Design, transform and control of optical field in discrete optical system: an example

A discrete optical system can broaden the spatial distribution of the input light through optical coupling in array waveguides, just like diffraction in continuous media. Here, we theoretically demonstrate several kinds of control methods of optical field propagation in a discrete optical system, which is composed of an Airy fiber with two perpendicular arrayed cores. A brief transform mechanism between Gaussian and Airy beam propagation in such a fiber is presented. The wavefront of the output beam from the Airy fiber is actually dependent on the phased arrayed modulation of coupling array cores. Except the optical wavelength changing, we propose two new methods, including fiber length and bending-induced refractive-index changing, to accomplish that modulation. The calculation results show that these new methods are very effective for the Airy phase modulation. By combining these methods and controlling the corresponding parameters, the Gaussian beam, the one-dimension Airy beam, and the two-dimension Airy beam can be obtained by one same Airy fiber. These methods are also generally applicable to the other discrete optical system and can be extended to generate any other types of optical beams, such as Bessel beams and Mathieu beams.

outermost 1 st to 4 th order side lobes, as shown in Fig. 1(a). There are two perpendicular arrayed cores arranged in the x-axis (namely, core 1, 2, 3, 4, 5) and the y-axis (namely, core 1, 2′, 3′, 4′, 5′) embedded in a common cladding. The main core 1 is located at the center of the fiber, where the origin O of the Cartesian coordinate system (X, Y, Z) is taken to be. To observe optical field along a 45-degree direction, another coordinate system (X1,Y1, Z) is also established by rotating the (X, Y, Z) coordinate system 45 degrees clockwise around the z-axis. The refractive-index (RI) distribution of the arrayed-core is represented by the solid line in Fig. 1(b) (see the supplementary material). In order to ensure the stability of the propagating wave, every core of the fiber can only support one mode. And the center of every core of the fiber is located at the intensity peaks of corresponding optical lobes of Airy field |ψ(x)| 2 from equation (2) for λ = 980 nm, m 0 = 5 μm, m 1 = 4.79 μm, a m = 0.06, v m = 0, y = z = 0. Moreover, a Gaussian beam (GB) is coupled into the center main core of the Airy fiber by fusion splicing a single mode fiber (SMF), as shown in Fig.1(a).
Based on the supermode theory in arrayed-waveguide (see eq. (12)), we calculate the optical coupling power in the main core and side cores of the Airy fiber while Gaussian beam inputs, as shown in Fig. 1(c) and (d). One can find that the magnitude of the coupling power decreases gradually from inside out. And the light coupling among arrayed-core is periodical, whose coupling period T 0 is 4.4 mm depicted in Fig. 1(c). Figure 1(a) also shows the transverse field patterns (left side) at different fiber lengths Z = (0.5 + m/6)T 0 . Here m is any integers among −3 to 3, that is corresponding to fiber lengths Z A , Z B , …, Z G , respectively. In the first half period (from Z A to Z D ), as fiber length increases, the intensity of propagating light in the center main core is decreased and at the meantime side cores' are increased. In the second half period (from Z D to Z G ), light powers in side cores are gradually coupled into the center main core with fiber length increasing and finally one can obtain a Gaussian-like beam at Z G . Consequently, the Gaussian-like beam and the Airy-like beam can be converted each other using a half period T 0 /2 length of the Airy fiber. And that means, as a Gaussian beam input, one can obtain an Airy-like beam and a Gaussian-like beam with (2 m − 1)T 0 /2 and mT 0 length of Airy fiber (m is any positive integer), respectively. Figure 2(a) shows a schematic diagram of a bent Airy fiber. Note that the light propagation in the arrayed-core of the bent Airy fiber is sensitive to the magnitude and direction of the bending radius due to its asymmetry structure. Therefore, for the sake of simplicity, the bending direction of the Airy fiber is assumed along the X1-aixs. In this case, we give the equivalent RI distribution of bent Airy fiber using eq. (15), as shown in Fig. 2(b). From the figure, we can find that the equivalent RI distribution of bent fiber is tilted with respect to the original without bending. And the corresponding linear change can be determined by the magnitude and direction of the bending. Figure 2(c) gives the calculation result of the transverse mode field propagation in Core 1 with a bending radius of 35 mm using eq. (13). We can clearly see that the mode field shifts away from the core center, which causes a distortion along the bending direction. For comparison, the simulation results are obtained by using commercial COMSOL Multiphysics software. From Fig. 2(c)-(e), we can find that the calculation results are very agree with the simulation results.
Transform mechanism in the Airy fiber. Using the coupling matrix M in eq. (9), we can calculate the propagation constants β ′ i of the guided supermodes in the arrayed-core of the Airy fiber, which are given by dashed line marked circle in Fig. 3(a). One can find that there are four pairs of degenerate modes whose mode number are 2m−1 and 2 m (m = 1, 2, 3, 4). However, not all supermodes can be excited in Airy fiber. Based on the absolute values of mode amplitude depicted by the dashed line marked square in Fig. 3(a), we can see that mainly excited supermodes are 3-, 5-, 7-, 9-order modes, especially the 9-order mode which contains a large percentage of the total power (more than 80% in our case). The transverse intensity patterns of mainly excited supermodes are shown in Fig. 4(c)-(f). Note that all the mode fields are symmetric to the X1-axis for Gaussian beam input.
As mentioned above, we have demonstrated that an Airy-like beam can be generated by a (2 m − 1)T 0 /2 length of Airy fiber after periodical amplitude and phased arrayed modulation through supermodes interference. In Fig. 4(a) and (b), we can find that there is almost no phase difference between such the Airy-like beam and the ideal Airy beam with the launch angle θ = 0. However, the additional phase should be introduced into the Airy-like beam if we change the length of the Airy fiber. In Fig. 4(b), one can find that the phase differences (Δϕ 1 , Δϕ 2 , …, Δϕ 5 ) between the two beams are increased from the main lobe to side lobes, just like an ideal Airy beam with the launch angle θ = 12 mrad.
From eq. (11), we can find that the modal amplitude ′  a i (see Fig. 3(a)) and field distributions ′ E i (see Fig. 3(c)-(f)) can provide Airy amplitude modulation, so that Airy-like intensity profiles can be obtained along the fiber arrayed-core, as shown in Fig. 1(a). However, how to obtain Airy phase modulation (see Fig. 4(a))? The different propagation constants β ′ i of excited supermodes during propagation in fiber arrayed-core can induce phase differences. Thus, a phased array is formed in a discrete optical system of the Airy fiber. We can change parameters of the fiber arrayed-core to control the phased array. Particularly, in Fig. 3(b), we design a special arrayed cores (see Fig. 1(a) and (b)) to form a phased array whose phase differences between two adjacent excited supermodes are just π at Z = (2 m − 1)T 0 /2, just like cubic phase modulation with a Gaussian beam in the process of the ideal Airy beam generation. Thus, by the supermodes interference along the arrayed-core of the (2 m − 1)T 0 /2 length of the Airy fiber, both Airy amplitude and phased array modulation with a Gaussian beam are implemented to generate a 2D Airy-like beam, which almost has the same amplitude and phase distributions of ideal Airy beam, as shown in Fig. 4(a) and (b). Remarkably, to ensure the stable and high-quality beam output, a short Airy fiber (a few millimeters length in our case) is chosen by us. Figure 5(a) shows the transverse patterns of the generated Airy-like beam during propagation in free space. Here the calculated results are based on the angular spectrum method 19,20 . For simplicity, the origin coordinate is located at the fiber end-face for observing the propagation behavior of the generated Airy-like beam from the Airy fiber. We can clearly see that the internal lobes of the Airy-like beam are gradually reborn during propagation due to self-healing, just like a complete ideal 2D Airy beam (see Fig. 5(b)). Furthermore, if we block the main lobe of the Airy-like beam, this  incomplete Airy-like beam also shows an extraordinary self-healing property to form its main lobe after propagating dozens of micrometers, as shown in Fig. 5(c). In the Y1-axis direction, the width of the main lobe of the incomplete Airy-like beam is always less than the ideal Airy beam during short-range propagation (hundreds of micrometers), as shown in dotted and dashed lines of Fig. 5(d). However, in the X1-axis direction, the width of the beam's main lobe can not remain invariant compared with the ideal Airy beam during propagation due to lack of the internal lobes, as shown in Fig. 5(b) and (c).

Properties of the output beam from different length of Airy fiber.
As mentioned above, an Airy-like beam or a Gaussian-like beam can be obtained by changing Airy fiber length. Therefore, the corresponding output beam properties for a certain length of the Airy fiber, especially the width of the main lobe, could be similar to Airy beam or Gaussian beam. From Fig. 5(d), we can find that the generating Airy-like beams for the Airy fiber length near to (2 m − 1)T 0 /2 remain almost nonspreading during short-range propagation and the widths of their main lobes along the Y1-axis can keep almost invariant compared From Fig. 5(a), we can find that the Airy-like beam has a remarkable ability to accelerate its main lobe along the direction of the negative X1-axis (see the trajectory marked with the dashed line in Fig. 5(a)). Actually, the transverse shift of the Airy beam depends on the initial launch angle θ m , which is determined by the phase changing trend compared with an ideal Airy beam with θ m = 0 21, 22 . In our case, the slope of the linear fitting curve of the additional phase differences as an equivalent launch angle can evaluate such phase changing trend 23 . From Fig. 5(e), we can find that the equivalent initial launch angle of the generated Airy-like beam is increased with the Airy fiber length. Thus, the first type of phased array modulation is based on changing the fiber length can be implemented to generate an Airy beam. As increasing the fiber length, the transverse shift of the output beam is enhanced during propagation, as shown in Fig. 5(f). Therefore, one can choose a suitable length of Airy fiber to generate an Airy-like beam with an expected phase distribution and an expected equivalent initial launch angle that can cause the main lobe following a certain curved trajectory. Notably, the generated Airy-like beam has almost the same acceleration ability as the ideal Airy beam. As a result, the propagation trajectory of the Airy-like beam (see the solid line marked D in Fig. 5(f)) matches pretty well with the ideal Airy beam (see the dashed line in Fig. 5(f)). From Fig. 5(f), we note that a Gaussian-like beam is generated by a Z G length of the Airy fiber and there is almost no deflection during propagation during short-range propagation (see the solid line marked G in Fig. 5(f)). For the output beam from the Airy fiber with 3.68 mm (Z F ) length in the inset of Fig. 5(f), we can clearly see that the intensity of the Airy-like beam decreases rapidly during propagation until its main lobe disappears after propagating about 0.4 mm (see the solid line marked F in Fig. 5(e)). The cause of this unexpected behavior is that the optical power of the main lobe heals the side lobes and internal lobes so that it can not recovery due to too strong transverse shift, even at the beginning, the beam has a strong main lobe (see Fig. 1(a)). Therefore, by increasing the Airy fiber length, one can obtain an Airy-like beam which has enhanced capacity for the transverse shift but weak ability to remain nondiffraction. Note that at the beginning the beams from Airy fiber do not accelerate and their deflections go to the negative value, as shown in Fig. 5(f). It is because that a part of the beam energy is used to form the internal lobes so that the total power flow is along the direction of the positive X1-axis, which is just opposite to the desired acceleration direction.
Phased-array control effect in the bent Airy fiber. As we described in the previous section, a discrete optical system is designed by using an Airy fiber with two perpendicular arrayed cores to implement transform between Gaussian beam and Airy beam through Airy amplitude and phased array modulation. In this section, we give another type of phased array modulation based on the bending effect. Unlike the previous type of phased array modulation by changing the length of a straight Airy fiber, this type is modulated by the magnitude and the direction of fiber bending. Figure 6 shows the output powers from arrayed cores of the bent Airy fiber with Z D = 2.2 mm length. We can clearly see that the output powers from all arrayed cores are monotone changing with the bending radius when the bending radius is greater than 0.1 m depicted by the dashed line in Fig. 6(a) and (b). From eq. (15), we can find that the equivalent RI is inversely proportional to the bending radius. In the case of the positive bending along the X1-aixs, as the bending radius decreases the equivalent RI of side cores in two perpendicular arrayed-cores increases compared with the original without bending, resulting in weakening optical power coupling between main core and side cores. Therefore, the output powers of the main core increases but side cores' decrease while more and more power is kept in the main core with the bending radius decreasing, as shown in Fig. 6(a). On the contrary, as the bending radius decreases more optical power from the main core is coupled into side cores. As a result, we can clearly see that such optical coupling causes the output power of the main core decreasing but side cores' increasing, as shown in Fig. 6(b). Figure 7(a) and (b) give the results of Airy amplitude and phased array modulation for a generated Airy beam from the bent Airy fiber with Z D = 2.2 mm length. We can clearly see that the output beam's intensity and phase distributions can be easily changed by bending the fiber. When the bending radius of Airy fiber is positive, there is Scientific RepoRts | 7: 5171 | DOI:10.1038/s41598-017-05414-w much greater intensity for the main lobe of the output Airy beam but smaller for side lobes, as shown in Fig. 7(a). In this case, the equivalent initial launch angle (the slope of the trend line of the phase profile shown in Fig. 7(b)) is always negative so that the transverse shift is stronger than the general Airy beam from Airy fiber without bending, as shown in Fig. 7(c). On the other hand, there are much stronger side lobes but weaker main lobe when we bend the fiber along negative X1-axis. And the positive equivalent initial launch angle (see Fig. 7(b)) causes the output Airy beam has a weak transverse shift but an enhanced capacity to remain nondiffraction during propagation, as shown in Fig. 8(c) and (d). Note that there is almost propagation-invariant intensity for the main lobes of the output Airy beam and we can also clearly see that an internal lobe reborn due to self-healing, as shown in Fig. 7(d).
In order to reveal wavelength response of the Airy fiber, we calculate the output spectrum of nine cores of the 2.2 mm (Z D ) length of Airy fiber with different bending radius, as shown in Fig. 8(a)-(c). An Airy beam generation zone is defined for evaluating the relationship between the output spectrum and the bending radius. In the Airy beam generation zone, note that the output powers of nine cores are almost monotone changing with optical wavelength. As mentioned above, from eq. (12) an Airy beam can be generated through suitable phase modulation, which can be expressed as: where ′ n i is the effective RI of the supermode propagation in Airy fiber, z 0 is the length of the fiber. From eq. (1), we can find that the phase modulation depends on the optical wavelength λ, the effective RI of the supermode ′ n i , and the fiber length z 0 . For a length of the Airy fiber, the effective RI ′ n i is proportional to the optical wavelength λ for ensuring the same modulation phase ϕ. Actually, the effective RI ′ n i is determined by the RI distribution of fiber. Thus, we can control the effective RI ′ n i by changing the equivalent RI of the bent fiber in our case. And that means the equivalent RI profile changing in fiber causes the effective RI ′ n i increases (decreases) when we positively (negatively) curve the Airy fiber. Therefore, from Fig. 8(a)-(c), we can clearly see that the Airy beam generation zone of Airy fiber with the positive and negative bending radius is shifted to longer and shorter wavelengths, respectively. What is also quite interesting is the fact that the curved trajectory of the main lobe of the Airy-like beam from the negatively bent Airy fiber varies sharply with the wavelength, as shown in Fig. 9(a). For the negatively bent Airy fiber, however, it is not sensitive. This remarkable characteristic is named "rainbow effect" due to the transverse shift control induced by phased array modulation 23 . As the wavelength increases, the phase modulation is weakened from equation (1) so that the equivalent initial launch angle decreases, resulting in weakening the transverse shift of the beam, as shown in Fig. 9(a). And this phenomenon is more evident for the 6.6 mm (3T 0 /2) length of the Airy fiber, as shown in Fig. 9(b). The reason is quite obvious that the phase modulation is enhanced when the fiber length z 0 increases from eq. (1). We can see that the curved trajectories of the beams from the negatively bent Airy fiber are obviously separated by wavelength manipulation in Fig. 9(b). However, the generated beams from the positively bent Airy fiber are still not sensitive. And the main lobe may vanish due to too strong transverse shift in wavelength manipulation, as shown in the dashed line of Fig. 9(b) for R b + . For a shorter wavelength, such as 950 nm, the side lobes of the generated beam could disappear and only the main lobe can be preserved. And the transverse shift of the beam is very weak (see the solid line of Fig. 9(b) for R b + ). And its propagation trajectory is similar to the Gaussian beam, as shown in the inset of Fig. 9(b). In general, by changing any of the parameters, including fiber length, optical wavelength, and bending-induced refractive-index changing, one can obtain a desired Airy-like beam or Gaussian-like beam from an Airy fiber.
From eq. (15), we can find that the equivalent RI of the bent Airy fiber is directly affected by the angle θ  b between the arrayed-core and the bending direction. Therefore, we can obtain asymmetric RI distribution of two perpendicular arrayed cores of an Airy fiber by changing bending direction. Figure 10(a) and (b) respectively give the output power profiles of two perpendicular arrayed cores along the X-and the Y-axis when we curve the Airy fiber along the positive X-axis direction. In this case, the equivalent RI along the X-axis increases but is invariant along Y-axis. As a result, the output powers of the arrayed-core along the X-and the Y-axis are different. The former (except the output power in the main core 1, which is located in the origin) are much smaller than the latter, as shown in Fig. 10(a) and (b). Remarkably, a one-dimensional Airy beam is generated when the bending radius is 93.24 mm, as shown in Fig. 10(c). From Fig. 10(d), we can clearly see that the output beam has an Airy distribution along the Y axis but almost no energy in the side cores of the arrayed-core along the X-axis. Therefore, we can obtain a desirable 1D or 2D Airy beam by choosing a suitable bending radius of the Airy fiber along an appropriate direction.

Discussion
We have provided several effective approaches to control the propagation dynamics of light in a discrete optical system of Airy fiber. The supermode theory and refractive-index equivalent method have been applied to analyze propagation properties in the straight and the bent Airy fibers. The calculated results showed that the amplitude and phased array modulation on a Gaussian beam are dependent on the fiber parameters, such as fiber length, refractive-index distribution, and the incident wavelength. By changing these parameters, we can perfectly control the wavefront of light propagation in the Airy fiber and obtain an output Airy beam with desirable abilities to remain quasi-nondiffraction propagating, self-healing and transverse accelerating. We also obtain the one-dimension Airy beam or the Gaussian beam in one same Airy fiber through parameter control. Therefore,   a and b) are the output powers of Airy fiber arrayed cores on the X-and the Y-axis as a function of bending radius, respectively. (c and d) are the transverse output field and its intensity profiles along the X-and the Y-axis, respectively, when the bending radius is 93.24 mm. Here R b + is the bending radius along the positive X1-axis.
we not only give a tool of a beam generator but also proposed a technology to control the propagation dynamics of light in a general discrete optical system. This technology might apply to other interesting areas, including integration optics and nonlinear optics.

Airy beam. Considering the input field distribution
m x y m mm mm , the two-dimension (2D) finite energy Airy beam envelope expresses as follows 3 : Supposing only the fundamental mode LP 01 (Gaussian beam) of the central Core 1 as input (see Fig. 1(a)), each amplitude of supermodes of arrayed-core can be solved by eq. (9):  (10) and (11), we obtain the total electric field along arrayed-core: Mode fields calculation in the bent Airy fiber. To analysis the optical characteristics in a bent fiber, we introduce a simplified approximation model in Fig. 11. The z-components of the electromagnetic field (  E p , ∼ H p ) guided in Core p and cladding regions of the bent fiber (see Fig. 11) can be expressed in terms of cylinder functions in the coordinate system r, θ 25