1. Introduction

Imaging and sensing in the mid-wave infra-red (MWIR) band ($3\text { to }5\; \mu$m) are useful for molecular fingerprinting, low-light level night vision, tracking missiles, surveillance, emergency response, biomedical imaging, astronomy and environmental monitoring [1–3]. In particular, $4\; \mu$m offers high transparency through our atmosphere, which makes it useful for free-space optical communications. MWIR imaging is also useful for non-destructive testing of integrated circuits. In long range imaging applications, MWIR systems provide exceptional clarity in various weather and light conditions.

Metalenses for the MWIR have been demonstrated recently, but only with a very small aperture diameter less than $0.5$ mm, and requires relatively complex fabrication in silicon due to the minimum feature width of $0.36\; \mu$m and aspect ratios as high as 12.6 [4]. Recently, sophisticated numerical approaches for inverse design have succeeded in designing achromatic MWIR metalens in silicon, [5] but no experiments have been demonstrated as the fabrication is challenging. The simulated focusing efficiency (defined as fraction of transmitted light inside a focal spot of diameter several times the diffraction-limited spot) is limited to about $30\%$ at $\lambda =4\; \mu$m. Metalenses with achromatic focusing in the MWIR for one polarization have also been designed, but not experimentally demonstrated. The focusing efficiency remains below $30\%$[6]. An MWIR metalens in reflection was demonstrated and a focusing efficiency of $80\; \%$ was claimed, but this measurement was erroneous as a focal spot region of $3$ mm was used for power measurement, which was 18 times the $1/e^2$ focal spot diameter [7]. Such a large focal spot combines the zero and first diffraction orders, and therefore over-estimates focusing efficiency. It is useful to point the reader to the seminal paper by Buralli and Morris that discussed the intricacies of focusing efficiency in flat lenses [8].

Multi-level diffractive lenses (MDLs) utilize propagation phase and can be considered as modified versions of the classical fresnel zone plate (FZP). Previously, MDLs have been demonstrated over the UV, [9] visible, [10] near-infra-red [11] and long-wave infra-red [12,13] bands. In general, the MDL is designed using optimization-based inverse design where the pupil is generally divided into concentric rings of equal width (determined primarily by lithographic constraints) and the phase shift of each ring serves as the optimization variable. Upper and lower bounds on the ring heights as well as an upper bound on the number of phase levels are imposed for ease of fabrication. Since optimization is used for design, the objective function can be tailored for specific desirable properties such as enhanced depth-of-focus [14] or increased operating bandwidth, [15] for example. We also note that these approaches can be used for non-rotationally-symmetric lenses as well [16]. In this paper, we designed an MDL for $\lambda =4\; \mu$m with diameter of $25$ mm and focal length of $25$ mm. Further, we fabricated this MDL in silicon using multiple aligned lithography and etch steps, and then optically characterized its performance. We also fabricated an FZP with very similar specifications as the MDL and compared their performance, which provides useful insights into potential advantages and disadvantages of the MDL. The main contributions of this work are the simulation and experimental demonstrations of inverse-designed MDL in the MWIR band with enhanced field-of-view (FOV) and depth-of-focus (DOF) when compared to conventional flat lenses.

2. Design

Inverse design of the MDL used a direct-binary-search method to maximize the fraction of power incident on the MDL that is focused into a spot of diameter $\approx 12\; \mu$m, averaged over a FOV of $\pm 10^{\circ}$. Note that the diffraction limited full-width at half-maximum (FWHM) of the focal spot is $0.5\times \lambda /\text {NA}=4.44\; \mu$m where NA = 0.45 for our lens. Specifically, the objective function attempts to maximize power concentration within a spot of diameter $2.7 \times$ FWHM, averaged over the incident angles of 0°, 5°and 10°. Such an objective function also implicitly attempts to minimize off-axis aberrations including distortion as the off-axis PSF is computed over a window that is centered on the geometric off-axis spot, following the expression $f\times \tan (\theta )$, where $f$ is the focal length and $\theta$ is the incident field angle.

For inverse design, we assumed material refractive index of silicon, incorporated fabrication constraints of ring-width $=4\; \mu$m, maximum ring-height $=6\; \mu$m and 8 gray levels. The resulting geometry, represented by its radial cross-section is illustrated in Fig. 1(a). The design contains 3,125 rings, and the bottom insets show the innermost and outermost 100 rings on the left and right, respectively. For comparison, the geometry of an FZP with very similar specifications (diameter = $25.4$ mm, focal length = $25.4$ mm, $\lambda =4\; \mu$m) is illustrated in Fig. 1(b). The intensity distributions in the vicinity of the focal plane were simulated using the first Rayleigh Sommerfeld diffraction integral, and are shown in Figs. 1(c) and (d) for the MDL and FZP, respectively. Although the FZP offers a smaller focal spot, (Fig. 1(d)) the MDL achieves sharply focused spots at oblique incidence, which is much sharper than what is possible with the FZP (Fig. 1(e)). This is the result of the inverse design algorithm. Surreptitiously, we also observed that the MDL provides a larger depth-of-focus. These simulations suggest and are later confirmed by our experiments that the inverse-designed MDL offers a wider FOV and a larger DOF when compared to the conventional FZP.

 

Fig. 1. Design and simulations of (a) inverse-designed MDL (diameter = $25$ mm, focal length = $25$ mm) and (b) FZP (diameter = $25.4$ mm, focal length = $25.4$ mm), both for $\lambda =4\; \mu$m. Simulated axial point-spread functions of (c) the MDL and (d) the FZP. Magnified views of each are shown in the corresponding right panels. (e) Simulated transverse point-spread functions of the MDL (left column) and the FZP (right column) for incident angles of $0^{\circ}$, $5^{\circ}$ and $10^{\circ}$.

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3. Fabrication and metrology

A silicon MDL was fabricated on an intrinsic silicon wafer with a $100$ mm-diameter, $525\; \mu$m-thickness, <100> orientation, and resistivity $>10,000\; \Omega$ cm (MSE Supplies, LLC). Three sequential lithography and reactive-ion etching (RIE) steps were used to pattern the samples. Details of the fabrication process are in section 2 of Supplement 1 and shown in Fig. 2(a). After fabrication, we measured the heights of 92 rings close to the center of the MDL using an optical profilometer (Zygo Newview-9000). The distance between each measured point was over a quarter the width of each ring, so only the center point was considered in the error calculation. The absolute height error averaged over the measured rings was $504$ nm, with standard deviation of $1.08\; \mu$m.

We fabricated the FZP using grayscale optical lithography, details of which are also included in section 2 of Supplement 1. The FZP geometry consisted of 5,680 concentric rings. The designed ring-width of $2.23\; \mu$m was discretized to fit the positioning grid on the lithography tool to $2.25\; \mu$m. This change causes negligible effect on the FZP performance. We used a confocal microscope (Olympus LEXT OLS5000) to measure the heights of the outermost 51 rings of the FZP. The measured data compared to the expected heights is shown in Fig. S2 (Supplement 1). The error between these two data sets was calculated by subtracting the designed heights from the measured heights. To mitigate the error values caused by the inevitable sloping between large height changes in the design, we averaged the values in the center of each measured ring (equal to one third the width of a ring) to find the most accurate value. The absolute height error averaged over all the measured rings was $778$ nm, with standard deviation of $1.49\; \mu$m. We simulated the impact of these ring-height errors and summarized the results in section 3 of Supplement 1. In the case of the MDL, these errors tend to increase power diffracted into the background. The ring-height errors have almost no effect on the FZP performance. It is noteworthy then that the inverse-designed MDL, despite being more sensitive to fabrication errors, still offers a larger FOV and DOF when compared to the FZP.

 

Fig. 2. (a) MDL fabricated in silicon using multiple lithography and etch steps. (b) FZP fabricated in photoresist atop a sapphire wafer using grayscale optical lithography. The weight of each device is dominated by the substrate and is $2.81$ g (MDL) and $3.63$ g (FZP). In both panels, the right insets show the 3D optical profilometer data and cross-section through the white line indicating 8 discrete height levels spanning a height range of $6\; \mu$m.

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4. Optical characterization

In order to measure the point-spread functions (PSFs) of the lenses, each lens was illuminated by a collimated beam from a laser ($\lambda =4\; \mu$m). A MWIR focal plane array (FPA) (SCD Fast Pelican InSb 640 x 512 pixels with $15\; \mu$m pixel pitch) was placed in the focal plane of each lens. The distance between the FPA and the lens was adjusted slightly to measure PSFs at a few defocus values. A dark frame was also captured in each case, and this was subtracted from each raw frame. The results shown in Figs. 3(a) and 3(b) for the MDL and the FZP, respectively confirm that the MDL exhibits a much higher depth-of-focus (DOF) compared to the FZP. Note that the diffraction-limited DOF is $\lambda /(\text {NA})^2\approx 20\; \mu$m. At the focal plane, both the MDL and FZP exhibit spot size that is smaller than the pixel size of the FPA.

 

Fig. 3. Optical characterization results. Measured point-spread function vs defocus for (a) the Si MDL and (b) the FZP. In all cases, the exposure time was $50\; \mu$s. In both cases, the spot size at focus is smaller the sensor pixel size of $15\; \mu$m, but the MDL exhibits much sharper focusing over a large depth. Illumination was a collimated beam from a laser with $\lambda =4\; \mu$m. (c) Imaging a soldering iron and an Air Force (AF) resolution chart using the MDL. (d) Image of the AF resolution chart using the FZP (same scale as (c)). (e) Comparing field-of-view of the MDL and the FZP. Distance between target and lens was $66.12$ mm. Arrow points to edge of circle, which is resolved in the MDL, but not for the FZP.

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In order to test imaging performance, we first placed a soldering iron (see Fig. 3(c)) at a distance of $1.8$ m from the MDL and the distance between the MDL and the FPA was $\approx 95$ mm. The resulting image clearly resolves the tip of the soldering iron, whose size is $\approx 2$ mm, which corresponds to an angular resolution of $\approx 1.11$ mrad. Subsequently, we placed an Air Force (AF) resolution chart at a distance of $38.1$ mm away from the MDL and back-illuminated it using a black-body source (Santa Barbara Infrared, see Supplement 1 for additional details). In this case, the distance between the MDL and the FPA was $\approx 38$ mm, which indicates the large depth-of-focus of this lens. Space between lines as small as $1.21$ mm are very well resolved (this corresponds to an angular resolution of $\approx 32$ mrad). The contrast in X and Y directions were measured as 39% and 50%, respectively (see Fig. S6 in Supplement 1). The difference in X and Y contrast is attributed to the uncorrected non-uniformity contrast error and the associated Narcissus effect, which is common in IR detectors [17]. The corresponding image from the FZP is shown in Fig. 3(d). The contrast for the FZP image in X and Y directions were measured at 55% and 33%, respectively (see Fig. S6 in Supplement 1). We note that at least for the smaller angular frequencies (measurement here at $\approx 32$ mrad), the contrast of the MDL and the FZP are comparable.

In order to explore the relative field-of-view (FOV) of the MDL versus that of the FZP, we captured the image of a target placed $66.12$ mm away from the lens (see Supplement 1) and the resulting images are shown in Fig. 3(e). The on-axis contrast of the FZP is higher, as expected, but the edges of the circles away from the optical axis are better resolved by the MDL (see yellow arrows at FOV of 8.8°). These results confirm that the inverse-designed MDL offers a larger FOV than the FZP.

We also measured the transmission efficiency, defined as the fraction of incident light transmitting the lens by placing a power meter (Ophir 30A-BB-18) in front of, and after the lens, while the lens is illuminated by the collimated laser beam. The measured transmission efficiency was $86\%$ and $88\%$ for the MDL and FZP, respectively (see Table S1 in Supplement 1). The transmission losses arise primarily from fresnel reflections at the interfaces and silicon is expected to have higher losses than sapphire. In addition, the patterned surface in both cases act as an effective-medium anti-reflection coating (something we have observed before in the visible [18]). We also estimated integrated focused power as the fraction of transmitted light that is focused to a spot following the guidelines in Ref. [8]. This definition separates fresnel reflection losses (which may be mitigated with effective anti-reflection coatings) from the focusing of the flat lens itself. The power meter was placed in the focal plane, which measures the combined focused and zero-order powers (details in section 5 of Supplement 1). Then, the power meter was placed in a far-field plane, which measures the zero-order power. An iris of approximately the same size as the lens was used. The resulting integrated focusing efficiencies are $43\%$ and $55\%$ for the MDL and FZP, respectively. A large fraction of transmitted light is diffracted into the 0 and higher (>1) diffraction orders. At present, we are investigating potential strategies to increase this integrated focusing efficiency, such as improved design algorithms and minimizing fabrication errors.

5. Discussion

Although the objective of our inverse design was to enhance the FOV of the MDL, it also resulted in an increase in the DOF. As far as we are aware, this connection between FOV and DOF has not been explored theoretically even though great advances have been made in each metric separately, see for example Ref. [19]. Further theoretical investigation is clearly needed to clarify the causes of this improvement. We also note that even larger lens apertures are possible, and the key challenges are the complexity of computation for inverse design due to the much larger number of rings, and the associated fabrication challenges to maintain high precision over a larger area. An important point for the future is that for efficient off-axis focusing, larger diffraction angles are required when compared to on-axis focusing as discussed in section 9 of Supplement 1. Since we did not consider this requirement here, it is one of the reasons for reduced focusing efficiencies and can be mitigated in the future. The approach for inverse design described here can be readily extended to other spectral bands as well. We also simulated the axial PSFs for both lenses under normal and oblique incidence (Fig. S16 in Supplement 1). These simulations indicate that interestingly the field curvature of the MDL is also reduced relative to the FZP. Our conjecture is that by forcing tight focusing for oblique incidence, the variation in focal length with angle is also minimized, thereby reducing field curvature.

MWIR imaging systems are critical for many defense applications including surveillance and targeting, both on ground, in sea as well as in air. For wide FOV imaging, large FPAs are required, and in order to collect sufficient light, large aperture lenses are also required. In order to correct chromatic aberrations and off-axis aberrations, multiple refractive lenses need to be combined [20]. Furthermore, large aperture fast lenses also suffer from a reduced depth-of-focus, which necessitates very precise alignment and sophisticated optomechanics, especially for field deployment. All these conditions dramatically increase the weight and cost of such imaging systems. By employing flat optics, it may be possible to mitigate these disadvantages. In prior work, flat optics in the form of gradient-index lenses have demonstrated their potential [21]. Here, we demonstrate multi-level diffractive optics as an alternative approach for correcting off-axis and defocus aberrations. Although our design was performed at a single wavelength, $\lambda =4\; \mu$m, our imaging experiments (see Fig. 3) suggest useful broadband performance. MDLs offer significantly more degrees of freedom when compared to alternatives such as FZPs, GRIN lenses or refractives. Although not demonstrated here, MDLs could be manufactured at low cost and high volume using imprint lithography, something we have demonstrated previously in the visible band [22]. This could be performed with IR-transparent polymers to significantly reduce the cost of such optics in the future. Additional work is required to extend the FOV further and overall operating bandwidth, as well as to minimize the loss in contrast that is sometimes associated with MDLs. Nevertheless, our preliminary experiments reported here show feasibility of this approach for MWIR imaging.