# Characteristic Analysis of Compact Spectrometer Based on Off-Axis Meta-Lens

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Design Principle of Spectrometer Based on Off-Axis Meta-Lens

_{d}imparted by it follows

_{d}is the design wavelength (corresponding to λ

_{2}in Figure 1) and f is the focal length.

_{1}, λ

_{2}and λ

_{3}indicated in Figure 1 as an illustration) has a skew orientation angle with the optical axis. The output plane (OP) where the detector is placed does not always coincide with the actual focal plane in practice. A local coordinate system (u, v, w) centered at the focal point of the design wavelength (f sinα, 0, f cosα) is introduced at a relation with (x, y, z) by

_{i}(x, y, 0) is written as

_{o}(u, v, 0) is expressed by [41]

## 3. Results and Discussion

#### 3.1. Simulation Results of a Typical Meta-Lens-Based Spectrometer

_{d}= 1550 nm, as shown in Figure 2. It corresponds to a small numerical aperture NA = 0.035 (see Equation (5) below). The focal line profile along the u-axis as λ varies in 1520–1580 nm is shown in Figure 2a. The dispersion-caused spatial focal spot distribution ensures the technical practicability to develop a meta-lens based spectrometer. One can see the focal line gets widened obviously as wavelength significantly deviates from the design wavelength due to the chromatic aberration and the mismatch of the actual focal plane and the OP. In this spectral range, the displacement Δu is nearly linearly dependent on wavelength, suggesting a linear dispersion du/dλ = 9.95 μm/nm. A diffraction-limited spatial full width at half-maximum (FWHM) of the focal spot of 19.6 μm (12.6 λ

_{d}) at λ

_{d}= 1550 nm is observed from Figure 2b. It raises slowly as wavelength deviates from λ

_{d}. We introduce the effective spectral range Δλ to evaluate the performance of the spectrometer. Within this spectral range, the FWHM of the spot line is less than 1.2 times $\mathrm{FWHM}{|}_{\lambda ={\lambda}_{\mathrm{d}}}$. In fact, this criterion is much stringent compared with previous studies about optical spectrometers and the commercial products [21]. For the configuration proposed here, Δλ = 1574 nm − 1526 nm = 48 nm. The spectral resolution δλ as a function of wavelength define by the FWHM value of the U

_{o}(λ) curve is also shown in Figure 2b. The inset of Figure 2b plots the U

_{o}(λ) intensity along the dashed line in Figure 2a at u = 0. Over the spectral range 1524–1574 nm, the spectral resolution is within 1.8 nm. Figure 2c plots the field intensity (normalized by the maximum of the field intensity at the design wavelength λ

_{d}) profile along the u-axis at different wavelengths at an interval of 1.5 nm. Different spectral lines are distinguishable in this interval. These results are acquired without considering the actual pixel size of the detector.

#### 3.2. Relationship between Structural Parameters and Evaluation Indexes of Spectrometer

#### 3.3. Two Practical Device for Different Applications

_{d}= 1550 nm. Its focal line along the u-axis and spectral resolution as a function of wavelength are shown in Figure 5a,b, respectively, indicating a remarkably wide spectral range. This device will work at the band 800–1800 nm with the spectral resolution of 2–5 nm. The dimension of this configuration is estimated as 70 × 20 × 10 mm

^{3}. This configuration is compact and its spectral range and spectral resolution is comparable to the traditional commercial mini-spectrometer that has the similar working parameters, for example, Hamamatsu Photonics C11482GA [44]. The structural parameters for the narrowband spectrometer with high spectral resolution are f = 30 mm, D = 6 mm, α = 45°, β = −65° at λ

_{d}= 850 nm. Its focal line along the u-axis and the spectral resolution as a function of wavelength are shown in Figure 5b,c, respectively, indicating a narrow spectral range 780–920 nm. Within this spectral range, the spectral resolution is within 0.15–0.6 nm. Its dimension is approximately 40 × 30 × 10 mm

^{3}. This results are also comparable to the traditional commercial spectrometer such as Hamamatsu Photonics C13054MA [44].

#### 3.4. Influence of the Fabrication Error Analysis

_{m}of the focal spot. The latter two are taken at the design wavelength. In the calculation, a random noise using the built-in function of the MATLAB is applied to the phase profile in Equation (1), whose value obeys a normal distribution $\mathcal{N}\left(0,{\sigma}^{2}\right)$. Here, σ denotes the standard deviation. It is seen that the maximum intensity I

_{m}decreases when the value σ increases, indicating the weakened focusing efficiency at larger phase distortion. However, the effective spectral range and the spectral resolution at the design wavelength almost remain stable as σ changes because they are solely contributed by the zeroth diffraction order. These effects could be more clearly understood from the focusing pattern plotted in Figure 6b–d at σ = 0°, 90° and 120°, respectively. The spatial characteristics of the zeroth diffraction order are hardly influenced when the phase distortion gets more deteriorated. Its intensity decreases as the unwanted higher diffraction orders become stronger. For practice, we can expect that the proposed off-axis meta-lens can suffer a spatial phase distortion with the largest deviation less than 48° corresponding to a 50% amplitude reduction where the higher diffraction orders are thought weakly enough and have no influence on the performance of the spectrometer. This is relatively a large tolerance and allows the device to be precisely implemented.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Staude, I.; Schilling, J. Metamaterial-Inspired Silicon Nanophotonics. Nat. Photonics
**2017**, 11, 274–284. [Google Scholar] [CrossRef] - Liu, L.; Zhang, X.; Kenney, M.; Su, X.; Xu, N.; Ouyang, C.; Shi, Y.; Han, J.; Zhang, W.; Zhang, S. Broadband Metasurfaces with Simultaneous Control of Phase and Amplitude. Adv. Mater.
**2014**, 26, 5031–5036. [Google Scholar] [CrossRef] [PubMed] - Ni, X.; Emani, N.K.; Kildishev, A.V.; Boltasseva, A.; Shalaev, V.M. Broadband Light Bending with Plasmonic Nanoantennas. Science
**2012**, 335, 427. [Google Scholar] [CrossRef] [PubMed] - Pors, A.; Nielsen, M.G.; Eriksen, R.L.; Bozhevolnyi, S.I. Broadband Focusing Flat Mirrors Based on Plasmonic Gradient Metasurfaces. Nano Lett.
**2013**, 13, 829–834. [Google Scholar] [CrossRef] [PubMed] - Arbabi, A.; Horie, Y.; Bagheri, M.; Faraon, A. Dielectric Metasurfaces for Complete Control of Phase and Polarization with Subwavelength Spatial Resolution and High Transmission. Nat. Nanotechnol.
**2015**, 10, 937–943. [Google Scholar] [CrossRef] [PubMed] - Balthasar Mueller, J.P.; Rubin, N.A.; Devlin, R.C.; Groever, B.; Capasso, F. Metasurface Polarization Optics: Independent Phase Control of Arbitrary Orthogonal States of Polarization. Phys. Rev. Lett.
**2017**, 118, 113901. [Google Scholar] [CrossRef] [PubMed] - Khorasaninejad, M.; Crozier, K.B. Silicon Nanofin Grating as a Miniature Chirality-Distinguishing Beam-Splitter. Nat. Commun.
**2014**, 5, 5386. [Google Scholar] [CrossRef] [PubMed] - Arbabi, A.; Arbabi, E.; Kamali, S.M.; Horie, Y.; Han, S.; Faraon, A. Miniature Optical Planar Camera Based on a Wide-Angle Metasurface Doublet Corrected for Monochromatic Aberrations. Nat. Commun.
**2016**, 7, 13682. [Google Scholar] [CrossRef] [PubMed] - Wen, D.; Yue, F.; Ardron, M.; Chen, X. Multifunctional Metasurface Lens for Imaging and Fourier Transform. Sci. Rep.
**2016**, 6, 27628. [Google Scholar] [CrossRef] [PubMed] - Yang, H.; Li, G.; Su, X.; Cao, G.; Zhao, Z.; Chen, X.; Lu, W. Reflective Metalens with Sub-Diffraction-Limited and Multifunctional Focusing. Sci. Rep.
**2017**, 7, 12632. [Google Scholar] [CrossRef] [PubMed] - Khorasaninejad, M.; Zhu, A.Y.; Roques-Carmes, C.; Chen, W.T.; Oh, J.; Mishra, I.; Devlin, R.C.; Capasso, F. Polarization-Insensitive Metalenses at Visible Wavelengths. Nano Lett.
**2016**, 16, 7229–7234. [Google Scholar] [CrossRef] [PubMed] - Khorasaninejad, M.; Chen, W.T.; Zhu, A.Y.; Oh, J.; Devlin, R.C.; Rousso, D.; Capasso, F. Multispectral Chiral Imaging with a Metalens. Nano Lett.
**2016**, 16, 4595–6000. [Google Scholar] [CrossRef] [PubMed] - Chen, W.T.; Zhu, A.Y.; Khorasaninejad, M.; Shi, Z.; Sanjeev, V.; Capasso, F. Immersion Meta-Lenses at Visible Wavelengths for Nanoscale Imaging. Nano Lett.
**2017**, 17, 3188–3194. [Google Scholar] [CrossRef] [PubMed] - Groever, B.; Chen, W.T.; Capasso, F. Meta-Lens Doublet in the Visible Region. Nano Lett.
**2017**, 17, 4902–4907. [Google Scholar] [CrossRef] [PubMed] - Ni, X.; Kildishev, A.V.; Shalaev, V.M. Metasurface Holograms for Visible Light. Nat. Commun.
**2013**, 4, 2807. [Google Scholar] [CrossRef] - Chen, W.T.; Yang, K.Y.; Wang, C.M.; Huang, Y.W.; Sun, G.; Chiang, I.D.; Liao, C.Y.; Hsu, W.L.; Lin, H.T.; Sun, S.; et al. High-Efficiency Broadband Meta-Hologram with Polarization-Controlled Dual Images. Nano Lett.
**2014**, 14, 225–230. [Google Scholar] [CrossRef] [PubMed] - Wen, D.; Yue, F.; Li, G.; Zheng, G.; Chan, K.; Chen, S.; Chen, M.; Li, K.F.; Wong, P.W.H.; Cheah, K.W.; et al. Helicity Multiplexed Broadband Metasurface Holograms. Nat. Commun.
**2015**, 6, 8241. [Google Scholar] [CrossRef] [PubMed] - Zheng, G.; Mühlenbernd, H.; Kenney, M.; Li, G.; Zentgraf, T.; Zhang, S. Metasurface Holograms Reaching 80% Efficiency. Nat. Nanotechnol.
**2015**, 10, 308–312. [Google Scholar] [CrossRef] [PubMed] - Zhang, C.; Yue, F.; Wen, D.; Chen, M.; Zhang, Z.; Wang, W.; Chen, X. Multichannel Metasurface for Simultaneous Control of Holograms and Twisted Light Beams. ACS Photonics
**2017**, 4, 1906–1912. [Google Scholar] [CrossRef] - Khorasaninejad, M.; Chen, W.T.; Oh, J.; Capasso, F. Super-Dispersive Off-Axis Meta-Lenses for Compact High Resolution Spectroscopy. Nano Lett.
**2016**, 16, 3732–3737. [Google Scholar] [CrossRef] [PubMed] - Zhu, A.Y.; Chen, W.T.; Khorasaninejad, M.; Oh, J.; Zaidi, A.; Mishra, I.; Devlin, R.C.; Capasso, F. Ultra-Compact Visible Chiral Spectrometer with Meta-Lenses. APL Photonics
**2017**, 2, 036103. [Google Scholar] [CrossRef] - Minovich, A.E.; Miroshnichenko, A.E.; Bykov, A.Y.; Murzina, T.V.; Neshev, D.N.; Kivshar, Y.S. Functional and Nonlinear Optical Metasurfaces. Laser Photonics Rev.
**2015**, 9, 195–213. [Google Scholar] [CrossRef] - Walter, F.; Li, G.; Meier, C.; Zhang, S.; Zentgraf, T. Ultrathin Nonlinear Metasurface for Optical Image Encoding. Nano Lett.
**2017**, 17, 3171–3175. [Google Scholar] [CrossRef] [PubMed] - Zhang, L.; Mei, Z.L.; Zhang, M.R.; Yang, F.; Cui, T.J. An Ultrathin Directional Carpet Cloak Based on Generalized Snell’s Law. Appl. Phys. Lett.
**2013**, 103, 151115. [Google Scholar] [CrossRef] - Ni, X.; Wong, Z.J.; Mrejen, M.; Wang, Y.; Zhang, X. An Ultrathin Invisibility Skin Cloak for Visible Light. Science
**2015**, 349, 1310–1314. [Google Scholar] [CrossRef] [PubMed] - Yang, Y.; Jing, L.; Zheng, B.; Hao, R.; Yin, W.; Li, E.; Soukoulis, C.M.; Chen, H. Full-Polarization 3D Metasurface Cloak with Preserved Amplitude and Phase. Adv. Mater.
**2016**, 28, 6866–6871. [Google Scholar] [CrossRef] [PubMed] - Born, M.; Wolf, E. Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed.; Cambridge University Press: Cambridge, UK, 1999; ISBN 978-0-521-64222-4. [Google Scholar]
- Bao, J.; Bawendi, M.G. A Colloidal Quantum Dot Spectrometer. Nature
**2015**, 523, 67–70. [Google Scholar] [CrossRef] [PubMed] - Henstridge, M.; Zhou, J.; Guo, L.J.; Merlin, R. Wavelength Scale Terahertz Spectrometer Based on Extraordinary Transmission. Appl. Phys. Lett.
**2017**, 111, 063503. [Google Scholar] [CrossRef] - Yuan, L.; He, Z.; Lv, G.; Wang, Y.; Li, C.; Xie, J.; Wang, J. Optical Design, Laboratory Test, and Calibration of Airborne Long Wave Infrared Imaging Spectrometer. Opt. Express
**2017**, 25, 22440–22454. [Google Scholar] [CrossRef] [PubMed] - Cook, T. Anamorphic Integral Field Spectrometer for Diffuse Ultraviolet Astronomy. Appl. Opt.
**2013**, 52, 8765–8770. [Google Scholar] [CrossRef] [PubMed] - Cataldo, G.; Hsieh, W.-T.; Huang, W.-C.; Moseley, S.H.; Stevenson, T.R.; Wollack, E.J. Micro-Spec: An Ultracompact, High-Sensitivity Spectrometer for Far-Infrared and Submillimeter Astronomy. Appl. Opt.
**2014**, 53, 1094–1102. [Google Scholar] [CrossRef] [PubMed] - Zavvari, A.; Islam, M.T.; Anwar, R.; Abidin, Z.Z.; Asillam, M.F.; Monstein, C. Analysis of Radio Astronomy Bands Using CALLISTO Spectrometer at Malaysia-UKM Station. Exp. Astron.
**2016**, 41, 185–195. [Google Scholar] [CrossRef] - Udeigwe, T.K.; Young, J.; Kandakji, T.; Weindorf, D.C.; Mahmoud, M.A.; Stietiya, M.H. Elemental Quantification, Chemistry, and Source Apportionment in Golf Course Facilities in a Semi-Arid Urban Landscape Using a Portable X-Ray Fluorescence Spectrometer. Solid Earth
**2015**, 6, 415–424. [Google Scholar] [CrossRef] - Buzan, E.M.; Beale, C.A.; Boone, C.D.; Bernath, P.F. Global Stratospheric Measurements of the Isotopologues of Methane from the Atmospheric Chemistry Experiment Fourier Transform Spectrometer. Atmos. Meas. Tech.
**2016**, 9, 1095–1111. [Google Scholar] [CrossRef] - Mantouvalou, I.; Lachmann, T.; Singh, S.P.; Vogel-Mikuš, K.; Kanngießer, B. Advanced Absorption Correction for 3D Elemental Images Applied to the Analysis of Pearl Millet Seeds Obtained with a Laboratory Confocal Micro X-ray Fluorescence Spectrometer. Anal. Chem.
**2017**, 89, 5453–5460. [Google Scholar] [CrossRef] [PubMed] - Shaltout, A.; Liu, J.; Kildishev, A.; Shalaev, V. Photonic Spin Hall Effect in Gap Plasmon Metasurfaces for On-Chip Chiroptical Spectroscopy. Optica
**2015**, 2, 860–863. [Google Scholar] [CrossRef] - Maguid, E.; Yulevich, I.; Veksler, D.; Kleiner, V.; Brongersma, M.L.; Hasman, E. Photonic Spin-Controlled Multifunctional Shared-Aperture Antenna Array. Science
**2016**, 352, 1202–1206. [Google Scholar] [CrossRef] [PubMed] - Ding, F.; Pors, A.; Chen, Y.; Zenin, V.A.; Bozhevolnyi, S.I. Beam-Size-Invariant Spectropolarimeters Using Gap-Plasmon Metasurfaces. ACS Photonics
**2017**, 4, 943–949. [Google Scholar] [CrossRef] - Zhou, Y.; Chen, R.; Ma, Y. Design of Optical Wavelength Demultiplexer Based on Off-Axis Meta-Lens. Opt. Lett.
**2017**, 42, 4716–4719. [Google Scholar] [CrossRef] [PubMed] - Goodman, J.W. Introduction to Fourier Optics, 3rd ed.; Roberts & Company Publishers: Englewood, IL, USA, 2005; ISBN 978-0-9747077-2-3. [Google Scholar]
- Noll, R.J. Zernike Polynomials and Atmospheric Turbulence. J. Opt. Soc. Am. A
**1976**, 66, 201–211. [Google Scholar] [CrossRef] - Welford, W.T. Aberrations of Optical Systems, 1st ed.; IOP Publishing: Bristol, UK, 1986; ISBN 978-0-85274-564-9. [Google Scholar]
- Hamamatsu Mini-Spectrometers Product Page. Available online: http://www.hamamatsu.com/eu/en/product/category/5001/4016/index.html (accessed on 29 November 2017).

**Figure 1.**Schematic illustration showing the configuration of the spectrometer based on an off-axis meta-lens. The design wavelength is λ

_{2}and the off-axis angle is α. The actual focal plane denoted by the dashed line has a skew orientation angle with the optical axis. The angle between the u-axis and the x-axis is defined as the orientation angle of the output plane (OP) β. The inset shows the definition of its sign. In addition to the global coordinate system (x, y, z) based on the meta-lens, a local coordinate system (u, v, w) based on the OP is adopted to simplify the deduction.

**Figure 2.**Dispersive characteristics of a meta-lens-based spectrometer: (

**a**) The focal line along the u-axis as a function of wavelength for the configuration with f = 20 mm, aperture diameter D = 2 mm, α = 45° and β = 45°; (

**b**) FWHM and spectral resolution δλ of the focal line as a function of wavelength. Inset shows the field profile along the dashed line in (

**a**); (

**c**) Electric field intensity normalized by the maximum of the field intensity at the design wavelength on the OP at different wavelengths (from 1526 to 1574 nm at a constant interval of 1.5 nm).

**Figure 3.**Influences of the orientation angle of the OP β on the dispersive properties of the meta-lens: (

**a**) The effective spectral range Δλ and spectral resolution δλ as a function of β. Arrows ascribe the curves to the left or right y-axis. The data for β close to −45° (indicated by the gray box) has low accuracy because the OP is parallel to the optical axis at this specific case; (

**b**) Ray tracing calculations for the meta-lens at wavelengths of 1525, 1550 and 1665 nm, respectively, indicating an optimal orientation angle β = −70°; (

**c**) Displacement Δu and FWHM of the focal line as a function of wavelength at β = −70°; (

**d**) Spectral resolution δλ as a function of wavelength at β = −70°.

**Figure 4.**Effective spectral range Δλ and spectral resolution δλ as a function of aperture diameter D when f = 20 mm and α = 45° (

**a**); focal length f when D = 2 mm and α = 45° (

**b**); off-axis angle α when D = 2 mm and f = 20 mm.

**Figure 5.**Two practical configurations for the actual application prospects: (

**a**) The focal line along the u-axis as a function of wavelength for the configuration with f = 30 mm, D = 2 mm, α = 15° and β = −89° at λ

_{d}= 1550 nm; (

**b**) The focal line along the u-axis as a function of wavelength for the configuration with f = 30 mm, D = 6 mm, α = 45° and β = −65° at λ

_{d}= 850 nm; (

**c**) Spectral resolution δλ as a function of wavelength of the configuration in (

**a**); (

**d**) Spectral resolution δλ as a function of wavelength of the configuration in (

**b**).

**Figure 6.**Influence of error analysis: (

**a**) The effective spectral range Δλ (black), the spectral resolution δλ (red) and the maximum intensity I

_{m}(green) of focal spot changing as a function of the random phase error factor σ. The values are normalized by those at σ = 0. Here the added spatial phase distortion value obeys a normal distribution $\mathcal{N}\left(0,{\sigma}^{2}\right)$ realized by the built-in random function in MATLAB; (

**b**–

**d**) The focus pattern of the proposed off-axis meta-lens at σ = 0°, 90° and 120°, respectively.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhou, Y.; Chen, R.; Ma, Y.
Characteristic Analysis of Compact Spectrometer Based on Off-Axis Meta-Lens. *Appl. Sci.* **2018**, *8*, 321.
https://doi.org/10.3390/app8030321

**AMA Style**

Zhou Y, Chen R, Ma Y.
Characteristic Analysis of Compact Spectrometer Based on Off-Axis Meta-Lens. *Applied Sciences*. 2018; 8(3):321.
https://doi.org/10.3390/app8030321

**Chicago/Turabian Style**

Zhou, Yi, Rui Chen, and Yungui Ma.
2018. "Characteristic Analysis of Compact Spectrometer Based on Off-Axis Meta-Lens" *Applied Sciences* 8, no. 3: 321.
https://doi.org/10.3390/app8030321