Abstract

A multiple-order diffractive engineered surface (MODE) lens is introduced, in which focal position change with wavelength exhibits both refractive and diffractive characteristics. Engineering calculations are provided that indicate Strehl ratio and encircled energy performance over a large range of focal length and aperture diameter design space. A prototype lens is designed and constructed for the astronomical R-band (589 nm to 727 nm) wavelength range. Test results show that measured full-width-at-half-maximum focal spot diameter is 2.1 times larger than the ideal Airy spot diameter, and focal position versus wavelength is nearly identical to the design. The 48 mm aperture diameter, $f/{3.12}$ prototype telescope exhibits angularly resolved features in natural scenes at 0.006°, with subtense of the Airy spot diameter at 0.002°. Applications include eventual use in large aperture, ultralightweight space telescopes.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. INTRODUCTION

The term Fresnel lens is commonly thought of in several ways. Lighthouse lenses are the first historical example of Fresnel lenses designed for making refractive lenses lighter by sectioning lenses in radial zones and removing unnecessary lens material. Lighthouse lenses are useful for illumination, but they are not designed with high-quality imaging in mind. Similarly, there are large Fresnel-type lenses used for illumination in projector systems made from molded plastic. Like lighthouse lenses, these optical components are not designed for high-quality imaging. A third type of Fresnel lens is the diffractive Fresnel lens (DFL), which is fundamentally based on diffraction from a Fresnel zone plate. DFLs, unlike their lighthouse-lens cousins, are typically very small and have extreme chromatic focal variation. However, for a narrow band of wavelengths, like from a laser, the DFL can produce good quality imaging. DFLs are attractive, because they can be made very thin and lightweight. This paper discusses a new type of lightweight lens that initially may appear as a lighthouse lens. However, the design is a hybrid refractive/diffractive lens on one side and a diffractive structure on the second side. Motivation for development of the multiple-order diffractive engineered surface (MODE) lens is imaging for large-aperture space telescopes, where launch mass and alignment tolerances are of primary importance. This paper describes engineering principles of MODE lenses and discusses results from a simple prototype.

Use of DFLs in space applications is not new, and the motivation to produce Gossamer-class (${\gt}{20}\;{\rm m}$ diameter) space telescopes for exoplanet exploration is an inspiring goal [1]. Research has been reported for two large transmissive ultralight space telescopes based on DFLs. The Eyeglass telescope consists of a deployable 25 m to 100 m diameter, 1 km focal length DFL primary coupled with a complex separated eyepiece containing a relay system and a negative power diffractive Fresnel lens for chromatic correction [2]. The MOIRE telescope demonstrated a similar design with a smaller $f$-number using a segmented membrane diffractive optic [3]. Both Eyeglass and MOIRE telescope primary lenses are based on single-harmonic DFLs, in which the lens microstructure is designed to provide a maximum of one wavelength of optical path difference in transmission over the surface. These ultralight systems enjoy important system-level advantages over mirror-based systems [4]. However, the primary lenses are extremely wavelength dispersive, where the axial focus point changes with wavelength according to

$$f\!\left(\lambda \right) = {f_0}\frac{{{\lambda _0}}}{\lambda},$$
where ${\lambda _0}$ is the design wavelength and $\lambda $ is the operating wavelength. Equation (1) can be roughly interpreted in terms of an effective Abbe number ${\sim} -\! {3}$ over the visible wavelength region. Correction for imaging over a wavelength band greater than 100 nm in the visible (${\sim} {20}\%$ spectral bandwidth) is challenging due to high focal dispersion. For practical implementation of an ultralight transmissive DFL primary lens, the focal change with wavelength must be dramatically reduced.

Hybrid refractive–diffractive lenses offer substantially reduced focal change compared to single-harmonic DFLs, where the lens element is composed of structures larger than those of single-harmonic devices to modify the transmitted wavefront. These devices exhibit characteristics of both diffractive and refractive focal dispersion. Diffractive properties of a small-aperture-diameter, planar multiple-order diffractive (MOD) singlet lens are described in [5,6]. MODE lenses are comprised of a front-surface MOD structure and a rear-surface diffractive structure, like a DFL, as shown in Fig. 1. The longitudinal chromatic aberration (LCA) is the change in focus over the operating wavelengths, and these lenses have both a refractive component and a diffractive component of LCA.

 

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 ($M \sim {1000}$) produces a small value for the diffractive component of the longitudinal chromatic aberration (LCA). The DFL reduces the refractive part of the total focal dispersion, making each MOD zone achromatic.

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This paper is divided into several sections. In Section 2, background is presented for MOD-type lenses with respect to their chromatic focal behavior. In Section 3, simulation results are presented for a wide range of focal length and aperture diameter design space. In Section 4, a prototype MODE lens is analyzed for spot and image quality. Sections 5 and 6 present a short discussion of space telescopes and the conclusions, respectively.

2. BACKGROUND

For a MOD lens, the surface is sculpted in the shape shown on the front surface of the element in Fig. 1, such that each circularly symmetric zone acts like a separate lens element directed to a common design focus ${f_0}$. The Fresnel-lens-like transitions are meticulously designed, such that an integer $M$ waves of optical path difference in transmission is at each transition for the design wavelength ${\lambda _0}$. Under these conditions, the ideal MOD surface has the interesting diffractive property that wavelengths given by

$${\lambda _p} = M{\lambda _0}/p,$$
where $p$ is an integer representing the focal order, focus to the design focal length ${f_0}$ with diffraction-limited performance and 100% diffraction efficiency. At the design wavelength, $p = M$. Although [5] only discusses $M \sim {10}$, much larger values are practical using commercially available precision diamond-turning technology for fabrication. For example, $M \sim {1000}$ requires a transition height of only $Mh = {1000}{\lambda _0}/(n - {1}) \sim {1}\;{\rm mm}$ for visible wavelengths in common optical glasses and plastics with refractive index $n$ and $h$ is the glass thickness of index $n$ that produces one wave of optical path difference in transmission. The additional weight burden caused by an increased substrate thickness compared to a membrane optic, as in the MOIRE device [3], is not prohibitive, given the advantages of a stable optical surface compared to a membrane optic. In fact, the 5 m diameter aperture Eyeglass prototype used a glass substrate thickness of 0.75 mm [7].

A primary difference between an $M = {1}$ DFL and a MOD surface with higher $M$ is that multiple focal orders occur with different diffraction efficiencies around the design focal length ${f_0}$ and focal orders given by Eq. (2). As described by [5], the diffraction efficiency follows a narrow ${({\sin}(x)/x)^{2}}$ distribution as a function of $x \propto {1/}\lambda$, where no more than two focal orders have significant diffraction efficiency for any wavelength. Irradiance in the focal region as a function of wavelength appears as sketched in Fig. 2, where, for large $M$ and $\lambda$ close to ${\lambda _p}$, the focal spot from order $p$ moves toward the lens as the wavelength increases according to Eq. (1). However, the axial range of the movement is limited to $\Delta\! f \sim {f_0}/M$, due to the decrease in diffraction efficiency away from the co-focal wavelengths defined by Eq. (2). Near the end of the wavelength cycle range defined by $\Delta \lambda \sim{\lambda _0}/M$, irradiance of the $M$th order is approximately equal to the $M - {1}$ order, which now enters the axial focal range on the side away from the lens. Peak diffraction efficiency occurs in the center of the diffractive focal range. The wavelength cycle range increases slightly for wavelengths longer than ${\lambda _0}$ and decreases slightly for shorter wavelengths. For example, at $M = {1000}$, a MOD surface with ${\lambda _0} = {658}\;{\rm nm}$ and ${f_0} = {150}\;{\rm mm}$ exhibits a diffractive wavelength cycle range of $\Delta \lambda \sim{0.66}\;{\rm nm}$ and diffractive focal range of $\Delta\! f \sim {0.15}\;{\rm mm}$.

 

Fig. 2. Cyclic behavior of MOD focal orders along a horizontal optical axis due to diffraction. With high $M$, the change of focus position with wavelength occurs over diffractive focal range $\Delta\! f \sim {f_0}/M$, where ${f_0}$ is the design focus position. Starting with the design wavelength ${\lambda _0}$ at focal order $p = M$ (top) maximum diffraction efficiency is achieved at ${f_0}$. As the wavelength increases, this focal order moves toward the lens, and diffraction efficiency (gray shading–middle) reduces. Toward the end of the diffractive cycle (bottom), focal order $p = M - {1}$ exhibits nearly equal diffraction efficiency to order $M$, and efficiency of the $M - {1}$ order increases as it moves toward ${f_0}$ as the wavelength increases. The diffractive focal range $\Delta\! f$ is the diffractive component of the secondary spectrum $SS$. (See Visualization 1, which is a colormap irradiance profile in the focal zone of a ${f_0} = {1}\;{\rm m}$, $M = {1000}$, $f/{4.17}$ mode lens designed for the R-band and displayed for individual wavelengths from 648 nm to 668 nm.)

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The current generation of MODE lenses includes a long-focal-length DFL on the back surface and a high-$M$-number ($M = {1000}$) MOD lens element front surface, as shown in Fig. 1, which forms a refractive achromat for each zone and thus limits the refractive component of LCA due to refractive index dispersion [8]. Although combination of a DFL and a refractive surface is known to affect an achromat [9], the combination of a high-$M$-number MOD surface and a DFL surface has not been shown experimentally. The combination of a low-$M$-number ($M \sim {21}$) MOD surface and a DFL was suggested in [6], although neither design data nor experimental results were reported for the combination. Although it may be possible to use a low-$M$-number MOD lens for chromatic correction on the back surface [10], only a $M = {1}$ DFL is considered here. A more general phase-altering engineered surface structure can also be used to perform specific functions, other than simple focusing. For example, engineered diffractive structures used to provide more complete chromatic control [11] could be designed in combination with the MOD surface. Also, meta-lens structures could be used to provide polarization functionality of the device [12]. Therefore, the combination of the MOD surface and the engineered back surface is called a MODE lens.

3. SIMULATION

Several aspects of the MODE-lens focus cone due to diffractive focal dispersion are approximated through simulation. One important metric is the Strehl ratio (SR), which is the ratio of the peak on-axis focus irradiance to the ideal peak value. Another metric is encircled energy (EE), which is the percentage of energy within a specified diameter divided by the total focused power. This section discusses estimation of the Strehl ratio, focal spot distribution and encircled energy with a scalar Fresnel-zone model. In these simulations, the calculations are made over one $\Delta \lambda$ cycle of the diffractive dispersion around the design wavelength ${\lambda _0}$. Refractive dispersion is also an important factor, as discussed in the next section, but it is not considered here.

The calculation geometry is shown in Fig. 3, where the lens is focal length ${f_0}$ away from the observation plane at the design wavelength ${\lambda _0}$. As the wavelength increases, the focal spot moves toward the lens distance $\delta\! f$, according to the diffractive focal dispersion of Eq. (1). Distance from the shifted focal point to the lens is ${z_{{\rm src}}}$. The out-of-focus distribution at the observation plane is modeled as a simple Fresnel ring problem, where the Fresnel number and the marginal ray determine the observed irradiance pattern.

 

Fig. 3. Calculation geometry for estimation of SR and focus spot characteristics.

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For the SR calculation in the diffractive cycle around ${\lambda _0}$, on-axis irradiance given by

$${I_{{\rm axis}}}\left({\delta\! f} \right) \propto {I_\infty}\left({\delta\! f} \right)\!{\sin ^2}\!\left({\frac{{\pi\! {N_{\!f}}}}{2}} \right)$$
is integrated over the bandwidth $\Delta \lambda$. In Eq (3),
$$\delta\! f = - {f_0}\frac{{\delta \lambda}}{{{\lambda _0}}},$$

$\delta \lambda = \lambda - {\lambda _0}$, ${N_f}$ is the Fresnel number given by

$${N_f} = - \frac{{{D^2} \delta \lambda}}{{4{f_0} \lambda _0^2}},$$
and ${I_\infty}$ is the on-axis irradiance without a lens aperture that is proportional to ${({{\lambda _0}/{f_0}\delta \lambda})^2}$. Combining the above terms yields on-axis irradiance
$${I_{{\rm axis}}}\left({\delta \lambda} \right) \propto \frac{{{{\sin}^2}\left({\alpha \delta \lambda} \right)}}{{{{\left({\alpha \delta \lambda} \right)}^2}}} ,$$
where
$$\alpha = \frac{{\pi\! D}}{{8f\# \lambda _0^2}}$$
and $f\# = {f_0}/D$.

Integration of Eq. (5) leads to

$$\begin{split}{\rm SR}& = \frac{1}{{\Delta \lambda}}\int_a^b {\frac{{{{\sin}^2}\left({\alpha \delta \lambda} \right)}}{{{{\left({\alpha \delta \lambda} \right)}^2}}}{\rm d}\left({\delta \lambda} \right)}\\ &= \frac{1}{{\alpha \Delta \lambda}}\left[{{\rm Si}\left({2\alpha \delta \lambda} \right) - \frac{1}{{2\alpha \delta \lambda}} + \frac{{\cos \left({2\alpha \delta \lambda} \right)}}{{2\alpha \delta \lambda}}} \right]_a^b,\end{split}$$
where limits of integration are $a = - {\lambda _0}/{2}M$ and $b = {\lambda _0}/{2}M$. The sinint function (Si) is defined as
$${\rm Si}\!\left(x \right) = \int_0^x {\frac{{\sin \left(t \right)}}{t}{\rm d}t} .$$

Note that Eq. (5) is not the same as the ${({\sin}(x)/x)^\wedge{2}}$ diffraction efficiency described in [5], because Eq. (5) describes on-axis irradiance at fixed position ${f_0}$.

Calculation results for a limited range of $D$ and ${f_0}$ are shown in Fig. 4 (left), with ${\lambda _0} = {658}\;{\rm nm}$ and $M = {1000}$. Higher SR is achieved with larger $f\#$ and longer focal length, where $f\#$ is displayed as light dashed lines. Figure 4 (right) displays the number of MOD radial zones required over the same range of parameters. The gray dot is the design point for the prototype described in Section 4, for which there are three MOD zones, ${f_0} = {150}\;{\rm mm}$ and $D = {48}\;{\rm mm}$. ${\rm SR} \sim\! {0.3}$ for the prototype, according to this calculation.

 

Fig. 4. Left: contours of calculated Strehl ratio (SR) as dark lines for ${\lambda _0} = {658}\;{\rm nm}$ and $M = {1000}$ as a function of aperture diameter $D$ and focal length ${f_0}$. The gray dot indicates the design point of the MODE lens prototype described in Section 4. Right: number of MOD radial zones required. Both: five values of $f\#$ are shown as thin dashed lines.

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The focal spot distribution is calculated using a Hankel transform with a uniform amplitude distribution and phase factor:

$${U_s}\left({{\rho _s}}\! \right) = \exp\! \left({j\pi\! {N_f}\rho _s^2} \right),$$
where ${\rho _s}$ is the lens radial coordinate normalized by its maximum radius. Irradiance is added for each of 100 wavelengths over $\Delta \lambda$ with ${\lambda _0} = {658}\;{\rm nm}$. The 90% encircled energy diameter ${d_{90}}$ is divided by the Airy spot diameter ${d_{{\rm Airy}}} = {2.44}\lambda\! f\#$ to produce data for Fig. 5 (left), which indicates that the encircled energy at the prototype design point (gray dot) is approximately 4 times the Airy spot diameter due to diffractive dispersion. Figure 5 (right) shows the ratio of the diffractive focal range $\Delta\! f$ versus the classical depth of focus ${z_{{\rm DOF}}} = {2}\lambda\! f{\# ^2}$. At the prototype design point, $\Delta\! f \sim {12}{z_{{\rm DOF}}}$.
 

Fig. 5. Left: contours of constant 90% encircled energy diameter ${d_{90}}$ divided by the Airy spot diameter ${d_{{\rm Airy}}}$ with $M = {1000}$ and ${\lambda _0} = {658}\;{\rm nm}$ as a function of aperture diameter $D$ and focal length ${f_0}$. Right: contours of diffractive focal range $\Delta\! f$ divided by classical depth of focus ${z_{{\rm DOF}}}$ versus $D$ and ${f_0}$. Both: five values of $f\#$ are shown as thin dashed lines.

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Tables Icon

Table 1. Design Parameters of the MODE Prototypea

4. PROTOTYPE

The prototype is a 48 mm diameter aperture, $f/{3.12}$ three-zone MODE lens with a 150 mm focal length and is designed for evaluation with an object at infinity, as shown in Table 1, where the DFL is designed with a Sweatt surface [13] that extends over all MOD zones. The prototype is constructed in two parts. First, the MOD surface with $M = {1000}$ ($Mh = {1.338}\;{\rm mm}$) is made with single-point diamond turning from 8 mm thick bulk PMMA. Second, a DFL mold is made in RSA 905 aluminum [14] and is coated with Super Hydroseal [15] as a release agent. The DFL is then replicated onto a simple, 4 mm thick BK7 window with UV adhesive (Norland, NOA61), and the mold is released. Finally, the two parts are cemented together with NOA61 for a total physical thickness of 12 mm.

The MODE lens is mounted in a lens tube with a CMOS camera (Thorlabs, DCC1545M, 5.2 µm pixels) at the focus plane, as shown in Fig. 6, for imaging experiments. Due to the size of the CMOS detector, the full angular range of the lens and camera is 2.54° by 2.03°, with a pixel separation of 0.002°. The lens tube also contains a filter that transmits only R-band wavelengths.

 

Fig. 6. MODE lens layout (left) includes a plastic MOD lens with thickness ${t_a} = {8}\;{\rm mm}$ and a ${t_b} = {4}\;{\rm mm}$ glass plate with a DFL on the back surface. The MODE lens is mounted in a lens tube with a CMOS camera at the focal plane (right) for imaging experiments.

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Fig. 7. (a)–(c) Narrowband polychromatic focal spot measurements at $\lambda = {658}\;{\rm nm}$ with a bandwidth of about 2 nm. (a) Peak-normalized 2D spot irradiance distribution with a black circle representing the Airy spot diameter; (b) profiles of the spot irradiance and comparison with the ideal Airy irradiance and a commercial achromatic doublet with about the same optical parameters; (c) encircled energy (EE) as a function of diameter; and (d) R-band filtered image of the moon (cropped).

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Experimental evaluation of the MODE prototype is accomplished with a star test, measurement of focal dispersion versus wavelength, and imaging various scenes. The star test setup consists of a fiber-coupled supercontinuum laser (NKT Photonics, SC-400) with a tunable high-contrast single-line filter (NKT Photonics, LLTF Contrast). The filter output has an optical bandwidth of about 2 nm and is tuned over the R-band of wavelengths (589 nm to 727 nm) for which the prototype is designed. The filter output is coupled into a single-mode optical fiber, the output of which is placed at the focus of a large off-axis parabola. This light source is effectively an incoherent point source, with each wavelength in the transmitted spectrum independently forming an irradiance pattern at the camera. The parabola is aligned with an alignment telescope and a shear plate to provide a collimated on-axis beam into the MODE lens, which is removed from the R-band filter and CMOS camera assembly. Instead of the CMOS camera, a high-quality CCD monochrome camera (Andor iKon-934, 13 µm pixels) is used with a ${10} \times$ objective lens microscope tube to measure the spot image at focus.

The measured polychromatic focus spot is shown Fig. 7(a) as a normalized irradiance colormap, where the Airy spot diameter of 5.02 µm at ${\lambda _0} = {658}\;{\rm nm}$ is shown as a black ring. Since the bandwidth of the LLTF filter used with the source is about 2 nm, the measured spot irradiance shown in Fig. 7(a) is an integration of the focal orders over several $\Delta \lambda \sim {0.66}\;{\rm nm}$ diffractive wavelength cycle ranges. Figure 7(b) displays peak-normalized measured line profiles, comparison to profiles from the Fresnel calculation described in Section 3, the calculated ideal Airy spot distribution, and the point-spread function calculation of a commercial achromatic doublet (Edmund Optics 32886). The calculated curves show narrow central peaks with approximately the same widths. A slight error in the edge of the DFL occurred due to releasing the DFL from the master mold, which results in additional wavefront aberration on one side of the DFL and causes the flare shown in the measured $y$-direction spot profile and the larger spot width as compared to the calculated profile. Otherwise, the central peak of the measured focus spot is much smaller than the blur circle $B = D/M = {48}\;{\unicode{x00B5}{\rm m}}$ due to marginal rays of the focus cone over $\Delta \lambda$. This difference is due to the nature of Fresnel zone patterns comprising the focus cone for individual wavelengths and the change in diffraction efficiency of the focal orders. Note that the commercial doublet calculation displays a narrow central peak but higher sidelobe energy due to spherical aberration than the MODE calculated profile. The encircled energy shown in Fig. 7(c) indicates that measured EE corresponds well to the calculated value using the techniques of Section 3, and near 100% EE is obtained near the simple value of $B = {48}\;{\unicode{x00B5}{\rm m}}$. Figure 7(d) shows a photo of the moon through the prototype assembly displayed in Fig. 6.

Numerical values for the polychromatic measurement compared to ideal Airy values are shown in Table 2. The calculated spot profile in Fig. 7(b) indicates that the full width at half-maximum (FWHM) of the calculation is nearly the same as for the Airy spot. The average FWHM diameter for the measured spot is 2.1 times bigger than the ideal Airy spot, which is due to a small fabrication error in the plastic MOD surface. The ratio ${d_{90}}/{d_{{\rm Airy}}} \sim {3.6}$, which is nearly the same as that predicted by Fig. 5 (left).

Tables Icon

Table 2. Measured MODE, Calculated MODE, and Ideal Airy Values

Measured, simulated, and design focal dispersion versus wavelength are shown in Fig. 8. Measured LCA of the MOD surface by itself, before the DFL is bonded to the lens, is shown as the dotted curve (MOD), with over 1 mm of focal dispersion. The refractive design curve for the achromat of the central MODE zone is shown as the solid red curve (Design), which displays a refractive focal dispersion of about 0.2 mm. The combined diffractive and refractive LCA chromatic focal shift found from a simulation based on data from a Code V ray-trace design program is shown as a light blue line (Simulated). The wavelength cycle range of 0.66 nm produces a diffractive focal range ${\sim} {0.15}\;{\rm mm}$, which is centered around the design curve. Measurement of the MODE change in focal position is shown as the heavy black line, which follows the center of the design curve and extends beyond the R-band.

 

Fig. 8. Measured, simulated, and design change in focal length versus wavelength. MOD: measurement without the DFL back surface; MODE: measurement with the complete MODE lens; Simulated: diffraction simulation using data from a Code V ray-trace program; and Design: design curve for the achromat of the central MODE zone.

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Two unprocessed scenes are shown in Fig. 9. Figure 9 (left) shows the image of a tri-bar resolution target displayed on a high-resolution monitor and viewed at an object distance of 6 m. The circle represents the designed 1° full field of view. With ${f_0} = {150}\;{\rm mm}$ and Airy ${\rm diameter} = {5.02}\;{\unicode{x00B5}{\rm m}}$, the angular Airy diameter is 0.002°. The inset shows a magnified portion of the image, where the Group 0, Element 4 bar spacing is 0.006°. Figure 9 (right) shows a full-field image of a softball field fence and nearby building that is 185.8 m from the camera. A magnified section of a wire mesh is shown as an inset, where the period of the mesh is visible at an angular subtense of 0.006°. By the sampling theorem of Fourier optics, the best sampled resolution is 0.004° due to pixel spacing. Even though ${d_{90}}$ in Table 2 indicates an angular resolution of ${\sim} {0.01}^\circ$, the actual resolution is about a factor of 2 better, with pixel sampling being a non-negligible factor.

 

Fig. 9. R-band images of different scenes. Left: image of the central portion of a tri-bar resolution target displayed a high-resolution monitor viewed at 6 m. The circle represents the 1° designed field of view. The inset shows a magnified portion of the image, where the Group 0, Element 4 bar spacing is 0.006°. Right: image of a softball field in daylight over the full field of the camera (2.54° by 2.03°), where the screen mesh shown in the inset is resolvable at an angular separation of 0.006°.

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5. DISCUSSION

Development of MODE lenses is motivated by a recently introduced concept for constructing an array of large-aperture space telescopes for exoplanet transit studies [16,17]. The long-range goal is 8.5 m diameter apertures on each telescope in the array. The use of a MODE lens as the primary optical element offers an attractive alternative to mirror-based telescopes, with an order of magnitude launch-mass reduction in the primary component of the optical system and up to 2 orders of magnitude reduction in alignment sensitivity. The results presented here provide a foundation for further study. Certainly, the minimum thickness of the MOD front surface and DFL back surface can be much less than the prototype thickness of 12 mm for a 1. 338 mm transition depth at $M = {1000}$. We are now applying glass molding technology to form both surfaces in the same material in one step without the need to bond substrates. For example, the practical thickness of a MODE lens in the prototype configuration described in this work might be 2 mm in a low-temperature glass, like L-BSL7. In that case, the mechanical aspect ratio of a lens similar to the prototype described here is 24:1. Design details of improved molded-glass MODE lenses will be reported soon in separate publications, including improving performance parameters of MODE lenses and how to reduce diffractive blur for larger telescope diameters.

6. CONCLUSIONS

In conclusion, a new type of hybrid lens, which is called a MODE lens, is discussed that exhibits properties of both refractive and diffractive focal shift as a function of wavelength. A prototype plastic lens is designed over the astronomical R-band wavelength range (589 nm to 727 nm) and is fabricated and tested. Results indicate that it produces a FWHM spot size about twice the Airy spot size and produces clear images of natural and artificial scenes.

Funding

Gordon and Betty Moore Foundation (7728).

Acknowledgment

We thank R. O. Liang, M. Spires Esparza, and H. Choi, all of Wyant College of Optical D. Sciences Apai of Steward Observatory, and T. Lavoie of the National Optical Astronomy Observatory (NOAO) for contributions to this work.

Disclosures

The authors declare no conflicts of interest.

REFERENCES

1. A. B. Chmielewski, C. Moore, and R. Howard, “The Gossamer initiative,” in IEEE Aerospace Conference Proceedings (Cat. No. 00TH8484) (IEEE, 2000), Vol. 7, pp. 429–438.

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

3. P. Atcheson, J. Domber, K. Whiteaker, J. A. S. Britten, N. Dixit, and B. Farmer, “MOIRE: ground demonstration of a large aperture diffractive transmissive telescope,” Proc. SPIE 9143, 91431W (2014). [CrossRef]  

4. 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,” UCRL-JC-148223 (Lawrence Livermore National Laboratory, 2002).

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

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

7. R. A. S. Hyde, N. A. Dixit, H. Weisberg, and M. C. Rushford, “Eyeglass: a very large aperture diffractive space telescope,” Proc. SPIE 4849, 28–39 (2002). [CrossRef]  

8. T. D. L. Milster, C. Johnson, and D. Apai, “Multi-order diffractive Fresnel lens (MOD-DFL) and method for enhancing images that are captured by the MOD-DFL,” U.S. Provisional Patent Series 62695531 (9 July 2018).

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

10. M. Rossi, R. E. Kunz, and H. P. Herzig, “Refractive and diffractive properties of planar micro-optical elements,” Appl. Opt. 34, 5996–6007 (1995). [CrossRef]  

11. P. Wang, N. Mohammad, and R. Menon, “Chromatic-aberration-corrected diffractive lenses for ultra-broadband focusing,” Sci. Rep. 6, 21545 (2016). [CrossRef]  

12. E. Arbabi, S. M. Kamali, A. Arbabi, and A. Faraon, “Full-Stokes imaging polarimetry using dielectric metasurfaces,” ACS Photonics 5, 3132–3140 (2018). [CrossRef]  

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

14. http://www.rsp-technology.com/site-media/user-uploads/rsp_alloys_optics_2018lr.pdf.

15. https://www.nacl.com/nacl-super-hydroseal-hydrooleophobic-coating-technical-data-sheet/super-hydroseal-tds/.

16. D. Apai, T. D. D. Milster, 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. 83, 158 (2019). [CrossRef]  

17. D. Apai, T. D. D. Milster, W. Kim, A. Bixel, G. Schneider, B. V. Rackham, R. Liang, and J. Arenberg, “Nautilus observatory: a space telescope array based on very large aperture ultralight diffractive optical elements,” Proc. SPIE 11116, 1111608 (2019). [CrossRef]  

References

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  1. A. B. Chmielewski, C. Moore, and R. Howard, “The Gossamer initiative,” in IEEE Aerospace Conference Proceedings (Cat. No. 00TH8484) (IEEE, 2000), Vol. 7, pp. 429–438.
  2. R. A. Hyde, “Eyeglass 1. Very large aperture diffractive telescopes,” Appl. Opt. 38, 4198–4212 (1999).
    [Crossref]
  3. P. Atcheson, J. Domber, K. Whiteaker, J. A. S. Britten, N. Dixit, and B. Farmer, “MOIRE: ground demonstration of a large aperture diffractive transmissive telescope,” Proc. SPIE 9143, 91431W (2014).
    [Crossref]
  4. 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,” UCRL-JC-148223 (Lawrence Livermore National Laboratory, 2002).
  5. D. Faklis and G. M. Morris, “Spectral properties of multiorder diffractive lenses,” Appl. Opt. 34, 2462–2468 (1995).
    [Crossref]
  6. D. W. Sweeney and G. E. Sommargren, “Harmonic diffractive lenses,” Appl. Opt. 34, 2469–2475 (1995).
    [Crossref]
  7. R. A. S. Hyde, N. A. Dixit, H. Weisberg, and M. C. Rushford, “Eyeglass: a very large aperture diffractive space telescope,” Proc. SPIE 4849, 28–39 (2002).
    [Crossref]
  8. T. D. L. Milster, C. Johnson, and D. Apai, “Multi-order diffractive Fresnel lens (MOD-DFL) and method for enhancing images that are captured by the MOD-DFL,” U.S. Provisional Patent Series62695531 (9July2018).
  9. T. Stone and N. George, “Hybrid diffractive-refractive lenses and achromats,” Appl. Opt. 27, 2960–2971 (1988).
    [Crossref]
  10. M. Rossi, R. E. Kunz, and H. P. Herzig, “Refractive and diffractive properties of planar micro-optical elements,” Appl. Opt. 34, 5996–6007 (1995).
    [Crossref]
  11. P. Wang, N. Mohammad, and R. Menon, “Chromatic-aberration-corrected diffractive lenses for ultra-broadband focusing,” Sci. Rep. 6, 21545 (2016).
    [Crossref]
  12. E. Arbabi, S. M. Kamali, A. Arbabi, and A. Faraon, “Full-Stokes imaging polarimetry using dielectric metasurfaces,” ACS Photonics 5, 3132–3140 (2018).
    [Crossref]
  13. W. C. Sweatt, “Describing holographic optical elements as lenses,” J. Opt. Soc. Am. 67, 803–808 (1977).
    [Crossref]
  14. http://www.rsp-technology.com/site-media/user-uploads/rsp_alloys_optics_2018lr.pdf .
  15. https://www.nacl.com/nacl-super-hydroseal-hydrooleophobic-coating-technical-data-sheet/super-hydroseal-tds/ .
  16. D. Apai, T. D. D. Milster, 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. 83, 158 (2019).
    [Crossref]
  17. D. Apai, T. D. D. Milster, W. Kim, A. Bixel, G. Schneider, B. V. Rackham, R. Liang, and J. Arenberg, “Nautilus observatory: a space telescope array based on very large aperture ultralight diffractive optical elements,” Proc. SPIE 11116, 1111608 (2019).
    [Crossref]

2019 (2)

D. Apai, T. D. D. Milster, 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. 83, 158 (2019).
[Crossref]

D. Apai, T. D. D. Milster, W. Kim, A. Bixel, G. Schneider, B. V. Rackham, R. Liang, and J. Arenberg, “Nautilus observatory: a space telescope array based on very large aperture ultralight diffractive optical elements,” Proc. SPIE 11116, 1111608 (2019).
[Crossref]

2018 (1)

E. Arbabi, S. M. Kamali, A. Arbabi, and A. Faraon, “Full-Stokes imaging polarimetry using dielectric metasurfaces,” ACS Photonics 5, 3132–3140 (2018).
[Crossref]

2016 (1)

P. Wang, N. Mohammad, and R. Menon, “Chromatic-aberration-corrected diffractive lenses for ultra-broadband focusing,” Sci. Rep. 6, 21545 (2016).
[Crossref]

2014 (1)

P. Atcheson, J. Domber, K. Whiteaker, J. A. S. Britten, N. Dixit, and B. Farmer, “MOIRE: ground demonstration of a large aperture diffractive transmissive telescope,” Proc. SPIE 9143, 91431W (2014).
[Crossref]

2002 (1)

R. A. S. Hyde, N. A. Dixit, H. Weisberg, and M. C. Rushford, “Eyeglass: a very large aperture diffractive space telescope,” Proc. SPIE 4849, 28–39 (2002).
[Crossref]

1999 (1)

1995 (3)

1988 (1)

1977 (1)

Apai, D.

D. Apai, T. D. D. Milster, 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. 83, 158 (2019).
[Crossref]

D. Apai, T. D. D. Milster, W. Kim, A. Bixel, G. Schneider, B. V. Rackham, R. Liang, and J. Arenberg, “Nautilus observatory: a space telescope array based on very large aperture ultralight diffractive optical elements,” Proc. SPIE 11116, 1111608 (2019).
[Crossref]

T. D. L. Milster, C. Johnson, and D. Apai, “Multi-order diffractive Fresnel lens (MOD-DFL) and method for enhancing images that are captured by the MOD-DFL,” U.S. Provisional Patent Series62695531 (9July2018).

Arbabi, A.

E. Arbabi, S. M. Kamali, A. Arbabi, and A. Faraon, “Full-Stokes imaging polarimetry using dielectric metasurfaces,” ACS Photonics 5, 3132–3140 (2018).
[Crossref]

Arbabi, E.

E. Arbabi, S. M. Kamali, A. Arbabi, and A. Faraon, “Full-Stokes imaging polarimetry using dielectric metasurfaces,” ACS Photonics 5, 3132–3140 (2018).
[Crossref]

Arenberg, J.

D. Apai, T. D. D. Milster, 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. 83, 158 (2019).
[Crossref]

D. Apai, T. D. D. Milster, W. Kim, A. Bixel, G. Schneider, B. V. Rackham, R. Liang, and J. Arenberg, “Nautilus observatory: a space telescope array based on very large aperture ultralight diffractive optical elements,” Proc. SPIE 11116, 1111608 (2019).
[Crossref]

Atcheson, P.

P. Atcheson, J. Domber, K. Whiteaker, J. A. S. Britten, N. Dixit, and B. Farmer, “MOIRE: ground demonstration of a large aperture diffractive transmissive telescope,” Proc. SPIE 9143, 91431W (2014).
[Crossref]

Bixel, A.

D. Apai, T. D. D. Milster, W. Kim, A. Bixel, G. Schneider, B. V. Rackham, R. Liang, and J. Arenberg, “Nautilus observatory: a space telescope array based on very large aperture ultralight diffractive optical elements,” Proc. SPIE 11116, 1111608 (2019).
[Crossref]

D. Apai, T. D. D. Milster, 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. 83, 158 (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,” UCRL-JC-148223 (Lawrence Livermore National Laboratory, 2002).

Britten, J. A. S.

P. Atcheson, J. Domber, K. Whiteaker, J. A. S. Britten, N. Dixit, and B. Farmer, “MOIRE: ground demonstration of a large aperture diffractive transmissive telescope,” Proc. SPIE 9143, 91431W (2014).
[Crossref]

Chmielewski, A. B.

A. B. Chmielewski, C. Moore, and R. Howard, “The Gossamer initiative,” in IEEE Aerospace Conference Proceedings (Cat. No. 00TH8484) (IEEE, 2000), Vol. 7, pp. 429–438.

Dixit, N.

P. Atcheson, J. Domber, K. Whiteaker, J. A. S. Britten, N. Dixit, and B. Farmer, “MOIRE: ground demonstration of a large aperture diffractive transmissive telescope,” Proc. SPIE 9143, 91431W (2014).
[Crossref]

Dixit, N. A.

R. A. S. Hyde, N. A. Dixit, H. Weisberg, and M. C. Rushford, “Eyeglass: a very large aperture diffractive space telescope,” Proc. SPIE 4849, 28–39 (2002).
[Crossref]

Domber, J.

P. Atcheson, J. Domber, K. Whiteaker, J. A. S. Britten, N. Dixit, and B. Farmer, “MOIRE: ground demonstration of a large aperture diffractive transmissive telescope,” Proc. SPIE 9143, 91431W (2014).
[Crossref]

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,” UCRL-JC-148223 (Lawrence Livermore National Laboratory, 2002).

Faklis, D.

Faraon, A.

E. Arbabi, S. M. Kamali, A. Arbabi, and A. Faraon, “Full-Stokes imaging polarimetry using dielectric metasurfaces,” ACS Photonics 5, 3132–3140 (2018).
[Crossref]

Farmer, B.

P. Atcheson, J. Domber, K. Whiteaker, J. A. S. Britten, N. Dixit, and B. Farmer, “MOIRE: ground demonstration of a large aperture diffractive transmissive telescope,” Proc. SPIE 9143, 91431W (2014).
[Crossref]

George, N.

Herzig, H. P.

Howard, R.

A. B. Chmielewski, C. Moore, and R. Howard, “The Gossamer initiative,” in IEEE Aerospace Conference Proceedings (Cat. No. 00TH8484) (IEEE, 2000), Vol. 7, pp. 429–438.

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,” UCRL-JC-148223 (Lawrence Livermore National Laboratory, 2002).

Hyde, R. A.

Hyde, R. A. S.

R. A. S. Hyde, N. A. Dixit, H. Weisberg, and M. C. Rushford, “Eyeglass: a very large aperture diffractive space telescope,” Proc. SPIE 4849, 28–39 (2002).
[Crossref]

Johnson, C.

T. D. L. Milster, C. Johnson, and D. Apai, “Multi-order diffractive Fresnel lens (MOD-DFL) and method for enhancing images that are captured by the MOD-DFL,” U.S. Provisional Patent Series62695531 (9July2018).

Kamali, S. M.

E. Arbabi, S. M. Kamali, A. Arbabi, and A. Faraon, “Full-Stokes imaging polarimetry using dielectric metasurfaces,” ACS Photonics 5, 3132–3140 (2018).
[Crossref]

Kim, W.

D. Apai, T. D. D. Milster, 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. 83, 158 (2019).
[Crossref]

D. Apai, T. D. D. Milster, W. Kim, A. Bixel, G. Schneider, B. V. Rackham, R. Liang, and J. Arenberg, “Nautilus observatory: a space telescope array based on very large aperture ultralight diffractive optical elements,” Proc. SPIE 11116, 1111608 (2019).
[Crossref]

Kunz, R. E.

Liang, R.

D. Apai, T. D. D. Milster, 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. 83, 158 (2019).
[Crossref]

D. Apai, T. D. D. Milster, W. Kim, A. Bixel, G. Schneider, B. V. Rackham, R. Liang, and J. Arenberg, “Nautilus observatory: a space telescope array based on very large aperture ultralight diffractive optical elements,” Proc. SPIE 11116, 1111608 (2019).
[Crossref]

Menon, R.

P. Wang, N. Mohammad, and R. Menon, “Chromatic-aberration-corrected diffractive lenses for ultra-broadband focusing,” Sci. Rep. 6, 21545 (2016).
[Crossref]

Milster, T. D. D.

D. Apai, T. D. D. Milster, 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. 83, 158 (2019).
[Crossref]

D. Apai, T. D. D. Milster, W. Kim, A. Bixel, G. Schneider, B. V. Rackham, R. Liang, and J. Arenberg, “Nautilus observatory: a space telescope array based on very large aperture ultralight diffractive optical elements,” Proc. SPIE 11116, 1111608 (2019).
[Crossref]

Milster, T. D. L.

T. D. L. Milster, C. Johnson, and D. Apai, “Multi-order diffractive Fresnel lens (MOD-DFL) and method for enhancing images that are captured by the MOD-DFL,” U.S. Provisional Patent Series62695531 (9July2018).

Mohammad, N.

P. Wang, N. Mohammad, and R. Menon, “Chromatic-aberration-corrected diffractive lenses for ultra-broadband focusing,” Sci. Rep. 6, 21545 (2016).
[Crossref]

Moore, C.

A. B. Chmielewski, C. Moore, and R. Howard, “The Gossamer initiative,” in IEEE Aerospace Conference Proceedings (Cat. No. 00TH8484) (IEEE, 2000), Vol. 7, pp. 429–438.

Morris, G. M.

Rackham, B. V.

D. Apai, T. D. D. Milster, W. Kim, A. Bixel, G. Schneider, B. V. Rackham, R. Liang, and J. Arenberg, “Nautilus observatory: a space telescope array based on very large aperture ultralight diffractive optical elements,” Proc. SPIE 11116, 1111608 (2019).
[Crossref]

Rossi, M.

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,” UCRL-JC-148223 (Lawrence Livermore National Laboratory, 2002).

Rushford, M. C.

R. A. S. Hyde, N. A. Dixit, H. Weisberg, and M. C. Rushford, “Eyeglass: a very large aperture diffractive space telescope,” Proc. SPIE 4849, 28–39 (2002).
[Crossref]

Schneider, G.

D. Apai, T. D. D. Milster, W. Kim, A. Bixel, G. Schneider, B. V. Rackham, R. Liang, and J. Arenberg, “Nautilus observatory: a space telescope array based on very large aperture ultralight diffractive optical elements,” Proc. SPIE 11116, 1111608 (2019).
[Crossref]

D. Apai, T. D. D. Milster, 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. 83, 158 (2019).
[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,” UCRL-JC-148223 (Lawrence Livermore National Laboratory, 2002).

Sommargren, G. E.

Stone, T.

Sweatt, W. C.

Sweeney, D. W.

Wang, P.

P. Wang, N. Mohammad, and R. Menon, “Chromatic-aberration-corrected diffractive lenses for ultra-broadband focusing,” Sci. Rep. 6, 21545 (2016).
[Crossref]

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,” UCRL-JC-148223 (Lawrence Livermore National Laboratory, 2002).

Weisberg, H.

R. A. S. Hyde, N. A. Dixit, H. Weisberg, and M. C. Rushford, “Eyeglass: a very large aperture diffractive space telescope,” Proc. SPIE 4849, 28–39 (2002).
[Crossref]

Whiteaker, K.

P. Atcheson, J. Domber, K. Whiteaker, J. A. S. Britten, N. Dixit, and B. Farmer, “MOIRE: ground demonstration of a large aperture diffractive transmissive telescope,” Proc. SPIE 9143, 91431W (2014).
[Crossref]

ACS Photonics (1)

E. Arbabi, S. M. Kamali, A. Arbabi, and A. Faraon, “Full-Stokes imaging polarimetry using dielectric metasurfaces,” ACS Photonics 5, 3132–3140 (2018).
[Crossref]

Appl. Opt. (5)

Astron. J. (1)

D. Apai, T. D. D. Milster, 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. 83, 158 (2019).
[Crossref]

J. Opt. Soc. Am. (1)

Proc. SPIE (3)

D. Apai, T. D. D. Milster, W. Kim, A. Bixel, G. Schneider, B. V. Rackham, R. Liang, and J. Arenberg, “Nautilus observatory: a space telescope array based on very large aperture ultralight diffractive optical elements,” Proc. SPIE 11116, 1111608 (2019).
[Crossref]

P. Atcheson, J. Domber, K. Whiteaker, J. A. S. Britten, N. Dixit, and B. Farmer, “MOIRE: ground demonstration of a large aperture diffractive transmissive telescope,” Proc. SPIE 9143, 91431W (2014).
[Crossref]

R. A. S. Hyde, N. A. Dixit, H. Weisberg, and M. C. Rushford, “Eyeglass: a very large aperture diffractive space telescope,” Proc. SPIE 4849, 28–39 (2002).
[Crossref]

Sci. Rep. (1)

P. Wang, N. Mohammad, and R. Menon, “Chromatic-aberration-corrected diffractive lenses for ultra-broadband focusing,” Sci. Rep. 6, 21545 (2016).
[Crossref]

Other (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,” UCRL-JC-148223 (Lawrence Livermore National Laboratory, 2002).

T. D. L. Milster, C. Johnson, and D. Apai, “Multi-order diffractive Fresnel lens (MOD-DFL) and method for enhancing images that are captured by the MOD-DFL,” U.S. Provisional Patent Series62695531 (9July2018).

A. B. Chmielewski, C. Moore, and R. Howard, “The Gossamer initiative,” in IEEE Aerospace Conference Proceedings (Cat. No. 00TH8484) (IEEE, 2000), Vol. 7, pp. 429–438.

http://www.rsp-technology.com/site-media/user-uploads/rsp_alloys_optics_2018lr.pdf .

https://www.nacl.com/nacl-super-hydroseal-hydrooleophobic-coating-technical-data-sheet/super-hydroseal-tds/ .

Supplementary Material (1)

NameDescription
» Visualization 1       Dynamic simulation of focal spot irradiance over a range of wavelengths for MODE lenses.

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Figures (9)

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 ($M \sim {1000}$) produces a small value for the diffractive component of the longitudinal chromatic aberration (LCA). The DFL reduces the refractive part of the total focal dispersion, making each MOD zone achromatic.
Fig. 2.
Fig. 2. Cyclic behavior of MOD focal orders along a horizontal optical axis due to diffraction. With high $M$, the change of focus position with wavelength occurs over diffractive focal range $\Delta\! f \sim {f_0}/M$, where ${f_0}$ is the design focus position. Starting with the design wavelength ${\lambda _0}$ at focal order $p = M$ (top) maximum diffraction efficiency is achieved at ${f_0}$. As the wavelength increases, this focal order moves toward the lens, and diffraction efficiency (gray shading–middle) reduces. Toward the end of the diffractive cycle (bottom), focal order $p = M - {1}$ exhibits nearly equal diffraction efficiency to order $M$, and efficiency of the $M - {1}$ order increases as it moves toward ${f_0}$ as the wavelength increases. The diffractive focal range $\Delta\! f$ is the diffractive component of the secondary spectrum $SS$. (See Visualization 1, which is a colormap irradiance profile in the focal zone of a ${f_0} = {1}\;{\rm m}$, $M = {1000}$, $f/{4.17}$ mode lens designed for the R-band and displayed for individual wavelengths from 648 nm to 668 nm.)
Fig. 3.
Fig. 3. Calculation geometry for estimation of SR and focus spot characteristics.
Fig. 4.
Fig. 4. Left: contours of calculated Strehl ratio (SR) as dark lines for ${\lambda _0} = {658}\;{\rm nm}$ and $M = {1000}$ as a function of aperture diameter $D$ and focal length ${f_0}$. The gray dot indicates the design point of the MODE lens prototype described in Section 4. Right: number of MOD radial zones required. Both: five values of $f\#$ are shown as thin dashed lines.
Fig. 5.
Fig. 5. Left: contours of constant 90% encircled energy diameter ${d_{90}}$ divided by the Airy spot diameter ${d_{{\rm Airy}}}$ with $M = {1000}$ and ${\lambda _0} = {658}\;{\rm nm}$ as a function of aperture diameter $D$ and focal length ${f_0}$. Right: contours of diffractive focal range $\Delta\! f$ divided by classical depth of focus ${z_{{\rm DOF}}}$ versus $D$ and ${f_0}$. Both: five values of $f\#$ are shown as thin dashed lines.
Fig. 6.
Fig. 6. MODE lens layout (left) includes a plastic MOD lens with thickness ${t_a} = {8}\;{\rm mm}$ and a ${t_b} = {4}\;{\rm mm}$ glass plate with a DFL on the back surface. The MODE lens is mounted in a lens tube with a CMOS camera at the focal plane (right) for imaging experiments.
Fig. 7.
Fig. 7. (a)–(c) Narrowband polychromatic focal spot measurements at $\lambda = {658}\;{\rm nm}$ with a bandwidth of about 2 nm. (a) Peak-normalized 2D spot irradiance distribution with a black circle representing the Airy spot diameter; (b) profiles of the spot irradiance and comparison with the ideal Airy irradiance and a commercial achromatic doublet with about the same optical parameters; (c) encircled energy (EE) as a function of diameter; and (d) R-band filtered image of the moon (cropped).
Fig. 8.
Fig. 8. Measured, simulated, and design change in focal length versus wavelength. MOD: measurement without the DFL back surface; MODE: measurement with the complete MODE lens; Simulated: diffraction simulation using data from a Code V ray-trace program; and Design: design curve for the achromat of the central MODE zone.
Fig. 9.
Fig. 9. R-band images of different scenes. Left: image of the central portion of a tri-bar resolution target displayed a high-resolution monitor viewed at 6 m. The circle represents the 1° designed field of view. The inset shows a magnified portion of the image, where the Group 0, Element 4 bar spacing is 0.006°. Right: image of a softball field in daylight over the full field of the camera (2.54° by 2.03°), where the screen mesh shown in the inset is resolvable at an angular separation of 0.006°.

Tables (2)

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Table 1. Design Parameters of the MODE Prototypea

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Table 2. Measured MODE, Calculated MODE, and Ideal Airy Values

Equations (10)

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f ( λ ) = f 0 λ 0 λ ,
λ p = M λ 0 / p ,
I a x i s ( δ f ) I ( δ f ) sin 2 ( π N f 2 )
δ f = f 0 δ λ λ 0 ,
N f = D 2 δ λ 4 f 0 λ 0 2 ,
I a x i s ( δ λ ) sin 2 ( α δ λ ) ( α δ λ ) 2 ,
α = π D 8 f # λ 0 2
S R = 1 Δ λ a b sin 2 ( α δ λ ) ( α δ λ ) 2 d ( δ λ ) = 1 α Δ λ [ S i ( 2 α δ λ ) 1 2 α δ λ + cos ( 2 α δ λ ) 2 α δ λ ] a b ,
S i ( x ) = 0 x sin ( t ) t d t .
U s ( ρ s ) = exp ( j π N f ρ s 2 ) ,

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