## Abstract

Based on gray-tone optical lithography technology combined with the overlay alignment method, a spherical concave micro-mirror is fabricated at the end of a rectangular optical waveguide (ROW) for low vertical coupling loss. The optimal structures of the spherical concave micro-mirrors were designed through ray-tracing simulation. The results indicate that the minimal vertical coupling loss is only 1.02 dB for the ROW core size of 20 μm × 20 μm. The surface roughness of the micro-mirror is considered, and it should be less than 106 nm to ensure that the vertical coupling loss is less than 1.5 dB. The radius of the fabricated spherical concave micro-mirror was measured as 263.3 μm and the surface roughness of the micro-mirror is 29.19 nm. The vertical coupling loss induced by the micro-mirror was measured as 1.39 dB. 1-dB tolerances in the direction of *x*-, *y*-, and *z*-axes are calculated to be ± 6.9 μm, ± 6.3 μm, and 46.2 μm, respectively.

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

## 1. Introduction

In optical interconnect application systems, vertical optical coupling structures on optical printed circuit boards (OPCBs) have garnered considerable attention [1–6]. This is because they are benefit to the integration of optical devices, reasonable utilization of inter-board space, and flexible design of system structures. Various methods have been proposed to realize vertical coupling, such as grating couplers [7–10], flexible waveguides [11], and reflective micro-mirrors [12–14]. Among the aforementioned methods, the characteristics of grating couplers highly depend on the wavelength, and this method is mostly used for the optical waveguides with small core size on silicon substrate. Additionally, the fabrication of grating couplers requires high accuracy and vertical coupling may be unavailable when the off-vertical tilt angle is greater than 10°. The vertical coupling loss induced by the flexible waveguide is 3.01 dB, which is slightly large [11]. The reflective micro-mirror is an available and excellent choice because it is highly coupling-efficient, insensitive to wavelength and polarization, and easy to fabricate [6].

The 45° tilted micro-mirror has been extensively studied because its simple structure is easy to fabricate. There are several technologies used to achieve fabrication, such as laser ablation [15], reactive ion etching (RIE) [16], micro-dicing [13,17], ultraviolet (UV) photolithography [6,18], hard mold [12,19], manual polishing [20], deep proton writing [21], and gray-tone optical lithography [22,23]. Among the aforementioned fabrication technologies, photolithography technology is commonly used in printed circuits because it has an excellent ability to fabricate high-precision microstructures. Gray-tone optical lithography inherits the advantages mentioned above and it is also more suitable for fabricating three-dimensional (3D) irregular microstructures. The 45° tilted micro-mirror plays an important role when the numerical aperture (NA) and the size of the optical waveguide match those of the optical components [24]. However, in most cases, the NA and the size of the optical waveguides do not satisfy the above condition. Owing to their beam-focusing ability, several concave micro-mirrors have been proposed to solve the problem of mismatch between NAs and sizes of two optical components. A concave micro-mirror based on laser ablation was fabricated with a loss of 1.83 dB, but its surface was cylindrical, implying that the micro-mirror bends only in one direction [25]. A spherical concave micro-mirror has more potential because it realizes a beam-focusing function in more directions. A 45° curved micro-mirror based on gray-tone optical lithography with a loss of 3.2 dB has been proposed and it bends in two directions. However, a large cavity needs to be prepared in advance for the integration of the micro-mirror resulting in more fabrication steps [26,27]. If a spherical concave micro-mirror is formed on the under-cladding during the fabrication of the optical waveguide, the fabrication steps can be reduced, and the micro-mirror can be more accurately aligned with the core. This fabrication scheme of spherical concave micro-mirrors on FR4 substrates for OPCBs has not been investigated, therefore, it is an innovative fabrication method with great significance.

In this study, a spherical concave micro-mirror is proposed based on gray-tone optical lithography for highly efficient vertical optical coupling between a rectangular optical waveguide (ROW) and a multi-mode fiber (MMF). First, a theoretical model of the spherical concave micro-mirror for vertical optical coupling was constructed. The optimal structure of a spherical concave micro-mirror was determined through ray-tracing simulation, and the influence of the surface roughness on the vertical coupling loss was discussed in detail. Thereafter, a spherical concave micro-mirror was fabricated based on gray-tone optical lithography combined with the overlay alignment method. Subsequently, the profile and surface roughness of the spherical concave micro-mirror were measured using a 3D optical profiler. Finally, an experimental measurement setup for vertical coupling was established. The vertical coupling loss induced by the micro-mirror and the misalignment tolerances of the receiving tapered MMF were obtained experimentally.

## 2. Theory and simulation

#### 2.1 Structural design of spherical concave micro-mirror

The analysis model of the spherical concave micro-mirror is shown in Fig. 1. Optical polymer materials are used to fabricate ROW with the most popular combination of EpoCore and EpoClad (Micro Resist Technology, Germany). At a wavelength of 850 nm, the refractive indices of ROW core and cladding are 1.583 and 1.571, respectively, so that the NA of ROW is 0.1945. In the simulation model shown in Fig. 1, the O-point (0, 0, 0) is the origin of the 3D coordinate system. A ray from the B-point (*x _{B}*,

*y*,

_{B}*z*) on the ROW end-face is reflected by the spherical concave micro-mirror at the C-point (

_{B}*x*,

_{C}*y*,

_{C}*z*), and subsequently passes through the interface between the ROW over-cladding and air at the D-point (

_{C}*x*,

_{D}*y*,

_{D}*z*), and finally enters the receiving MMF at the E-point (

_{D}*x*,

_{E}*y*,

_{E}*z*). The F-point (

_{E}*x*,

_{F}*y*,

_{F}*z*) is the center of the sphere. Given the structural symmetry of the spherical concave micro-mirror, the value of

_{F}*x*is 0, and

_{F}*R*denotes the radius of the sphere. In addition, the M-point (0, 0,

*z*) is the bottom point of the spherical concave micro-mirror. The structure of the spherical concave depends on the radius

_{M}*R*and the parameter

*β*

_{0}, which is the angle between the MF-line and

*z*-axis. To obtain the E-point (

*x*,

_{E}*y*,

_{E}*z*), the auxiliary N-point (

_{E}*x*,

_{N}*y*,

_{N}*z*) is introduced as the foot point of the B-point on the normal line.

_{N}To ensure that rays enter the MMF and are transmitted stably, the core radius and NA of the MMF must satisfy the following expression:

where the P-point (*x*,

_{P}*y*,

_{P}*H*+

*h*) represents the center of the receiving MMF, and

*a*denotes the radius of the MMF core.

*n*

_{0}denotes the refractive index of the space between MMF and ROW, and it can generally be regarded as the refractive index of air.

*H*represents the thickness of ROW, whereas

*h*represents the distance between the receiving MMF and over-cladding.

*φ*denotes the incident angle of the rays entering the MMF. The procedure for calculating the coordinates of the points is provided in the Appendix. From the end-face of the ROW core, the optical intensity follows a Gaussian distribution, which means that thousands of rays follow a Gaussian distribution. The optical intensity of each ray is equal, and the divergence angle of each ray is limited by the NA of the ROW. The rays are reflected by the micro-mirror, and travel into the receiving MMF. Therefore, the vertical coupling loss induced by the spherical concave micro-mirror is expressed as follows: where

*I*denotes the optical intensity of the total rays, and

_{all}*I*denotes the optical intensity of the effective rays that satisfy Eq. (1).

_{s}In the simulation, the NA of the receiving MMF was 0.1955, *h* was set as 10 μm, the thickness of the ROW cladding *H*_{1}/*H*_{3} was set as 20 μm, and *H*_{0}, which represents the distance between bottoms of the micro-mirror and the ROW core, was set as 0 μm and 5 μm, respectively. The light beam from the end-face of the ROW was considered to be a Gaussian distribution owing to its multimode structure. The maximal divergence angle of the light beam was set as 7.11°, which was limited by the NA of the ROW, and it matched the NA of the receiving MMF. The sizes of the ROW cores *W* × *H*_{2} were set as 10 μm × 10 μm, 20 μm × 20 μm, 30 μm × 30 μm, 40 μm × 40 μm, and 50 μm × 50 μm, respectively. The value of *H* (=*H*_{1}+*H*_{2}+*H*_{3}) was easily obtained. To verify the beam-focusing ability of the spherical concave micro-mirror, the diameters of the receiving MMF 2*a* were set as 10 μm, 20 μm, 30 μm, 40 μm, and 50 μm, respectively. The optimal structural parameters of the spherical concave micro-mirror are summarized in Table 1, with corresponding vertical coupling losses of 1.09 dB (1.09 dB), 1.02 dB (0.91 dB), 0.95 dB (0.86 dB), 1.18 dB (1.13 dB), and 1.13 dB (0.87 dB), respectively, when *H*_{0} is 0 μm (5 μm).

In order to make the simulation consistent with the fabrication of the spherical concave micro-mirror, the parameter *H*_{0} was set as 0 μm in the following. The dependences of the vertical coupling loss on parameters *R* and *β*_{0} are shown in Fig. 2, with the ROW core sizes of 10 μm × 10 μm, 20 μm × 20 μm, and 30 μm × 30 μm, respectively. The vertical coupling loss increases slightly when the two parameters change within a certain range. The ROW core with the size of 20 μm × 20 μm is taken as an example, the loss increments are less than 0.2 dB when *R*∈(220 μm, 320 μm) and *β*_{0}∈(40°, 44°).

To verify the advantages of the spherical concave micro-mirror, the vertical coupling loss induced by the 45° tilted micro-mirror was also simulated with the ROW and MMF remained unchanged [25]. In Fig. 3(a), the vertical coupling losses induced by the 45° tilted micro-mirror are 0.12 dB, 0.45 dB, 0.59 dB, 0.42 dB, and 0.65 dB higher than those induced by the spherical concave micro-mirror for different sizes of ROW cores, respectively. In addition, the loss induced by the 45° tilted micro-mirror increases as the size of the core increases, whereas the loss induced by the spherical concave micro-mirror fluctuates slightly. The diameters of the light spots at the MMF receiving end-face reflected by the spherical concave and 45° tilted micro-mirrors are shown in Fig. 3(b). The differences between diameters of light spots reflected by the spherical concave and 45° tilted micro-mirrors are 3.4 μm, 6.0 μm, 9.1 μm, 10.6 μm, and 10.2 μm, respectively. The diameters of the light spots reflected by the spherical concave micro-mirror are slightly smaller than the width of the ROW core, while those reflected by the 45° tilted micro-mirror are not. It is confirmed that there is an enlarged light beam reflected by the 45° tilted micro-mirror that mismatches the core size of the receiving MMF. As an example, the vertical optical path conversions of two different micro-mirrors are shown in Fig. 4(a) and 4(b) for the ROW core with a size of 20 μm × 20 μm. In Fig. 4(c) and 4(d), the diameters of light spots, which are formed at the MMF receiving end-face by the spherical concave and 45° tilted micro-mirrors, are 17.5 μm and 23.5 μm, respectively. In conclusion, the spherical concave micro-mirror causes a low vertical coupling loss and it has a stronger beam-focusing ability.

#### 2.2 Surface roughness of spherical concave micro-mirror

Scattering occurs when the sample surface is rough. Roughness significantly influences the transmission loss of optical waveguides [28,29] and the performance of optical components [30,31]. Therefore, the surface roughness of the spherical concave micro-mirror is an important parameter that affects the vertical coupling loss. The sketched reflective scatter is shown in Fig. 5(a), where *I _{i}*,

*I*, and

_{r}*I*represent the incident, reflected, and scattered intensities, respectively. When the rough surfaces are symmetrical, the intensity of the light scattered into the solid-angle element

_{s}*dθ*in the direction (

_{s}*θ*,

_{s}*φ*) is expressed as follows [32]:

_{s}*θ*and

_{i}*θ*denote the incident and scattering angles, respectively,

_{s}*λ*denotes the radiation wavelength,

*Q*represents the polarization factor that is affected by the surface material and polarization parameters,

*W*

_{1}(

*p*) denotes the one-dimensional power spectral density of the surface roughness, and

*L*represents the relative length.

*Z*(

*x*) denotes the exponential radial surface profile and it is expressed as follows:

*σ*denotes the root-mean-square of the surface roughness. According to the geometric optics theory, the angle

*φ*

_{0}between the reflected ray and the optical axis of the MMF can be calculated. When the critical angle of the MMF is

*φ*, only those rays that satisfy the condition

_{max}*φ*

_{0}<

*φ*are effective. As shown in Fig. 5(b), the scattering rays are transmitted stably under the following condition:

_{max}*θ*denotes the scattering angle. Therefore, the total effective light intensity is determined as follows:

_{s}*c*denotes the scattering factor.

_{s}The effect of a rough spherical concave micro-mirror on vertical coupling was also simulated using the ray-tracing method. In simulation, *L* was set as 30 μm [33], and *φ _{max}* was set as 11.27°, which depends on the NA of the receiving MMF. The dependence of the loss on the surface roughness of the optimal spherical concave micro-mirror is shown in Fig. 6(a) for the ROW core with a size of 20 μm × 20 μm. The results indicate that the vertical coupling loss is 1.02 dB when the surface roughness of the micro-mirror is 0 nm under ideal conditions. Additionally, it can be found that a roughness of 50 nm, 100 nm, and 150 nm causes additional coupling losses of 0.10 dB, 0.42 dB, and 0.99 dB, respectively. To ensure that the vertical coupling loss is not greater than 1.5 dB, the surface roughness of the spherical concave should be less than 106 nm. In addition, incoherent irradiance distributions of the light spots, which are received at the MMF end-face, in the direction of the

*x*- and

*y*-axes are shown in Fig. 6(b), when the surface roughness of the micro-mirror is 0 nm, 100 nm, and 150 nm, respectively. The larger roughness causes the reflected beam to diverge, the irradiance intensity in the center of the light spot is lower, and the intensity distribution is not very concentrated. Therefore, the vertical coupling loss increases with an increase in the surface roughness of the spherical concave micro-mirror.

## 3. Fabrication of spherical concave micro-mirror

A combination of gray-tone optical lithography and the overlay alignment method were used to fabricate the designed spherical concave micro-mirror on the ROW under-cladding. A pair of negative optical polymer photoresists, EpoClad and EpoCore, was used for ROW cores with a size of 20 μm × 20 μm. The positive polymer photoresist, ma-P1275HV (Micro Resist Technology, Germany), was employed for the spherical concave structure. The complete fabrication process of the spherical concave micro-mirror is illustrated in Fig. 7: (a) the under-cladding layer is spin-coated onto the FR4 substrate with a sufficiently thick layer (∼20 μm), (b) the rectangular core is formed using photolithography after spin-coating the core layer, (c) the positive photoresist is spin-coated onto the core layer with a thickness slightly larger than that of the core layer (∼25 μm), (d) the 3D structure of the spherical concave is formed through gray-tone optical lithography using a maskless lithography machine (Heidelberg, MLA100, Germany), (e) a thin layer of metal is deposited on the micro-mirror surface using magnetron sputtering technology, and (f) the over-cladding layer is spin-coated to fill the cavities and encapsulate both micro-mirrors and cores.

For the overlay alignment process, a mask pattern for the ROW core and a grayscale mask pattern for the spherical concave structure were designed, as shown in Fig. 8(a) and Fig. 8(b), respectively. The sizes of the two patterns are the same. The A-point and B-point in Fig. 8(a) are the midpoints on the right side of the rectangles, and the A'-point and B'-point in Fig. 8(b) are the center points of the arcs. When two patterns overlap, the A-point coincides with the A'-point, and the B-point simultaneously coincides with the B'-point, as shown in Fig. 8(c). By the overlay alignment method, the same area of the sample is exposed twice under the two patterns with no position deviation. Therefore, the spherical concave micro-mirror is precisely integrated into the end-face of the ROW core. This fabrication procedure is simplified and compatible with the ROW fabrication process using photolithography.

Through repeated experiments and comparison, the results indicate that an exposure energy of 8000 mJ/cm^{2} is sufficient to expose the ∼25 μm-thick positive polymer photoresist. The positive polymer photoresist is originally insoluble in a developer. However, it becomes easily soluble due to chemical decomposition after exposure [34]. Figure 9(a) illustrates the principle of gray-tone optical lithography [35,36]. With the limit of the grayscale mask, the light intensity is determined by a grayscale of the mask pattern, which is a number between 0 and 255. A higher grayscale value indicates a lighter shade (closer to white), whereas a lower grayscale value indicates a darker shade (closer to black). The lower the grayscale value is, the weaker the light intensity is. The upper part of the positive polymer photoresist is always exposed firstly, so that it is dissolved in a developer and cleaned up first, while the unexposed part remains. In the experiment, a positive photoresist layer with a thickness of 26.23 μm was formed. The dependence of the lithography depth on the grayscale under an exposure energy of 8000 mJ/cm^{2} is obtained, as shown in Fig. 9(b). The average error of the experimental lithography depth is only 0.34 μm. A grayscale value of 250 is taken as an example, the measurement of the lithography depth is shown in the insets in Fig. 9(b). The left inset is the surface profile of the developed positive photoresist layer, and the right inset demonstrates the height difference between the A-point and the B-point measured by a 3D optical profiler (Sensofar-Tech, SNEOX, Spain), which means that the lithography depth is 25.89 μm. Thus, combined with the optimal spherical concave structure obtained through simulation, the width of the spherical concave structure is 24.5 μm when its height is taken as 25 μm. According to the simulation and experimental results, the grayscales for each part of the spherical concave structure are obtained. As shown in Fig. 9(c), the grayscale decreasing distribution is evaluated using the curve-fitting method, and it is expressed by the following equation:

*A*

_{0},

*A*

_{1}, and

*t*represent fitting parameters with values of 254.65864, -985.11756, and 86.91678, respectively. According to the fitting equation, the grayscale mask pattern can be drawn by stepped grayscale setting, as shown in Fig. 9(d). It is arched with the grayscale value gradually decreasing from the center to the edge. The smaller the step, the higher fitness between the stepped structure and the spherical concave structure. In grayscale mask patterns, the minimum step was set as 0.5 μm, because each pixel of the pattern corresponds to a length of 0.5 μm.

Figure 10(a) shows the scanning electron microscopy (SEM) (JEOL Ltd, JSM-7500F, Japan) images of the fabricated spherical concave micro-mirror. The surface profile of the micro-mirror is measured by a 3D optical profiler, as shown in Fig. 10(b). In Fig. 10(c), the curve of the fabricated micro-mirror nearly coincides with the circle equation *y*^{2} + *z*^{2} = 263.3^{2}, thus, the radius *R* is 263.3 μm. The parameter *β*_{0} is determined to be 40.6${^\circ }$ from the fitting result through calculation. The width and depth of the spherical concave micro-mirror are 24.24 μm and 26.23 μm, respectively. Moreover, the surface roughness of the micro-mirror is also measured to be 29.19 nm after the shape removal operation in Fig. 10(d). According to the simulation results, this structure of the micro-mirror causes a vertical coupling loss of 1.14 dB, and the roughness may cause an additional coupling loss of 0.04 dB. Therefore, the total loss is 1.18 dB in theory.

## 4. Vertical coupling experiment and analysis

The measurement setup for the vertical coupling loss of the spherical concave micro-mirror is shown in Fig. 11(a). A physical picture of the experimental setup is shown in Fig. 11(b). It is evident that the red light travels through the single-mode fiber (SMF), ROW, and MMF, and it can be clearly observed at the other end-face of the MMF. Therefore, the fabricated spherical concave micro-mirror achieves the basic function of vertical optical coupling. To measure the vertical coupling loss, an optical signal with a wavelength of 850 nm propagated from a light source (Thorlabs, S4FC852, USA), and was coupled into a SMF, followed by ROW. Then it was reflected by a spherical concave micro-mirror, and vertically coupled into a tapered MMF. The tapered MMF was fabricated by flame brushing technique, and the core diameter was measured as 20.67 μm. The output power at the other end of the MMF was measured using a dual-channel optical power and energy meter (Thorlabs, PM320E, USA). The insertion loss *L _{sum}* is measured to be 2.62 dB, which obeys the following formula:

*L*represents the transmission loss of the ROW,

_{w}*L*represents the input-side coupling loss between the SMF and ROW, and

_{c}*L*denotes the vertical coupling loss induced by the micro-mirror. To obtain the value of

_{mirror}*L*, ROW samples were fabricated with the same materials and sizes. By applying the cut-back method to the prepared sample, the propagation loss was measured to be 0.31 dB/cm. Accordingly,

_{w}*L*of the sample, with micro-mirrors and 1.2 cm-length ROW, is 0.37 dB. To obtain the value of

_{w}*L*, one side of the ROW sample was edge-coupled with the SMF through index matching, and the output power from the other side is detected. Thus,

_{c}*L*is measured to be 0.86 dB. According to Eq. (8), the vertical coupling loss

_{c}*L*induced by the fabricated spherical concave micro-mirror is 1.39 dB.

_{mirror}The misalignment tolerances of the receiving tapered MMF in the direction of the *x*- and *y*-axes were measured, as shown in Fig. 12(a). 1-dB tolerances of *x*- and *y*-axes are ± 6.9 μm and ± 6.3 μm, respectively. The normalized vertical coupling loss increases faster in the direction of the *y*-axis because the width of the micro-mirror in the *x*-axis direction is larger than that in the *y*-axis direction. As a result, the width of the light spot reflected by the micro-mirror in the *y*-axis direction is a little shorter than that in the *x*-axis direction, which can be seen in Fig. 4(c). The misalignment tolerance of the *x*-axis is nearly symmetrical because of the symmetrical structure of the spherical concave micro-mirror. In Fig. 12(b), the misalignment tolerance in the direction of the *z*-axis was also measured. 1-dB misalignment tolerance of the *z*-axis is 46.2 μm, which is significantly larger than that of the *x*- and *y*-axes. The separation *h* should be 12 μm for minimum loss.

The obtained excess vertical coupling loss induced by the micro-mirror is slightly higher than the predicted value in the simulation. The main reason for this difference is the structure of the fabricated micro-mirror. There is a small deviation between the simulation and measurement values of parameters *R* and *β*_{0}. In our future studies, the grayscale mask pattern needs to be further optimized for a lower vertical coupling loss. Moreover, the surface roughness of the micro-mirror may be further reduced after the thermal reflow process.

## 5. Conclusion

In this study, a spherical concave micro-mirror was fabricated using gray-tone optical lithography technology for highly efficient vertical coupling. The optimal structures of the micro-mirror for ROW with different core sizes were obtained through ray-tracing simulation, as well as the corresponding minimum vertical coupling loss. For the ROW core with the size of 20 μm × 20 μm, the results indicate that the parameters *R* and *β*_{0} of the optimal spherical concave micro-mirror are 280 μm and 42°, respectively. The vertical coupling loss is 1.02 dB when the surface roughness of the micro-mirror is 0 nm. A roughness of 50 nm causes an additional coupling loss of 0.10 dB, and it should be maintained below 106 nm to ensure that the vertical coupling loss is less than 1.5 dB. The micro-mirror was fabricated successfully, with the parameters *R* and *β*_{0} of 263.3 μm and 40.6°, respectively. The surface roughness of the micro-mirror is 29.19 nm, which causes an additional vertical coupling loss of 0.04 dB in theory. The vertical coupling loss induced by the fabricated micro-mirror is 1.39 dB. 1-dB tolerances of *x*-, *y*-, and *z*-axes are ± 6.9 μm, ± 6.3 μm, and 46.2 μm, respectively. A combination of gray-tone optical lithography technology and overlay alignment method is an innovative and effective way of fabricating spherical concave micro-mirrors, which have significant potential for vertical coupling on OPCBs.

## Appendix

Spherical concave micro-mirror | |
---|---|

For E-point, x can be expressed as_{E}, y_{E}, z_{E} | For D-point, x can be expressed as_{N}, y_{N}, z_{N} |

$\begin{aligned} {x_{E}} &= \frac{{{n_{clad}}h({{x_D} - {x_C}} )}}{w} + {x_D}\\ {y_E} &= \frac{{{n_{clad}}h({{y_D} - {y_C}} )}}{w} + {y_D}\\ {z_E} &= H + h \end{aligned}$ | $\begin{array}{l} {x_D} = \frac{{{s_1}}}{{{s_0}}}({z_D} - {z_C}) + {x_C}\\ {y_D} = \frac{{{s_2}}}{{{s_0}}}({z_D} - {z_C}) + {y_C}\\ {z_D} = H \end{array}$ |

where | where |

$w = \sqrt {{{({z_C} - {z_D})}^2} - (n_{clad}^2 - n_{air}^2)[{{{({x_C} - {x_D})}^2} + {{({y_C} - {y_D})}^2}} ]}$ | $\begin{array}{l} {s_0} = 2{z_N} - {z_C} - {z_B}\\ {s_1} = 2{x_N} - {x_C} - {x_B}\\ {s_2} = 2{y_N} - {y_C} - {y_B} \end{array}$ |

For N-point, x can be expressed as_{D}, y_{D}, z_{D} | For C-point, x can be expressed as_{C}, y_{C}, z_{C} |

$\begin{array}{l} {x_N} = \frac{1}{{{R^2}}}({{m_1} + {m_2} - m{}_3} )\\ {y_N} = \frac{{({y_C} - {y_F})({x_N} - {x_C})}}{{({x_C} - {x_F})}} + {y_C}\\ {z_N} = \frac{{({z_C} - {z_F})({x_N} - {x_C})}}{{({x_C} - {x_F})}} + {z_C} \end{array}$ | $\begin{array}{l} {x_C} = \frac{{{f_0}}}{{\cos \alpha }} + {x_B}\\ {y_C} = \frac{{{f_0}}}{{\cos \beta }} + {y_B}\\ {z_C} = \frac{{{f_0}}}{{\cos \gamma }} + {z_B} \end{array}$ |

where | where |

$\begin{array}{l} {m_1} = {({x_C} - {x_F})^2}{x_B}\\ {m_2} = [{{{({y_C} - {y_F})}^2} + {{({z_C} - {z_F})}^2}} ]{x_C}\\ {m_3} = ({x_C} - {x_F})[{({z_C} - {z_F})({z_C} - {z_B}) + ({y_C} - {y_F})({y_C} - {y_B})} ]\end{array}$ | $\begin{array}{l} {f_0} = \frac{{ - {f_2} + \sqrt {f_2^2 - 4{f_1}{f_3}} }}{{2{f_1}}}\\ {f_1} = \frac{1}{{{{\cos }^2}\alpha }} + \frac{1}{{{{\cos }^2}\beta }} + \frac{1}{{{{\cos }^2}\gamma }}\\ {f_2} = \frac{{2({{x_B} - {x_F}} )}}{{\cos \alpha }} + \frac{{2({{y_B} - {y_F}} )}}{{\cos \beta }} + \frac{{2({{z_B} - {z_F}} )}}{{\cos \gamma }}\\ {f_3} = {({{x_B} - {x_F}} )^2} + {({{y_B} - {y_F}} )^2} + {({{z_B} - {z_F}} )^2} - {R^2} \end{array}$ |

For F-point, x can be expressed as_{F}, y_{F}, z_{F} | |

$\begin{array}{l} {x_F} = 0\\ {y_F} ={-} R\sin {\beta _0}\\ {z_F} = R\cos {\beta _0} + {H_3} - {H_0} \end{array}$ |

In above equations, *α*, *β*, *γ* is included angle of BC-line and *x*-, *y*-, *z*-axes, respectively. *n _{clad}* and

*n*are the refractive index of the ROW cladding and the air, respectively, while the other parameters are all shown in Fig. 1.

_{air}## Funding

National Natural Science Foundation of China (61735009, 62027818); Science and Technology Commission of Shanghai Municipality (16511104300).

## Disclosures

The authors declare that there are no conflicts of interest related to this article.

## 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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