where the object distance is a function of wavelength. Ideally, the output image location created by the relay does not vary as a function of wavelength. Therefore, the secondary spectrum of the CC (${{\rm SS}_2}$) should be opposite in sign compared to ${{\rm SS}_1}$. As shown in Fig. 5, ${{\rm SS}_1}$ is achromatic with total focal range of about 0.55 mm. To achieve apochromatic output at the image plane, ${{\rm SS}_2}$ must satisfy the relationship [13]
where ${f_1}$ and ${f_2}$ are focal lengths of the first and second lenses, respectively, and ${y_1}$ and ${y_2}$ are marginal ray heights at principal planes of the lenses.One important task of this design is reducing the size of the CC, which is the second lens in Fig. 5. A short color-corrector focal length ${f_2}$ is required to lower ${y_2}$, which requires large ${{\rm SS}_2}$ to satisfy Eq. (3), for example, if ${f_1} = {1}\;{\rm m}$, ${f_2} = {112}\;{\rm mm}$, ${y_1} = {120}\;{\rm mm}$, ${y_2} = {40}\;{\rm mm}$, and ${{\rm SS}_1} = {0.55}\;{\rm mm}$. ${{\rm SS}_2} = - {2.22}\;{\rm mm}$, which is extremely difficult to produce in a compact space with only refractive elements. To provide the large dispersion for ${{\rm SS}_2}$, a DFL is used in combination with refractive doublets. Reduction of the CC diameter is accomplished by redirecting the marginal ray through distributing optical power with doublets A and B. The solution shown in Fig. 6, which is the CC design without the AZTECC lens, is a first group of elements to collimate the light from the field lens (A and B) and second group of elements to refocus the beam (C and D). This system provides a collimated space between B and C, and the principal plane image of the primary lens is usually within or close to the collimated space. This collimated region provides room for placing diffractive optical elements, including the DFL for type 1 correction and the AZTECC lens for type 2 correction, as explained in the next section. Table 1 lists the CC optical design associated with Fig. 6.
Fig. 6. Color corrector for the design listed in Table 1, without the AZTECC lens. Extension of the marginal ray is shown to the principal plane to illustrate the difference between the effective radius ${y_2}$ in Eq. (3) and the and physical radii of the lenses.
Table 1. Optical Design A of the Color Corrector Without AZTECC Lens Detailsa
The design in Fig. 6 is nearly symmetric, with doublets A and D and doublets B and C having the same prescriptions, but doublets C and D are reversed in the converging cone. Spacings between the doublets are optimized for optical performance, as discussed in the next section. Simple spherical-surface doublets are used for the initial conceptual test, due to their well-known properties for ease of fabrication and control of spherical aberration and coma. Doublets A and D provide almost no optical power, but they are very dispersive. Doublets B and C provide the great majority of the optical power to collimate and then refocus the light cone. Since ${{\rm SS}_1}$ is very large and an even larger ${{\rm SS}_2}$ is required with a small-diameter CC, the amount of dispersion required from the positive doublets is also very large. To form the required achromatic behavior from the CC, it is very difficult to form both large positive and negative dispersions from only refractive elements. Addition of the DFL allows one-sided dispersion from the refractive doublets and large ${{\rm SS}_2}$. The effective ${y_2}$ for use in Eq. (3) is given by the projection of the marginal ray to the principal plane in the collimated space between B and C. Therefore, the physical lens diameter is smaller than the effective diameter used for color correction.
These concepts are displayed graphically with the dispersion curves in Fig. 7, where large refractive dispersion of the CC doublets is shown as a red dotted curve, and DFL dispersion is shown as a red dashed line. The combination of these two type 1 LCA functions results in the achromatic LCA form of the CC, which is shown as a solid red curve and compensates for the opposite LCA curve of the MODE primary lens that is shown as a blue solid line. The result is an apochromatic type 1 LCA design. Results from two CC designs, labeled as design A and design B, are illustrated with dashed green and solid green lines, respectively. These designs are discussed in more detail in Section 4. Note that all the type 1 LCA curves in Fig. 7 are evaluated at zone 1, while the power and dispersion of each MODE zone are slightly different. However, chromatic performance of the remaining four zones is similar to zone 1.
Fig. 7. Scaled type 1 LCA due to individual components of the color corrector (CC) and combinations of components. All LCA values are evaluated in zone 1 of the MODE primary and color corrector. Smaller LCA values are scaled up to ${10} \times$ or ${100} \times$ for presentation purposes.
Fig. 8. LCA plot of the MODE lens before color correction (blue lines) and after color correction (green lines) with the color corrector design A (CC A) that includes the AZTECC lens. The solid blue line is the center of the total LCA before correction, which corresponds to type 1 LCA, and dotted blue lines are boundaries of type 2 variation around the center. The color-corrected result shows more than 10 times less type 1 LCA and almost no type 2 LCA compared to before correction. This figure shows LCA including all zones of the system.
Although there are some similarities with the Schupmann design, such as the existence of a field lens, the configuration shown in Fig. 6 is distinctly different. Where the Schupmann configuration uses a negative-power element with the same dispersion as the primary, the new design uses a combination of refractive and diffractive components to affect a secondary spectrum that cancels with the secondary spectrum of the primary.
As the focal length of the primary MODE lens increases, ${{\rm SS}_1}$ also increases. As described by [14], ${{\rm SS}_1}$ due to a lens made from BK7-like glass, such as L-BSL7, with an Abbe number of 64.1 combined with a long focal length HOE/DFL on one surface is about 0.6% of the focal length over the visible spectrum. Unlike the high-power HOEs discussed in Section 2, the low-power DFL used in our system has a relatively low spatial frequency, which enables practical fabrication with our current technology [15]. In the case of CC design A, the minimum period of the DFL is 80 µm. ${{\rm SS}_1}$ for the design example over the astronomical R-band is about 0.55% of the focal length, or ${-}{550}\;\unicode{x00B5}{\rm m}$. If the focal length of the MODE primary lens increases by a factor of 10, ${{\rm SS}_1}$ increases to about 6 mm. A less dispersive low-temperature glass used for the MODE primary, such as N-PK51 with an Abbe number of 76.98, will produce smaller ${{\rm SS}_1}$. For example, based on the expressions in [14], an N-PK51 glass MODE primary with a 1 m focal length would produce ${{\rm SS}_1} = - {478}\;{\unicode{x00B5}{\rm m}}$, which is a reduction in magnitude of 72 µm from the current L-BSL7 design.
As ${{\rm SS}_1}$ increases, the CC must be modified to provide for high-quality imaging. Although ${f_2}$ could be increased, it is not a desirable option, due to the increased diameter of the CC. Instead, both the refractive dispersion of the CC and the DFL dispersion must increase. At some point, the divergence of the DFL will force the solution to use asymmetric doublet designs on both sides of the DFL, with the addition of more low-optical-power doublets to increase the refractive dispersion and aspherics on the higher-power surfaces. In addition, as the diameter of the MODE primary increases, the marginal ray height ${y_2}$ must also increase for a fixed ${f_2}$. To keep the same corrector lens diameter, ${f_2}$ must decrease, again forcing an increase in ${{\rm SS}_2}$. Also, as the $f$-number of the primary lens increases, the $f$-number of the CC must also increase.
B. Type 2 LCA
Type 2 LCA is due to the abrupt $mh$ change in surface height on the MOD surface, as shown in Fig. 1, which produces a discrete jump in OPD at the transition points [5–7]. This change results in a step profile of OPD versus radius for any wavelength, in addition to any focus or aberrational functional dependence. The step profile produces a variation in focus of the MODE primary lens, as shown in Fig. 8, which is a cyclic variation in focus with period $\Delta f = {f_0}/M$ around the design wavelength of 658 nm. Dashed lines indicate the boundary of the type 2 $\Delta f$ variation, and the solid blue line is ${{\rm SS}_1}$ of type 1 LCA. The solid green line is the total LCA after correction with the design A CC and AZTECC lens. There are no dashed-line boundaries associated with the solid green line, because the residual type 2 LCA after full correction is less than a few micrometers.
Fig. 9. (a) Generation and compensation for abrupt OPD changes between MODE zones. Optical elements between the MODE primary lens and AZTECC type 2 LCA corrector, as well as elements after AZTECC are not presented for the sake of simplicity. (b) CAD model of AZTECC element used in the MODE system.
The solution for type 2 LCA used to calculate Fig. 8 is to compensate for OPD jumps between zones with flat plates in the collimated section of the CC, which are close to the image of the primary lens. For example, the zonal transition between the first and second zones results in an abrupt change of OPD by $M$ waves. In the CC, a type 2 LCA corrector plate in the collimated space is designed with zone 1 thicker than zone 2, so the light transmitting through zone 1 obtains additional $M$ wavelengths of OPD. The idea of using powerless stepped plates has been applied for reducing the OPD caused by thermal effects, but not for reducing type 2 LCA [16]. In the example MODE primary lens design, $M = {2196}$ corresponds to an L-BSL7 glass thickness change at each zone boundary of $Mh = {2.812}\;{\rm mm}$ at the 658 nm wavelength. Because the AZTECC lens uses the same glass as the primary, OPDs of all wavelengths are compensated for, which solves the type 2 LCA problem. As shown in Fig. 9, OPD between MODE zones is caused by both front and back surfaces in the design example, where transition heights between the two zones are ${t_1}$ and ${t_2}$, respectively. MODE transitions of both sides are designed for eliminating zonal field shift, which is a special type of aberration of zonal lenses, and it results in the outer zones shifting toward the focus direction, as described in [10]. OPDs between each zone for all wavelengths are
Since the AZTECC uses the same type of glass as the MODE primary lens, the OPD is compensated for all wavelengths with
A limitation to the AZTECC type 2 LCA corrector is due to the field angle in object space. In the design example, doublets A and B collimate finite-conjugate fields focused near the field lens. The collimated on-axis field is parallel to the optical axis.
Fig. 10. Formation of zonal confusion. This illustration is simplified to show zonal confusion for only one field angle of one zone. On-axis fields intercept the AZTECC perpendicularly, which results in correct OPD compensation for all zones (black). Off-axis fields have oblique incidence to the AZTECC and may enter undesirable zones, which results in undesired OPD values (blue).
4. DESIGN EXAMPLE
This section presents both raytracing and physical optics modeling of the MODE telescope system with the design A CC. The MODE primary lens design given in Code 1, Ref. [11], is used for the analysis in all cases. This MODE primary lens design is a spherical-front shape, as described in [10], which results in low off-axis aberration.
During Zemax optimization, the color-corrector DFL is modeled using an aspheric Sweatt model [17], as shown in Table 1, which calculates the OPD as a function of aperture. In the range of optical conjugates used here, the Sweatt model results in an adequate model of the DFL [18]. Detailed design files illustrating our Sweatt model are included in Code 1, Ref. [11]. When the condition of Eq. (3) is satisfied, an apochromatic system is produced, as shown in the CC design B green curve in Fig. 7, which illustrates total type 1 LCA. Compared to type 1 LCA of the MODE primary, the total LCA is a factor of ${560.8/11.2}\;\sim{50} \times$ smaller. Type 2 LCA is assumed to be compensated for by the AZTECC lens. However, a slightly different solution with different spacings of the doublets, CC design A, was found by allowing slightly more type 1 LCA and produced a slightly better root-mean-square (rms) spot size. The overall geometrical performance of the system includes not only chromatic aberration, but also monochromatic aberrations such as astigmatism and coma, which are characteristic for off-axis field angles. The resulting spot diagram of the MODE system with CC design A is shown in Fig. 11. The uncorrected system spot diagrams in Figs. 11(a) and 11(b) are nearly ${15} \times$ the Airy spot diameter at 658 nm of 6.7 µm. There is an insignificant effect due to a non-zero field angle in Fig. 11(b). The corrected system spot diagrams shown in Figs. 11(c) and 11(d) display ray intercepts well within the Airy spot diameter for most rays, with a slight increase in diameter at the full-field angle of 0.125°. Figures 11(c) and 11(d) indicate that the corrected system is diffraction limited in the geometrical sense.
Fig. 11. Spot diagrams of the MODE telescope with and without color corrector design A at three wavelengths in the astronomical R-band: (a) MODE primary lens only, on axis; (b) MODE primary lens only, full field; (c) MODE primary with color corrector, on axis; and (d) MODE primary with color corrector, full field. (a) and (b) show near-diffraction-limited spot diagrams.
Design with raytracing in Zemax results in a color-corrected system with the consideration of refractive aberrations including coma, spherical aberration, astigmatism, and type 1 LCA, but not diffractive properties. A full model of the MODE system requires diffractive analysis that includes calculation of Fresnel propagation of field amplitudes using Hankel transforms from optical path lengths (OPLs) across the exit pupil, as described in [10]. OPLs of multiple wavelengths are calculated by raytracing, which is performed by a customized macro in Code V raytracing software after converting the Zemax sequential model to a Code V non-sequential model. Then, the optical phase in the exit pupil is calculated with 1000 sample points across the pupil radius, and Fresnel propagation using a Hankel transform is used to calculate field amplitude of the point-spread function (PSF) versus focal-plane distance and radius for each wavelength. PSF irradiance is calculated by the magnitude squared of the field amplitude. PSF profiles, encircled energy, Strehl ratio, and modulation transfer function (MTF) properties are calculated from the polychromatic PSF data. 10,000 sample points in the radial direction at each focal plane location are used to obtain sufficient range and resolution for the MTF.
It is sufficient to test five uniformly distributed wavelengths across the astronomical R-band for the slowly varying type 1 LCA, which are 589 nm, 623 nm, 658 nm, 692 nm, and 727 nm. Around each test wavelength, a small band of finely sampled wavelengths is used to evaluate type 2 LCA. Since $\Delta \lambda \;\sim{0.3}\;{\rm nm}$ in the design example, the PSF is evaluated in a 0.3 nm range near each test wavelength with a wavelength increment of 0.05 nm. This wavelength sampling produces polychromatic PSF data from a total of 35 wavelengths.
Polychromatic PSFs across the astronomical R-band as calculated with the physical optics model are shown in Fig. 12(a), and the corresponding encircled energies are shown in Fig. 12(b). The encircled energy is calculated by integrating the PSFs as a function of radius. In Fig. 12(a), spot irradiance profiles are shown normalized to the peak of the ideal polychromatic PSF with no aberrations, including type 1 and type 2 LCAs. The linear irradiance scale shows an indistinguishable difference between the polychromatic PSFs of the ideal system and the color-corrected MODE system, with a Strehl ratio ${\sim}{0.99}$. The MODE primary without any color correction (MODE) exhibits a Strehl ratio of about 0.12, and the system with type 1 correction [MODE w/CCA (no AZTECC)] exhibits a Strehl ratio of only about 0.2. As with the PSF profiles, the encircled energy plot in Fig. 12(b) is nearly perfect for the corrected system. Figure 12(c) displays polychromatic MTFs of the MODE telescope before and after correction, with the corrected system displaying diffraction-limited performance.
Fig. 12. (a) PSFs of the MODE telescope system before correction, after correcting only type 1 LCA, after correcting both types of LCA with CC design A, and the diffraction limit ideal case. (b) Corresponding encircled energy and (c) modulation transfer functions (MTFs) based on the PSFs in (a).
5. DISCUSSION
Development of color correcting systems for high-harmonic diffractive lenses is motivated by a recently introduced concept for constructing an array of large-aperture space telescopes for exoplanet transit studies [1]. The long-range goal is to realize 8.5-m-diameter apertures on each telescope in the array. The results presented here provide a color-correction methodology for optical systems with harmonic diffractive lenses, and a preliminary design for further study is listed. Prototypes of both the CC and the corresponding primary MODE lens are currently being fabricated and tested to verify the theories presented here. Results on tolerancing, fabrication, alignment, and imaging will be discussed in future works. Currently, preliminary tolerancing analysis shows that the impact of manufacturing errors can be easily compensated for by air gaps, while monochromatic performance is sensitive to alignment errors.
The design sequence documented in this paper is to first design a sequential-surface model in Zemax using a multiconfiguration editor, then transfer to a non-sequential Code V model, and finally use a MATLAB model for physical optics analysis. This sequence was a consequence of how understanding of MODE lenses and color correction matured in our research group. Going forward, advanced features of modern lens design software such as Zemax and Code V could be used to shortcut the design sequence. In addition, the aspheric Sweatt model used for the back surface of the MODE primary and the DFL in the CC could be replaced with more advanced surface models.
The mass of the CC reported here is too large for the space telescope application if the system is scaled up for a MODE primary that is 8.5-m-diameter. However, lightweight CC design can be achieved while maintaining near-diffraction-limited performance by using aspherical surfaces on the doublets, as well as using a DFL with more optical power. Design experiments show that the mass of the CC can be reduced at least by a factor of four in comparison with the current design, making it much lighter. We have found that the CC diameter does not scale directly with the diameter of the primary MODE lens. In fact, the diameter of lightweight designs is typically a factor of two or more smaller than the current design. In addition, color correction methods described in Section 3 are robust over broader spectra than the astronomical R-band, so the color corrected wavelength range could be expanded with minor adjustments to the optical system.
6. CONCLUSION
In conclusion, a new type of color correcting method is described for reducing unique dispersion characteristics of high-harmonic diffractive lenses, which are a combination of classical (type 1) and MOD (type 2) dispersion. Design examples with practical considerations are presented with both raytracing and physical simulation models. The resulting design successfully reduces dispersion from an existing harmonic diffractive lens design and results in diffraction-limited performance over the astronomical R-band up to 0.25° full field of view.
Funding
Gordon and Betty Moore Foundation (7728).
Acknowledgment
We thank D. Apai of Steward Observatory for motivation to do this work and useful comments.
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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