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Full-view imaging on dynamic closed surface by curved-to-flat conversion lens

Si Chen, Fei Sun, Yichao Liu, Yibiao Yang, and Zhihui Chen

Open Access Open Access

Abstract

Conventional full-view imaging systems, which often need complicated image processing algorithms to reconstruct full-view images captured by motional/multiple cameras from different views, cannot have good real-time imaging capability. We design curved-to-flat conversion lens (CFCL) based on optic-null medium, which can directly project/image optical patterns from closed object surface onto image plane (e.g., the focal plane of microscopy), and shows good real-time full-view imaging performance. To realize the CFCL, the reduced optic-null medium is designed by subwavelength metal channels filled with homogeneous isotropic dielectrics. Numerical simulation results verify the function of the designed CFCL, which can image various dynamic optical patterns from the closed object surface to the finite-view image plane. The designed CFCL may have many applications in real-timely observing dynamic closed surfaces in full view, e.g., living tissue/cell and soft material’s surface.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

In recent years, the researches about imaging the full-view of closed surfaces developed extensively for the need of various fields, such as 3D reconstruction of human body [1], 3D profile of material surface [2,3], biomedical measurement of time-varying surfaces [4,5], dynamic strain measurement on soft biological tissues [6,7], skin shape and deformation measurement [8], and etc. To measure geometrical profiles/deformations or observe optical patterns on dynamic closed surfaces (e.g., living cell and soft material’s surface), it requires real-time full-view imaging of closed surfaces. However, current methods to achieve full-view imaging of closed surfaces often need motional/multiple cameras and image processing algorithms, which restrict the real-time imaging capability. For instance, multi-view systems need stereo calibration and subset matching [4,9], which spend additional time to process a large amount of imaging data. In many occasions, to image dynamic micro-structures on dynamic closed surfaces, it needs a full-view imaging system that can directly image full-view of closed surfaces real-timely. However, these current methods need to first capture images of closed surfaces from several different viewpoints by motional/multiple cameras, and then processed by digital image processing algorithms to reconstruct full-view images, which are unable to directly image closed surfaces in full view and unsuitable for real-time imaging.

Based on invariance-form of Maxwell equations under coordinate transformations, transformation optics (TO) provides an effective method to control electromagnetic fields by designing appropriate coordinate transformations [10–12], which has been used to design invisible cloak [13], compact lens [14], superlens [15], etc. Optical surface transformation (OST) [16], stemmed from TO, can be used to design most devices (e.g., invisible cloak [17], hyper-lens [18], and concentrators [19], etc.) that previously could only be designed by TO. Compared with TO, OST is a graphical design method without involving complex mathematical calculations, and all devices of various functions designed by OST only need the same medium, i.e., optic-null medium (ONM). ONM is a highly anisotropic medium (i.e., the relative permittivity and permeability are extremely large along its main axis and almost zero in all other orthogonal directions), which can be regarded as perfect ‘endoscope’ that can project the electromagnetic field distribution along its main axis direction without phase delay and amplitude attenuation [16,20–27]. With the help of ONM’s perfect directional projecting feature, it provides a new way to achieve full-view imaging on dynamic closed surfaces.

In this study, a full-view curved-to-flat conversion lens (CFCL) is designed based on OST and ONM, which can directly image various dynamic optical patterns from closed object surface onto image plane real-timely in full view. When observer directly views the sample of closed surface (e.g., living cell) by traditional microscope, only micro-structures/patterns on the focal plane can be captured by the observer (patterns on other sides of the sample are lost; see Fig. 1(a)). If the designed CFCL is applied around the sample, patterns on the whole closed surface of the sample can be imaged directly onto the focal plane of the microscope, which means full-view image of the closed surface can be observed real-timely (see Fig. 1(b)). Compared with previous methods on full-view imaging of closed surfaces, the designed CFCL can directly achieve full-view imaging without complex calibration and additional image processing, which can avoid error of image stitching and improve the real-time performance of imaging.

 figure: Fig. 1.

Fig. 1. Illustration of observing the minute star-shaped sample of closed surface, which has different patterns on each side, by microscope without CFCL (a) and with CFCL (b), respectively. When the sample is observed by microscope without the designed CFCL as shown in Fig. 1(a), only finite-viewed patterns on the focal plane can be observed. However, if the sample is placed in the CFCL (colored blue) as shown in Fig. 1(b), full-view image of the sample can be observed directly.

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2. Theoretical design

The design process of OST is relatively straight [16]: the first step is to determine the shape of the input/output surfaces of the device (e.g., the object/image surfaces of the lens, which may have different shapes), and the second step is to design the main axis of the ONM filling between the surfaces. Based on directional projecting feature of ONM, optical field distribution on one surface (e.g., the CFCL’s curved object surface) can be identically projected onto another surface (e.g., the CFCL’s planar image surface) by designing the main axis of the ONM to establish one-to-one correspondence between the points on the object surface and the image plane.

As shown in Fig. 2(a), the designed CFCL is based on the ONMs of different axes in region I and region II. The optical patterns on the closed object surface S1 are directionally projected along the main axes of the ONM in the region I onto a transition surface S’, and then further projected along the x direction by the ONM in the region II onto the image plane S2. In principle coordinate system, the relative permittivity εI and relative permeability μI of the ideal ONM in the region I can be expressed as [12,16]:

$${\varepsilon _I} = {\mu _I} = \left[ {\begin{array}{ccc} {\Delta _1^{ - 1}}&0&0\\ 0&{\Delta _1^{}}&0\\ 0&0&{\Delta _1^{}} \end{array}} \right],\Delta _1^{} \to 0.$$

The local x-direction of the principle coordinate system is the same as the main axis of the ONM in region I, which is marked by the purple arrow in Fig. 2(a). Actually, the direction of the ONM’s main axis that connects S1 and S’ can be arbitrarily designed, once one-to-one corresponding is obtained between points on S1 and S’ [16]. In our design, to project the points on the closed cylindrical surface S1 continuously and smoothly onto a segment of circular curve S’, we first assume this mapping is a to-be-determined continuous function θ=θ(r,θ0), where r is the radius of the points in the region I, θ0 is the polar angles of the points on S1, and θ is the angle between the ONM’s main axis in the region I and the x axis. From the perspective of OST, any continuous function θ=θ(r,θ0) that satisfies the boundary conditions θ(r0,θ0)=θ0 (r0 is the radius of the closed object surface S1) and θ(R0,π)<π/2 (R0 is the radius of the transition surface S’) can be chosen. The simplest linear transformation can be used for the variable θ0, and considering that θ should gradually decrease as r increases from S1 to S’, the transformation relation is designed as θ=(Ar0/r + B)θ0, r0 ≤r ≤ R0, where A and B are coefficient to be determined. Considering the boundary condition θ(r0,θ0)=θ0 and θ(R0,π)<π/2, the relation between A and B can be determined: A + B=1 and Ar0/R0+B<1/2. Since the second inequality can be satisfied by designing the sizes of the CFCL (i.e., r0 and R0) properly, the two parameters A and B in the to-be-determined function θ=(Ar0/r + B)θ0 only need to satisfy the first equation A + B=1, and therefore the final transformation is designed as:

$$\theta ({r,{\theta_0}} )= \left( {\frac{{5{r_0}}}{{6r\,}} + \frac{1}{6}} \right){\theta _0},{r_0} \le r \le {R_0}.$$

 figure: Fig. 2.

Fig. 2. (a) The structure of CFCL based on ideal ONM. The CFCL’s closed object surface is designed as a cylindrical surface S1 (colored yellow) whose radius is r0. The optical patterns on the closed object surface S1 are directionally projected onto the transition cylindrical surface S’ (colored black) with radius R0 by the ONM in region I, and then further projected onto the image plane S2 (colored green) by the ONM in region II. In this way, the designed CFCL helps to achieve full-view imaging from the closed object surface S1 onto the finite-view image plane S2. The original of the coordinate system is set at center of the cylindrical object surface S1. The main axes of ONMs are indicated by purple arrows. (b) Numerical simulation result when optical patterns on the closed object surface S1 is chosen as Hz=sin(4θ): we plot the total normalized magnet field distribution Hz inside the whole ideal CFCL with R0=0.467λ0 and r0=0.140λ0 (λ0=2.141 m is the designed working wavelength). The distance between the coordinate origin and the image plane is chosen as d=0.514λ0. (c) Normalized magnet field distribution on the object surface (yellow solid line) and the image plane (green solid line) with the width 2w=0.902λ0, respectively.

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With the help of the ONM in region I, the full-view optical pattern on the closed object surface S1 is ideally projected onto the finite-view transition surface S’. In this way, optical patterns on the closed object surface in full view can be captured directly from a single viewpoint.

Then, the ONM in the region II can further project the optical patterns from the curved transition surface S’ onto the image plane S2. In Cartesian coordinate, the relative permittivity εII and permeability μII of the ideal ONM in the region II can be expressed as [12,16]:

$${\varepsilon _{II}} = {\mu _{II}} = \left[ {\begin{array}{ccc} {\Delta _2^{ - 1}}&0&0\\ 0&{\Delta _2^{}}&0\\ 0&0&{\Delta _2^{}} \end{array}} \right],\Delta _2^{} \to 0.$$

The main axis of the ONM in region II is designed as the x axis. As shown in Fig. 2(b), the CFCL can project the magnetic field distribution from the object surface along the main axes of the ideal ONMs onto the image plane without any phase delay. The magnetic fields on the closed object surface and the image plane of the ideal CFCL are always the same at the same moment, which is plotted in Fig. 2(c). The simulated results in Figs. 2(b) and 2(c) show the CFCL based on ideal ONMs can perfectly project the full-view optical patterns from the closed object surface onto the finite-view image plane, which verify the ideal CFCL can directly image the full-view of closed surfaces without any image stitching.

3. Realization design

The ideal ONMs described by Eqs. (1) and (3) are extremely high anisotropic media, which can be realized after some reductions in various ways [17,18,20–27]. One simple way to realize reduced ONMs is filling homogeneous isotropic dielectrics in 2N (N is an integer) subwavelength channels surrounded by 2N+1 metal plates (see Figs. 3(a) and 3(b)) [25]. The optical path in each subwavelength channel satisfies Fabry-Perot resonance condition, and local orientations of subwavelength channels are the same as the ONMs’ main axes.

 figure: Fig. 3.

Fig. 3. (a) The structure of the reduced CFCL by subwavelength metal channels filled with homogeneous isotropic dielectrics (colored gray). The 2N metal channels are surrounded by 2N+1 metal plates (black solid lines) which are ordered as the number k from top to bottom and modeled as perfect electric conductors in simulations. (b) The distribution of the refractive index in each subwavelength metal channel of the reduced CFCL. (c) Numerical simulation result when optical patterns on the closed object surface S1 is chosen as Hz=sin(4θ): we plot the total normalized magnet field distribution Hz inside the whole reduced CFCL with R0=0.467λ0, r0=0.140λ0, d=0.514λ0, N=30 and m=1 (λ0=2.141 m is the designed working wavelength). (d) The normalized Hz on the object surface (yellow solid line) and the image plane (green dots) with the width 2w=0.902λ0, respectively.

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To describe the structure of reduced CFCL in Fig. 3(a), the metal plates from top to bottom are numbered as the k-th plate (0≤k≤2N). Moreover, the geometrical length of the k-th subwavelength channel (1≤k≤2N), which is surrounded by (k-1)-th and k-th metal plates, is approximated to the mean of these two adjacent metal plates’ lengths in our design. For the central planar metal plate (i.e., the N-th plate), which is parallel to x axis in both region I and II, its geometrical length can be expressed as lN=d-r0. For other metal plates (k≠N), which consist of the planar part in region II and the curved part Ck in region I, their parametric equations can be derived from Eq. (2) by setting θ0 as π(1-k/N):

$${\theta _k}(r )= \pi \left( {1 - \frac{k}{N}} \right)\left( {\frac{{5{r_0}}}{{6r\,}} + \frac{1}{6}} \right),{r_0} \le r \le {R_0},k \ne N,$$
and their geometrical lengths can be expressed as:
$${l_k} = \int_{{C_k}} {ds + d - {R_0}\cos [{{\theta_k}({{R_0}} )} ]} ,k \ne N,$$
where d is the distance between the coordinate origin and the image plane. In order to reduce the electromagnetic scatterings, the optical length of the k-th subwavelength channel needs to satisfy the Fabry-Perot resonance condition, which means the refractive index nk of the homogeneous isotropic dielectric in the k-th subwavelength should be designed as (see Fig. 3(b)):
$${n_k} = \frac{{2m{\lambda _0}}}{{{l_{k - 1}} + {l_k}}},$$
where λ0 is working wavelength and m is an integer.

The full-view imaging effect by the reduced CFCL is verified by the numerical simulation in Figs. 3(c) and 3(d). As shown in Fig. 3(c), the reduced CFCL can still project optical patterns (i.e., the magnetic field distribution) from the object surface (colored yellow) along the main axes of the reduced ONMs onto the image plane (colored green) with a phase delay of exactly 2mπ. The magnetic fields on the closed object surface and the image plane of the reduced CFCL can still remain consistent at the same moment, which is plotted in Fig. 3(d). The simulated results in Figs. 3(c) and 3(d) verify the reduced CFCL can also directly project the full-view optical patterns from the closed object surface onto the finite-view image plane without any image stitching.

The designed CFCL is based on ONM, whose permittivity and permeability are extremely large along its main axis and close to zero in other directions (corresponds to an extremely flat elliptical dispersion curve). The designed CFCL has a super-resolution imaging capability like a hyperlens, which can convert the evanescent waves to propagating ones on the object surface, and convert them back to the evanescent components on the image plane. The super-resolution imaging capability of the CFCL can be verified by numerical simulations in Fig. 2(c) and Fig. 3(d): the distance between adjacent peaks and valleys on the object plane is on the subwavelength order (e.g., 0.053λ0), which are projected onto the image plane and can be clearly resolved.

To check the universality of the reduced CFCL, various patterns on the closed object surface are simulated. As shown in Fig. 4, the magnetic field on the object surface (colored yellow) and the image plane (colored green) can always keep the same distributions when magnetic fields on the object surface are different, which verify the reduced CFCL can always make a full-view imaging on the closed surface with various patterns. To further verify the real-time performance of the reduced CFCL, time-varying patterns are set on the object surface in Fig. 5 (also see Visualization 1 and Visualization 2). As shown in Fig. 5, if the magnetic fields on the object surface are time-varying (corresponds to the dynamic closed surface), the reduced CFCL can still provide a real-time imaging (i.e., the magnetic fields on the image plane are coincident with the fields on the object surface at any observation time).

 figure: Fig. 4.

Fig. 4. The simulated normalized magnetic field’s z component Hz on the object surface and the image plane of the reduced CFCL at the same time when the patterns on the object surface are chosen as (a) Hz = sin(θ), (b) Hz = cos(θ/2), (c) Hz = (θ/π)2, and (d) Hz = (θ/π)4. For different magnetic field distributions on the object surface, all corresponding magnetic fields on the image plane of the reduced CFCL keep the same distributions as the fields on the object surface. Other settings in simulations consist with the ones in Fig. 3.

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 figure: Fig. 5.

Fig. 5. The simulated normalized magnetic field’s z component Hz on the object surface and the image plane of the reduced CFCL at the same time when the patterns on the object surface are time-varying Hz=sin(4θ+2π×106t). The observation times are (a) 250 ns, (b) 500 ns, (c) 750 ns and (d) 1000 ns. At these times, all corresponding magnetic fields on the image plane of the reduced CFCL are coincident with the fields on the object surface (also see Visualization 1 and Visualization 2). Other settings in simulations consist with the ones in Fig. 3.

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In Fig. 5, the modulation frequency of the magnetic field on the object plane is randomly chosen as f=106Hz (i.e., Hz=sin(4θ+2π×106t)). If the modulation frequency increases (e.g., approaches to the higher frequency f=107Hz), the reduced CFCL can still keep its performance (see Fig. 6 and Visualization 3).

 figure: Fig. 6.

Fig. 6. The simulated normalized magnetic field’s z component Hz on the object surface and the image plane of the reduced CFCL at the same time when the patterns on the object surface are time-varying Hz=sin(4θ+2π×107t). The observation times are (a) 25 ns, (b) 50 ns, (c) 75 ns and (d) 100 ns (also see Visualization 3). Other settings in simulations consist with the ones in Fig. 3.

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All simulations in this work (including the frequency domain simulations in Figs. 2–4 and the time domain simulations in Fig. 5–6 and Visualization 1, Visualization 2 and Visualization 3) were performed by the wave optics module of COMSOL Multiphysics 5.4.

4. Conclusion

In summary, CFCL is designed by OST and ONMs, which can directly image various dynamic optical patterns from the closed object surfaces to the image plane. Compared with conventional full-view imaging systems, the designed CFCL can provide a full-view imaging on dynamic closed surface without image stitching and complex image processing algorithms. To realize the CFCL, the reduced CFCL is designed by subwavelength metal channels filled with homogeneous isotropic dielectrics, which also shows very good real-time and full-view imaging performance of dynamic closed surface. The simulated results show that the reduced CFCL can directly image various time-varying patterns from the closed object surface onto a finite-view image plane real-timely. Note that the design method based on OST and ideal ONM can theoretically be extended to other shapes of closed image surfaces (e.g., 3D closed surface such as sphere). However, it is still challenging to implement 3D ONMs, and hyperbolic metamaterials/metasurfaces may be the future direction to realize 3D ONM. This work puts forwards a new way to image dynamic closed object surface real-timely in full view, which may have potential applications on the real-time observation for living tissue/cell and soft material’s surface.

Funding

National Natural Science Foundation of China (61971300, 11674239, 61905208); College Student Innovation Project in Shanxi Province (20210094).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Supplementary Material (3)

NameDescription
Visualization 1       Visualization 1
Visualization 2       Visualization 2
Visualization 3       Visualization 3

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (6)
Fig. 1.
Fig. 1. Illustration of observing the minute star-shaped sample of closed surface, which has different patterns on each side, by microscope without CFCL (a) and with CFCL (b), respectively. When the sample is observed by microscope without the designed CFCL as shown in Fig. 1(a), only finite-viewed patterns on the focal plane can be observed. However, if the sample is placed in the CFCL (colored blue) as shown in Fig. 1(b), full-view image of the sample can be observed directly.

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Fig. 2.
Fig. 2. (a) The structure of CFCL based on ideal ONM. The CFCL’s closed object surface is designed as a cylindrical surface S1 (colored yellow) whose radius is r0. The optical patterns on the closed object surface S1 are directionally projected onto the transition cylindrical surface S’ (colored black) with radius R0 by the ONM in region I, and then further projected onto the image plane S2 (colored green) by the ONM in region II. In this way, the designed CFCL helps to achieve full-view imaging from the closed object surface S1 onto the finite-view image plane S2. The original of the coordinate system is set at center of the cylindrical object surface S1. The main axes of ONMs are indicated by purple arrows. (b) Numerical simulation result when optical patterns on the closed object surface S1 is chosen as Hz=sin(4θ): we plot the total normalized magnet field distribution Hz inside the whole ideal CFCL with R0=0.467λ0 and r0=0.140λ0 (λ0=2.141 m is the designed working wavelength). The distance between the coordinate origin and the image plane is chosen as d=0.514λ0. (c) Normalized magnet field distribution on the object surface (yellow solid line) and the image plane (green solid line) with the width 2w=0.902λ0, respectively.

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Fig. 3.
Fig. 3. (a) The structure of the reduced CFCL by subwavelength metal channels filled with homogeneous isotropic dielectrics (colored gray). The 2N metal channels are surrounded by 2N+1 metal plates (black solid lines) which are ordered as the number k from top to bottom and modeled as perfect electric conductors in simulations. (b) The distribution of the refractive index in each subwavelength metal channel of the reduced CFCL. (c) Numerical simulation result when optical patterns on the closed object surface S1 is chosen as Hz=sin(4θ): we plot the total normalized magnet field distribution Hz inside the whole reduced CFCL with R0=0.467λ0, r0=0.140λ0, d=0.514λ0, N=30 and m=1 (λ0=2.141 m is the designed working wavelength). (d) The normalized Hz on the object surface (yellow solid line) and the image plane (green dots) with the width 2w=0.902λ0, respectively.

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Fig. 4.
Fig. 4. The simulated normalized magnetic field’s z component Hz on the object surface and the image plane of the reduced CFCL at the same time when the patterns on the object surface are chosen as (a) Hz = sin(θ), (b) Hz = cos(θ/2), (c) Hz = (θ/π)2, and (d) Hz = (θ/π)4. For different magnetic field distributions on the object surface, all corresponding magnetic fields on the image plane of the reduced CFCL keep the same distributions as the fields on the object surface. Other settings in simulations consist with the ones in Fig. 3.

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Fig. 5.
Fig. 5. The simulated normalized magnetic field’s z component Hz on the object surface and the image plane of the reduced CFCL at the same time when the patterns on the object surface are time-varying Hz=sin(4θ+2π×106t). The observation times are (a) 250 ns, (b) 500 ns, (c) 750 ns and (d) 1000 ns. At these times, all corresponding magnetic fields on the image plane of the reduced CFCL are coincident with the fields on the object surface (also see Visualization 1 and Visualization 2). Other settings in simulations consist with the ones in Fig. 3.

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Fig. 6.
Fig. 6. The simulated normalized magnetic field’s z component Hz on the object surface and the image plane of the reduced CFCL at the same time when the patterns on the object surface are time-varying Hz=sin(4θ+2π×107t). The observation times are (a) 25 ns, (b) 50 ns, (c) 75 ns and (d) 100 ns (also see Visualization 3). Other settings in simulations consist with the ones in Fig. 3.

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Equations (6)

Equations on this page are rendered with MathJax. Learn more.

ε I = μ I = [ Δ 1 − 1 0 0 0 Δ 1 0 0 0 Δ 1 ] , Δ 1 → 0.
θ ( r , θ 0 ) = ( 5 r 0 6 r + 1 6 ) θ 0 , r 0 ≤ r ≤ R 0 .
ε I I = μ I I = [ Δ 2 − 1 0 0 0 Δ 2 0 0 0 Δ 2 ] , Δ 2 → 0.
θ k ( r ) = π ( 1 − k N ) ( 5 r 0 6 r + 1 6 ) , r 0 ≤ r ≤ R 0 , k ≠ N ,
l k = ∫ C k d s + d − R 0 cos ⁡ [ θ k ( R 0 ) ] , k ≠ N ,
n k = 2 m λ 0 l k − 1 + l k ,
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