Wide-band achromatic flat focusing lens based on all-dielectric subwavelength metasurface

Shaowu Wang; Jianjun Lai; Tao Wu; Changhong Chen; Junqiang Sun

doi:10.1364/OE.25.007121

1. Introduction

The shaping of the wavefront of light is an important part in the optical field. Conventional optical refractive components such as lenses, as well as diffractive elements such as grating, rely on gradual phase shift accumulated along the optical path, therefore these devices with large structure size or long optical path length are not suitable for integration. Besides, both refractive and diffractive optical components have fundamentally characteristic of dispersion. For some optical splitting systems this characteristic is advantageous, but for the optics imaging systems, it is disadvantageous which can cause chromatic aberrations. To solve this problem, conventional optical system usually applied complicated multiple bulky optical components or diffractive-refractive combination to achieve chromatic aberrations. In infrared waveband, the situation also leads to costly budget due to expensive and limited materials have to be used, so there is always a large demand to implement cheap and compact infrared optical systems in applications such as imaging cameras or light concentrators in photovoltaic cells systems [1].

In past years, metasurface which consists of subwavelength structures array in the thin film, as the new way to manipulate wavefront in subwavelength scale, attracts more and more attention [2–7]. By constructing different two-dimensional arrays, amount of functions about wavefront molding have been realized such as beam focusing [8–11], deflection [12–14], holography [15–17] and polarization [18, 19]. But these operations still remain wavelength dependent, in other words, the chromatic phenomenon also exists in metasurfaces structures. Recently, Francesco Aieta et.al demonstrated a new metasurface design based on coupled rectangular dielectric resonators array, which can realize achromatic wavefront control at three telecommunication wavelengths [20, 21], then achromatic metasurface structure became a research hotspot [22–25]. However, most of reported structures for achromatization are only limited in two or three wavelengths, although Yang Li et.al proposed an achromatic metasurface designed for sprectrum range from 1μm to 2μm based on surface plasmon polaritons (SPPs) metal-insulator-metal (MIM) waveguides [26], but owing to the characteristics of metallic material, the designed structure supports the function of achromatization only in limited infrared sprectrum range and has enormous energy loss. Therefore, the way to realize the achromatic wavefront control for wide-band spectral range with low energy loss is still a problem.

In this paper, we propose and investigate a method to design achromatic metasurface structure and realize wide-band achromatic wavefront control based on arrayed subwavelength all-dielectric slit waveguides with varied slit widths at long-wave infrared waveband. Each slit waveguide is designed to transmit light with specific phase shift controlled by the slit width, and appropriate period can cause amazing structural characteristic to support achromatic wavefront control. By selecting appropriate structural parameters, we design and simulate a wide-band achromatic flat focusing lens, and the vertically incident electromagnetic waves in spectral range from 8 $μ m$ to 12 $μ m$ can be focused on the designed litter spot. Because of the material properties of silicon and air, the designed structure presents low energy loss and high transmissivity in this infrared waveband. To our knowledge, this is the first time to realize achromatic wavefront control in such wide spectral range based on subwavelength all-dielectric metasurface.

2. Structural design and theoretical analysis

The designed achromatic flat focusing lens is schematically shown in Fig. 1, which is composed of arrayed subwavelength silicon-air slit waveguides. The whole structure includes amount of unit cells as shown in the inset, and each unit has uniform period (P) and length (L) but different slit width (w). When a TM-polarized (electric field along the x direction) plane electromagnetic wave normally illuminates the metasurface structure, the electromagnetic wave cannot be strongly confined in the high-index slabs (silicon) in the designed structure and inherent leaky modes will appear in the low-index slits (air). Furthermore, if the width of the slit is smaller than the characteristic decay length of the wave inside the slit, the wave can be confined in the air slits relying on the external reflections provided by the interference effects between the leaky modes, and transmits through each slit waveguide with low loss as shown in Fig. 2(a). In this situation, the slit waveguide eigenmode can be seen as being formed by the interaction between the fundamental eigenmodes of the individual slab waveguides, which is consistent with the theory about slit waveguide introduced by V. R. Almeida et al. in 2004. According to this theory, the effective refraction index of slit waveguide for fundamental TM eigenmode can be calculated by solving the transcendental characteristic equation [27]:

\begin{matrix} \tan [κ_{H} (P - w) - ϕ] = \frac{γ_{S} n_{H}^{2}}{κ_{H} n_{S}^{2}} \tan h (γ_{S} w / 2) \\ k_{0}^{2} n_{H}^{2} - κ_{H}^{2} = k_{0}^{2} n_{S}^{2} + γ_{S}^{2} = β^{2} n_{e f f} = β / k_{0} \\ ϕ = arc \tan [γ_{S} n_{H}^{2} / (κ_{H} n_{S}^{2})] \end{matrix}

where,

κ_{H}

is the transverse wave number in the high-index slabs (silicon),

γ_{S}

is the field decay coefficient in the slits (air), and

β

is the eigenmode propagation constant, n_H = n_si and n_S = n_air are the refractive index of high-index material (silicon) and low-index material (air),

n_{e f f}

is the effective refractive index of the slit waveguide, k₀ = 2π/λ is the vacuum wave number, λ is the wavelength of incident electromagnetic wave. From the Eq. (1), the effective refractive index (

n_{e f f}

) is directly related to the wavelength (λ) of incident electromagnetic wave, the period (P) and slit width (w) of unit cell. Figure 2(b) shows the effective refractive index of slit waveguide in the infrared spectral range, when the slit width (w) is changed from 0.1μm to 0.3μm and the period (P) is fixed at 1μm. As can be observed, the slit waveguide has different effective refraction index (

n_{e f f}

) when the wavelength varies from 5μm to 15μm, furthermore, as the increase of the width w, the effective refraction index will also increase. Therefore, simply tuning the slit width is a potential way to manipulate effective refraction index.

Fig. 1 Schematic of achromatic flat focusing lens based on all-dielectric subwavelength metasurface.

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Fig. 2 (a) Normalized electric field intensity distribution for slit waveguide with the period P = 1μm, the slit width w = 0.3μm, the length L = 15μm, at λ = 10μm. (b) Eeffective refractive index of unit cell with differernt slit widths in the spectral range from 5μm to 15μm.

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The phase of electromagnetic wave transmitted through the slit waveguide can be expressed as [9,14]:

φ = φ_{0} + n_{e f f} k_{0} L + Δ φ_{1} + Δ φ_{2} - θ

where L is the length of the slit waveguide,

φ_{0}

is the initial phase at the entrance of slit waveguide.

n_{e f f} k_{0} L

is the accumulated phase caused by electromagnetic wave propagating in the slit, and it is the main factor of the phase retardation in slit waveguide.

Δ φ_{1} = \arg [(n_{1} - n_{e f f}) / (n_{1} + n_{e f f})]

and

Δ φ_{2} = \arg [(n_{e f f} - n_{2}) / (n_{2} + n_{e f f})]

indicate the phase shift at the entrance and exit interface respectively, where n₁ and n₂ are the refractive indices of the media outside the slit waveguide. In the designed structure, n₁ = n₂ = n_air = 1, therefore ∆φ₁ and ∆φ₂ have the same value but opposite signs which can be cancelled together. The last term

θ = \arg {1 - {[\frac{1 - n_{e f f}}{1 + n_{e f f}}]}^{2} \exp (i 2 n_{e f f} k_{0} L)}

originating from the multiple reflections between the entrance and exit. According to our calculation, the value of θ is very small compared with the value of

n_{e f f} k_{0} L

, and can be seen as a perturbation. For physical analysis, this term is not the main factor of the phase shift and can be ignored. Therefore, the equation of electromagnetic wave through the slit waveguide can be simplified as:

φ = φ_{0} + n_{e f f} k_{0} L

From the above equation, the phase shift for two different slit widths is determined by

Δ φ = 2 π L [n_{e f f} (w_{2}) - n_{e f f} (w_{1})] / λ

, which means that adjusting the slit width can manipulate the phase shift to realize wavefront control for single wavelength. For different wavelengths of incident electromagnetic wave (for example λ₁, λ₂), their phase shift

Δ φ

caused by varied widths (w₁, w₂) can be expressed as:

Δ φ_{1, 2} = 2 π L [n_{e f f} (w_{2}, λ_{1, 2}) - n_{e f f} (w_{1}, λ_{1, 2})] / λ_{1, 2}

∆φ_1,2 are the phase shift for λ₁ and λ₂. As can be seen, ∆φ_1,2 are non-uniform even the slit widths are set as the same value. The influence of wavelength must be considered for controlling the wavefront with multi-wavelengths, such as achromatic flat focusing. To realize the function of focusing, the required phase shift distribution of electromagnetic wave along x direction from the center of the lens has to be:

Δ φ_{f} = 2 π (f - \sqrt{f^{2} + x^{2}}) / λ

where f is the focus length of the lens, x is the distance from the center of the lens. The designed phase shift ∆φ must be consistent with the required phase shift

Δ φ_{f}

along x direction to realize the focusing function (∆φ = ∆φ_f). If the focus length of lens is fixed for emitted electromagnetic waves with multi-wavelengths, it means that the designed structure has no chromatic for the required spectral range. In this case, in the same position along x direction, we can get the equation:

Δ φ_{f_{1, 2}} λ_{1, 2} / 2 π = (f - \sqrt{f^{2} + x^{2}}) =con s t

where

Δ φ_{f 1, 2} λ_{1, 2} / 2 π

are the achromatic phase shift for λ₁ and λ₂. Then through combining Eq. (4) and Eq. (6), we can obtain:

Δ φ_{1, 2} λ_{1, 2} / 2 π = Δ φ_{f 1, 2} λ_{1, 2} / 2 π = [n_{e f f} (w_{2}, λ_{1, 2}) - n_{e f f} (w_{1}, λ_{1, 2})] L = c o n s t

In this equation, w₁ is the slit width of the center position of the lens, w₂ is the slit width of one position along x direction. From the Eq. (7) we will get another important equation:

[n_{e f f} (w_{2}, λ_{2}) - n_{e f f} (w_{1}, λ_{2})] L - [n_{e f f} (w_{2}, λ_{1}) - n_{e f f} (w_{1}, λ_{1})] L = 0

To achieve the above equation, the relationship between effective refractive index and wavelength must satisfy one of the following conditions: (1) the effective refractive index is independent of wavelength; (2) the difference of effective refractive index caused by different slit widths is independent of wavelength.

As we know from Eq. (1), the period of unit cell is another important structural parameter for determining the effective refractive index of the slit waveguide. Figure 3 shows the effective refractive index of unit cell with different periods. The period is set as 0.5μm, 1.0μm and 2.0μm respectively, but the fill ratio F = w/P (width / period) is fixed at 0.2 to ensure the small difference of effective refractive index because of varied periods. In Fig. 3, all of the curves become more and more flat as the increase of the wavelength. The effective refractive index almost has no change in spectrum range from 5μm to 15μm when the period is 0.5μm (P = 0.5μm). The same phenomenon appears in spectrum range from 10μm to 15μm when the period is 1.0μm (P = 1.0μm). But when the period is 2.0μm (P = 2.0μm), the effective refractive index changes as the variation of wavelengths. It is known that the material dispersion of silicon dielectric is small enough and can be ignored at mid-wave and long-wave infrared spectral range (λ>3μm). Moreover, the structure dispersion of the subwavelength silicon-air slit waveguide will also disappear if the period of unit cell is far less than the incident wavelength (generally $P \leq λ / 10$ ), therefore the whole designed structure almost has no dispersion.

Fig. 3 Effective refractive index of unit cell with different periods in the spectral range from 5μm to 15μm.

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Figure 4(a) presents the effective refractive index of unit cell as varied slit widths when the period is set as 0.5μm, 1.0μm and 2.0μm while the wavelength of incident electromagnetic wave is changed from 8μm to 12μm, to verify the conclusions of our research about the dispersion characteristics of whole designed structure. By setting P = 0.5μm (here P = λ_min /16, less than λ_min /10, and λ_min = 8μm is the minimum wavelength in the operating wavelength range), the curves of the effective refractive index with different wavelengths are completely coincident, in other words, the effective refractive index is independent of wavelength which meets the first condition mentioned above to control the wavefront with multi-wavelengths. When the P = 1.0μm (P = λ_min /8, close to λ_min /10), the curves of the effective refractive index do not coincide completely especially in the width range from 0.1μm to 0.5μm, but these curves are parallel to each other in the width range from 0.1μm to 0.35μm shown in Fig. 4(b), and the difference value of effective refractive index between each wavelength is fixed in this width range, therefore the difference of effective refractive index caused by different slit widths is independent of wavelength, which is consistent with the second condition of manipulating the wavefront with multi-wavelengths. If the period is 2.0μm (P = λ_min /4, larger than λ_min /10), the curves of the effective refractive index for different wavelengths do not coincide or parallel. In this situation, the effective refractive index or the difference of effective refractive index are dependent of wavelength and it is hardly to achieve the excellent effect of achromatic wavefront manipulating.

Fig. 4 Effective refractive index of unit cell vary as slit widths for the cell with non-uniform period of unit at different wavelengths of incident electromagnetic wave(a), and the cell with uniform period P = 1.0μm at different wavelengths of incident electromagnetic wave (b).

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Next, a wide-band achromatic flat focusing lens composed by 141 unit cells with uniform period, length (P = 1.0μm, L = 15μm) and varied widths is investigated. The designed focus length is 200μm from the central of lens, and the operating wavelength range is from 8μm to 12μm. Figure 5 shows the phases shift (∆φ) of slit waveguide with different slit widths [Fig. 5(a)] and wavelengths [Fig. 5(b)] compared with the phase of the central position of the designed lens, where the slit width is w = 0.1μm. As the slit width increases, a clear trend can be observed that the phase shift will increase. This phenomenon is consistent with the relation between the effective refractive index and slit width, indicating the wider slit widths have smaller $n_{e f f}$ and cause larger difference values of $n_{e f f}$ and ∆φ. An opposed trend will appear as the increase of wavelength and can be explained by Eq. (4) that ∆φ shows inverse ratio with the wavelength. Therefore, in the spectral range from 8μm to 12μm, ∆φ at λ = 12μm has the smallest value under the same width variation. when the width w is changed from 0.1μm to 0.33μm, the ∆φ at λ = 12μm can reach 2π, and in this situation the necessary condition for full control of the wavefront in the sprectrum range from 8μm to 12μm can be achieved.

Fig. 5 (a) Phase shift of unit cell with differernt silt widths. (b) Phase shift of unit cell with differernt wavelengths of incident electromagnetic wave.

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The achromatic phase shift (∆φλ/2π) of slit waveguides varying with wavelength for different slit widths, as well as its varation with slit width at diffrerent wavelengths, are obtained and shown in Fig. 6. In Fig. 6(a), the curves with differernt slit widths are nearly parallel to each other in the spectral range from 8μm to 12μm, and the fluctuation of curves is caused by the perturbation term θ in the Eq. (2). Furthermore, in Fig. 6(b), all of the curves with differernt wavelengths are almost coincidence in the width range from 0.1μm to 0.35μm. These phenomena show that the phase shifts caused by different slit widths are independent of wavelengths, which match the theoretical analysis, and indicate the designed structure has the foundation to realize wide-band achromatic flat focusing lens.

Fig. 6 (a) Achromatic phase shift of unit cell with differernt silt widths. (b) Achromatic phase shift of unit cell with differernt wavelengths of incident electromagnetic wave.

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3. Simulation results and discussion

According to Eq. (6), we can calculate the required achromatic phase shift in different position along x direction for a broadband focusing with certain focal length. Furthermore, based on the results shown in Fig. 6(b), we can get the slit widths distribution of various slit waveguides along x direction corresponding to the achromatic phase shift data. The designed slit widths distribution (in the range from 0.1μm to 0.33μm) along x direction are shown in Fig. 7(a). Figures 7 (b) and 7(c) present the phase shift and achromatic phase shift for different wavelength (8μm, 9μm, 10μm, 11μm, 12μm) along x direction respectively. The designed phase shift distribution for different wavelengths is consistent with the required phase shift through Eq. (5). Besides, the curves of the designed achromatic phase shift for different wavelengths are almost coincident to support achromatic focusing.

Fig. 7 (a) Slit widths distribution along x direction. (b) Phase shift distribution for different wavelengths along x direction. (c) Achromatic phase shift distribution for different wavelengths along x direction.

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When the structure data of various slit waveguides is obtained, we can simulate the focusing properties of this planar metasurface at interested infrared waveband. When the TM-polarized vertically incident plane electromagnetic wave transmits through the designed structure, the electromagnetic waves will be focused on one point as shown in Fig. 8(a)-8(e), which are the normalized electric field intensity distribution for different wavelengths (8μm, 9μm, 10μm, 11μm, 12μm). Besides, the average transmissivity of designed lens can reach 70% in spectral range from 8μm to 12μm. In Fig. 8(a)-8(e), these focus points are roughly in the same position and consistent with the designed focus point about 200μm away from the central of lens, but the sizes of focus spots gradually increase with the wavelengths of incident electromagnetic wave, as can be seen from the Fig. 8(f), which show the normalized intensity peaks in the cross section of focus spot in x direction. Figure 8(g) indicates that focus points for wavelengths in spectral range from 8μm to 12μm are all close to the position in 200μm and the function of wide-band achromatic flat focusing is realized.

Fig. 8 (a-e) Normalized electric field intensity distribution for different wavelengths. (f) Cross section of focus spot in x direction for different wavelengths. (g) Focus position in spectral range from 8μm to 12μm.

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In the practical fabrication of the metasurface infrared lens, the required air slit array can be implementd by electron beam lithography (EBL) and dry etching on a silicon wafer. However, due to the limitation of micro/nano manufacturing technology, the fabricated device maybe differs from the designed one, and obtains different effect compared with above simulation results. Therefore, the influence of fabrication error needs to be considered.We just give a simple analysis. The most possible error may came from the deviation of widths of air slit array, which can directly affect the effective refractive index of unit cell ( $n_{e f f}$ ). From the Fig. 4(b) the curves of the effective refractive index for different wavelength show approximately linear relations with the width of air slit, and the achromatic phase shifts have the same relationships with the width of air slit in Fig. 6(b). Therefore, if the whole widths of air slits increase or decrease with a same deviation value such as 10nm, the difference of effective refractive index caused by varied slit widths and achromatic phase shift almost have no change. That is to say, this kind of small deviation have no serious influence on the performance of lens. However, since the actual fabrication condition is complicated, a comprehensive fabricaiton error modeling is required to fulfill an accurate analysis.

4. Conclusion

In summary, we present a way for realizing achromatic wavefront control based on all-dielectric subwavelength metasurface in mid-infrared and far-infrared spectral range. Research results indicate that the designed subwavelength silicon-air slit waveguide array has the characteristic of non-dispersive when the period of unit cell is far less than the wavelength of incident electromagnetic wave (about λ/10), furthermore, the effective refraction index of unit cell and the desired phase shift distribution along the exit facet of the lens can be adjusted just by the slit width of silicon-air slit waveguide. Therefore, the wide-band achromatic flat focusing lens composed by amount of arraied subwavelength silicon-air slit waveguides with appropriate period, uniform length and varied slit widths (P = 1.0μm, L = 15μm, w = 0.1μm~0.33μm) is shown and investigated. Through simulation calculation, the electromagnetic waves in spectral range from 8μm to 12μm can be focused on one point and these focus points are all close to designed focus point (about 200μm away from the central of lens), which means the function of wide-band achromatic flat focusing is realized in this spectrum range, furthermore the average transmissivity of designed lens can reach 70% in spectral range from 8μm to 12μm, and the designed structure presents the characteristic of low energy loss and high transmissivity. Compared with conventional dielectric flat lenses optical system or metasurface lenses, the designed lens based on simple structure has realized the function of achromatic wavefront control in such wide spectral range (from 8μm to 12μm). This work provides an effective solution for wide-band achromatic flat optical elements and shows potential applications in integrated achromatic optics imaging systems especially in infrared sensor fields.

Funding

National Natural Science Foundation of China (NSFC) (61474051); Fundamental Research Funds for the Central Universities (HUST: 2016YXMS022).

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Wide-band achromatic flat focusing lens based on all-dielectric subwavelength metasurface

Abstract

1. Introduction

2. Structural design and theoretical analysis

3. Simulation results and discussion

4. Conclusion

Funding

Reference and links

Cited By

Figures (8)

Equations (8)

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