Design optical antenna and fiber coupling system based on the vector theory of reflection and refraction

Ping Jiang; Huajun Yang; Shengqian Mao

doi:10.1364/OE.23.026104

1.Introduction

Geometric optical design methods based on numerical calculation of lenses and mirrors have been thoroughly studied in the past [1–10 ]. Most are based on two-dimensional (2-D) optical path calculation and combined with the optical design software to perform the aberrations evaluation. There-dimensional (3-D) ray tracing by using geometric numerical calculation [11], full finite-element methodology [12] and freeform optical surfaces design [13–16 ] always with large amount of calculation, low speed, hence make it difficult to trace every ray in the optical system design and aberration analysis process.

An improved 3-D ray tracing method based on the vector theory of reflection and refraction is capable of calculating spatial rays transmitting through a complex optical system which contains multiple lenses or mirrors, without conversion of the trigonometric functions and is convenient for computer programming, hence greatly improves the calculation efficiency. In this article a Cassegrain antenna system and an optical fiber coupling system which consists of a plano-concave lens and a plano-convex lens are designed by using this method. Aberrations which caused by on-axial defocusing, off-axial defocusing and deflection of the receiving antenna are calculated based on their physicals properties rather than using complex aberration formulas. Experimental test results of the image-forming characteristics of the optical antenna-fiber coupling system are comprehensively analyzed.

2. Vector Theory of Reflection and Refraction

A spatial ray can be regarded as a vector, assume a vector ray transmits through an 3-D optical system which contains continuously m reflective or refractive surfaces, as shown in Fig. 1 .

Fig. 1 Schematic diagram of a vector ray transmits through a 3-D optical system which contains m reflective or refractive surfaces

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The relationship between the unit vector $\overset{⇀}{A_{0}}$ of the first incident ray and the unit vector $\overset{⇀}{{A^{'}}_{m}}$ of the last emerging ray can be written as

{n^{'}}_{m} \overset{⇀}{{A^{'}}_{m}} = n_{0} \overset{⇀}{A_{0}} + \sum_{k = 1}^{m} Γ_{k} \overset{⇀}{N_{k}}

where

n_{0}

and

{n^{'}}_{m}

are the refractive indexes of the first incident space and the last emerging space, respectively.

Γ_{k}

indicates the deviation coefficient of the k-th surface, for a reflective system

Γ_{k} = - 2 n_{k} (\vec{N_{k}} \cdot \vec{A_{k}})

and for a refractive system

Γ_{k} = {n^{'}}_{k} \vec{N_{k}} \cdot \vec{{A^{'}}_{k}} - n_{k} \vec{N_{k}} \cdot \vec{A_{k}}

.

\vec{A_{k}}

and

\vec{{A^{'}}_{k}}

indicate the unit vectors of incident ray and the reflective or refractive ray of the k-th surface, respectively.

n_{k}

and

{n^{'}}_{k}

are the refractive indexes of the incident space and the reflective or refractive space of the k-th surface, respectively. The direction cosine of the unit normal vector

\vec{N_{k}}

of the k-th surface at the projecting point M is

(\cos α_{N_{k}}, \cos β_{N_{k}}, \cos γ_{N_{k}}) = (\frac{- {F^{'}}_{_{k} (x)}}{\sqrt{{F^{'}}_{_{k} (x)}^{2} + {F^{'}}_{_{k} (y)}^{2} + {F^{'}}_{_{k} (z)}^{2}}}, \frac{- {F^{'}}_{_{k} (y)}}{\sqrt{{F^{'}}_{_{k} (x)}^{2} + {F^{'}}_{_{k} (y)}^{2} + {F^{'}}_{_{k} (z)}^{2}}}, \frac{- {F^{'}}_{_{k} (z)}}{\sqrt{{F^{'}}_{_{k} (x)}^{2} + {F^{'}}_{_{k} (y)}^{2} + {F^{'}}_{_{k} (z)}^{2}}})

where

{F^{'}}_{_{k} (x)}

,

{F^{'}}_{_{k} (y)}

,

{F^{'}}_{_{k} (z)}

are the partial derivatives of the k-th surface equation

F_{k} (x, y, z)

, respectively.

Every spatial ray can be traced transmitting through a 3-D optical system which contains multiple surfaces including quadric surfaces (i.e. parabolic, hyperbolic, ellipsoid surface, etc.), multi-quadric surfaces and even freeform surfaces by using Eq. (1). This method based on the vector theory of reflection and refraction is very efficient for 3-D optical system design and aberration calculation.

3. Optical Communication System Model Establishment and Aberrations Calculation

A typical optical communication system consists of an optical antenna system and an optical fiber network. Optical antenna system is the key component of the optical communication system, and an efficient fiber coupling technology is the important guarantee to realize transformation from space transmission to optical fiber network.

3.1 Establish the antenna system model based on vector theory of reflection

Suppose an optical antenna system consists of two Cassegrain antennas with distance of L, and the single-mode optical fiber end face is located at the Gaussian image plane of the receiving antenna. Some thermal expansion damages or mechanical assembling errors may cause defocusing and deflection of the antenna, hence impact the transmission performance of the antenna system and the coupling efficiency into the single-mode fiber [1] [11]. Schematic diagram of an optical communication antenna system with a deflected receiving antenna is shown in Fig. 2 .

Fig. 2 Schematic diagram of an optical communication antenna system with a deflected receiving antenna

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Assume the optical axis of the optical antenna system is along x axis, and the receiving antenna rotates around z axis with an angle of η and the spin center is at the left focus of its sub-reflector. The new coordinate frame $(x^{'}, y^{'}, z^{'})$ of the receiving antenna can be written as

{\begin{cases} x^{'} = x \cos η - y \sin η + L \\ y^{'} = x \sin η + y \cos η \\ z^{'} = z \end{cases}

where (x, y, z) is the coordinate frame of the transmitting antenna. Four reflector surface equations of the optical antenna system in the same coordinate frame are shown in Table 1 .

Table 1. Four reflector surface equations of the optical antenna system

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where (x ₀₁, y ₀₁, z ₀₁) = (-c ₁, 0, 0), (x ₀₂, y ₀₂, z ₀₂) = (-p ₁/2, 0, 0), (x ₀₃, y ₀₃, z ₀₃) = (p ₁/2 + ∆x, ∆y, 0), (x ₀₄, y ₀₄, z ₀₄) = (c ₁, 0, 0) are the apex and center coordinates of the four reflectors, respectively. $c_{1} = \sqrt{a_{1}^{2} + b_{1}^{2}}$ is the focal length of the hyperboloid, ∆x and ∆y stand for on-axial and off-axial defocusing amounts between the primary reflector and the sub-reflector of the receiving antenna, respectively. 3-D ray tracing simulation result of the Cassegrain antenna system is shown in Fig. 3 .

Fig. 3 3-D ray tracing simulation result of Cassegrain antenna system with L = 0.5 m, p ₁ = 140.14 mm, a ₁ = 60.00mm and b ₁ = 42.63 mm

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For a confocal Cassegrain antenna system without deflection (i.e. ∆x = 0, ∆y = 0, η = 0), the emitting beam from the transmitting antenna is a collimated hollow beam and form an ideal image point with very tiny size at the Gaussian image plane of the receiving antenna. Defocusing and deflection of antenna may result in aberrations. The spot diagrams at the single-mode optical fiber end face which is located at the Gaussian image plane of the receiving antenna with on-axial defocusing ∆x = 0.7 μm, off-axial defocusing ∆y = 0.5 μm and deflection angle η = 8 μrad, respectively, are shown in Fig. 4 .

Fig. 4 Spot diagrams at single-mode optical fiber end face which is located at the Gaussian image plane of the receiving antenna with (a) on-axial defocusing ∆x = 0.7 μm, (b) off-axial defocusing ∆y = 0.5 μm and (c) deflection angle η = 8 μrad

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As shown in Fig. 4, the blue circle denotes the fiber core with diameter 10 μm. It is shown that on-axial defocusing of receiving antenna results in spherical aberration, off-axial defocusing and deflection of receiving antenna produce coma aberrations. The maximum defocusing and deflection amounts under which spots enter into the core of the single-mode optical fiber correspond to ∆x = 0.7 μm, ∆y = 0.5 μm and η = 8 μrad, respectively.

3.2 Design Single-mode Fiber Coupling System based on Vector Theory of Refraction

For the sake of aberration correction and coupling efficiency improvement, an optical fiber coupling system which consists of a plano-concave lens and a plano-convex lens is designed (as shown in Fig. 5 ).where f ₁ and f ₂ are the focal lengths of the plano-concave lens and the plano-convex lens, respectively. n ₀ and n are the refractive indexes of the air and the lens, respectively. θ is the divergence angle of incident light. The focal point (denotes by point A) of the plano-concave lens overlaps the ideal image point of the confocal receiving antenna. The plano-concave lens converts the incident convergent ray into a collimated beam, and the plano-convex lens focuses the collimated beam into an ideal image point at the center of the fiber core. The hyperboloidal surface equations of the plano-concave lens and plano-convex lens are

F_{5} (x, y, z) = \frac{{(x - x_{05})}^{2}}{a_{2}^{2}} - \frac{{(y - y_{05})}^{2}}{b_{2}^{2}} - \frac{{(z - z_{05})}^{2}}{b_{2}^{2}} - 1 = 0

F_{6} (x, y, z) = \frac{{(x - x_{06})}^{2}}{a_{3}^{2}} - \frac{{(y - y_{06})}^{2}}{b_{3}^{2}} - \frac{{(z - z_{06})}^{2}}{b_{3}^{2}} - 1 = 0

where

a_{2} = \frac{n_{0} f_{1}}{n - n_{0}}, b_{2} = f_{1} \sqrt{\frac{n + n_{0}}{n - n_{0}}}, a_{3} = \frac{n_{0} f_{2}}{n + n_{0}}, b_{3} = f_{2} \sqrt{\frac{n - n_{0}}{n + n_{0}}},

(x_{05}, y_{05}, z_{05}) = (L + 2 c_{1} - \frac{n f_{1}}{n - n_{0}}, 0, 0), (x_{06}, y_{06}, z_{06}) = (L + 2 c_{1} - a_{3}, 0, 0)

are center coordinates of the two hyperboloidal surfaces, respectively.

Fig. 5 The diagrammatic sketch of optical fiber coupling system structure consists of a plano-concave lens and a plano-convex lens

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Figure 6 shows ray tracing result of optical antenna-fiber coupling system and spot diagrams at single-mode optical fiber end face which locates at the Gaussian image plane of the optical fiber coupling system.

Fig. 6 (a) Ray tracing simulation result of optical antenna-fiber coupling system. (b) Spot diagrams at single-mode optical fiber end face which located at the Gaussian image plane of the optical fiber coupling system with f ₁ = 5.80 mm, a ₂ = 11.15mm, b ₂ = 12.77 mm, f ₂ = 15.29 mm, a ₃ = 6.07 mm, b ₃ = 6.95 mm, the refractive index of the BK7 lens is n = 1.52. Four spot diagrams correspond to confocal receiving antenna without deflection, on-axial defocusing ∆x = 0.7 μm, off-axial defocusing ∆y = 0.5 μm and deflection angle η = 8 μrad of the receiving antenna, respectively

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Spot diagrams as shown in Fig. 6 (b) are more concentrated to the center of the single-mode optical fiber core compared with Fig. 4, indicating that the aberrations caused by defocusing and deflection of receiving antenna are well corrected by the optical fiber coupling system. We calculate the aberrations based on their physicals properties rather than complex aberration formulas. Longitudinal spherical aberration $δ L^{'}$ is the distance from the ideal image point to the axial interaction of the ray. And the transverse spherical aberration $Δ Y^{'}$ is measured in the vertical direction. The spatial rays loss of symmetry and pass through the edge portions of the optical system with off-axis defocusing or deflection may be imaged at different heights than those pass through the center, this image fault called Coma aberration, including meridional coma K _t and sagittal coma K _s. Aberration curves calculated at the Gaussian image plane of antenna system and antenna-fiber coupling system are shown in Fig. 7 .

Fig. 7 Aberration curves calculated at the Gaussian image plane of optical antenna system and optical antenna-fiber coupling system (a) Spherical aberrations caused by on-axis defocusing (∆x = 0.7 μm) of the receiving antenna. (b) Coma aberrations caused by off-axis defocusing (∆y = 0.5 μm) of the receiving antenna. (c) Coma aberrations caused by deflection (η = 8 μrad) of the receiving antenna

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The aberration correction of the margin ray by using the fiber coupling system is shown in Table 2 .

Table 2. The aberration correction of the margin ray (h/h_m = 1) by using the fiber coupling system

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It is shown that the aberrations of the margin ray are well corrected by using the optical fiber coupling system. The correction effect of the aberration caused by the deflection of the receiving antenna is most significant.

Because of the fact that the refractive index varies with the wavelength of light, the image-forming properties of the optical fiber coupling system also vary with wavelength. Lateral chromatic aberration (LCA) is the dispersion that occurs because the magnification of the image differs with wavelength. LCA can be defined by the difference between the image heights for different wavelengths. LCA curves of blue, green, red and near-infrared wavelengths calculated at the Gaussian image plane of the optical antenna-fiber coupling system are shown in Fig. 8 . Four wavelengths and their corresponding refractive indexes in BK7 lens are λ _B = 445nm (n _b = 1.5258), λ _G = 532nm (n _g = 1.5195), λ _R = 632.8nm (n _r = 1.515), and λ _IR = 1550nm (n _IR = 1.5), respectively.

Fig. 8 LCA curves of blue, green, red and near-infrared wavelengths calculated at the Gaussian image plane of the optical antenna-fiber coupling system. (a) LCA curves in case of on-axis defocusing (∆x = 0.7 μm) of the receiving antenna. (b) LCA curves in case of off-axis defocusing (∆y = 0.5 μm) of the receiving antenna. (c) LCA curves in case of deflection (η = 8 μrad) of the receiving antenna

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It can be seen from Fig. 8 that the LCA curves of the red wavelength 632.8 nm and near-infrared wavelength 1550 nm have the same sign and distribute approximately. In the following part we make experimental tests of the actual optical system.

4. Experimental Test

Photograph of the experimental test system which consists of optical antenna-fiber coupling system and NanoScan laser beam profiling system is shown in Fig. 9 . By adjusting the defocusing amounts and deflection angle of the receiving antenna, aberrations were visualized by using NanoScan laser beam profiling system.

Fig. 9 Photograph of the experimental test system

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In this experimental test system, fiber coupling lens material is BK7(n = 1.517), aperture is 5.00mm, working distance is 14mm, measuring distance is 50 mm. Focal lengths of the plano-concave lens and the plano-convex lens are f ₁ = 5.80 mm and f ₂ = 15.29 mm, respectively, and the corresponding center thicknesses are 0.80 mm and 2.20 mm, respectively. The laser beam wavelength is 632.8 nm, instead of using the optical communication wavelength 1550 nm due to the limitation of experimental condition. Since the similar chromatic dispersion properties of the red wavelength 632.8 nm and near-infrared wavelength 1550 nm in BK7 lens, the experimental result may has certain practical significance to the optical communication field.

Figure 10 shows energy distributions and tested power profiles taken at the detecting plane of the antenna system and optical antenna-fiber coupling system, respectively.

Fig. 10 Comparison of energy distribution and tested power profile taken at the detecting plane of the antenna system and optical antenna-fiber coupling system. (a) Spherical aberration caused by on-axis defocusing of the receiving antenna, (b) off-axis aberration caused by off-axis defocusing of the receiving antenna, (c) off-axis aberration caused by deflection of the receiving antenna

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The experiment test results, which is similar with the simulation results depicted in Section 3, show that the aberrations caused by assembling errors of antenna can be well corrected by proposed fiber coupling system. In fact, spherical aberration and off-axis aberrations exist simultaneously in the experimental test.

5. Conclusion

An improved method of 3-D ray tracing and aberration calculation based on the vector theory of reflection and refraction is used to design an optical antenna-fiber coupling system. Results of simulation and the experimental tests show that the aberrations caused by on-axial defocusing, off-axial defocusing and deflection of receiving antenna are well corrected by the fiber coupling system.

Acknowledgments

Project supported by the National Natural Science Foundation of China (Grant No. 61271167 and No. 11574042), the Young Scientists Fund of the National Natural Science Foundation of China (Grant No. 61307093), the National Defense Pre-Research Foundation of China (Grant No. 9140A07040913DZ02106) and the Fundamental Research Funds for the Central Universities of Ministry of Education of China (Grant No. ZYGX2013J051).

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Reflector	Reflector surface equation	Eq.
Hyperboloidal sub-reflector of the transmitting antenna	$F_{1} (x, y, z) = \frac{{(x - x_{01})}^{2}}{a_{1}^{2}} - \frac{{(y - y_{01})}^{2}}{b_{1}^{2}} - \frac{{(z - z_{01})}^{2}}{b_{1}^{2}} - 1 = 0$	(4)
Paraboloidal primary reflector of the transmitting antenna	$F_{2} (x, y, z) = 2 p_{1} (x - x_{02}) - {(y - y_{02})}^{2} - {(z - z_{02})}^{2} = 0$	(5)
Paraboloidal primary reflector of the receiving antenna	$F_{3} (x, y, z) = 2 p_{1} [y \sin η + (x - L) \cos η - x_{03}] + {[y \cos η - (x - L) \sin η - y_{03}]}^{2} + {(z - z_{03})}^{2} = 0$	(6)
Hyperboloidal sub-reflector of the receiving antenna	$F_{4} (x, y, z) = \frac{{[y \sin η + (x - L) \cos η - x_{04}]}^{2}}{a_{1}^{2}} - \frac{{[y \cos η - (x - L) \sin η - y_{04}]}^{2}}{b_{1}^{2}} - \frac{{(z - z_{04})}^{2}}{b_{1}^{2}} - 1 = 0$	(7)

Reasons causing aberrations	On-axis defocusing (∆x = 0.7 μm) of receiving antenna		Off-axis defocusing (∆y = 0.5 μm) of receiving antenna		Deflection (η = 8 μrad) of receiving antenna
Aberration type	$δ L^{'}$	$Δ Y^{'}$	K_t	K_s	K_t	K_s
Correction	65.98%	39.03%	51.07%	53.22%	84.48_%	84.05_%

Reflector	Reflector surface equation	Eq.
Hyperboloidal sub-reflector of the transmitting antenna	$F_{1} (x, y, z) = \frac{{(x - x_{01})}^{2}}{a_{1}^{2}} - \frac{{(y - y_{01})}^{2}}{b_{1}^{2}} - \frac{{(z - z_{01})}^{2}}{b_{1}^{2}} - 1 = 0$	(4)
Paraboloidal primary reflector of the transmitting antenna	$F_{2} (x, y, z) = 2 p_{1} (x - x_{02}) - {(y - y_{02})}^{2} - {(z - z_{02})}^{2} = 0$	(5)
Paraboloidal primary reflector of the receiving antenna	$F_{3} (x, y, z) = 2 p_{1} [y \sin η + (x - L) \cos η - x_{03}] + {[y \cos η - (x - L) \sin η - y_{03}]}^{2} + {(z - z_{03})}^{2} = 0$	(6)
Hyperboloidal sub-reflector of the receiving antenna	$F_{4} (x, y, z) = \frac{{[y \sin η + (x - L) \cos η - x_{04}]}^{2}}{a_{1}^{2}} - \frac{{[y \cos η - (x - L) \sin η - y_{04}]}^{2}}{b_{1}^{2}} - \frac{{(z - z_{04})}^{2}}{b_{1}^{2}} - 1 = 0$	(7)

Reasons causing aberrations	On-axis defocusing (∆x = 0.7 μm) of receiving antenna		Off-axis defocusing (∆y = 0.5 μm) of receiving antenna		Deflection (η = 8 μrad) of receiving antenna
Aberration type	$δ L^{'}$	$Δ Y^{'}$	K_t	K_s	K_t	K_s
Correction	65.98%	39.03%	51.07%	53.22%	84.48_%	84.05_%

Design optical antenna and fiber coupling system based on the vector theory of reflection and refraction

Abstract

1.Introduction

2. Vector Theory of Reflection and Refraction

3. Optical Communication System Model Establishment and Aberrations Calculation

3.1 Establish the antenna system model based on vector theory of reflection

3.2 Design Single-mode Fiber Coupling System based on Vector Theory of Refraction

4. Experimental Test

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (10)

Tables (2)

Equations (5)

Optics Express