1. Introduction

In an ideal optical fiber with perfect translation symmetry, modes are not coupled mutually. The light launched into a certain mode propagates along this mode perpetually with the group velocity and dispersion determined by the mode structure. Longitudial periodic perturbations of a fiber can couple selected modes [1,2]. Coupling without changing of propagation direction is performed by long period gratings (LPGs) [2], while coupling with reflection is performed by fiber Bragg gratings (FBGs) [1]. A set of LPGs and FBGs introduced in an optical fiber form a photonic circuit, in which modes behave as the photonic wires and gratings transfer light from one wire to another. This circuit can serve as a filter, dispersion compensator, amplifier, sensor, etc. [1–4]. Design of such all-fiber photonic circuit includes design of a fiber (photonic wires) and design of periodic perturbations (wire connections). Theory and applications of single-period gratings (LPGs or FBGs), which convert individual pairs of modes, are well developed [1–4]. More complex fiber perturbations, which are composed of spatially superimposed gratings of different periods, can perform conversion between more than two modes simultaneously. Superimposed gratings are often preferable to those positioned in series due to enhanced functionality and compactness. While the theory of superimposed FBGs has been addressed and applied to fabrication of various devices [5,6], little research has been done on understanding the properties of superimposed LPGs (SLPG) and their possible applications. On the other hand, recently, LPG was used to demonstrate a very accurate conversion between the fundamental mode (FM) and desired higher order modes (HOMs) [7]. With this development, fabrication of SLPG is straightforward since it can be based on similar technology.

 

Fig. 1. Illustration of the superimposed LPG mode converters.

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In this paper, we investigate the SLPG performing simultaneous conversion between the FM mode and HOMs of an optical fiber. Schematically, the SLPG converters of our interest are shown in Fig. 1(a) and (b). These converters are used to transform an input laser beam into the FM of an optical fiber and vice versa. The single LPG versions of such devices, which, in contrast to SLPG, are unable to perform efficient conversion, have been experimentally demonstrated previously [8–12]. Section 2 introduces coupled wave equations that describe transitions of light in SLPG. Section 3 demonstrates how the SLPG converters shown in Fig. 1 can completely transform several arbitrary coherent modes into a single mode and vise versa. Section 4 briefly discusses conversion of an incident coherent beam into the FM with the SLPG shown in Fig. 1(a). This conversion is important, for example, for efficient coupling of a pump laser beam into the FM of an optical fiber. In Subsection 5.1, we consider a similar but inverted SLPG converter shown in Fig. 1(b) that can perform efficient focusing and shaping of a light beam emerging from a single mode optical fiber. The beam is composed by optimization of the amplitudes and phases of the first six LP_0j modes generated by SLPG from the FM. We demonstrate how to optimize the output beam in order to focus it at a certain distance (0.5 mm and 1 mm) from the fiber. Next, in Section 5.2, we optimize the same linear combination of LP_0j modes to form a beam that is uniform within 4⁰ with an accuracy of 0.2%. In Section 6, we present a design of SLPG that generates the uniform beam of Section 5.2. Section 7 summarizes our results.

2. Coupled wave equations for SLPG

Let us assume that SLPG is introduced in the core of an optical fiber by a perturbation of the refractive index,

where x and y are the transverse coordinates, z is the longitudinal coordinate, θ(s) is a Heaviside step function, ρ_core is the core radius, and Λ_jk are the periods of harmonics. In the coupled wave theory of a weakly guiding fiber, the field can be written in the scalar form E(x,y,z)=Σ_jA_j(z)exp(iβ_jz)e_j(x,y), where e_j(x,y) are the transverse components of eigenmodes and β_j are the propagation constants. It is assumed that the periods Λ_jk approximately match the differences between the propagation constants of the fiber modes, i.e., 2π/Λ_jk≈β_j-β_k, so that the harmonic (j,k) couples together modes j and k. The coupled mode equation for A_j(z) can be derived from the general coupled mode theory [1,2] in the form:

where κ_jk are the coupling coefficients defined by the following equations:

Here, λ is the wavelength of light in free space and the transverse eigenmodes e_j(x,y) are normalized, ∫^∞ _-∞∫^∞ _-∞ dxdye ² _j(x,y)=1. In Eq. (2), it is assumed that the periods Λ_jk are positive for j>k and the propagation constants are monotonically decreasing with their number, i.e. β^j<β_k for j>k. In addition, ϕ_jj=0, 1/Λ_jj=0, Λ_jk=-Λ_kj, ϕ_jk=-ϕ_kj, and, as follows from Eq. (3), κ_jk=κ_kj. It is convenient to remove the diagonal terms in the right-hand side of Eq. (2) by introducing the amplitudes Ā_j(z)=A_j(z)exp(-iκ_jjz), which obey the coupled wave equations:

 

Fig. 2. Diagram of mode conversion.

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As opposed to the single-grating case [1,2], the general form of Eq. (4) cannot be reduced to a system of linear differential equations with constant coefficients. However, under the phase match condition,

Eq. (4) is simplified to

A particular case of SLPG, which is important in practice and, at the same time, allows complete analytical solution of Eq. (6), is the subject of this paper.

3. Conversion between several modes and a single mode

In a single mode fiber, conversion between the FM and HOMs is of special importance for the applications [1–3]. Consequently, it is interesting to consider SLPG that couples the FM (having, for determinacy, number j=1) with HOMs (having j>1). In this case, illustrated in Fig. 2,

and Eq. (4) can be reduced to a system of differential equations with constant coefficients. Here we consider only the special case of phase matched SLPG that obey Eq. (5) and Eq. (7). The coupled wave equations for these SLPG, Eq. (6), have the general solution:

The SLPG illustrated in Fig. 1(a) convert N modes entering the SLPG with amplitudes |A_j(0)| and phases α_j=arg(A_j(0)) into the single FM at the SLPG exit. A similar converter that couples a number of HOMs and the FM was suggested in [13]. However, coupling of HOMs with the FM was treated in [9] independently and indirect transitions between HOMs through the FM were not considered. In the case shown in Fig. 1(a), solutions of Eq. (8) can be found under the assumption that A_j(0)=κ _1jexp(-iϕ _1j)C ₀. After substitution of the latter expression into Eq. (8), simple calculations yield the condition of the full conversion of N modes, which have the amplitudes |A_j(0)| and phases α_j at the SLPG entrance, z=0, into the single mode 1 at the SLPG exit,z=L. This condition is

where µ is defined in Eq. (8). Here, the first equation determines the phase shifts of LPGs, ϕ _1j, through the relative phases of incoming modes, α_j-α ₁. The second equation determines the coupling coefficients κ _1j, which are proportional to the amplitudes of the incoming modes. Finally, the third equation determines the length of SLPG, L. Thus, Eqs. (9) solve the problem of conversion of several arbitrary fiber modes into a single mode, which can be performed by an SLPG mode converter shown in Fig. 1(a).

Transformation of the FM into N modes with arbitrary ratios of amplitudes and phase differences can be performed with an SLPG mode converter shown in Fig. 1(b), which is inverted as compared to that shown in Fig. 1(a). Now, the input amplitudes of all modes, except for the FM, are equal to zero, i.e. A_j(0)=0, for j=2,3,…,N, and the requested amplitude ratios and phase differences of the output modes are |A_j(L)|/|A ₁(L)| and γ_j-γ ₁=arg(A_j(L)/A ₁(L)). From Eq. (8), this conversion is accomplished by the SLPG with the following parameters:

Similar to Eq. (9), this equation determines the phase shifts, ϕ _1j, the relative values of coupling coefficients, κ _1j/κ ₁₂, and the corresponding SLPG length L. The detailed design of an SLPG for a homogenized output beam is given in Section 6.

In conclusion of this Section, we have shown that the set of N-1 SLPG can convert a single mode (e.g. FM) into N modes with arbitrary amplitudes ratios and phase differences, and vice versa. Simple expressions for the parameters of these SLPG are given.

4. Optimization of conversion of an incident beam into the FM of a fiber

In this paper we are concerned in optimization of a beam emerging from a single mode optical fiber illustrated in Fig. 1(b), which is considered in Sections 5 and 6. However, a short remark on optimization of conversion of an incident coherent beam into the FM illustrated in Fig. 1(a) is worth mentioning. The part of the beam that is trapped by the fiber can be expressed as a linear combination of fiber modes:

The power of this beam is

In this case, the SLPG parameters are determined by Eq. (9). The values of SLPG periods and total number N of gratings are limited by the accuracy of grating inscription, maximum amplitude of photo-induced refractive index variation, and fiber uniformity. From Eq. (12), the most complete conversion is performed if SLPG couple the FM with modes that have the maximum values of intensities |A ⁽⁰⁾ _j|².

5. Optimization of a beam emerging from a fiber in the near field region

Consider a coherent beam emerging from the SLPG converter positioned at the end of a single mode optical fiber shown in Fig. 1(b). The simplest single-LPG version of this converter was suggested in [8] and further developed in [12]. The SLPG version of this converter is described by Eq. (10). As an example, we considered SLPG consisting of five axially symmetric LPGs that couple six LP_0j modes. The output beam generated by a linear combination of these modes is determined using the Fresnel diffraction integral in the form of the Hankel transform:

Here R is the fiber radius, E ₀(ρ,L) is the field distribution at the end-wall of the fiber, z=L, which in this case is a linear combination of six normalized transverse LP_0j modes, e^LP_oj(ρ):

Modes e^LP_oj(ρ) are uniquely determined by the refractive index profile of a fiber. In our modeling, we considered an SMF-28 fiber (R=62.5 µm, ρ_core=4.1 µm, refractive index difference 0.36%), for which these modes were calculated numerically. Their amplitudes are shown in the upper row of plots in Fig. 3. In the next two Subsections, the coefficients A_j in Eq. (13) are optimized to focus the beam in the near field region (Subsection 5.1) and to approach the homogeneous beam profile in the far field region (Subsection 5.2).

 

Fig. 3. Distribution of the field amplitude. Top plots - along the end-wall of the fiber. Bottom plots - at distances 0.5 mm (red solid curves) and 1 mm (blue dashed curves) from the fiber. All fields are normalized to the same input power.

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5.1. Focusing the beam in the near-field region

In order to increase the peak intensity and to suppress the sidelobes, the beam profile determined by Eq. (13) and (14) should be optimized by variation of the complex-valued coefficients A_j. In our modeling, the objective function is chosen in the form:

where we set A ₁=1 in the Eq.(14) for E_out(σ,z). Minimization of F(A ₂,A ₃,…A ₆) was performed at fixed z₀-L=0.5 mm and z ₀-L=1mm by variation of 5 complex parameters A ₂,A ₃,…A ₆. The parameter σ_m defines the region outside the central peak of the emerging beam where the sidelobes are suppressed. In our modeling, we chose σ_m=15 µm. The

Table 1. Optimum coefficients in the linear combination of LPoj modes, Eq. (14), for the beams focused at distances 0.5 and 1 mm from the fiber end and the homogenized beam.

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Table 2. Fraction of the total beam power inside the 15 µm radius circle at 0.5 mm from the fiber end (second row) and inside the 25 µm radius circle at 1 mm from the fiber end for the beams generated by the LP_oj modes and for the optimized beams.

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Fig. 4. Distribution of the field amplitude of the beams generated by the optimized linear combinations of the LP_0j modes and by the individual LP_0j modes. All fields are normalized to the same input power.

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obtained optimum values of A_j are given in Table 1. The corresponding field amplitude profiles at z ₀-L=0.5 mm and z ₀-L=1mm are shown as red solid and blue dashed curves, respectively, in the lower row of plots in Fig. 3. The axial cross-sections of the field amplitude profiles of the corresponding optimized beams and the beams generated by individual LP_0j modes are compared in Fig. 4. All fields are normalized to the same input power. Figures 3 and 4 demonstrate significant improvement of the optimized profiles as compared to the profiles of beams generated by individual LP_0j modes. Note that, for better visibility, in Fig. 3 and 4 we show the field amplitude rather that the field intensity distribution. The relative improvement of the optimized field intensity is much more dramatic. In particular, row 2 of Table 2 compares the fractions of power contained inside the 15 µm radius circle at 0.5 mm from the fiber end for LP_0j modes and for the beam optimized at 0.5 mm. For any of the LP_0j modes this fraction does not exceed 32%. However, it approaches 99% for the optimized beam. Similarly, row 3 of Table 2 compares the fractions of power contained inside the 25 µm radius circle at 1 mm from the fiber end for LP_0j modes and for the beam optimized at 1 mm. Again, while for the LP_0j modes this fraction varies between 16% and 43%, it is equal to 98.3% for the optimized beam. Comparison given by Figs. 3, 4 and Table 2 clearly indicates that the suggested SLPG mode converter can serve as an efficient beam focuser.

5.2. Shaping the beam in the far field region

In the far field region, which is defined by the inequality z-L≫R ²/λ, the integral in Eq. (13) is simplified to

Here the scattering amplitude f(θ) and the scattering angle θ are introduced. In numerous applications (e.g. materials processing, laser printing, micromachining in the electronics industry, optical processing) it is desirable to uniformly illuminate a specific volume of space with a laser beam. For this purpose, several methods of forming the beams that have a profile with a uniform central region had been developed [12–14]. In this Section we demonstrate that the SLPG converter shown in Fig. 1(b) can be used as simple, robust, and efficient all-fiber beam homogenizer. We optimize the sum of LP_0j modes generated by SLPG, Eq. (14), to form a beam that has a very uniform central region. In order to solve the problem of homogenizing of the beam in the far-field region, the objective function that for this problem was chosen in the form

The function F(A ₁,A ₂,A ₃,…A ₆) was minimized by numerical variation of six real variables A ₁,A ₂,A ₃,…A ₆, choosing the homogenized beam radius, θ_m, and the field amplitude, E ₀ manually. Figure 5 compares the far-field amplitude distributions for the first six LP_0j modes and their optimized sum. The homogenized beam profile, which is shown in Fig. 5, was obtained for parameters A_j given in column 4 of Table 1. The central peak of the optimized sum has the diameter of 6.8° and 91% of the total beam power. A homogeneous part of this peak, where the relative amplitude nonuniformity does not exceed ±0.2%, has the diameter 4⁰ and 52% of the total beam power. Thus, the considered example demonstrates that SLPG consisting of a reasonable number of gratings can produce extremely homogeneous light beams.

 

Fig. 5. Distribution of the field amplitudes in the far-field region for the beams generated by the individual LP_0j modes (blue curves) and by the homogenized linear combinations of the LP_0j modes (red curve). All fields are normalized to the same input power.

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6. Design of SLPG

Coefficients A_j in Eq. (14) determine the superposition of output fiber modes that forms a beam of the desired shape. In the previous Section, we have found these coefficients for the beams focused at certain distances from the fiber and for the beam with the uniform amplitude within a certain angle in the far-field region. From Section 3, it is possible to determine SLPG that generate a linear combination of fiber modes with arbitrary coefficients A_j. In this Section, we demonstrate the design of SLPG using the example of the homogenized beam considered in Subsection 5.2. The design is done by determination of the SLPG refractive index variation given by Eq. (1), which in our particular case has the form

with parameters Λ_1k, ϕ _1k, δ _1k, and δn ₀. Coefficients A_j for this example are given in column 4 of Table 1. Other parameters that determine the SLPG are summarized in Table 3. We calculated them as follows. First, we calculate the propagation constants of the LP_0j modes of an SMF-28 at wavelength λ=1.55 µm given in row 2. The overlap integrals I _1j and I_jj for these modes are calculated using Eq. (3). The values of these integrals are given in rows 3 and 4. Next, from the second equation of Eqs. (10), the coupling coefficients κ _1j

Table 3. Design of the SLPG generating the homogenized beam

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should be proportional to A_j, i.e. κ _1j=CA_j with a constant C to be determined. In theory, Eqs. (10) allow to choose the length of SLPG, L, independently of other parameters. However, smaller L requires stronger gratings and includes less LPG periods. Here we choose a reasonable value L=50 mm. Then with A_j from Table 1, column 4, we find C=1.4331·10^-5 µm^-1 and the values of coupling coefficients κ _1j=CA_j (j>1) given in row 5 of Table 3. With the known κ _1j and I _1j, from Eq. (3), we find δn _1j=λκ _1j/(πI _1j) at λ=1.55 µm, which is given in row 6. After that, we determine index δn ₀ from the condition that the overall introduced index variation should be positive: δn ₀=Σ_j>1|δn _1j|=1.8141·10^-4. This value of δn ₀ together with I_jj from row 4 determine the self-coupling coefficients given in row 7. The periods of gratings, Λ_1k, in row 8 are found from Eq. (5), where the self-coupling coefficients, κ_jj, and propagation constants, β_j, are given in row 6 and 1, respectively. Finally, the LPG phase shifts given in row 9 are calculated from ϕ _1j=(κ_jj-κ ₁₁)L+π/2. Fig. 6 shows the refractive index variation of the designed SLPG.

 

Fig. 6. The refractive index profile of the SLPG generating the homogenized beam considered in Section 6.

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7. Summary

The theory of superimposed LPG mode converters, which transform a single fiber mode into several modes with arbitrary phases and power ratios, and vice versa, is developed. Simple expressions are obtained for the parameters of LPGs that perform the requested mode transformation. Performance of the converters is demonstrated by solution of the optimization problem of focusing the output beam at certain distances from the fiber end. In addition, the SLPG output is optimized to produce an extremely homogeneous beam in the far field region. The amplitude of this beam is uniform with a 0.2% accuracy in a 4⁰ angle. The detailed design of the corresponding SLPG is presented. Further investigation is needed to improve the performance and, in particular, the bandwidth of these devices by optimization of both the SLPG and optical fiber parameters. The SLPG, which can be fabricated using well developed fiber grating fabrication methods, have the potential of becoming inexpensive and efficient devices for fiber based beam shaping.