Monte Carlo simulation for coupling between single

Monte Carlo simulation for coupling between single-mode fiber and slab waveguide with spherical fiber microlenses Shuping Li Yuzhou Sun Jingping Zhu Tiantong Tang

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Optical Engineering 50(11), 115001 (November 2011)

Monte Carlo simulation for coupling between single-mode fiber and slab waveguide with spherical fiber microlenses Shuping Li Tibet University for Nationalities School of Information Engineering Xianyang, Shaanxi, 712082, China and Xi’an Jiaotong University Key Laboratory for Physical Electronics and Devices of the Ministry of Education Xi’an 710049, China E-mail: [email protected]

Abstract. For the butt-joint coupling between a single mode fiber (SMF) and a single mode dielectric slab waveguide (SMDSW) with a spherical fiber microlens (SFML), because the SMF and the rectangular-section SMDSW have different symmetry, it is difficult to calculate the coupling efficiencies using the traditional analytic method. In this paper, a Monte Carlo model of the coupler is proposed. This model incorporates the geometrical surface of both the SFML and the SMDSW, the relative position of the SMF and the SMDSW, and both the aberration and diffraction effects into consideration. The propagation characteristics of the coupler are simulated. And the coupling efficiencies and misalignment loss are investigated.  C 2011 Society of Photo-Optical Instrumentation Engineers (SPIE). [DOI: 10.1117/1.3651812]

Yuzhou Sun Innolight Technology Corp. 328 Xinghu Str. 12-A3, Industrial Park Suzhou, 215123, China

Subject terms: Monte Carlo simulation; butt-joint coupling; spherical fiber microlens. Paper 110596R received Jun. 2, 2011; revised manuscript received Sep. 26, 2011; accepted for publication Sep. 27, 2011; published online Oct. 27, 2011.

Jingping Zhu Tiantong Tang Xi’an Jiaotong University Key Laboratory for Physical Electronics and Devices of the Ministry of Education Xi’an 710049, China

1 Introduction With the development of integrated optical technology, a variety of integrated waveguide devices are applied in communication systems. A major problem is improving the coupling efficiency between the single mode optical fiber (SMF) and the single mode dielectric slab waveguides (SMDSW) in guided wave circuits.1–3 The butt-joint coupling is the most popular coupling form in integrated optical systems. For a butt coupler of an SMF coupling to an SMDSW, the fiber is rotationally symmetric, and the slab waveguide is planar symmetric. The discrepancy in mode size and mode symmetry gives rise to increased coupling losses. This can be performed by fabricating a microlens on the fiber tip, or by appropriately adjusting the waveguide parameters.4–7 Among them, the tapered-cladding fiber microlenses (FEMLs) fabricated directly on the end of SMFs offer some distinct advantages over other coupling techniques. Because spherical FEML (SFML) fabrication is simpler and faster than other FEMLs, the SFML is often used in a coupling system. In the past decades, the butt-coupling characteristics of fibers with waveguides have been theoretically investigated by numerical methods such as finite difference time domain8 and the overlap integral method.9, 10 Based on the overlap integral method and using the Gaussian and modified Gaussian approximation of single mode fibers and waveguides, simple analytical formulas have been presented. However, this simple method cannot take the reflection loss

C 2011 SPIE 0091-3286/2011/$25.00 

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between the end face of the fibers and waveguides into consideration, the diffraction loss is not considered yet, and the influence of the spherical aberration on the efficiency cannot be calculated. Rigorous numerical methods require generally large computation resources, and a three-dimensional simulation is time consuming. Monte Carlo methods are widely used in simulating physical and mathematical systems, and the methods can be used to solve many problems that traditional deterministic methods cannot. Using a Monte Carlo simulation method, with the physical beam parameters as probability functions, to calculate the coupling effects from an SMF with a spherical lens to a rectangular-section waveguide is not easily found in previous works. In this paper, a Monte Carlo model of an SMF and an SMDSW coupling with an SFML is proposed. This model does not use the concept of wave phase difference and phase transformation. Instead, it uses the principles of reflection and refraction at a plane interface between two media, and the diffractions is taken into account also. This treatment can take the primary aberration, fifth order aberration, and even the higher order aberration and reflection losses into consideration. 2 Theories Since the refractive index profiles of SMFs are cylindrically symmetric, the mode power of radial distribution and angular distribution are11  β r 2ωμ |A|2 J02 (hr ) r ≤ a f (r ) = , (1) β |B|2 K 02 (qr ) r > a r 2ωμ

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Li et al.: Monte Carlo simulation for coupling between single-mode fiber...

from the probability distribution function by the method of acceptance–rejection. This gives the following procedure for generating a random number from the distribution: i. Generate a random number r from the interval [0, a], where a denotes the radius of the core or the cladding when a random photon is generated in the core or the cladding. ii. Generate a random number y from the interval [0, c] that is within the range of the probability distribution function. iii. If y ≤ f(r), then accept; otherwise, go back to step i. So the incident rays are generated randomly by the above procedure, and then initial coordinates of the incident ray are defined. To get an accurate approximation for this procedure, the inputs should be a large number and random. The random sample number of incident rays is 80,000 in our work.

Fig. 1 Propagation of an optical ray through the coupling systems.

f (ϑ) =

1 , 2π

where  h = n 21 k02 − β 2 , k0 =

ω . c

(2)

q=



β 2 − n 22 k02 ,

B=

A J0 (ha) , K 0 (qa) (3)

When a beam along the single mode fiber propagates through the SFML, according to the phase transformation formula, the intensity of the diffraction field emerging from the SFML can be written as    exp( jkz) I (x, y) =  f (x1 , y1 )P(x1 , y1 ) jλz ∞    jk  2 (x −x1 )2 +(y − y1 )2 x1 + y12 exp jk × exp − 2f 2z 2  (4) × d x1 dy1  . P(x1 , y1 ) is the formula for a function of the pupil. The expression f(x1 , y1 ) describes the radial distribution function in the rectangular coordinate system. 3 Monte Carlo Modeling The coupler of an SMF to an SMDSW coupling with an SFML is shown in Fig. 1. Based on the principles of reflection refraction and diffraction of light, modeling is performed according to the following steps: 3.1 Generate Incident Lights The incident light is the mode of the SMF, and mode field distribution obeys the probability distribution functions [Eqs. (1) and (2)]. This distribution function is continuous, and thus it is difficult to obtain the inverse function. To get around this problem, we generated the incident lights Optical Engineering

3.2 Light Rays Tracing When the incident rays propagate through the SFML, and we can define the direction of propagation and the coordinates of the exit ray via a ray trajectory formula; however, there is perturbation generated by the diffraction of the lens aperture. This perturbation changes the direction of propagation of the exit ray, and the perturbation terms are a random variable with probability distribution [Eq. (4)]. The trajectories of the exit ray are treated as the sum of the geometrical trajectories plus perturbation terms caused by the diffraction effects. These exit rays are incident on the end-face of the SMDSW and in the interface. According to the trajectories formula and Snell’s law, the direction and the coordinates of the refraction ray can be defined. 3.3 Aggregate the Results These refraction rays launch into the SMDSW, and only certain light that satisfies the wave guiding conditions can propagate in the waveguide. The light which can propagate in an SMDSW is counted in the simulation results. 4 Program Test The program consists of two main programs, and the program for random sampling of the incident photons is tested first. The statistical radial intensity distributions are shown in Fig. 2, and the results calculated by Eq. (1) are also given. It is clear that the simulation results are in excellent agreement with the calculated one. The second step tests the ray tracing program. For a classic example of a spherical aberration calculation, we assumed a uniform plane wave with wavelength 1.55 μm incident on a plano-convex lens with refractive index n = 1.46, and radius of curvature R = 8 μm. The meridional rays emerging from the lens are shown in Fig. 3. One of the random sampling rays exits at a distance 4 μm from the optical axis, the calculating axial spherical aberration by formula is 4.0014,12 and the simulated axial spherical aberration is about 4 μm. These simulation results are in excellent agreement with the calculated data. It is clearly shown that the program is both numerically stable and accurately calculated. The optical propagation characteristics of the fiber end-face microlens are simulated using these programs.

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Li et al.: Monte Carlo simulation for coupling between single-mode fiber...

Fig. 2 Radial relative intensity of the incident beam in the incident plane.

Fig. 4 Spot sizes as a function of the curvature radius of the fiber microlens.

5 Propagation Characteristics of SFMLs The random sample number of incident rays is 80,000. The spot size is defined as the radius within which 84% of the photon energy is enclosed. In order to estimate the spot size and the waist position accurately, five trials are used in each case and the average results of the five trials. The spot sizes as a function of the curvature radius of the fiber microlens is shown in Fig. 4. For the lens with spherical aberration case, with the curvature radius increasing, the influence of the spherical aberration will reduce, and the influence of the diffraction effect will increase. So for both the aberration and diffraction effects, the spot size will have a minimum value. And with the curvature radius increasing, the spherical aberration effect decreases, and diffraction effect grows. The spherical aberration microlens is close to the aberration-free microlens, and the two spot size curves are close. It has been confirmed in Fig. 4. To show the effects of spherical aberration on the radial intensity distributions, numerical calculations for various curvature radii lens are carried out. Figure 5 illustrates the intensity distributions for lenses with different radii. Because

of the spherical aberration effect, the peak intensities are not located in the optical axis, and the intensity curves have two maxima. The position of the axial maximum intensity does not coincide with the waist position and the geometrical focus, and the intensity distribution curves should not match each other [as shown in Figs. 5(a) and 5(b)]. In the geometrical focal plane, the intensity undergoes more divergence, and thus the spot size will be larger [as shown in Figs. 5(c) and 5(d)].

Fig. 3 Simulation result of the axis spherical aberration.

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6 Coupling Efficiencies and Misalignments As stated in the above discussion, for an SFML, it is very clear that the intensity distribution curve of the focused beam is absolutely different from that of the SMF. Because of this significant difference, it is not valid to calculate the coupling efficiency using an overlap integral method. In the coupler, the ratio of the number of rays is satisfied by the wave guiding conditions and the number of incident rays is defined as the coupling efficiency. The coupling efficiencies of the SMFs to the LiNbO3 and silicon-based SMDSWs are calculated using this method, respectively. In the Monte Carlo model, the refractive indices of core and substrate of the LiNbO3 SMDSW are 2.148 and 2.138, respectively, and that of the SiO2 /Si SMDSW are 1.454 and 1.445, respectively. For both two slab waveguides, refractive indices of the cladding layers are 1. In the alignment case, the coupling efficiencies of a specific SMF coupling to the LiNbO3 SMDSW with different core thicknesses are shown in Fig. 6. When the core thickness is in the region of 3 to 5 μm, the end-face of the waveguide is in the geometrical focal plane and the coupling efficiencies are in the region of 45% to 67%. If the end-face is in the waist plane, the coupling efficiencies are in the region of 55% to 77%. Figure 7 shows the coupling efficiencies for the coupler of the SMF to the SiO2 /Si SMDSW. It is can be seen that the coupling efficiency variation tendency is similar as that of LiNbO3 SMDSW but the curves are flatter than that of LiNbO3 SMDSW. From the above discussions, when the end-face of the SMDSW lies in the waist plane, the coupling efficiencies are higher compared to the end-face lying in the geometric focal

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Fig. 5 Radial intensity distribution in the observed plane: (a) R = 5 μm; (b) R = 20 μm; (c) in the focal plane; (d) in the waist plane.

plane. The coupling efficiency curves are also shown to be flatter of end-face waveguides in a geometric plane. Even if the coupler is aligned, the coupling efficiencies are still not high, and the reasons may be as follows: the reflective

loss of the interface is one of the major sources. And another source is that the energy of the light emerging from the microlens cannot be launched into the SMDSW completely, because the spot size of the microlens with a large radius

Fig. 6 Coupling efficiencies for the SMF to LiNbO3 SMDSW when the end-face of the SMDSWs are in the (a) focal plane and (b) waist plane, respectively.

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Fig. 7 Coupling efficiencies for the SMF to the SiO2 /Si SMDSW when the end-face of the SMDSWs are in the (a) focal plane and (b) waist plane, respectively,

Fig. 8 Misalignment loss caused by displacement from y axis for the end-face of the SMDSW lying in the (a) focal plane and (b) waist plane, respectively.

Fig. 9 Loss caused by angular misalignment for the end-face of the SMDSW lying in the (a) focal plane and (b) waist plane, respectively.

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of curvature is larger than the thickness of the waveguide core. Compared with the focused beam by the microlens with larger radius of curvature, the beam divergence angle of the focused beam by the microlens with a smaller radius of curvature is larger, even if the spot size is small enough to launch into the SMDSW; however, these beams do not satisfy the wave guiding conditions, and thus it is not possible to achieve high coupling efficiencies. The major disadvantage of FEMLs is that their coupling efficiency is alignment sensitive. To investigate the loss caused by misalignment, as an example, the misalignment loss of an SMF to a LiNbO3 SMDSW coupling with a spherical FEML is discussed. The radius of curvature of the SFML is 14 μm, and the thickness of the SMDSW core is 5 μm. This is assuming the loss is 0 dB when they are in alignment. The misalignment loss of butt-joint coupling of an SMF and an SMDSW due to transverse and angular misalignments are shown in Figs. 8 and 9, respectively. In Fig. 8, when the offset distance is ± 1 μm, the corresponding losses are 0.31 and 0.3 dB, respectively, in the focal plane, and the losses are 0.38 and 0.36 dB in the waist plane, respectively. The loss curves are asymmetric due to the asymmetric SMDSW. The loss curves are also asymmetric in Fig. 9. When the waveguide end-face locates the focal plane of the SFML, the angular misalignment is ± 4 deg, and the losses reach 3.2 and 3 dB, respectively. However, when the end-face of the SMDSW is in the waist plane, the losses are 2.4 and 2.2 dB, respectively. It is clear that the bottom of the loss curves in the waist plane are flatter, in the region of –1.5 to 1.5 deg, and the losses caused by the angular misalignment are almost zero. As the angular misalignment increases, the losses rapidly rise. From the above results and discussion, the loss caused by the angular misalignment is more significant than that caused by the transverse offset. This is because of the angular misalignment directly effected on the propagation direction of the coupling beam in the SMDSW, and does not satisfy the propagation conditions of the guiding wave in the SMDSW. However, the higher coupling efficiencies can be gained when the angular misalignment is in the region of − 4 to 4 deg. The z-dependence of the coupling loss is shown in Fig. 10. The loss is 0 dB when the end-face of the SMDSW is

Fig. 10 z-dependence of the coupling loss.

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lying in the focal plane. The lowest loss is not at the focus of the microlens, but close to the microlens due to the spherical aberration and the diffraction effects. As the offset distance is in the region of − 5 to 5 μm, the loss is only less than 2 dB. It is clearly seen that the z-dependence of the coupling efficiencies is more tolerable to longitudinal positioning than displacement from the y axis and angular misalignment. And the curve is also asymmetric due to the spherical aberration. 7 Conclusions A Monte Carlo model for the coupler is proposed in this paper. This model does not use the concept of wave phase difference or phase transformation, preferring to use the principles of reflection and refraction at a plane interface between two media since reflection loss can be calculated by the Fresnel equations. This treatment takes the primary aberration, fifth order aberration, higher order aberrations, and reflection loss into consideration. And it can take the geometrical surface of both the SFML and the SMDSW, the relative positions of the fibers, and SMDSWs, and both aberration and diffraction effects into consideration. The propagation characteristics of the coupler are simulated using this Monte Carlo model, and the coupling efficiencies and misalignment loss are investigated. The simulation results show that higher coupling efficiencies can be obtained in this coupler due to a small coupling loss. And the misalignment loss is tolerable when the angular misalignment is in the region of − 4 to 4 deg and the offset distance from the focus is in the region of − 5 to 5 μm. It is clear that this method makes it easy to calculate the coupling efficiencies, electromagnetic field distribution in the arbitrary observed plane, as well as the misalignment losses. Thus, the Monte Carlo model is universally applicable to the coupler of an SMF to a slab waveguide butt coupling with lens.

References 1. T. Paatzsch, I. Smaglinski, M. Abrham, H. D. Bauer, U. Hempelmann, G. Neumann, G. Mrozynski, and W. Kerndlmaier, “Very low-loss passive fiber-to-chip coupling with tapered fibers,” Appl. Opt. 36(21), 5129–5133 (1997). 2. K. Shiraishi, M. Kaqaya, K. Muro, H. Yoda, Y. Koqami, and S. T. Chen, “Single-mode fiber with a plano-convex silicon for an integrated butt-coupling scheme,” Appl. Opt. 47, 6345–6349 (2008). 3. S. Peng and R. M. Reano, “Cantilever coupler for intra-chip coupling to silicon photonic integrated circuits,” Opt. Express 17, 4565–4574 (2009). 4. C. W. Barnard and J. W. Y. Lit, “Single-mode fiber microlens with controllable spot size,” Appl. Opt. 30, 1958–1962 (1991). 5. F. Schiappelli and R. Kumar, “Efficient fiber-to-waveguide coupling by a lens on the end of the optical fiber fabricated by focused ion beam milling,” Microelectron. Eng. 73–74, 397–404 (2004). 6. M. Lehtonen, G. Genty, and H. Ludvigsen, “Tapered microstructured fibers for efficient coupling to optical waveguides: A numerical study,” Appl. Phys. B 81, 295–300 (2005). 7. J. K. Doylend and A. P. Knights, “Design and simulation of an integrated fiber-to-chip coupler for silicon-on-insulator waveguides,” IEEE J Sel. Topics Quantum Electron. 12, 1363–1370 (2006). 8. L. Yang, D. Dai, B. Yang, Z. Sheng, and S. He, “Characteristic analysis of tapered lens fibers for light focusing and butt-coupling to a silicon rib waveguide,” Appl. Opt. 48, 672–678 (2009). 9. R. Van Aken, “Single mode fiber to planar waveguide coupling with ball lenses,” Technische Universiteit Eindhoven, Eindhoven (1991). 10. S. R´ıos, R. Srivastava, and C. G´omez-Reino, “Coupling of single-mode fibers to single-mode GRIN waveguides by butt-joining,” Opt. Commun. 119, 517–522 (1995). 11. A. Yariv, Optical Electronics in Modern Communications, 5th Ed., p. 97, Publishing House of Electronics Industry, Beijing (2002). 12. B. Max and W. Emil, Principles of Optics, 7th ed. University of Cambridge Press, Cambridge, England (1999).

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Shuping Li received a BEng degree in optical information science and technology from Changchun University of Science and Technology, Changchun, China, in 1990. She has received her PhD degree in the Department of Electronic Science and Technology, Xi’an Jiaotong University, Xi’an, Shaanxi, China. Her current research interests include integrated optic circuits and passive components. Biographies and photographs of the other authors are not available.

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