It is generally thought that correcting chromatic aberrations in imaging requires multiple surfaces. Here, we show that by allowing the phase in the image plane of a flat lens to be a free parameter, it is possible to correct chromatic aberrations over a large continuous bandwidth with a single diffractive surface. In contrast to conventional lens design, we utilize inverse design, where the phase in the focal plane is treated as a free parameter. This approach attains a phase-only (lossless) pupil function, which can be implemented as a multi-level diffractive flat lens that achieves achromatic focusing and imaging. In particular, we experimentally demonstrate imaging using a single flat lens of diameter > 3 mm and focal length = 5 mm (NA = 0.3, f/1.59) that is achromatic from λ = 450 nm (blue) to 1 μm (NIR). This simultaneous achievement of large size, NA, and broad operating bandwidth has not been demonstrated in a flat lens before. We experimentally characterized the point-spread functions, off-axis aberrations, and broadband imaging performance of the lens.

The lens is considered the most fundamental element for imaging. Imaging is information transfer from the object to the image planes. This can be accomplished via a conventional lens that essentially performs a one-to-one mapping,1 via an unconventional lens (such as one with a structured point-spread function or PSF) that performs a one-to-many mapping, or via no lens, where the light propagation essentially performs a one-to-all mapping. In the first case, the image is formed directly. In the second case, the image is formed after computation and can be useful when encoding spectral2,3 or depth4 or polarization5 or other information into the geometry of the PSF itself. Note that the modification of the PSF may be at the same scale as the diffraction limit,2–5 or it can even be much larger.6–9 The image can be recovered in many cases in the optics-less scenario as well.10–12 However, the conventional lens approach is preferred in many cases due to the high signal-to-noise ratio achievable at each image pixel (resulting from the 1:1 mapping). When this conventional lens is illuminated by a plane wave, it forms a focused spot at a distance equal to its focal length.

Now, if we appeal to the fact that in the vast majority of imaging applications, only the intensity is measured, the spatial distribution of the phase (here, we refer to the phase of a scalar electromagnetic field, but the argument is equally valid for vector fields) in the focal plane can be an arbitrary function. Then, it is easy to see, for instance, via the inverse-diffraction transform,13 that the spatial-phase distribution of the plane wave after it transmits the lens (henceforth, we refer to this as the pupil function) can have multiple forms. In conventional optics, the positive lens is a device that converts incident plane waves into converging spherical waves, whose pupil function, therefore, follows a hyperbolic relationship to the radial coordinate (see the black lines in the insets of Fig. 1). However, our argument above shows that this picture is incomplete, and the set of pupil functions for an ideal lens is, in fact, infinite. Another way to visualize this is that the converging spherical wave is just one of an infinite set of waves that can converge to a focus, i.e., an intensity distribution that is sharply localized in space. This critical insight can be exploited to search for pupil functions that enable achromatic focusing,14–18 extreme depth-of-focus,19 or even high efficiency at a high-NA.20 In this paper, we utilize this concept and inverse design to create a flat lens with diameter = 3.145 mm, focal length = 5 mm, NA = 0.3, and an operating bandwidth of λ = 0.45 μm–1 μm [simulated point-spread functions, all at a distance of 5 mm from the multi-level diffractive lens (MDL) are shown in Fig. 1]. In comparison, although there are several reports of achromatic metalenses21 with a broad bandwidth in the visible band (see, e.g., Refs. 22–28), the broadest bandwidth visible metalens that we are aware of has a diameter of only 44 μm, NA = 0.125, and an operating bandwidth of λ = 0.4 μm–0.65 μm.28 Summarized in Table I is a literature survey of broadband achromatic metalenses in the visible band. Furthermore, metalenses require high refractive-index dielectrics (GaN is used in the example above) that necessitate expensive semiconductor processing, which makes it impractical for larger-area optics.29 In contrast, the flat multi-level diffractive lenses (MDLs) described here can be inexpensively replicated into low-index polymers, even over large areas with high precision.

FIG. 1.

Simulated point-spread functions (PSFs) of the MDL for different wavelengths. The insets show the heights [labeled (h)] and corresponding phase distributions [labeled ϕ(λ)] for the innermost rings (within a radius of 130 μm). See the top view of the fabricated lens in Fig. 2(a). The ideal-lens complex pupil function (hyperbolic function) is plotted for each wavelength with black lines for comparison.

FIG. 1.

Simulated point-spread functions (PSFs) of the MDL for different wavelengths. The insets show the heights [labeled (h)] and corresponding phase distributions [labeled ϕ(λ)] for the innermost rings (within a radius of 130 μm). See the top view of the fabricated lens in Fig. 2(a). The ideal-lens complex pupil function (hyperbolic function) is plotted for each wavelength with black lines for comparison.

Close modal
TABLE I.

Literature survey of experimental broadband achromatic metalenses in the visible and comparison to this work.

ReferencesMaterialWavelength rangeBandwidthNAFocal length/diameter
Chen et al.22  TiO2 460 nm–700 nm 240 nm 0.2 67 μm/26.4 μ
Chen et al.23  TiO2 470 nm– 670 nm 200 nm 0.2 63 μm/25 μ
Liang et al.24  TiO2 560 nm–800 nm 240 nm up to 0.8 2 to 14 μm/5.4 μ
Ye et al.25  GaN 435 nm–685 nm 250 nm 0.17 20 μm/7 μ
Wang et al.26  a-Si 470 nm–658 nm 188 nm 0.35 400 μm/300 μ
Shi et al.27  TiO2 490 nm–550 nm 60 nm 0.2 485 μm/200 μ
Wang et al.28  GaN 400 nm–660 nm 260 nm 0.125 235 μm/50 μ
This work photoresist 450 nm–1000 nm 550 nm 0.3 5 mm/3.145 mm 
ReferencesMaterialWavelength rangeBandwidthNAFocal length/diameter
Chen et al.22  TiO2 460 nm–700 nm 240 nm 0.2 67 μm/26.4 μ
Chen et al.23  TiO2 470 nm– 670 nm 200 nm 0.2 63 μm/25 μ
Liang et al.24  TiO2 560 nm–800 nm 240 nm up to 0.8 2 to 14 μm/5.4 μ
Ye et al.25  GaN 435 nm–685 nm 250 nm 0.17 20 μm/7 μ
Wang et al.26  a-Si 470 nm–658 nm 188 nm 0.35 400 μm/300 μ
Shi et al.27  TiO2 490 nm–550 nm 60 nm 0.2 485 μm/200 μ
Wang et al.28  GaN 400 nm–660 nm 260 nm 0.125 235 μm/50 μ
This work photoresist 450 nm–1000 nm 550 nm 0.3 5 mm/3.145 mm 

In Fig. 1, we present the simulated point-spread functions (PSFs) of the MDL for λ = 0.45 μm to 0.85 μm. In the inset, we also plotted the radial-ring-height distribution (top row, labeled h) and the corresponding phase-shift distributions for λ = 0.45 μm to 0.85 μm for the inner 201 rings (radius = 130 μm). The hyperbolic phase function for each wavelength is plotted (solid black lines) for comparison. It is defined as ϕ = −2π/λ*r2/(2f), where r is the radial coordinate and f is the focal length, and as expected varies with the wavelength, which makes it infeasible to design achromatic diffractive lenses.

The MDL is comprised of 2419 concentric rings of fixed width (0.65 μm) and varying heights (0–2.6 μm). Details of our design procedure have been described elsewhere.29,30 Briefly, we performed nonlinear optimization to maximize the power that is focused to a circle of radius equal to 3 X the diffraction-limited FWHM, averaged over the spectral band of interest. The design parameters were ring-width = 0.65 μm, maximum ring-height = 2.6 μm, and number of levels = 100. The MDL was patterned using high-resolution optical-grayscale lithography in a positive-tone photoresist (maP1200G), whose dispersion was used for design.20 An optical micrograph of the fabricated device and the measured point-spread functions at λ = 0.45 μm–1 μm in increments of 50 nm are shown in Figs. 2(a) and 2(b), respectively. All PSFs in Fig. 2 are measured at a distance of 5 mm from the MDL. We also measured the PSFs (Fig. S12) and the on-axis intensities as a function of distance (Fig. S11) from the MDL, which peaks at 5 mm for all wavelengths, proving achromaticity. Additional confirmation is provided via simulated PSFs with finer z-spacing around the designed focus (Fig. S17). In contrast to conventional multi-order or harmonic diffractive lenses, our MDLs operate over a continuous spectrum, which we proved by measuring the PSFs with a wavelength interval of 25 nm (Figs. S13 and S14). The measured Strehl ratio as a function of wavelength is also plotted in Fig. S15, revealing a wavelength-averaged value of 0.75. In all cases, the MDL was illuminated by a collimated beam from a super-continuum source coupled to tunable filters.14 The encircled power of the MDL was noted to be lower than that in the simulations (see Fig. S16). Although a complete explanation is not available, we attribute this discrepancy to fabrication errors.

FIG. 2.

Experimental achromatic focusing. (a) Optical micrograph of the fabricated lens. Hm is the maximum ring height and W is the minimum ring width. (b) Measured PSFs in the visible-NIR band. The illumination bandwidth of each wavelength was 15 nm.

FIG. 2.

Experimental achromatic focusing. (a) Optical micrograph of the fabricated lens. Hm is the maximum ring height and W is the minimum ring width. (b) Measured PSFs in the visible-NIR band. The illumination bandwidth of each wavelength was 15 nm.

Close modal

The imaging performance of the MDL was characterized next by measuring the off-axis PSFs under broadband illumination (0.45 μm–0.85 μm). The angle of incidence of the plane wave illumination was adjusted from 0° to 30° (Fig. S1), and the corresponding PSFs were recorded on a monochrome image sensor [see Fig. 3(a)]. Off-axis aberrations become apparent at angles >= 15°. The modulation-transfer functions (MTFs) were extracted from the measured PSFs [Fig. 3(b)] and indicate that a contrast of 10% is maintained for all angles less than ∼15°, from which we can conclude that the estimated full field of view is ∼30°. Note that no special effort was made to correct for off-axis aberrations in this design, which, if appropriately performed, should significantly increase this value. The linearity of the lens is confirmed by plotting the centroid of the measured PSFs as a function of the incident angle [Fig. 3(c)]. Finally, in Figs. 3(d) and 3(e), we show the images formed by this lens in the visible and NIR bands of the Macbeth color chart and the Airforce resolution chart, respectively.

FIG. 3.

Experimental imaging performance. (a) Off-axis PSFs under broadband (0.45 μm–0.85 μm) illumination. (b) MTF curves at various angles of incidence corresponding to the PSFs in (a). (c) Centroid of the PSF as a function of incident angle demonstrating linearity. (d) Visible (white LED illumination) and NIR (850 nm LED flashlight illumination) images of the Macbeth chart. (e) Broadband image of the resolution chart. The experimental details are in shown in Fig. S2.

FIG. 3.

Experimental imaging performance. (a) Off-axis PSFs under broadband (0.45 μm–0.85 μm) illumination. (b) MTF curves at various angles of incidence corresponding to the PSFs in (a). (c) Centroid of the PSF as a function of incident angle demonstrating linearity. (d) Visible (white LED illumination) and NIR (850 nm LED flashlight illumination) images of the Macbeth chart. (e) Broadband image of the resolution chart. The experimental details are in shown in Fig. S2.

Close modal

In this Letter, we demonstrated that a single MDL could be made achromatic over a continuous spectrum from 0.45 μm to 1 μm with a diameter > 3 mm and NA = 0.3. Such a lens is made possible via a new approach to design that invokes the non-uniqueness of the lens pupil function. To re-iterate this point, we show a similar performance from a second MDL with the same design specifications as the one above, but with a different lens pupil function (Figs. S3 and S4). In addition, we have fabricated other visible-NIR MDLs and characterized their aberration performance (Figs. S6–S8).16 All our experimental results confirm that a single diffractive surface is sufficient to correct for aberrations, including chromatic aberrations, for a large majority of imaging applications over a continuous spectrum.

It is important to note that the theoretical work by Stamnes and co-workers identified the “perfect” wave for maximal concentration of power is, in fact, the diverging electric-dipole wave,31,32 which is not strictly spherical. In our work, rather than opting for the global maximum of concentration, we design our MDLs to achieve sufficient light concentration at the focus to enable imaging performance. This allows us to incorporate fabrication constraints into the design process, enabling useful devices.

See the supplementary material for the details and data related to the design, fabrication, and characterization of the MDLs.

We thank Brian Baker, Steve Pritchett, and Christian Bach for the fabrication advice, and Tom Tiwald (Woollam) for measuring the dispersion of materials. We would also like to acknowledge the support from Amazon AWS (No. 051241749381). R.M. and B.S.R. acknowledge the funding from the Office of Naval Research Grant No. N66001-10-1-4065 and from an NSF Award Nos. 1351389, 1828480, and 1936729, respectively.

R.M. is the co-founder of Oblate Optics, Inc., which is commercializing the technology discussed in this manuscript. The University of Utah has filed for patent protection for the technology discussed in this manuscript.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

1.
M.
Born
and
E.
Wolf
,
Principle of Optics
, 7th ed. (
Cambridge University Press
,
Cambridge
,
1999
).
2.
P.
Wang
and
R.
Menon
, “
Computational multi-spectral video imaging
,”
J. Opt. Soc. Am. A
35
(
1
),
189
199
(
2018
).
3.
R.
Menon
,
P.
Rogge
, and
H.-Y.
Tsai
, “
Design of diffractive lenses that generate optical nulls without phase singularities
,”
J. Opt. Soc. Am. A
26
(
2
),
297
(
2009
).
4.
S. R. R.
Pavani
,
M. A.
Thompson
,
J. S.
Biteen
,
S. J.
Lord
,
N.
Liu
,
R. J.
Tweig
,
R.
Piestun
, and
W. E.
Moerner
, “
Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix point-spread function
,”
Proc. Natl. Acad. U. S. A.
106
(
9
),
2995
2999
(
2009
).
5.
M.
Khorasaninejad
,
W. T.
Chen
,
A. Y.
Zhu
,
J.
Oh
,
R. C.
Devlin
,
C.
Roques-Carmes
,
I.
Mishra
, and
F.
Capasso
, “
Visible wavelength planar metalenses based on titanium dioxide
,”
IEEE J. Sel. Top. Quantum Electron.
23
(
3
),
4700216
(
2017
).
6.
M.
Meem
,
A.
Majumder
, and
R.
Menon
, “
Multi-plane, multi-band image projection via broadband diffractive optics
,”
Appl. Opt.
59
,
38
44
(
2020
).
7.
N.
Mohammad
,
M.
Meem
,
X.
Wan
, and
R.
Menon
, “
Full-color, large area, transmissive holograms enabled by multi-level diffractive optics
,”
Sci. Rep.
7
,
5789
(
2017
).
8.
G.
Kim
,
N.
Nagarajan
,
E.
Pastuzyn
,
K.
Jenks
,
M.
Capecchi
,
J.
Sheperd
, and
R.
Menon
, “
Deep-brain imaging via epi-fluorescence computational cannula microscopy
,”
Sci. Rep.
7
,
44791
(
2017
).
9.
G.
Kim
,
N.
Nagarajan
,
M.
Capecchi
, and
R.
Menon
, “
Cannula-based computational fluorescence microscopy
,”
Appl. Phys. Lett.
106
,
261111
(
2015
).
10.
G.
Kim
and
R.
Menon
, “
Computational imaging enables a “see-through” lensless camera
,”
Opt. Express
26
(
18
),
22826
22836
(
2018
).
11.
G.
Kim
,
K.
Isaacson
,
R.
Palmer
, and
R.
Menon
, “
Lensless photography with only an image sensor
,”
Appl. Opt.
56
(
23
),
6450
6456
(
2017
).
12.
G.
Kim
,
S.
Kapetanovic
,
R.
Palmer
, and
R.
Menon
, “
Lensless-camera based machine learning for image classification
,” arXiv:1709.00408.
13.
G. C.
Sherman
, “
Integral-transform formulation of diffraction theory
,”
J. Opt. Soc. Am. A
57
(
12
),
1490
1498
(
1967
).
14.
M.
Meem
,
S.
Banerji
,
A.
Majumder
,
P.
Hon
,
J. C.
Garcia
,
B.
Sensale-Rodriguez
, and
R.
Menon
, “
Imaging from the visible to the longwave infrared via an inverse-designed flat lens
,” arXiv:2001.03684.
15.
M.
Meem
,
S.
Banerji
,
A.
Majumder
,
C.
Dvonch
,
B.
Sensale-Rodriguez
, and
R.
Menon
, “
Imaging across the short-wave infra-red (SWIR) band via a flat multi-level diffractive lens
,”
OSA Continuum
2
(
10
),
2968
2974
(
2019
).
16.
S.
Banerji
,
M.
Meem
,
A.
Majumder
,
B.
Sensale-Rodriguez
, and
R.
Menon
, “
Imaging over an unlimited bandwidth with a single diffractive surface
,” arXiv:1907.06251.
17.
M.
Meem
,
S.
Banerji
,
A.
Majumder
,
F. G.
Vasquez
,
B.
Sensale-Rodriguez
, and
R.
Menon
, “
Broadband lightweight flat lenses for longwave-infrared imaging
,”
Proc. Natl. Acad. Sci. U. S. A.
116
,
21375
(
2019
).
18.
N.
Mohammad
,
M.
Meem
,
B.
Shen
,
P.
Wang
, and
R.
Menon
, “
Broadband imaging with one planar diffractive lens
,”
Sci. Rep.
8
,
2799
(
2018
).
19.
S.
Banerji
,
M.
Meem
,
A.
Majumder
,
B.
Sensale-Rodriguez
, and
R.
Menon
, “
Diffractive flat lens enables extreme depth-of-focus imaging
,”
Optica
7
(
3
),
214
2017
(
2020
).
20.
M.
Meem
,
S.
Banerji
,
A.
Majumder
,
C.
Pies
,
T.
Oberbiermann
,
B.
Sensale-Rodriguez
, and
R.
Menon
, “
Large-area, high-NA multi-level diffractive lens via inverse design
,”
Optica
7
(
3
),
252
253
(
2020
).
21.
H.
Liang
,
A.
Martins
,
B. H. V.
Borges
,
J.
Zhou
,
E. R.
Martins
,
J.
Li
, and
T. F.
Krauss
, “
High performance metalenses: Numerical aperture, aberrations, chromaticity, and trade-offs
,”
Optica
6
(
12
),
1461
1470
(
2019
).
22.
W. T.
Chen
,
A. Y.
Zhu
,
J.
Sisler
,
Z.
Bharwani
, and
F.
Capasso
, “
A broadband achromatic polarization-insensitive metalens consisting of anisotropic nanostructures
,”
Nat. Commun.
10
(
1
),
355
(
2019
).
23.
W. T.
Chen
,
A. Y.
Zhu
,
V.
Sanjeev
,
M.
Khorasaninejad
,
Z.
Shi
,
E.
Lee
, and
F.
Capasso
, “
A broadband achromatic metalens for focusing and imaging in the visible
,”
Nat. Nanotechnol.
13
,
220
226
(
2018
).
24.
Y.
Liang
,
H.
Liu
,
F.
Wang
,
H.
Meng
,
J.
Guo
,
J.
Li
, and
Z.
Wei
, “
High-efficiency, near-diffraction limited, dielectric metasurface lenses based on crystalline titanium dioxide at visible wavelengths
,”
Nanomaterials
8
(
5
),
288
(
2018
).
25.
M.
Ye
,
V.
Ray
, and
Y. S.
Yi
, “
Achromatic flat subwavelength grating lens over whole visible bandwidths
,”
IEEE Photonics Technol. Lett.
30
(
10
),
955
958
(
2018
).
26.
H. C.
Wang
,
C. H.
Chu
,
P. C.
Wu
,
H. H.
Hsiao
,
H. J.
Wu
,
J. W.
Chen
,
W. H.
Lee
,
Y. C.
Lai
,
Y. W.
Huang
,
M. L.
Tseng
,
S. W.
Chang
, and
D. P.
Tsai
, “
Ultrathin planar cavity metasurfaces
,”
Small
14
,
1703920
(
2018
).
27.
M.
Khorasaninejad
,
Z.
Shi
,
A. Y.
Zhu
,
W. T.
Chen
,
V.
Sanjeev
,
A.
Zaidi
, and
F.
Capasso
, “
Achromatic metalens over 60 nm bandwidth in the visible and metalens with reverse chromatic dispersion
,”
Nano Lett.
17
(
3
),
1819
1824
(
2017
).
28.
S.
Wang
,
P. C.
Wu
,
V.-C.
Su
,
Y.-C.
Lai
,
M.-K.
Chen
,
H. Y.
Kuo
,
B. H.
Chen
,
Y. H.
Chen
,
T.-T.
Huang
,
J.-H.
Wang
,
R.-M.
Lin
,
C.-H.
Kuan
,
T.
Li
,
Z.
Wang
,
S.
Zhu
, and
D. P.
Tsai
, “
A broadband achromatic metalens in the visible
,”
Nat. Nanotechnol.
13
,
227
(
2018
).
29.
S.
Banerji
,
M.
Meem
,
A.
Majumder
,
B.
Sensale-Rodriguez
, and
R.
Menon
, “
Imaging with flat optics: Metalenses or diffractive lenses?
,”
Optica
6
(
6
),
805
810
(
2019
).
30.
S.
Banerji
and
B.
Sensale-Rodriguez
, “
A computational design framework for efficient, fabrication error-tolerant, planar THz diffractive optical elements
,”
Sci. Rep.
9
,
5801
(
2019
).
31.
J. J.
Stamnes
and
V.
Dhayalan
, “
Focusing of electric-dipole waves
,”
Pure Appl. Opt.
5
,
195
225
(
1996
).
32.
J. J.
Stamnes
and
V.
Dhayalan
, “
Focusing of electric-dipole waves in the Debye and Kirchoff approximations
,”
Pure Appl. Opt.
6
,
347
372
(
1997
).

Supplementary Material