Decoupling and asymmetric coupling in triplecore photonic crystal fibers

Transcription

1 1488 J. Opt. Soc. Am. B/ Vol. 5, No. 9/ September 008 Yan et al. Decoupling and asymmetric coupling in triplecore photonic crystal fibers Yan Yan, 1, * Jean Toulouse, 1 Iavor Velchev, and Slava V. Rotkin 1 1 Department of Physics, Lehigh University, 16 Memorial Drive E., Bethlehem, Pennsylvania 18015, USA Laser Center, Fox Chase Cancer Center, 43 Rhawn Street, Philadelphia, Pennsylvania 19111, USA *Corresponding author: yay@lehigh.edu Received May 0, 008; revised July 8, 008; accepted July 9, 008; posted July 1, 008 (Doc. ID 9640); published August 19, 008 We systematically investigate the intercore coupling in triple-core photonic crystal fibers (PCFs). Coupledmode equations are developed and solved analytically in the linear approximation. We derive the eigenmodes in triple-core PCFs, in particular we model the decoupling mode and test its stability against perturbations. The coupling coefficients and the coupling length are determined experimentally and found to agree extremely well with the calculated results. We describe the propagation of light in triple-core PCFs for different launching conditions. 008 Optical Society of America OCIS codes: , INTRODUCTION In recent years, photonic crystal fibers (PCFs) have attracted substantial interest [1 4]. The PCFs have been shown to support endlessly single-mode guidance [5], tunable chromatic dispersion properties [6,7], strong confinement [8], and high nonlinearities [9,10]. Multicore PCFs, because of their unique properties, offer additional advantages and provide new opportunities for original applications [11 14] such as in-fiber frequency shifters and directional couplers. Multicore fiber lasers [15] or multicore fiber amplifiers [16,17] that use evanescent field-coupled arrays are also potential applications. So far, most of the research on multicore PCFs has focused on theory and computer simulations [18 0]. The few experimental results that have been reported are mainly on dual-core PCFs. Blanchard et al. proposed a two-dimensional bend sensor with a single microstructured fiber [1]. Fogli et al. analyzed dual-core PCF couplers with full vectorial beam propagation methods []. The properties of dual-core PCF were experimentally characterized by Mangan et al. [3]. Directional coupling in a dual-core PCF was realized by Kakarantzas et al. using heat treatment [4]. Yet we anticipate that triple-core PCFs, because of their unique structure, possess distinct characteristics and open new possibilities beyond those of dual-core fibers. In triplecore PCFs more eigenmodes are possible and therefore more complex light patterns in propagation, which offers new opportunities for versatile manipulation and control of guided light, e.g., switching on the basis of polarization. Nonlinear switching with a pump beam also combines intercore coupling with nonlinear aspects in PCFs. In summary, the triple-core fibers represent the simplest example of multicore fibers, allowing us to demonstrate specific coupling effects present only in fibers with more than two cores. Here we systematically investigate the coupling behavior in triple-core PCFs by means of both theory and experiments. First, the particular PCF of interest is presented. Second, we establish the coupled-mode equations that describe the propagation of light in triple-core PCFs. We then study propagation of light for different launching conditions, in particular launching into the central core. From our simulations and calculations, we also determine the coupling length of the triple-core PCF under test. Experimental results are then reported and shown to be in excellent agreement with theoretical predictions.. PHOTONIC CRYSTAL FIBER The PCF used in our theoretical analysis and experiments consisted of three solid cores with an index of refraction of n=1.46, embedded in a triangular lattice of airholes with a diameter of d=1.08 m, as sketched in Fig. 1(a). The three central missing airholes provided the PCF with the necessary defect to guide light. The pitch of the periodic structure (the hole spacing) was =.5 m. The distance between the centers of the cores was 3. Amicroscope image of the actual cross section of the fiber used in the experiments is shown in Fig. 1(b), which shows that the fiber was well fabricated and cleaved. The central part, including the core area and the periodic microstructure zone, did not reveal any obvious sign of deformation. Nevertheless, it is important to note that even small variations in the shape of microcores can affect the overlap between optical modes that propagate in neighboring cores and therefore the coupling between these cores, even though the coupling constant is barely affected by such imperfections [5]. 3. THEORY In multicore fibers, the neighboring cores must be sufficiently close for the optical modes that propagate in the adjacent core to overlap the separating region. Such evanescent wave coupling between optical modes associated with an adjacent core can lead to the transfer of power /08/ /$ Optical Society of America

2 Yan et al. Vol. 5, No. 9/September 008/J. Opt. Soc. Am. B 1489 Fig. 1. (Color online) (a) Fiber geometry; cores are marked with large circles for illustration purpose only. (b) A microscope image of the cross section of the fiber used in the experiments. (c) The cross section profile. 1 and are the coupling coefficients between neighboring cores. from one core to the another under suitable conditions. Since the optical modes tend to be localized within each core, only nearest neighbors communicate with each other [6]. Neglecting direct coupling between cores 1 and 3, the coupled-mode equations in the linear approximation for triple-core PCFs can therefore be written as [7] A 1 z + 1 A 1 v g t + i A 1 t = i 1A, A z + 1 A v g t + i A t = i 1A 1 + i A 3, A 3 z + 1 A 3 v g t + i A 3 t = ia, 1 where A 1, A, and A 3 denote the amplitudes in cores 1,, and 3, respectively, at position z along the fiber; v g is the group velocity; is the group velocity dispersion (GVD); 1 and are the coupling coefficients between cores 1 and and cores and 3, respectively, as illustrated in Fig. 1(c). For a perfect fiber, 1 and should be equal; however, we found that was not the case in our fibers. The coupling length, which is the main characteristic parameter of the coupling system, is defined as the shortest distance over which the signal fully transfers from one waveguide to an adjacent one. For triple-core fibers, if the beam is initially launched into only the central core, it will fully transfer to both side cores as it propagates over a coupling length along the fiber. For a pair of coupled waveguides, the coupling length is simply given by L = /, where designates the coupling coefficient between the two waveguides. For the triple-core PCF used in our experiments, the coupling length is found to be L c = / 1 +, where we have assumed for generality that coupling coefficients 1 and between neighboring cores might not be equal. Starting from z=0, all the power is transferred from the central core to the two side cores over a distance L c or vice versa. Since we are considering a low-power cw beam launched into the triple-core fiber, the time-dependent terms can be set to zero. Therefore, Eq. (1) can be simplified as da 1 dz = i 1A, Starting with trial solutions da dz = i 1A 1 + i A 3, A i z = C i1 e i + C i e i + C i3, da 3 dz = ia. where i=1,,3, substituting them into Eqs. () and rearranging the terms, we obtain A 1 z = 1 C + e i + 1 C e i + C 0, A z = 1 + C + e i 1 + C e i, A 3 z = C + e i + C e i 1 C 0. 3 By putting A i z=0 =A i0, we determine the three independent constants of the problem C 0, C +, and C in terms of A 10, A 0, A 30, the initial amplitudes in cores 1,, and 3, respectively, C 0 = 1 + A A 30, 1 C + = 1 + A 0 1A 10 + A , 1 C = 1 + A 0 1A 10 + A To determine the eigenmodes for the triple-core fiber, we rewrite Eqs. (3) in matrix form T J A 0=A. The matrix T J is thus found to be 4 = 1 cos 1 + z T J i sin 1 + z cos 1 + z 1 i sin 1 + z cos 1 + z i 1 + sin 1 + z cos 1 + z 1 i 1 + sin 1 + z. 5 cos 1 + z

3 1490 J. Opt. Soc. Am. B/ Vol. 5, No. 9/ September 008 Yan et al. Fig.. Initial amplitudes in three cores of the eigenmodes. 1 and are equal. It is worth noting here that the factor i in the off-diagonal terms of the matrix indicates that the fiber introduces a relative phase shift between the coherent beams that propagate in the three cores, which can be applied to the design of fiber interferometers. Solving the determinant of this 3 3 matrix, we obtain three eigenvalues of the system: =1, e i 1+ z, e i 1+ z. The corresponding eigenvectors are then easily found to have the form 0 1, After normalization, we obtain A d A 0 1 +, ,. 6 A s A + z 1 0 ei 1 1 +, A a A + z 1 0 e i 1 1 +, where A 0 =P 0 is the total initial power launched into the fiber and A d, A s, A a are the eigenmodes, or so-called supermodes, of the propagation. They are labeled as decoupling, symmetric, and antisymmetric, respectively. An arbitrary optical amplitude distribution between the three cores can then be written as a superposition of these three eigenmodes. For example, launching light into the central core would correspond only to a superposition of two eigenmodes: A s A a, launching into core 1 corresponds to 1/ A s+a a +A d and launching into core 3 corresponds to 1/ A s+a a A d. The amplitude distributions that correspond to these three eigenmodes at z=0 are shown in Fig., where 1 and are set equal for illustration purpose. By definition, light launched according to one of these eigenmodes will propagate undisturbed with a constant ratio of the intensities in the three cores, i.e., A i z =const. We note that, for the decoupling mode A d in Eq. (6), neither the amplitudes nor the phases of the beams in the three cores change with distance. Decoupling here means that no energy transfer occurs between cores. When no beam is launched into the central core, and the ratio of the amplitudes in the two side cores is equal to that of the coupling coefficients but out of phase, these amplitudes will remain constant as they propagate along the fiber, with no light being transferred to the central core, as shown in the first graph in Fig.. Figure 3(a) shows the amplitudes in the three cores along the fiber when launching A d by itself. 1 / is set to 3/4 for illustration purpose. In practice, there can be slight variations along the fiber due to imperfections. Hence we have studied the stability of the decoupling mode against perturbations. Supposing that the ratio of the amplitudes in the two side cores suffers a minor change due, for example, to unequal fiber loss, we can write the launching condition as A 1 = A 0 A, A =0, A 3 = 1 A 0, which is equivalent to launching into the decoupling mode except for a small additional amplitude A in core 1. Using Eqs. (3), we then calculate the amplitudes in the three cores after a propagation length z: 1 A A 1 z = A cos 1 A + z 1 +, A z = i 1 A 1 + sin 1 + z, Fig. 3. (Color online) (a) Amplitudes in the three cores for the decoupling mode. (b) Amplitudes in the three cores for the decoupling mode with small perturbations.

4 Yan et al. Vol. 5, No. 9/September 008/J. Opt. Soc. Am. B 1491 Fig. 4. (Color online) Decoupling versus coupling. 4. SIMULATION To study the guided propagation of light with coupling in multicore PCFs, we developed an efficient and versatile numerical program [8]. Although many methods have been employed, such as the modal field expansion method [9,30], multipole method [31], and beam propagation method [3,33], our model utilizes a fully threedimensional (3-D) numerical method to solve Maxwell s equations in vector form. The code is based on applying the dispersion and propagation operators separately for each single propagation step in a finite-difference scheme. We use the fast Fourier transform (FFT) technique for the calculation of diffraction effects in k space. The calculations are run on a rectangular mesh with a square cross section. A super-gaussian instead of a Gaussian model is used to eliminate the light reflected back from the boundaries of the fiber structure. This 3-D code allows for the determination of coupling coefficients for various fiber geometries. The high precision of our method is confirmed by the consistency of our results with values reported in the literature []. A 3 z = 1 A 0 1 A 1 + cos 1 + z + 1 A The propagated amplitudes are plotted in Fig. 3(b). This result shows that the decoupling mode is stable against small perturbations. We have also checked the perturbation that is due to small changes in the coupling coefficients and have found that they affect the decoupling mode in a similar way. It is useful to note that, for the other two eigenmodes, symmetric and antisymmetric at any propagation length z, we always have A 1 z A 3 z = P 1 P 3 = 1, 10 where P 1 and P 3 are the intensities in core 1 and core 3, respectively. Equation (10) shows that the ratio of the two coupling coefficients is equal to the square root of the ratio of the output powers in the two side cores. Note here that both the symmetric and the antisymmetric modes maintain constant intensities in all three cores, whereas the phases change as they propagate along the fiber. A. Decoupling Mode We first tested the theoretical results on the decoupling mode by modeling two antiphase beams of equal intensity, simultaneously incident into the two side cores, as in decoupling mode A d. A symmetric fiber, in which the two coupling coefficients were equal, was used for this simulation. The results are shown in Fig. 4. After 10 cm, the intensity distribution indeed remains unchanged. We compared this with the result obtained for two in-phase beams in the same launching conditions (not an eigenmode). After 10 cm we see observed that 59% of the energy that propagates initially in the side cores was transferred to the central core. B. Propagation in Guided Cores In the experiments, to measure the coupling lengths and describe the propagation, we found it simple and sufficient to launch light into the central core, i.e., core. In this case, the normalized initial amplitudes are Fig. 5. (Color online) Amplitudes in the three cores versus the propagation length, with the initial light in core. (a) 1 / =1 (b) and 1 / =3/4

5 149 J. Opt. Soc. Am. B/ Vol. 5, No. 9/ September 008 Yan et al. As can be seen from Eqs. (11), the amplitudes in the two side cores vary sinusoidally while the amplitude in the central core varies like a cosine wave, as shown in Fig. 5, where we plotted the amplitudes in the three cores along the propagation length. For illustration purpose, 1 / is set to 1 in Fig. 5(a) and to 3/4 in Fig. 5(b). It is worth noting that the amplitudes in the two side cores reach their maxima or minima at the same points along the fiber. The difference between the two coupling coefficients 1 and, if there is any, simply results in different maximum amplitudes in the two side cores rather than different coupling lengths. In another words, 1 and do not result in different interaction lengths but only different amplitudes. In the simulation, we modeled the propagation of light launched into the central core. The first complete energy transfer from the central core to the two side cores is found at z=67.54 nm, which is therefore the coupling length. The result is shown in Fig. 6. Similarly, we obtained the propagation equations for launching with one side core, core 1 or core 3. The amplitudes in the three cores along the propagation length are plotted in Fig. 7: 1 / is set to 3/4 for illustration purpose. Fig. 6. (Color online) At z=67.54 mm, the light in the central core is entirely transferred to the two side cores. and the coefficients are A10 A 0 A 30 = 0 0 1, C 0 =0, C + = C = Equation (3) there becomes A A 1 z = i 1A sin 1 + z, A z = A 0 cos 1 + z, A 3 z = ia sin 1 + z. 11 C. Offset Sensitivity To test the sensitivity of the latter experiment to the launching conditions, we performed a simulation in which the light was launched again into the central core, but witha1 m offset from the center of the core. As expected from theory, this resulted in higher optical modes being excited, although these were found to die out within 5 cm as the beam propagated in the fiber. The results are shown in Fig EXPERIMENTS We now present the experimental technique followed by the experimental results from which we determined the coupling coefficients and the coupling length. The coupling length is estimated using an empirical formula as well. A. Experimental Setup To probe the linear coupling characteristics of the fiber, we launched light from a helium neon laser into a single core. To do this, the beam was first expanded through a Galilean expansion system, and the divergence and spot size of the beam matched to the diameter and numerical aperture (NA) of the microcore, using an apochromatic aberration and flat field optical correction microscope objective. The cross section of the fiber was illuminated through a glass plate and was projected onto a television screen along with the focal spot of the launched beam. In this way proper launching conditions were accurately monitored. At the opposite end of the fiber, another microscope objective was placed at its focal distance from the fiber tip. A magnified image of the near field of the output was produced through a CCD camera. In this way we managed to record the output and measured the respective intensities in each of the three cores. Figure 9 is a schematic of the experimental setup. Fig. 7. (Color online) Amplitudes in the three cores versus the propagation length with the initial light in one side core. Light launched into (a) core 1 and (b) core 3.

6 Yan et al. Vol. 5, No. 9/September 008/J. Opt. Soc. Am. B 1493 Fig. 8. (Color online) Offset dies out at 5 cm through a simulation program. For a final test that the intensity output from the three cores would not be affected by small offsets of the launched beam from the center of a particular core, we launched light into the central core. The launching spot was slightly displaced along the x and the y axis [parallel with and perpendicular to the orientation of the three cores, respectively, as shown in Fig. 1(c)] so as to test the result of offsets in launching. The normalized transmitted power through each core was found to be insensitive to x-axis offsets to within 8 m and y-axis offsets to within 7 m. This provides convincing background for later experiments. The results are shown in Fig. 10. B. Experimental Results Using the setup described above, we conducted experiments on the light propagation in a triple-core three-pitch PCF to study the intercore linear coupling. The lengths of the tested fibers ranged from 7 to 65 cm. For longer fibers, although nominally isotropic, small twists, bends, and other stresses can induce unknown and uncontrolled birefringence on the fiber so that the output becomes unpredictable. Shorter fibers gave stable results. No visible beam coupling was observed in fibers shorter than 9 cm, which indicated longer coupling lengths. The results reported below were obtained on a 5. cm fiber. The output intensity distributions for light launched successively into each of the three cores are shown in Fig. 11. Comparison of these intensity distributions reveals some asymmetry in the intercore coupling, most clearly seen in Figs. 11(a) and 11(c): very little power was found in core in Fig. 11(c); some light was transferred into core in Fig. 11(a). When light was launched into one of the side cores, that is, core 1 or core 3, the intensity distributions at the output were not symmetric. This asymmetry persisted when light was launched into the central core; the intensities transferred to the two side cores also being unequal. After a certain propagation distance, the power in the central core was coupled more into one side core than into the Fig. 9. (Color online) A schematic of the experimental setup. other. This asymmetric coupling phenomenon was observed for all sections of the triple-core fiber and for different launching conditions, indicating that the coupling asymmetry was intrinsic to the fiber, i.e., the coupling coefficients 1 and were not equal. The coupling coefficients and the coupling length were determined in the following experiments. As mentioned earlier, a simple and efficient method for obtaining the coupling length is to launch light into the central core and measure the relative output intensities from the three cores. Substituting the output intensities on the left-hand side of Eqs. (11) and the input intensities on the righthand side, we were able to calculate the coupling coefficients and the corresponding coupling length: 1 = m 1, 1 =.1778 m, L c = / 1 + =54.56 cm. C. Estimate of the Coupling Length with an Empirical Formula For a symmetric coupler, the following empirical formula can also be used to estimate the coupling coefficient [34]: V = k 0 n 0 a exp c 0 + c 1 d + c d, 1 where c 0, c 1, and c are constants: c 0 = V V, c 1 = V 0.015V, c = V V. V is the fiber parameter, so-called normalized frequency V=k 0 a n 1 n 1/ = / 0 ana in which a is the core radius and d is the normalized centerto-center spacing between neighboring cores d. Equation (1) has been shown to be accurate to within 1% for values of V and d in the range of 1.5 V.5 and Fig. 10. (Color online) Normalized transmitted power through cores versus offsets along the x- and y-axes.

7 1494 J. Opt. Soc. Am. B/ Vol. 5, No. 9/ September 008 Yan et al. Fig. 11. (Color online) Basic result of the asymmetric coupling between cores. The fiber was 5. cm long..0 d 4.5. The NA of the fiber was calculated to be NA=sin 0.0 by illuminating the entire input end of the fiber and measuring the diverging angle of the outcoming beam. Therefore, for the triple-core three-pitch PCF under test, we find the coupling coefficients and the coupling length, as shown in Table 1, where the values obtained from the simulations and the experiments are listed as well. Considering the discretization errors in the simulations and the approximation in the empirical formula, they are in good agreement. Further confirmation of the agreement between modeling and experiments was obtained by launching light into one side core. For example, with all the other input conditions fixed but light launched into core 3 and using the propagation in Eq. (3), our simulation predicted the normalized output intensity in core 3 to be 73.% of the total while the experiment yielded 7.49%. Experiments were also conducted with triple-core fourpitch PCFs (hole diameter of d=1.10 m, hole spacing =.5 m, core spacing of 4 ). Similar asymmetric coupling and even longer coupling lengths were found in these fibers. 6. SUMMARY We have analyzed the mechanism of linear coupling in triple-core PCFs and identified and calculated the eigenvalues and eigenmodes or supermodes. The amplitudes in the three cores were found to change in a periodic fashion. In particular, the decoupling mode was simulated and measured in great detail. The coupling parameters of the triple-core PCF were measured and found to be in good agreement with those from theory. The PCFs studied were all found to exhibit asymmetric coupling, possibly due to slight variations in fiber parameters or fiber geometry along the length, even though no obvious structural variations or deformations were visible by observation Table 1. Coupling Length and the Coupling Coefficients Obtained from the Estimation, the Simulation, and the Experiments Coupling Length (mm) Coupling Coefficient mm 1 Estimation NA= NA= Simulation Experiments , 0.00 under an optical microscope. Future research will focus on the polarization dependence of the intercore coupling and the effect of fiber nonlinearities for higher launched powers. ACKNOWLEDGMENTS Special thanks to Ivan Biaggio for his useful suggestions with regard to the experimental setup. We acknowledge a grant from the Optoelectronics Industry Development Association (OIDA) that enabled us to acquire the photonic crystal fibers used in this study. The research was made possible by a grant from the Electrical Engineering Division of the National Science Foundation, NSF ECS Additional support was also provided by the Center for Optical Technologies at Lehigh University funded by a grant from the State of Pennsylvania Department of Community and Economic Development. REFERENCES 1. P. St. J. Russell, Photonic-crystal fibers, J. Lumin. 4, (006).. J. A. West, N. Venkataramam, C. M. Smith, and M. T. Gallagher, Photonic crystal fibers, in 7th European Conference on Optical Comunication (IEEE, 001), paper Th. A Y. Jiang, Z. Shi, B. Howley, and R. T. Chen, Highly dispersive photonic crystal fibers for true-time-delay modules of an x-band phased array antenna, Proc. SPIE 5360, (004). 4. J. G. Rarity, J. Fulconis, J. Duligall, W. J. Wadsworth and P. St. J. Russell, Photonic crystal fiber source of correlated photon pairs, Opt. Express 13, (005). 5. J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, All-silica single-mode optical fiber with photonic crystal cladding, Opt. Lett. 1, (1996). 6. L. P. Shen, W.-P. Huang, G. X. Chen, and S. S. Jian, Design and optimization of photonic crystal fibers for broad-band dispersion compensation, IEEE Photon. Technol. Lett. 15, (003). 7. G. K. L. Wong, A. Y. H. Chen, S. W. Ha, R. J. Kruhlak, S. G. Murdoch, R. Leonhardt, J. D. Harvey, and N. Joly, Characterization of chromatic dispersion in photonic crystal fibers using scalar modulation instability, Opt. Express 13, (005). 8. J. C. Knight, T. A. Birks, P. St. J. Russell, and J. P. de Sandro, Properties of photonic crystal fiber and the effective index model, J. Opt. Soc. Am. A 15, (1998). 9. A. I. Siahlo, L. K. Oxenlowe, K. S. Berg, A. T. Clausen, P. A. Andersen, C. Peucheret, A. Tersigni, P. Jeppesen, K. P. Hansen, and J. R. Folkenberg, A high-speed demultiplexer based on a nonlinear optical loop mirror with a photonic crystal fiber, IEEE Photon. Technol. Lett. 15, (003).

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