Articleshttps://doi.org/10.1038/s41565-017-0034-6

A broadband achromatic metalens for focusing and imaging in the visibleWei Ting Chen   1, Alexander Y. Zhu1, Vyshakh Sanjeev1,2, Mohammadreza Khorasaninejad1, Zhujun Shi3, Eric Lee1,2 and Federico Capasso1*

1Harvard John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA. 2University of Waterloo, Waterloo, ON, Canada. 3Department of Physics, Harvard University, Cambridge, MA, USA. *e-mail: capasso@seas.harvard.edu

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1

Supplementary Information for: 1 2

A broadband achromatic metalens for focusing and imaging 3

in the visible 4

Wei Ting Chen1, Alexander Y. Zhu1, Vyshakh Sanjeev1,3, Mohammadreza Khorasaninejad1, 5

Zhujun Shi2, Eric Lee1,3 and Federico Capasso1,* 6

1Harvard John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, 7 Massachusetts 02138, USA 8

2Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA 9

3University of Waterloo, Waterloo, ON N2L 3G1, Canada 10 11

*Corresponding author: capasso@seas.harvard.edu 12

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2

S1. Strong dispersion from periodicity 14 15

Figure S1(a) shows a schematic of a conventional glass prism, with a constant refractive 16

index (dispersionless). A broadband chromatic beam is deflected by this prism to the same angle. 17

Figure S1(b) is its diffractive counterpart comprising of periodic miniature prisms. Once these 18

miniature prisms are arranged together in a periodic manner, they start showing strong dispersion. 19

The inset of Fig. S1(b) shows an example, assuming that a given green wavelength λg is 20

diffracted to an angle θ following the Bragg equation: 21

sin( ) gmθ λΛ ⋅ = (1) 22

where Λ is the periodicity and m is an integer. From Eq. 1, another wavelength, gλ δλ+ , is 23

forbidden from propagating to the same angle θ ; it would have to go to a larger angle because of 24

the increase in wavelength. This results in a strong negative dispersion property compared to 25

refractory optics. Alternatively, this can be intuitively understood from the fact that a constant 26

wavenumber from periodicity (1

Λ) is applied to different wavelengths of incident light, causing 27

each wavelength to be deflected to a different angle. 28

29

Figure S1: Schematic illustrating the origin of the chromatic effect in periodic meta-surfaces and 30

diffractive optics. Colors are representative of their respective wavelengths (red, green and blue). 31

(a) A conventional glass prism. (b) Diffractive counterpart of (a). The inset in (b) shows a 32

magnified view, assuming that the green wavelength is deflected to an angle θ. The optical path 33

difference between the two green beams is equal to an integer multiplied by sin( )θΛ ⋅ . 34

3

S2. Scanning microscope images 35 36

37 38 Figure S2: Scanning microscope images of an achromatic metalens. (a) and (b) Top view 39

images. Scale bar: 1 μm. The inset of (a) shows an oblique view with larger magnification. Scale 40 bar: 500 nm. 41

42

4

S3. Available choices of element at wavelength λ = 530 nm 43 44

45 46

Figure S3: Polarization conversion (PC) efficiency versus group delay for different 47

elements after filtering those with PC efficiencies lower than 5%. Each dot represents an 48

element with its x and y coordinates determined by the group delay and PC efficiency 49

respectively. The group delays are obtained by linearly fitting the phase plots at λ = 530 nm 50

within a 120 nm bandwidth. The elements with R-squared values lower than 0.98 are dropped. 51

Most of the high polarization conversion efficiency nano-fins have group delays between 2 to 5 52

femto-seconds. This is a result of the waveguiding effect being dominant. The observed group 53

delay values can be obtained by substituting the appropriate neff between 1 and 2.4 (the refractive 54

index of air and TiO2) into the first term of Eq. 5 in the main text. 55

56

5

S4. Focal spot characterization 57 58

(a) (b)

450 500 550 600 650 700Wavelength (nm)

0.8

0.85

0.9

0.95

1S

treh

l rat

io

59 60 Figure S4: (a) Strehl ratio and (b) full-width at half-maximum (FWHM) for the achromatic 61

metalens with a numerical aperture of 0.2. Their corresponding focal spot profiles are shown 62

in Fig. 3(f) of the main text. The dashed black line shows the theoretical FWHM of the Airy disk. 63

64

6

S5. Focal spot under incoherent illumination 65

66

67 68

Figure S5: Focal spot profile for the achromatic metalens (NA = 0.2) under broadband 69

incoherent illumination. (a) Focal spot profile taken by a color CCD camera (UI-1540SE, IDS 70

Inc.). Scale bar: 5 μm. (b) Normalized intensity of light source measured by a spectrometer 71

(USB4000, Ocean optics) for a Tungsten source coupled to a light guide (OSL1, Thorlabs Inc.). 72

The focal spot size is slightly beyond the diffraction limit because the light guide has ~ 5 mm in 73

diameter and the metalens is not corrected for the whole spectrum. A pair of crossed circular 74

polarizer was used to remove background. 75

76

7

S6. Focal spot profiles and images from a resolution target 77 78

79 80

81 Figure S6: A comparison for the measured focal spot profiles and images of the USAF 82

resolution target using the achromatic and diffractive metalenses (NA = 0.02, diameter = 83

220 μm) for different wavelengths. (a) and (c) Focal spot profiles of achromatic and diffractive 84

metalenses. Scale bar: 20 μm. These focal spot profiles were taken without re-focusing to 85

visualize chromatic focal length shift. The bottom rows show intensity along the horizontal cut 86

through the center of each focal spot. The illumination wavelength (with a bandwidth of about 5 87

nm) is denoted on the top. (b) and (d) Imaging using the metalenses. Schematic set-up is shown 88

in Fig. S11. These images were taken at the focal plane for wavelength λ = 470 nm. Scale bar: 89

100 μm. 90

91 92

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S7. Focal spot characterization under oblique incidence. 93 94

0 5 10 15 20Angle of incidence (deg)

15

16

17

18

19

20= 670 nm

0 5 10 15 20Angle of incidence (deg)

10

11

12

13

14

15= 470 nm

0 5 10 15 20Angle of incidence (deg)

12

13

14

15

16

17= 530 nm

0 5 10 15 20Angle of incidence (deg)

0.5

0.6

0.7

0.8

0.9

1= 670 nm

0 5 10 15 20Angle of incidence (deg)

0.5

0.6

0.7

0.8

0.9

1= 530 nm

0 5 10 15 20Angle of incidence (deg)

0.5

0.6

0.7

0.8

0.9

1= 470 nm

(a)

(b)

95 96

Figure S7: Analysis for the achromatic metalens (NA = 0.02, diameter = 220 μm) under 97

different angles and wavelengths of incidence. (a) and (b) Experimentally measured full-width 98

at half-maximum (FWHM) and Strehl ratio for different angles of incidence and wavelengths. 99

The theoretical values of FWHM are marked as dashed lines. The Strehl ratio indicates how 100

close a focal spot is compared to the theoretical Airy disk profile. A Strehl ratio larger than 0.8 is 101

commonly acknowledged as a requirement of diffraction-limited focusing. 102

103

9

S8. Efficiencies as a function of wavelength 104

105

106 107

Figure S8: Efficiencies of metalenses. (a) Measured (blue line) versus theoretical efficiencies 108

(red and yellow lines) for the achromatic metalens (NA = 0.02, diameter = 220 μm). The 109

efficiency is defined by the power of focal spot divided by that of light passing through an 110

aperture with the same diameter. The power of the focal spot was measured by placing a power 111

meter on the image plane of a custom-built microscope. An aperture with a size about the 112

diameter of Airy disk was used to filter out the background light. Theoretical values were 113

obtained by the averaged polarization conversion efficiency of each element on the achromatic 114

metalens. Although the coupling between each element and the diffraction effect are neglected, 115

this gives an upper limit of efficiency. (b) Measured efficiencies of metalenses with the same NA 116

of 0.02 but different diameters. The diameters are labeled in the legend. The efficiency becomes 117

higher for smaller diameters because the required group delay is smaller than our current realized 118

values. This avoids choosing those efficiency elements (see Fig. S3 for a plot of group delay 119

versus efficiency). 120

121 122

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S9. Correcting aberrations using dispersion-engineered metalens. 123

Using our approach, it is possible to correct not only monochromatic but also chromatic 124

aberrations of a refractive lens. As an example, we chose a commercially available and low-cost 125

plano-convex lens from Thorlabs Inc. Figure S9(a) shows a raytracing diagram at wavelength λ 126

= 530 nm. A metalens is attached to the planar side of the lens (depicted by the blue line). The 127

diameter of the entrance aperture is 5.4 millimeters, and the refractive/metalens doublet has a 128

numerical aperture of about 0.1. The frequency-dependent metalens was designed by the 129

principle described in Fig. 1 in the main text, i.e. it needs to provide various group delays and 130

group delay dispersions such that all wavepackets from different lens coordinates arrive at the 131

focus together and with the same pulse shape. The green and black curves in Fig. S9(b) show 132

focal spot intensity profiles with and without the metalens, respectively. The metalens corrects 133

the spherical aberration of the refractive lens by introducing a W-shaped phase profile similar to 134

the well-known Schmidt plate (see the green curve in Fig. S9(c)). Moreover, the phase profile of 135

the metalens changes as a function of wavelength to correct chromatic aberration simultaneously. 136

The required range of group delay and group delay dispersion for this frequency dependent phase 137

profile is shown in Fig. S9(d). These required values are only a few times larger than those 138

provided by our current library shown in Fig. S10(a) and can be readily achieved with 139

modifications of the nanostructure design. The focal length shift is about 700 μm for the 140

uncorrected lens, i.e. without the metalens (the orange curve of Fig. S9(e)). Intriguingly, if one 141

only corrects for group delay (see the red curve of Fig. S9(e)), the focal length changes by an 142

amount similar to that of an achromatic doublet. Taking the group delay dispersion into account 143

results in a performance close to a triplet lens, as shown in the blue curve of Fig. S9(e). The 144

11

metalens-corrected spherical lens is now achromatic and diffraction-limited from 450 nm to 700 145

nm, as seen in Fig. S9(f) with < 0.075·λ root-mean-square wavefront error. 146

147

Figure S9: Simulation results for aberration correction using a metalens. The refractive lens 148

is a generic spherical lens available from Thorlabs Inc. The metalens corrector was designed 149

with a frequency dependent phase profile, using the method described in the text. (a) A 150

raytracing simulation of the refractive/metalens doublet at λ = 530 nm. The simulation was done 151

using a commercial software OpticsStudio (Zemax Inc.). The layout of the refractive lens was 152

obtained from Thorlabs’ website. (b) Focal spot intensities with and without the metalens. The 153

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spherical lens suffers spherical aberrations resulting in low Strehl ratio (black curve) away from 154

the diffraction limit. (c) Phase profile across the center of the frequency-dependent metalens. The 155

chosen wavelengths are shown in the legend; units are in nanometers. (d) The required group 156

delay and group delay dispersion from the center to the edge of the metalens. (e) A comparison 157

between relative focal length shifts for the refractive lens (orange), the metalens with engineered 158

group delay only (red) and the metalens with simultaneously engineered group delay and group 159

delay dispersion. (f) Root-mean-square wavefront error of the refractive/metalens doublet. The 160

metalens provides group delay and group delay dispersion, showing the frequency-dependent 161

phase profile in (c). The black dashed line shows a wavefront error of 0.075·λ, corresponding to 162

Strehl ratio of 0.8. 163

164 165

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S10. Group delay and group delay dispersion plot of elements 166

-1 0 1 2 3 4 5 6-8

-4

0

4

8

12

Gro

up d

elay

dis

pers

ion

(fs2 )

Group delay (fs)

-15 -10 -5 0 5 10 15

0

1

2

3

4

Gro

up

dela

y (f

s)

Coordinate (μm)

-15 -10 -5 0 5 10 15

-2

-1

0

1

Gro

up

dela

y d

isp

ersi

on

(fs2

)

Coordinate (μm)

Required

Original data + offsets

Original data

(a) (b)

(c)

RequiredRealized

RequiredRealized

167 168 Figure S10: (a) Group delay (GD) and group delay dispersion (GDD) of elements (colored 169

circles) versus that of the required GD and GDD (black circles) for realizing a metalens with n = 170

2. The coordinate of each purple circle represents the group delay and group delay dispersion of 171

an element. The group delay and GDD were obtained by fitting the phase as a function of 172

angular frequency using a quadratic polynomial for a bandwidth of 120 nm centered at 530 nm. 173

Only the elements with R2 values larger than 0.99 are shown. Since only the relative group delay 174

and GDD of the metalens need to be fulfilled, one can introduce two different offsets to the 175

group delay and GDD, which is equivalent to translating the data for the best fit. We determine 176

the appropriate offsets using the particle swarm method, and the data after adding the offsets are 177

shown in green symbols. (b) and (c) Realized GDs and GDDs (green symbols) versus that of the 178

required (black lines). 179

180

181

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S11. Experimental setup used for imaging with metalenses. 182 183 For imaging, a pair of crossed circular polarizer was used to reduce background signals. The 184

images were projected on a semi-transparent screen to show the full field of view, without being 185

limited by the size of a camera sensor. The tube lens was removed to show the whole view given 186

in Fig. 4(c) and 4(d) in the main text. We adjust the distance between the resolution target and 187

the achromatic metalens such that the image is best focused on the screen at wavelength λ = 470 188

nm. This distance is maintained when changing laser center wavelength from 470 nm to 670 nm 189

in steps of 20 nm. For each wavelength, the laser bandwidth is 40 nm and a camera (Fujifilm X-190

T10) was utilized to take photos. The achromatic metalens was replaced by a diffractive 191

metalens for comparing the imaging quality. 192

Laser

Fiber Coupled Collimator

LP

10x

λ/4

Tube Lens

Resolution TargetLPλ/4

Condenser

Screen

CameraMetalens 193

Figure S11: Schematic diagram of the experimental setup used for imaging by the 194

achromatic metalens with a numerical aperture of 0.02. The laser beam is collimated by a 195

fiber collimator (Thorlabs, RC08APC-P01). The collimated beam then passes through a Glan-196

Thompson polarizer (Thorlabs, GTH10) and a quarter waveplate (Thorlabs, AQWP05M-600) to 197

generate circularly polarized light. The Mitutoyo objective (10× magnification, NA=0.28) was 198

used as a condenser for providing more intensive illumination on the target. The target was 199

placed at the focal plane of meta-lens under wavelength λ = 470 nm illumination. To reduce 200

background, we use another pair of quarter waveplate and wire or Glan-Thompson polarizer 201

aligned in crossed polarization with respect to that of the incident light. A tube lens with focal 202

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length f = 180 mm (Thorlabs, TTL180-A) was selectively used to form image on a semi-203

transparent screen. 204