Multimode fibers

CHAPTER 14

Multimode fibers Multimode fibers were first used for nonlinear optics [1] during the 1970s because most optical fibers available at that time supported multiple modes [2–4]. The situation changed in the 1980s when single-mode fibers were commercialized in view of their telecommunication applications. The interest in multimode fibers re-surfaced after 2005, partly motivated by their use for space-division multiplexing in optical communication systems. As a byproduct of this interest, the nonlinear effects in multimode fibers have been studied extensively since 2010. This chapter covers recent advances in a systematic fashion. Three types of multimode fibers are introduced in Section 14.1, where we also discus modes supported by them. Section 14.2 extends theory of Section 2.3 to obtain a set of coupled nonlinear equations for various fiber modes. These equations are used in Sections 14.3 to discuss modulation instability and soliton formation in multimode fibers. The focus of Section 14.4 is on the intermodal nonlinear effects that transfer energy among the modes. The spatio-temporal effects are discussed in Section 14.5 with emphasis on spatial beam cleanup and supercontinuum generation through multimode fibers. Section 14.6 is devoted to the nonlinear phenomena in multicore fibers.

14.1. Modes of optical fibers Conventional multimode fibers can be divided into three classes: step-index fibers, graded-index fibers, and multicore fibers; an example of each kind is shown in Fig. 14.1. Step-index fibers contain a cylindrical inner core of constant refractive index n1 that is surrounded by a cladding of constant refractive index nc such that nc < n1 . Graded-index fibers have a similar design but the refractive index inside their cores is not uniform and decreases radially from n1 to nc in a prescribed manner. Multicore fibers, as the name implies, contain multiple relatively narrow cores inside the same cladding. Optical modes of all three kinds of multimode fibers are discussed in this section. As fiber modes have been covered in many textbooks [2–4], the following discussion is intentionally brief.

14.1.1 Step-index fibers This case has been discussed in Section 2.2, where we derived an eigenvalue equation whose solutions provide the propagation constants of all modes supported by a step-index fiber. This eigenvalue equation is quite complex, but it applies for all values of n1 and nc as no approximation was made in its derivation. It can be simplified considerably in the weakly guiding approximation, valid for fibers in which the index Nonlinear Fiber Optics https://doi.org/10.1016/B978-0-12-817042-7.00021-X

Copyright © 2019 Elsevier Inc. All rights reserved.

621

622

Nonlinear Fiber Optics

Figure 14.1 Schematic cross-sections of three types of multimode fibers. In each case, a darker area represents the core region with a higher refractive index.

difference n1 − nc is relatively small. In terms of the parameter  = 1 − nc /n1 introduced in Eq. (1.2.1), the requirement is   1. This is the case in practice for most fibers as  is typically less than 0.01. The two axial components, Ez and Hz , used in Section 2.2 become so small in such fibers that the electric and magnetic fields are nearly transverse to the fiber’s axis (assumed to coincide with the z axis). As a result, all modes can be treated as being linearly polarized. Indeed, they are called LPlm modes, where LP is short for linearly polarized. In a cylindrically symmetric fiber, the choice of x and y axes is arbitrary as there is no preferred direction in the transverse plane. As a result, modes can be subdivided into two groups of orthogonally polarized modes. One group of modes has its electric field oriented along the x axis, and the other along the y axis, such that the two sets of modes are degenerate. Focusing on the x-polarized modes, we can find Ex by solving Eq. (2.2.1). Using the method of separation of variables, as was done in Section 2.2, the solution involves Bessel functions and can be written as [4] 

Ex =

AJl (pρ) cos(lφ)eiβ z ; CKl (qρ) cos(lφ)eiβ z ;

ρ ≤ a, ρ ≥ a,

(14.1.1)

where l ≥ 0 is an integer, p2 = n21 k20 − β 2 , and q2 = β 2 − n2c k20 . Notice that we have chosen cos(lφ) in place of eilφ appearing in Eq. (2.2.3). Another equally valid solution makes use of sin(lφ) in place of cos(lφ). These two solutions are referred to as being even and odd modes with respect to the azimuthal angle φ . The procedure for finding the propagation constant β remains the same. Boundary conditions requiring the continuity of the tangential components of the electric and magnetic fields across the core–cladding interface are used to derive an eigenvalue equation. We can deduce this equation from Eq. (2.2.8). In the weakly guiding approximation, the V parameter of the fiber can be approximated as √

V = k0 a(n21 − n2c )1/2 ≈ k0 an1 2,

(14.1.2)

Multimode fibers

where  = 1 − nc /n1 is a small parameter ( < 0.01). If we replace the ratio nc /n1 = 1 − with 1 in Eq. (2.2.8), it reduces to the following simpler eigenvalue equation: 



Jl (U ) Kl (W ) 1 1 + + = ±l , UJl (U ) WKl (W ) U2 W 2

(14.1.3)

where U and W are defined in terms of β as 

U = a n21 k20 − β 2 ,



W = a β 2 − n2c k20 .

(14.1.4)

The derivatives of Bessel functions in Eq. (14.1.3) can be removed using the well-known relations connecting each derivative to two Bessel functions of different orders [3]. This operation results in the relation U

Jl−1 (U ) Kl−1 (W ) +W = 0. Jl (U ) Kl (W )

(14.1.5)

The same eigenvalue equation is obtained when the solution in Eq. (14.1.1) involves sin(lφ), indicating the degenerate nature of the even and odd modes for any l = 0. The eigenvalue equation (14.1.5) must be solved numerically to find β for each integer value of l ≥ 0. Since multiple solutions may exist, it is customary to denote these solutions by βlm , where m = 1, 2, . . . in the decreasing order of β values. Each eigenvalue βlm corresponds to one specific mode, denoted by LPlm . The corresponding field distribution is obtained from Eq. (14.1.1) after noting that C = AJl (U )/Kl (W ). The constant A is set by the fraction of power carried by each specific mode when light is transmitted through the fiber. If we define the effective index of a mode as n = β/k0 , we can deduce that n lies between n1 and nc for all guided modes of a multimode fiber. The eigenvalue equation (14.1.5) should be solved numerically for a given value of V . It is useful to introduce a normalized effective index b as b=

β 2 /k20 − n2c n − nc ≈ . n1 − nc n21 − n2c

(14.1.6)

Fig. 14.2 shows a plot of b as a function of V for the first few LPlm modes. With the exception of the LP01 mode, a cut-off value of V exists below which each higher-order mode ceases to exist. We can find this cut-off value Vc by noting that the q parameter that governs the exponential decay of the mode amplitude outside the core vanishes at this value. Setting W = qa = 0 in Eq. (14.1.5), we obtain the condition Jl−1 (Vc ) = 0. Thus, the LP11 mode is cut-off at Vc = 2.405, the first zero of J0 (V ), the LP21 mode reaches its cut-off at Vc = 3.832, the first zero of J1 (V ), and so on. How many modes are supported by a specific multimode fiber depends on its V parameter at the operating wavelength. Since V scales inversely with wavelength, the

623

624

Nonlinear Fiber Optics

Figure 14.2 Normalized effective index of several low-order LPlm modes (indicated by the lm values) plotted as a function of the parameter V for step-index multimode fibers.

Figure 14.3 Spatial intensity patterns of the first four LP modes supported by a step-index fiber with V = 5. The spatial scale is in units of the core radius a.

same fiber supports more modes at shorter wavelengths. Also, mode degeneracy must be considered for any fiber mode. For example, only the LP01 mode exists in fibers designed such that V < 2.405. Such a fiber supports two modes when polarization degeneracy is included. The next mode in Fig. 14.2 is the LP11 mode. Since l = 0 for this mode, we must consider both the even and odd flavors of this mode, often denoted as LPe11 and LPo11 . In addition, each of them has its own two-fold polarization degeneracy. The net result is that, while all LP0m modes are 2-fold degenerate, all LPlm modes with l > 0 are 4-fold degenerate in the weakly guiding approximation. Fibers with V > 10 support a large number of modes, scaling as V 2 /2. For example, a multimode fiber with a = 25 µm and  = 5 × 10−3 has V 15 near λ = 1.5 µm and supports 112 modes. Fig. 14.3 shows the spatial intensity patterns of four low-order modes for a fiber with V = 5. Only even versions are shown for the LP11 and LP21 modes; odd versions can be obtained by rotating the spatial patterns. Note also that the phase is not spatially uniform for modes with l = 0. As we saw in Chapter 2, evolution of optical pulses inside single-mode fibers depended on several frequency derivatives of β(ω). In the case of a multimode fiber, these derivatives become different for different LP modes. As the group velocity of a mode is inversely related to the first derivative of β [see Eq. (1.2.9)], non-degenerate modes

Multimode fibers

travel at different speeds inside a multimode fiber. One consequence of this feature is that, unless the entire pulse energy is coupled onto one specific mode of the fiber, a single pulse excites several LP modes simultaneously that exit the fiber at different times, resulting in the so-called differential group (or mode) delay (DGD). The group-velocity dispersion (GVD) parameter β2 , related to the second frequency derivative of βlm , is also expected to vary from mode to mode. It is possible to calculate β1 and β2 for various fiber modes [4] in terms of the first two derivatives of the b(V ) curves shown in Fig. 14.1. The same process can be used for the higher-order dispersion parameters, if they are needed.

14.1.2 Graded-index fibers The refractive index of a graded-index (GRIN) fiber decreases radially from the core center, taking the value nc at ρ = a for a core of radius a. Its radial variation in most GRIN fibers is nearly parabolic and has the form n2 (x, y) = n21 [1 − 2(ρ/a)2 ] (ρ ≤ a),

(14.1.7)

with n(x, y) = nc for ρ ≥ a. The parameter  = (n1 − nc )/n1 is the same used earlier for step-index fibers. The modes are still obtained by solving the Helmholtz equation ˜ + n2 (x, y)k2 E ˜ ∇ 2E 0 = 0.

(14.1.8)

Although a numerical approach is necessary in general, considerable insight is gained by solving Eq. (14.1.8) analytically with the assumption that the index profile in Eq. (14.1.7) applies for all values of ρ . This assumption is reasonable for low-order modes in fibers designed with a core radius a  λ (a = 25 µm typically). Eq. (14.1.8) can be solved using either the Cartesian or cylindrical coordinates. In the cylindrical coordinates, the method of separation of variables leads to the following form of the electric field for the x-polarized even modes [3]: Ex (x, y, z) = F (x, y)eiβ z = R(ρ) cos(lφ)eiβ z ,

(14.1.9)

where F (x, y) is the spatial distribution, l = 0, 1, . . . is an integer, and the radial function R(ρ) satisfies 



 l2 ρ2  d2 R 1 dR 2 2 2 + k n 1 − 2  − R = 0. − β + 0 1 dρ 2 ρ dρ a2 ρ2

(14.1.10)

This equation has analytic solutions in terms of the associated Laguerre polynomials: 

Rlm (ρ) = Nlm

ρ ρ0

l



Lml −1

   ρ2 ρ2 exp − 2 , ρ02 2ρ0

(14.1.11)

625

626

Nonlinear Fiber Optics

√

where ρ0 = a/ V , and the parameter V is defined in Eq. (14.1.2). An important property of the modes for all fibers is that they are orthogonal to each other. If we use a single subscript p to denote mode profiles as Fp (x, y), the orthogonality of modes requires that



Fp∗ (x, y)Fq (x, y) dx dy = δpq ,

(14.1.12) 

where each mode is assumed to be normalized such that |Fp (x, y)|2 dx dy = 1 and all integrations extend from −∞ to +∞, unless specified otherwise. The constant Nlm can be found using this normalization. Similar to the case of step-index fibers, the integer m labels multiple solutions for each value of the integer l. Also, except for l = 0, the odd version of each mode with l = 0 exists with sin(lφ) in place of cos(lφ) in Eq. (14.1.9). The value of βlm for both sets of modes is given by [3] βlm (ω) = k0 n1 [1 − (l + 2m − 1)(4/V )]1/2 .

(14.1.13)

The main advantage of this analytic approach is that we have a closed-form expression for the effective index (n = β/k0 ) of each LPlm mode. It shows clearly that groups of modes for which the integer g = l + 2m − 1 has the same value are degenerate because n has the same value for the entire group. This degeneracy occurs in addition to the degeneracy discussed earlier for step-index fiber. Keeping this in mind, the g = 1 group has a single member with l = 0 and m = 1, and this LP01 mode is 2-fold degenerate. The next group with g = 2 also has a single member, the 4-fold degenerate LP11 mode. The next group with g = 3 has two members LP02 and LP21 and is 6-fold degenerate. The total group degeneracy increases rapidly as the integer g increases. If we label the modes with their group number g and use Eq. (14.1.13) in Eq. (14.1.6), we obtain an amazingly simple expression, b = 1 − 2g/V , which can be used to write the effective mode index in the form √

n ≈ n1 (1 − 2g/V ) = n1 − g 2/(k0 a).

(14.1.14)

Several differences between the GRIN and step-index fibers are now apparent. First, the curves in Fig. 14.2 for step-index fibers are replaced by the function b(V ) = 1 − 2g/V for GRIN fibers that is valid only for V > 2g. Fig. 14.4 shows the dispersion curves for mode groups up to g = 6. The values near V = 2g are not accurate since the boundary condition at the core-cladding interface has not been applied in this simple treatment. Second, the spatial mode profiles are different but they look similar in shape to those shown in Fig. 14.3 for step-index fibers. Third, the effective mode index n for GRIN √ fibers decreases by a constant value 2/(k0 a) from one mode group to the next. As we discuss later, this ladder-like structure of n leads to periodic self-imaging of optical beams

Multimode fibers

Figure 14.4 Normalized effective index b(V ) of several mode groups (indicated by the value of g) plotted as a function of the V parameter for GRIN fibers.

inside a GRIN fiber. Note also that the problem of finding modes of a GRIN fiber is analogous to finding the energy levels of a harmonic oscillator in quantum mechanics. Similar to the case of step-index fibers, Eq. (14.1.13) can be used to calculate the dispersion parameters for each mode by taking derivatives with respect to ω. For low-order modes for which 4g(/V )  1, the first derivative is found to be √ β1 = k0 n1g − g(a 2)−1 (d/dω),

(14.1.15)

where n1g is the group index of the core material. Physically, β1 represents the group delay and its inverse provides the group velocity. One is often interested in the DGD of a mode group with respect to the fundamental LP01 mode. Using this definition, DGD can be written as √ τg1 = (g − 1)(a 2)−1 (d/dω).

(14.1.16)

Typical values of DGD are ∼0.1 ps/m for g = 2 at wavelengths near 1.55 µm and they increase linearly for higher groups. The GVD parameter β2 for any group can also be found by calculating d2 β/dω2 . In commercial GRIN fibers, β2 for g = 1 vanishes at the zero-dispersion wavelength near 1.3 µm and is about −25 ps2 /km near 1.55 µm.

14.1.3 Multicore fibers Multicore fibers, as seen in Fig. 14.1, contain multiple cores within the same cladding. The cores are often designed to support a single mode. If the cores are relatively far apart, their individual modes overlap negligibly, and the multicore fiber behaves as a bundle of single-mode fibers. However, mode overlap is not negligible if the cores are closely spaced. One can find the modes of the whole structure by solving the Helmholtz

627

628

Nonlinear Fiber Optics

equation (14.1.8) numerically such that n(x, y) = n1 in the core regions and becomes nc in the cladding regions. Such modes are called supermodes and their number equals the number of single-mode cores within a multicore fiber. An approximate analytic technique makes use of the coupled-mode theory [6], used extensively for fiber couplers that constitute an example of a two-core fiber [5]. In this approach, the supermodes result from a linear combination of the modes supported by isolated individual cores [7–11]. Focusing on the x-polarized supermodes of a N-core fiber, the electric field has the form Ex (x, y, z) =

N

Cm (z)Fm (x, y) eiβ z ,

(14.1.17)

m=1

where Fm (x, y) is the modal field of the mth core, β is the propagation constant of the supermode, and Cm (z) is the amplitude that evolves with z because of mode coupling. Substituting the preceding form in Eq. (14.1.8) and following a standard procedure [5], we obtain a set of N coupled-mode equations: dCm = i(βcm − β)Cm + iκmn Cn , dz n=m N

(14.1.18)

where m = 1, 2, . . . , N and the coupling coefficient κmn is defined as k2 κmn = 0 2β



[n2 (x, y) − n2cm (x, y)]Fm∗ (x, y)Fn (x, y) dx dy.

(14.1.19)

Here ncm (x, y) is the refractive-index distribution when only the mth core is considered by itself and βcm is the corresponding modal propagation constant. The modal fields  Fm (x, y) are assumed to be normalized such that |Fm (x, y)|2 dx dy = 1. Eq. (14.1.18) shows that Cm is not constant when light is launched into one core of a multicore fiber. Its solution for a two-core fiber reveals that the power is transferred to the second core in a periodic fashion, a feature that is used to design fiber couplers [5]. For a supermode, Cm should not vary with z. Setting dCm /dz = 0, we obtain a matrix equation, MC = 0, where C is a column vector containing C1 , C2 , . . . , CN as its elements and M is a N × N matrix. A nontrivial solution exists when the determinant of M vanishes. This condition is used to find the propagation constants β for all supermodes. In general, they depend on the shapes, sizes, and geometrical locations of the cores and require a numerical approach. An analytical solution can be obtained when the cores are identical, equally spaced, and arranged in a circle such that each core couples only with its two nearest neighbors. In this case, M becomes a tridiagonal symmetric matrix, yielding the following solution [11]: β (n) = βc + 2κ cos(2π(n − 1)/N ],

Cmn = N −1/2 exp[2iπ(m − 1)(n − 1)/N ], (14.1.20)

Multimode fibers

Figure 14.5 Spatial distributions of the six supermodes of a multicore fiber with V = 5. The phase information is included by color-coding the field amplitudes from −1 to 1. The core regions are indicated by black circles. (After Ref. [9]; ©2011 OSA).

where β (n) is the propagation constant of the nth supermode and βc is its value for each isolated core. This technique can be generalized to fibers containing multiple circular rings of cores [11]. As an example, Fig. 14.5 shows the spatial field distributions of the six supermodes of a fiber containing a single ring of six cores [9]. The phase information is included by color-coding field amplitudes in the range of −1 to 1. Different spatial patterns result from different phases among the neighboring cores. We can apply Eq. (14.1.20) to the specific case of a two-core fiber that has been studied extensively in the context of fiber couplers [5,12]. If the two cores are separated by a distance d, they lie on a circle of diameter d centered in the middle. Using N = √ 2 in Eq. (14.1.20), the two supermodes are the even and odd combinations, (F1 ± F2 )/ 2, with the propagation constants βc ± 2κ , respectively. When an optical field is launched into a specific core, both supermodes are excited. The different phase shifts acquired by them during propagation results in a periodic transfer of power from one core to another, a feature that can be used to make directional couplers. Eq. (14.1.20) can also be applied to a fiber with three equidistant cores. Using N = 3, the propagation constants in this case become β (1) = βc + 2κ,

β (2) = βc + κ,

β (2) = βc − κ.

(14.1.21)

The corresponding supermodes are linear combinations of the three spatial profiles Fm (x, y) with coefficients Cmn obtained from Eq. (14.1.20). It is important to note that

629

630

Nonlinear Fiber Optics

Fm (x, y) is localized in the vicinity of the mth core, whereas supermodes extend over all cores.

14.1.4 Excitation of fiber modes As we have seen, a multimode fiber can support a large number of modes, depending on its core size. The important question from a practical standpoint is how many of these modes are exited when an optical beam is used to launch light into this fiber. In a typical experiment, the size, shape, and polarization of the input beam is controlled using lenses and other optical elements. Assuming that the beam is polarized along the x axis, the electric field at the input plane of the fiber can be written as Ex (x, y, t) = A0 (t)F0 (x, y), where A0 (t) is related to the temporal shape of each pulse and F0 (x, y) denotes spatial shape of the beam. To find which fiber modes are excited by this beam, we expand Ex in terms of all fiber modes as Ex (x, y, t) = A0 (t)

N

Cn Fn (x, y).

(14.1.22)

n=1

We can find the expansion coefficients by multiplying this equation by Fp∗ (x, y), integrating over the whole transverse plane, and using the mode-orthogonality relation in Eq. (14.1.12). The result is given by

Cp = 0

2π

0

∞

Fp∗ (ρ, φ)F0 (ρ, φ)ρ dρ dφ.

(14.1.23)

It shows that the extent of spatial overlap of each mode with the input beam determines how strongly that mode will be excited. The quantity |Cp |2 is called the coupling efficiency and its magnitude represents the fraction of total pulse energy carried by the pth mode. Consider the case of a Gaussian beam with F0 (ρ, φ) ∝ exp(−ρ 2 /w02 ) peaking precisely at the core center. Here, w0 is the spot size of the beam. Since this beam is radially symmetric (no φ dependence), it is easy to deduce that Cp vanishes all LPlm modes for which l = 0. As a result, such a Gaussian beam excites only the LP0m modes. Moreover, Cp is the largest for the fundamental LP01 mode because it has a central peak at ρ = 0 and overlaps most strongly with the input Gaussian beam. By careful adjusting the spot size w0 , it is possible to couple most of the pulse energy into this mode. On the other hand, many fiber modes can be excited simultaneously if we break the radial symmetry by shifting the intensity peak of the input beam from the core center. In this case pulse energy appears even in the modes with l = 0. For some applications, it is desirable to excite a single higher-order mode of the fiber. This is indeed possible, and many techniques have been developed for this purpose that make use of devices such as amplitude and phase masks, fiber gratings, spatial

Multimode fibers

light modulators, and photonic lanterns [13–19]. In particular, a spatial light modulator can be programmed to excite one or more specific modes of the fiber. Selective excitation of fiber modes has attracted attention because of the possibility of mode-division multiplexing for enhancing the capacity of optical communication system. Fiber modes for which l = 0 can be used to create a donut-shape spatial profile that carries orbital angular momentum (OAM) whose value depends on value of the integer l. This can happen in both the step-index and GRIN fibers and is a consequence of the factor cos lφ in Eq. (14.1.1). Recalling that each LPlm mode with l = 0 is four-fold degenerate, a superposition of these modes can lead to a spatial profile whose phase dependence is of the form R(ρ)e±ilφ such that R(ρ) vanishes at ρ = 0. The integer l is sometimes called the topological charge and represents the OAM of this beam [20,21]. In practice, it is not easy to maintain the OAM modes over significant fiber lengths because LPlm modes are not true modes of the fiber and represent linear combinations of several nearly degenerate HE-type modes discussed in Section 2.2. New kinds of fibers, whose high-index region is designed to be in the shape of a ring, have been fabricated for solving this problem [22]. In recent years, several techniques have been developed for generating and propagating OAM beams over considerable lengths of both the step-index and GRIN multimode fibers [23–26].

14.2. Nonlinear pulse propagation Although theory of nonlinear pulse propagation in multimode fibers attracted attention during the 1980s [27–29], it was only in 2008 that the coupled-mode equations were derived that included both the Kerr and the Raman nonlinearities for studying propagation of ultrashort pulses inside such fibers [30]. Since then, the same approach has been used in several other studies focused on different applications [31–36]. The single-mode NLS equation of Section 2.3 is extended in this section to multimode fibers.

14.2.1 Multimode propagation equations It was implicitly assumed in Section 2.3.1 that the input pulse was propagating inside a single-mode fiber with the spatial distribution F (x, y). In the case of a multimode fiber, unless special precautions are taken, pulse energy is likely to be distributed among several modes. In the coupled-mode approach, the Fourier transform of the electric field is expanded linearly in terms of all modes supported by the optical fiber and has the general form E˜ (r, ω) =



˜ m (z, ω) exp[iβm (ω)z], eˆ m Fm (x, y, ω)A

(14.2.1)

m

where Fm governs the shape of a specific mode, βm is the propagation constant of this mode, and eˆ m is its state of polarization. The sum should generally include both the

631

632

Nonlinear Fiber Optics

guided and radiation modes of the fiber. The analysis is much simpler if the sum is limited to include only the guided modes of a multimode fiber. The starting point is Eq. (2.3.1). Taking its Fourier transform and using Eq. (2.1.13), it can be written in the form ˜ + n2 (r, ω) ∇ 2E

ω2 ˜ ω2 ˜ E = − PNL (r, ω), c2 0 c 2

(14.2.2)

where n is the linear part of the refractive index. We substitute Eq. (14.2.1) into the preceding equation and note that each fiber mode satisfies ∇T2 Fm + n2 (r, ω)(ω2 /c 2 )Fm = βm2 Fm ,

(14.2.3)

˜m where the subscript T denotes the transverse part of the ∇ 2 operator. Assuming that A varies slowly with z and neglecting its second derivative, we obtain M n=1

2iβn

˜n ∂A ω2 eˆ n Fn (x, y, ω)eiβn z = − 2 P˜ NL (r, ω). ∂z 0 c

(14.2.4)

Multiplying the preceding equation with Fm∗ , integrating over the transverse plane, and using Eq. (14.1.12), the mode amplitudes are found to satisfy ˜ m iωe−iβm z ∂A = ∂z 2 0 cnm



Fm∗ (x, y, ω)ˆe∗m · P˜ NL (r, ω) dx dy,

(14.2.5)



where βm = nm ω/c and modes are normalized such that Fm∗ Fm dx dy = 1. We need to convert this equation to the time domain, following the method outlined in Section 2.3.1. This is a difficult task without making some simplifications. First, we ignore linear coupling among the modes that can occur in most fibers because of fabrication imperfections. Second, the input pulse is assumed to remain linearly polarized with no excitation of the orthogonally polarized fiber modes. Third, we assume that spatial mode profiles do not change significantly over the pulse bandwidth and replace Fm (x, y, ω) with its value at the carrier frequency ω0 . Fourth, we expand βm (ω) in a Taylor series [see Eq. (2.3.23)] and replace nm (ω) with its value at ω0 . Noting that ω − ω0 is replaced with i ∂∂t in the time domain, we obtain (see Appendix C) ∞

ik βkm ∂ k Am ∂ Am iω0 −i(β0m z−ω0 t) −i = e k ∂z k! ∂ t 2 0 cnm k=1

 i ∂ × 1+ Fm∗ (x, y)xˆ · PNL (r, t) dx dy, (14.2.6) ω0 ∂ t

where βkm = (dk βm /dωk )ω=ω0 is the kth-order dispersion parameter for the mth mode of the fiber with the effective mode index nm .

Multimode fibers

At this point we need to specify the nonlinear polarization PNL (r, t). For ultrashort pulses, one must use the general form given in Eq. (2.3.32) so that both the Kerr and Raman contributions are included. We initially neglect the Raman contribution and use the form given in Eq. (2.3.6) together with electric field, E(r, t) = xˆ



Fm (x, y)Am (z, t) exp[i(β0m z − ω0 t)],

(14.2.7)

m

and retain only the terms oscillating at the carrier frequency ω0 . When the result is substituted into Eq. (14.2.6), we obtain ∂ Am ∂ Am iβ2m ∂ 2 Am + β1m + = i γ fmnpq An A∗p Aq eiβmnpq z , ∂z ∂t 2 ∂ t2 n p q

(14.2.8)

where the triple sum extends over the number of modes M supported by the fiber, βmnpq = β0n − β0m + β0q − β0p

(14.2.9)

is the phase mismatch, and we kept only the dominant first two dispersion terms on the left side. We also neglected the self-steepening term containing the time derivative on the right side of Eq. (14.2.6). The preceding equation extends the single-mode NLS equation to the multimode case and reduces to it when M = 1. The parameter γ is defined as in Eq. (2.3.30) using the effective core area Aeff of the fundamental mode. The right side of Eq. (14.2.8) includes all the nonlinear effects related to the Kerr nonlinearity (SPM, XPM, and FWM). The relative strength of various intermodal terms is governed by the dimensionless overlap factors defined as



fmnpq = Aeff

Fm∗ (x, y)Fn (x, y)Fp∗ (x, y)Fq (x, y) dx dy.

(14.2.10)

As f1111 = 1 by definition, fmnpq involving four modes represents strength of that nonlinear term relative to the SPM term of the fundamental mode. The triple sum on the right side of Eq. (14.2.8) takes into account intermodal nonlinear coupling among various fiber modes. The magnitude of this coupling depends on fmnpq , or on the overlap of the four modes participating in the corresponding FWM process. The form of Eq. (14.2.8) indicates that the SPM and XPM phenomena can be interpreted as degenerate FWM processes that are self-phase matched. For example, f1122 corresponds to an intermodal XPM process that is phase matched because β1122 = 0.

14.2.2 Few-mode fibers It is clear from Eq. (14.2.8) that nonlinear propagation in multimode fibers is much more complex compared to single-mode fibers. The degree of complexity depends on

633

634

Nonlinear Fiber Optics

the number of modes excited by an input pulse, say M0 , which may be less than the total number M of modes supported by the fiber. Even when the linear mode coupling is ignored assuming an ideal fiber, one needs to solve M0 coupled NLS equations, each containing M03 nonlinear terms. Even when a single higher-order LP0m mode is excited using a device such as a spatial light modulator, some nonlinear terms in Eq. (14.2.8) couple to other modes of the fiber. However, these are not generally phase-matched, and the situation is similar to that of a single-mode fiber with an effective nonlinear parameter fmmmm γ . The case of few-mode fibers is interesting because it allows one to gain insight into the intermodal nonlinear effects by considering only two or three modes. The simplest case is that of a step-index fiber with 2.405 < V < 3.8 because such a fiber supports only the LP01 and LP11 modes (see Fig. 14.2). Since the LP11 mode is two-fold degenerate, let us denote its even and odd versions as modes 2 and 3, mode 1 being the fundamental mode. We ignore polarization degeneracy here, assuming that the optical fiber does not exhibit any birefringence. The first step is to find 81 overlap factors fmnpq resulting from 3 possible values of the 4 subscripts. Their magnitudes can be calculated using the mode profiles and carrying out the spatial integrals in Eq. (14.2.10). A numerical approach is necessary for step-index fibers, but approximate analytical expressions can be obtained for GRIN fibers [31]. Accurate numerical values has been provided in Ref. [37] for a specific GRIN fiber. It turns out that 60 out of 81 coefficients vanish. The remaining 21 elements take five distinct values. Three SPM coefficients are f1111 = 1 and f2222 = f3333 = 0.747 ≈ 3/4. The 12 intermodal XPM coefficients break into three groups: f1122 = f2211 = f1133 = f3311 = 0.496, f1212 = f2121 = f1313 = f3131 = 0.496, f2233 = f3311 = f2323 = f3232 = 0.249.

(14.2.11)

The remaining six coefficients represent intermodal FWM and have values f1221 = f2112 = f1331 = f3113 = 0.496,

f2332 = f3223 = 0.249.

(14.2.12)

These numerical values differ from 1/2 and 1/4 because the modes of the GRIN fiber were calculated more accurately by matching the boundary conditions at the core– cladding interface. As a simple example, we consider the two-mode case in which two input pulses at the same wavelength are injected into two specific LP modes of a few-mode fiber. Similar to the single-mode case, we use the reduced time T = t − β1p /z in a frame moving at the speed of the pth mode and obtain the following two coupled NLS equations:

   ∂ Ap iβ2p ∂ 2 Ap 2 2 2 ∗ 2iβ z + = i γ f | A | + 2f | A | A + f A A e , pppp p ppqq q p pqqp q p ∂z 2 ∂T 2

(14.2.13)

Multimode fibers

   ∂ Aq ∂ Aq iβ2q ∂ 2 Aq 2 2 2 ∗ −2iβ z + dqp + = i γ f | A | + 2f | A | A + f A A e , qqqq q qqpp p q qppq 1 2 ∂z ∂T 2 ∂T 2

(14.2.14) where β = β0q − β0p is the difference in the propagations constants of the two modes and dqp = β1q −β1p is the DGD parameter for them. These equations should be compared with those in Section 6.2 obtained for the two orthogonally polarized components of a single mode. Their forms are quite similar but the XPM and FWM coefficients of 2/3 and 1/3 are replaced with the nonlinear overlap factors involving the two spatial modes. If the two modes are not degenerate (β = 0) and the fiber length L is long enough that β L  1, the last term in Eqs. (14.2.13) and (14.2.14) does not contribute significantly because of its rapidly oscillating nature and can be neglected. In this specific case, these equations can be written as   ∂ Ap iβ2p ∂ 2 Ap 2 2 + = i γ f | A | + 2f | A | Ap , p q pp pq ∂z 2 ∂T 2

(14.2.15)

  ∂ Aq ∂ Aq iβ2q ∂ 2 Aq 2 2 + dqp + = i γ f | A | + 2f | A | Aq , qq q pq p ∂z ∂T 2 ∂T 2

(14.2.16)

where we used the contracted notation f pq ≡ fppqq . These equations should be compared to the coupled NLS equations in Section 7.2 for two pulses of different wavelengths propagating inside a single-mode fiber. Indeed, they reduce to them if all overlap factors are set to 1. The two-mode case provides physical insight into the relative importance of SPM, XPM, and FWM terms. If an input pulse excites more than two modes of the fiber, the number of nonlinear terms increases so rapidly that it is better to use the compact form given in Eq. (14.2.8).

14.2.3 Random linear mode coupling It is well known that a relatively long multimode fiber suffers from random linear coupling among its modes because of manufacturing variations in the shape and size of the fiber’s core along its length [33–35]. Such coupling also occurs if the fiber is bent or suffers from micro-bending. The extent of coupling between any two LP modes depends inversely on the difference in their propagation constants, δβ = βlm − βl m ; smaller is the difference, stronger becomes the coupling. For this reason, degenerate mode pairs (δβ = 0) are most strongly coupled in any multimode fiber. Random fluctuations in the fiber’s birefringence lead to coupling between the orthogonally polarized versions of the same LP mode because of their degenerate nature. Similarly, the even and odd versions of any LP mode with l > 0 become strongly coupled. Other mode pairs also exhibit some coupling depending on the magnitude of δβ for them. As fiber lengths can exceed hundreds or even thousands of kilometers for some telecommunication systems,

635

636

Nonlinear Fiber Optics

the power launched into a specific fiber mode is distributed among many modes of a multimode fiber, resulting in the linear as well as nonlinear crosstalk. The linear mode coupling not only requires that the full set of 2M coupled NLS equations must be solved for a fiber supporting M spatial modes (two accounts for polarization degeneracy), but also that this set be solved hundreds of times with different seeds to include the random nature of this coupling. In this section we extend the coupled multimode equations to include random coupling among fiber modes by replacing n2 (x, y) in Eq. (14.2.2) with n2 (x, y) + δ (x, y, z), where δ  n2 is a perturbation that varies with z in a random fashion because of fluctuations in the core’s shape and size (or stress) along the fiber. Using the expansion in Eq. (14.2.1), the modes are still obtained from Eq. (14.2.3) but Eq. (14.2.4) is replaced with M 

2iβn

n=1

˜n ω2  ω2 ˜ ∂A + δ 2 eˆ n Fn (x, y, ω)eiβn z = − 2 P NL (r, ω). ∂z c 0 c

(14.2.17)

Multiplying the preceding equation with Fm∗ , integrating over the transverse plane, and using the orthogonal nature of various modes, the mode amplitudes are found to satisfy −iβm z ˜m ∂A ˜ n ei(βn −βm )z = iωe −i κmn A ∂z 2 0 cnm n



Fm∗ (x, y, ω)ˆe∗m · P˜ NL (r, ω) dx dy,

(14.2.18)

where the coupling coefficients are defined as iω κmn (z) = nn c



δ (x, y, z)Fm∗ (x, y, ω)Fn (x, y, ω) dx dy.

(14.2.19)

The next step is to convert Eq. (14.2.18) to the time domain. However, we can no longer ignore the polarization effects since orthogonally polarized modes are strongly coupled because of their degenerate nature. Thus, we must use the vector form of the nonlinear polarization given in Eq. (6.1.3) together with Eq. (6.1.4). A simple way to include the polarization effects is to employ Jones vectors for each mode by defining a column vector as Am = [Amx Amy ]T . The final result can then be written in a vector form as [34] ∂ Am ∂ Am iβ2m ∂ 2 Am + β1m + =i κmn An ei(β0n −β0m )z 2 ∂z ∂t 2 ∂t n iγ T ∗ iβmnpq z + fmnpq [2(AH , p An )Aq + (An Aq )Ap ]e

3

n

p

(14.2.20)

q

where the superscripts H and T denote the Hermitian and transpose operations on a Jones vector. This equation is appropriate for relatively wide (> 1 ps) pulses; higherorder dispersive and nonlinear terms must be added for shorter pulses. It can only be

Multimode fibers

solved numerically. However, such an approach requires considerable computational resources. Because of the random nature of the coupling coefficients κmn , the preceding stochastic set of multimode equations must be solved many times before the average behavior of the underlying physical process can be predicted. It is thus natural to ask under what conditions random coupling among fiber modes can be ignored. Linear mode coupling is a well-understood effect in the context of fiber couplers [5]. If we consider only two modes (m = 1, 2) excited by a weak CW beam, neglect the nonlinear terms and time derivatives in Eq. (14.2.20), and assume Am = Bm exp(−iδa z) with δa = (β01 − β02 )/2, we obtain the following set of two coupled linear equations: dB1 = iδa B1 + iκ12 B2 , dz

dB2 = −iδa B2 + iκ21 B1 . dz

(14.2.21)

It is relatively easy to solve these equations. Assuming B2 (0) = 0 initially, we obtain [5] B2 (z) = (iκ21 /κe )B1 (0) sin(κe z),

κe =

 κ12 κ21 + δa2 .

(14.2.22)

This equation shows that power is transferred to the second mode in a periodic fashion with maximum transfer occurring for the first time at a distance Lc = π/(2κe ) known as the coupling length. The fraction of power transferred, η = |κ21 /κe |2 , depends on δa and becomes 100% when the two modes have the same propagation constants (δa = 0). Clearly, η is small if two modes are weakly coupled because of a small value of κ21 . However, η also depends on δa and decreases as 1/δa2 with increasing δa . Noting that δa = (π/λ)δ n, where δ n is the difference in the effective mode indices of the two modes involved, one can show that η becomes small for δ n > 10−5 . This is the reason why mode coupling is the strongest for two degenerate modes (δ n = 0) and becomes negligible for δ n > 10−4 . As discussed in Section 14.1, the degeneracy of the LPlm modes of step-index fibers depends on the integer l; modes with l = 0 are two-fold degenerate and those with l > 0 are four-fold degenerate. This suggests that launching light selectively into an LP0m mode may help in avoiding mode coupling. As any LP0m mode is radially symmetric, its coupling to nearby modes with l = 0 is not very strong because of a relatively small overlap between them. Moreover δ n between two neighboring LP0m modes increases with m and can easily exceed 10−5 for m > 5. It was found in 2006 that light launched into the LP07 mode of a step-index fiber with 86-µm core did not suffer from mode coupling over a 12-m length [38]. Since then, this approach to avoiding mode coupling has been used extensively. In essence, even a multimode fiber acts as a single-mode fiber over lengths ∼10 m when light is launched into a high-order LP0m mode. As the effective mode area is relatively large for such fibers, the nonlinear parameter γ is reduced considerably. This feature is useful for making high-power fiber amplifiers and lasers [39].

637

638

Nonlinear Fiber Optics

Random mode coupling cannot be avoided in mode-multiplexed communication systems in which multiple modes of fibers are intentionally excited to carry different data streams. Since fiber lengths can exceed hundreds of kilometers, all modes suffer from random coupling, and Eq. (14.2.20) must be used for modeling such systems. In this situation, an averaging procedure, similar to that used in Section 6.6 in the context of random birefringence in single-mode fibers, can be carried out for multimode fibers to obtain Manakov-like equations [33–35]. In the weak-coupling regime, one assumes that degenerate modes within each mode group are strongly coupled but no coupling occurs between the mode groups [34]. In the strong-coupling regime, all fiber modes are assumed to be equally coupled. The Manakov-like equations were first obtained in these two limiting cases. More recently, a more general approach was developed that provided such equations for all coupling regimes [40].

14.2.4 Graded-index fibers The coupled NLS equations given in Eq. (14.2.8) or (14.2.20) can be used for GRIN fibers by using the modal solution in Eq. (14.1.9) for calculating the nonlinear overlap factors fmnpq given in Eq. (14.2.10). However, a much simpler approach is available for a GRIN fiber whose core is wide enough that it supports a relatively large number of modes. When a pulsed Gaussian beam is launched into such a fiber, it excites many modes that propagate with different speeds and acquire different phase shifts. As a result, they interfere and create a spatial pattern in the transverse plane that changes considerably along the fiber length. Such a problem requires extensive numerical calculations if the multimode coupled NLS equations are used to solve it. However, the quadratic nature of the refractive index variations within the GRIN core simplifies the problem considerably. It turns out that this feature reduces the pulse-propagation problem to a single NLS equation that can be solved much faster numerically [41]. The basic idea is to solve Eq. (14.2.2) approximately with the variational method after including both the parabolic index profile given in Eq. (14.1.7) and the Kerr nonlinearity, but without paying attention to the fiber modes. As a further simplification, let us first consider a CW beam propagating along the z axis of a GRIN fiber, ignore its cladding completely, and assume that the CW beam remains confined to the fiber core because of the parabolic refractive index and Kerr self-focusing. We also neglect all polarization effects and assume that the CW beam remains polarized along the x axis. Using E˜ (r) = xB ˆ (r)eiβ0 z with β0 = n1 ω0 /c in Eq. (14.2.2) and assuming that B(r) is a slowly varying function of z, we obtain i

∂B 1 2 n2 ω0 2 ρ2 + ∇T B − β0  2 B + |B| B = 0, ∂ z 2β0 a c

(14.2.23)

where ∇T2 is the transverse part of the Laplacian operator. This equation was solved in 1992 with the variational method [42], assuming a Gaussian shape for the spatial

Multimode fibers

profile whose amplitude, width, and phase were allowed to change with z. The resulting solution can be written as B(r) = A0 F (r), where A0 is the amplitude at z = 0 and F (r) is given by F (r) =



(x2 + y2 ) w0 i φ( r ) . exp − + w (z) 2w2 (z)

(14.2.24)

Here φ(r) is a spatially varying phase whose functional form is known (but not needed) and w0 is the initial spot size at z = 0. The spatial width w evolves with z in a periodic fashion as w 2 (z) = w02 f (z),

f (z) = cos2 (π z/zp ) + Cg2 sin2 (π z/zp ).

(14.2.25)

The spatial period zp and parameter Cg in the preceding equation are defined as πa zp = √ , 2



Cg =

1 − pzp , πβ0 w02

(14.2.26)

where p is related to the initial amplitude A0 as p = n2 (β0 w0 A0 )2 /2n1 . The Kerr parameter n2 appears only through the dimensionless number p that is related to the beam collapse known to occur in any nonlinear Kerr medium [43]. The physical meaning of the CW solution in Eq. (14.2.24) is clear. It shows that a Gaussian CW beam, launched into a GRIN fiber at z = 0, maintains its Gaussian shape inside a GRIN fiber, but its amplitude and width (also phase) change in a periodic fashion such that it recovers its input shape periodically at distances z = mzp (m is a positive integer), a phenomenon known as self-imaging. At distances z = (m − 12 )zp , w (z) takes its minimum value Cg w0 , that is, Cg governs the extent of beam compression during each cycle. Typically zp < 1 mm for GRIN fibers. Using  = 0.01 with a = 25 µm, we find zp = 0.55 mm, a remarkably short distance. It is important to emphasize that self-imaging is a linear effect and occurs in GRIN fibers even in the absence of any nonlinearities. Eq. (14.2.25) shows that self-imaging is preserved even when the Kerr nonlinearity is included, as long as p < 1 so that Cg is a positive quantity. Indeed, p = 1 corresponds to the beam collapse induced by the Kerr nonlinearity [42]. In many cases of practical interest, p  1 at the power levels employed and we can replace 1 − p in Eq. (14.2.26) with 1. To apply the preceding CW solution to a pulsed Gaussian beam, we assume that the bandwidth of the pulse is narrow enough that the spatial profile F (r) of the beam does not vary with ω and write the solution of Eq. (14.2.2) at any frequency ω within the pulse bandwidth as ˜ (z, ω)F (r)eiβ(ω)z , E˜ (r, ω) = A

(14.2.27)

639

640

Nonlinear Fiber Optics

where β(ω) = n1 (ω)ω/c is the propagation constant at that frequency. We can now follow the procedure used in Section 2.3 for single-mode fibers, while keeping in mind that F (r) is not the spatial profile of a specific mode but changes with z along the fiber length as indicated in Eqs. (14.2.24) and (14.2.25). After eliminating the transverse coordinates through a spatial integration and going back to the time domain, the final result can be written in the form [41] ∂A ∂ A iβ2 ∂ 2 A α + β1 + + A = iγ (z)|A|2 A, ∂z ∂t 2 ∂ t2 2

(14.2.28)

where the nonlinear parameter is defined as  γ (z) =

ω0 n2 , cAeff (z)

Aeff (z) =

2 ∞ 2 dx dy | F ( r )| −∞  ∞ . 4 −∞ |F (r)| dx dy

(14.2.29)

This is a remarkable result. It shows that the pulse evolution inside a GRIN fiber can be studied by solving a single NLS equation, even though multiple spatial modes may be propagating simultaneously inside the fiber. The oscillating spatial width of a Gaussian beam, resulting from the GRIN nature of the refractive index, gives rise to an effective nonlinear parameter γ (z) that is periodic in z. This is not surprising since the intensity at a given distance z depends on the beam width, becoming larger when the beam compresses and smaller as it spreads. One can also interpret the same effect as a spatially varying, effective mode area. The spatial integrals appearing in Eq. (14.2.29) can be performed analytically using the functional form of F (r) in Eq. (14.2.24). The result can be written in the form γ (z) = γ /f (z), where γ is defined as in Section 2.3, except that the initial value of Aeff at z = 0 is used to calculate it. As a final step, if we neglect fiber losses and use the reduced time T = t − β1 z, Eq. (14.2.28) takes the form ∂ A iβ2 ∂ 2 A + = iγ f −1 (z)|A|2 A, ∂z 2 ∂T 2

(14.2.30)

where f (z) is given in Eq. (14.2.25). This equation reduces to the standard NLS equation when Cg = 1, resulting in f (z) = 1. We use Eq. (14.2.30) in later sections to discuss the nonlinear phenomena in GRIN fibers.

14.3. Modulation instability and solitons The set of coupled NLS Equations given in Eq. (14.2.8) or in Eq. (14.2.20) can be used to study how the nonlinear effects such as SPM, XPM, and FWM affect the evolution of optical fields inside multimode fibers. The multiple nonlinear terms within the triple sum in these equations include all intermodal couplings induced by the Kerr nonlinearity. In this section we first discuss the phenomenon of modulation instability

Multimode fibers

and then investigate whether soliton-like propagation of optical pulses is feasible in multimode fibers.

14.3.1 Modulation instability We have seen in Section 5.1 how modulation instability, occurring in the anomalousGVD region of a single-mode fiber, produces temporal oscillations on a CW beam that are reshaped by the Kerr nonlinearity into a train of optical solitons. It is thus natural to wonder whether modulation instability occurs in multimode fibers [44–47] and whether temporal solitons can form in such fibers, in spite of different group velocities associated with different modes [48,49]. This section is devoted to answering such questions. Consider first the simplest case in which a CW beam excites only two modes of a few-mode fiber that do not couple linearly to other modes but are coupled nonlinearly. In this situation, Eqs. (14.2.13) and (14.2.14) govern the stability of the CW solution. If the last nonlinear term in these equations is negligible because of a relatively large phase-mismatch, then Eqs. (14.2.15) and (14.2.16) can be used to analyze the CW instability with the approach outlined in Section 7.2. The results show that modulation instability can occur in the normal-GVD region only when the XPM dominates over the SPM and requires the following condition on the overlap factors: 

2f pq > f pp f qq .

(14.3.1)

These parameters can be calculated for any multimode fiber and are different for different mode pairs. As a result, the spectral sidebands appear around the CW-beam wavelength only for specific mode pairs. In a 2017 study, performed using a 40-cmlong GRIN fiber of 11-µm core radius [47], spectral sidebands were observed for the mode pair involving the LP01 and LP02 modes because the condition (14.3.1) holds for this pair (f pp = 1, f qq = fpq = 1/2). The observed frequency shift was close to 10 THz at power levels of 50 kW for 400-ps input pulses used in the experiment. In contrast, the condition (14.3.1) was not satisfied for the (LP01 , LP21 ) mode pair. The general case of M-mode fibers has been analyzed including both polarization components of each mode [46]. The linear stability analysis of the CW solution was based on Eq. (14.2.20) but it neglected any linear coupling among the modes. The analysis involving 2M modes is cumbersome as it leads to a 4M × 4M matrix whose eigenvalues provide the gain spectrum of modulation instability. The numerical results reveal that, depending on the fiber parameters, up to 4M spectral sidebands can form in different spectral regions, covering a wide bandwidth on both sides of the CW pump. As an example, Fig. 14.6 shows the gain spectrum of the modulation instability occurring in a four-mode fiber with 12-µm core radius (V ≈ 5 at λ = 1.55 µm), assuming that the 4-kW pump power is distributed equally among the four x-polarized modes. The

641

642

Nonlinear Fiber Optics

Figure 14.6 (A) Gain spectra of modulation instability for a specific four-mode fiber pumped with 4 kW power distributed equally among the four spatial modes. Parts (B) and (C) show magnified views covering different frequency ranges with labels marking different sidebands. (After Ref. [46]; ©2015 APS).

analysis included orthogonally polarized degenerate modes but ignored the odd versions of the LP11 and LP21 modes. The gain and frequency scales are normalized in Fig. 14.6 using the nonlinear length associated with the LP01 mode. All sidebands are shown in part (A), with two magnified views of the positive-side sidebands in parts (B) and (C). Both the peak gain and the bandwidth vary from one sideband to next. It is easy to conclude from this figure that the nature of modulation instability is quite complex in multimode fibers. The stability of a CW Gaussian beam inside a GRIN fiber has also been studied [50]. As discussed in Section 14.2.4, such fibers exhibit a self-imaging feature that results in periodic focusing of a Gaussian beam along the fiber’s length. Because of this feature, Eq. (14.2.30) can be used to find all features of modulation instability in GRIN fibers in an analytic form. A similar equation was encountered in Section 5.1.5 in the context of a fiber-optic communication link employing optical amplifiers periodically for compensating fiber losses. Since the peak power of optical pulses varies in a periodic fashion in such communication links, Eq. (14.2.30) with a different functional form of f (z) was used in 1993 to study modulation instability [51].

Multimode fibers

As discussed in Section 5.1.1, the stability of the CW solution is analyzed by perturbing the steady-state solution of Eq. (14.2.30) in the form



A = ( P0 + a) exp(iφNL ),

φNL = γ P0

z

f −1 (z) dz,

(14.3.2)

0

where a(z, t) is a small perturbation. Linearizing Eq. (14.2.30) in a, we obtain ∂a iβ2 ∂ 2 a iγ P0 =− + (a + a∗ ). ∂z 2 ∂ T 2 f (z)

(14.3.3)

We seek its solution in the form a = a1 e−iT + a2 eiT , where  is the modulation frequency and the amplitudes a1 and a2 satisfy: ∂ ak i i γ P0 = β2 2 ak + (ak + a∗3−k ), ∂z 2 f (z)

k = 1, 2.

(14.3.4)

Because of the periodic nature of f (z), the preceding coupled equations can be solved approximately by expanding f −1 (z) in a Fourier series as f −1 (z) =

∞

cm eimKz ,

m=−∞

cm =

1 zp



zp

f −1 (z)e−imKz dz,

(14.3.5)

0

where K = 2π/zp and zp is the self-imaging distance given in Eq. (14.2.26). Each term in this Fourier series can satisfy a phase-matching condition and results in one pair of sidebands located on opposite sides of the pump’s frequency. The frequencies at which the gain of each sideband pair becomes maximum are found to satisfy [51] 2m =

2π m 2c0 − , β2 zp β2 LNL

(14.3.6)

where m is an integer and the nonlinear length LNL = (γ P0 )−1 . The peak gain of the mth sideband depends on the Fourier coefficient cm and is given by gm = 2γ P0 |cm |. It is easy to show that c0 = 1/Cg . The other Fourier coefficients can also be computed numerically. The m = 0 sideband corresponds to the single-mode case discussed in Section 5.1. This sideband exists only if β2 < 0, i.e., GVD of the fiber must be anomalous at the pump wavelength. Eq. (14.3.6) shows that a CW Gaussian beam launched inside a GRIN fiber can become unstable to small perturbations, even when it propagates in the normal-GVD region of the fiber. The gain spectrum of modulation instability exhibits a rich structure with an infinite number of sideband pairs at frequencies that are not equally spaced (in contrast with the FWM sidebands). The peak gain of each sideband depends on the spatial pattern of the oscillating Gaussian beam through the function f (z). Since spatial

643

644

Nonlinear Fiber Optics

variations play a crucial role, this instability is also known as a geometric parametric instability, or as a spatio-temporal instability [52–54]. The numerical values of the sideband frequencies can be estimated using typical values for commercial GRIN fibers. We saw earlier that the self-imaging period zp is <1 mm in such fibers. Since γ is relatively small (< 0.1 W−1 /km), the nonlinear length exceeds 1 meter even at input power levels of 10 kW. The estimated value of c0 is < 5 for w0 < 20 µm. As a result, the first term in Eq. (14.3.6) dominates in most cases of practical interest, and we can approximate the sideband frequencies as 

fm = m /2π ≈ ± m/(2πβ2 zp ),

m = 1, 2, . . . .

(14.3.7)

As a specific example, using β2 = 20 ps2 /km and zp = 0.5 mm, the mth sideband is √ found to be shifted by 125 m THz from the pump frequency. For a pump wavelength near 1 µm with its central frequency at 300 THz, the m = 1 sidebands will be located at frequencies of 175 THz and 425 THz. The former is in the infrared region near 1700 nm, while the latter lies in the visible region near 700 nm. Moreover, the higher-order sidebands would lie in the ultraviolet and mid-infrared regions. Note that all sidebands exist even when β2 < 0 at the pump’s wavelength because the integer m can be negative in Eq. (14.3.6). In this case, frequency shift for the m = 0 sideband pair is typically below 1 THz. The sidebands at the frequencies predicted by Eq. (14.3.7) have been observed in several experiments in which relatively long (∼ 1 ns) pulses were launched into a GRIN fiber to realize a quasi-CW situation [52–54]. In one experiment, 900-ps pulses at 1064 nm emitted from a Nd:YAG microchip laser in the form of a Gaussian beam (FWHM 35 µm) were coupled into a 6-m-long GRIN fiber [52]. The bottom trace in Fig. 14.7 shows the experimentally observed spectrum when the peak power of input pulses was 50 kW. The top trace shows the numerically simulated spectrum based on Eq. (14.2.23). Dashed vertical lines mark the peak locations predicted by Eq. (14.3.7). The agreement seen in this figure indicates the usefulness of the NLS equation in Eq. (14.2.30). Its only drawback is that it cannot capture spatial changes that may occur in response to temporal changes.

14.3.2 Multimode solitons The question whether solitons can form inside multimode fibers attracted attention as early as 1980 [48,49]. It was realized that different group delays (or speeds) associated with different modes were likely to hinder the formation of such solitons. As intermodal group delays are relatively small for a GRIN fiber, it was natural to consider soliton formation in such a medium. Indeed, theoretical work indicated that formation of temporal solitons was feasible inside a GRIN medium [55–58], and it eventually led

Multimode fibers

Figure 14.7 Experimental (bottom) and numerically simulated (top) spectra when a 6-m-long GRIN fiber is pumped with 900-ps pulses with 50 kW peak power. Dashed vertical lines and circled numbers show the peak locations predicted by Eq. (14.3.7). (After Ref. [52]; ©2016 APS.)

in 2013 to the observation of multimode solitons in a GRIN fiber [59]. Since then, multimode solitons have remained of considerable interest [60–66]. The propagation of pulsed optical beams inside a GRIN fiber requires the solution of Eq. (14.2.2) for all frequency components of the pulse. We can solve this equation either using the modal expansion in Eq. (14.2.1), or ignore the modes and follow the approach of Section 14.2.4. In the latter case, we assume a solution of the form E˜ (r, ω) = xˆ B˜ (r, ω)eiβ(ω)z , where β(ω) = n1 (ω)(ω/c ) depends on frequency. The dispersive effects are included by expanding β(ω) in a Taylor series around ω0 . Returning to the time domain and retaining only terms up to the second order in the Taylor expansion, we obtain the so-called (3+1)D NLS equation: i

∂B 1 2 n2 ω0 2 β2 ∂ 2 B ρ2 + ∇T B − − β  B+ |B| B = 0, 0 2 2 ∂ z 2β0 2 ∂T a c

(14.3.8)

where T = t − β1 z is the reduced time. This equation should be compared with Eq. (14.2.23) obtained for a CW beam. Its new features are that B(r, t) is a function of both space and time and the dispersive effects are included through the GVD term containing β2 . Similar to the CW case discussed in Section 14.2.4, Eq. (14.3.8) can be solved with the variational method. Since we are looking for a soliton, we employ a trial function in the form [56,57] 

B(r, t) =

 T 

Ep i ρ2 2 2 sech + ( C ρ + C T ) + i φ , (14.3.9) exp − 1 2 2π w 2 T0 T0 2w2 2

645

646

Nonlinear Fiber Optics

where Ep is energy of the pulse, w (z) and T0 (z) are its spatial and temporal widths, and C1 (z) and C2 (z) are the corresponding chirp parameters. It turns out that a chirp-free (C1 = C2 = 0), soliton-like solution exists if β2 < 0 and the pulse energy Ep is below a certain value [56–58]. The temporal width of this solution is T0 = 4π w 2 |β2 |/(k0 n2 Ep ). We can deduce the steady-state value of w from f (z) given in Eq. (14.2.25). Clearly w(z) will stop changing and maintain its input value w0 when Cg = 1 and f (z) = 1. This occurs when the initial width is such that  z 1 − p 1/2 p . w = w0 = πβ0

(14.3.10)

The pulse energy appears in this equation through the parameter p. Clearly, the solution exists only for p < 1. At pulse energies such that p  1, the beam width depends only on the self-imaging period zp in addition to the input wavelength λ0 . Using zp = 0.5 mm with λ0 = 1.5 µm, w ≈ 6 µm and T0 ≈ 180 fs for such a soliton of 0.5-nJ energy. The preceding solution with constant spatial and temporal widths is called a spatiotemporal soliton [58]. It exists even in a homogeneous Kerr medium ( = 0) and is sometimes referred to as a “light bullet” as the optical pulse is confined in all three spatial dimensions [67]. The important question from a practical standpoint is whether this solution is stable against small perturbations. It turns out that light bullets are not stable in a homogeneous Kerr medium. However, a linear stability analysis performed in the case of a GRIN Kerr medium shows that small perturbations around the steady state may oscillate at specific frequencies but do not grow exponentially [57,64]. Eq. (14.3.8) also has pulse-like solutions whose temporal width remains constant, but the spatial width oscillates along the fiber’s length. Such a few-mode temporal soliton was first observed in a 2013 experiment by launching 300-fs pulses (wavelength near 1550 nm) into a 100-m-long GRIN fiber [59]. At a specific pulse energy of 0.5 nJ, the input and output pulses had nearly the same temporal shapes, but the spectra were different because of different spectral shifts of different modes that helped in minimizing DGDs and allowed the three-mode soliton to form and propagate at a common speed. This feature was confirmed through numerical solutions of the multimode NLS equations given in Eq. (14.2.8). Fig. 14.8 shows (A) spatial profiles, (B) temporal shapes, and (C) optical spectra for the LP01 , LP02 and LP03 , radially symmetric modes that were assumed to participate in the soliton formation. The (3+1)D equation (14.3.8) was also solved numerically over 52 m (requiring 20 days of computation time in 2012). The two sets of numerical results agreed reasonably well with each other and the experiment. It is important to emphasize that the experimental and numerical results in Ref. [59] differ from the variational solution in Eq. (14.3.9) in the sense that the spatial beam width does not remain constant. The temporal width did reach a constant value of about 200 fs after some initial oscillations. Also, a large faction of soliton energy was in the fundamental mode. Later numerical and experimental work included more modes

Multimode fibers

Figure 14.8 (A) Spatial profiles of three modes with their (B) temporal shapes and (C) spectra (right) at the end of a 100-m-long GRIN fiber pumped with 300-fs pulses with 0.5 nJ energy. Dashed vertical lines mark the expected DGD-induced delays of the three modes in the absence of soliton formation. (After Ref. [59]; ©2013 Macmillan.)

and presented evidence of multimode solitons composed of five or more modes [61]. However, a bullet-like spatiotemporal soliton had not been observed in GRIN fibers by early 2019. The numerical solutions of Eq. (14.3.8) suggest that we look for a solution that is periodic spatially but has a constant pulse width temporally. As we saw in Section 14.2.4, Eq. (14.2.30) governs just such a solution. It is clear from the presence of f (z) in this NLS equation that perfect solitons cannot form inside GRIN fibers. As Eq. (14.2.30) is not integrable by the inverse scattering method, the formation of ideal solitons is ruled out. However, an equation similar to Eq. (14.2.30) has been found in the case of single-mode fiber links, employing amplifiers periodically for compensating fiber losses. It was found in 1990 that a new kind of soliton (called the guiding-center soliton) can propagate inside such fiber links [68]. It is also known as a loss-managed soliton [69]. Based on this analogy, it was shown in 2018 that GRIN fibers support propagation of optical pulses that preserve their shape and behave like a soliton, even though their spatial width oscillates along the fiber’s length [65]. Such pulses are referred to as GRIN solitons to emphasize that a parabolic index profile is essential for their existence. Before solving Eq. (14.2.30) approximately, it is useful to normalize it in soliton units of Chapter 5 using the variables τ = T /T0 ,

ξ = z/LD ,



U = A/ P 0 ,

(14.3.11)

where T0 and P0 are the width and the peak power of input pulses and LD = T02 /|β2 | is the dispersion length. The normalized NLS equation takes the form i

∂ U 1 ∂ 2U N2 + + |U |2 U = 0, ∂ξ 2 ∂τ 2 f (ξ )

(14.3.12)

647

648

Nonlinear Fiber Optics

where we assumed β2 < 0 and introduced the soliton order as N = (γ P0 LD )1/2 . The periodically varying function f (z) given in Eq. (14.2.25) can be written as f (ξ ) = cos2 (π qξ ) + Cg2 sin2 (π qξ ),

q = LD /zp ,

(14.3.13)

where zp is the self-imaging period introduced in Eq. (14.2.26). As we discussed in Section 14.2.4, the numerical value of zp is < 1 mm for typical GRIN fibers. In contrast, the dispersion length LD exceeds 50 cm for T0 > 0.1 ps if we use β2 = −20 ps2 /km, a typical value near 1550 nm for silica fibers. As a result, q is a large number (q > 100) under typical experimental conditions. Physically, it represents the number of times the spatial beam width oscillates inside a GRIN fiber within one dispersion length. We saw in Chapter 5 that the dispersion length provides the scale over which solitons evolve [68]. Indeed, solitons cannot respond to beam width changes taking place on a scale of 1 mm or less when LD exceeds 10 cm. If we write the solution of Eq. (14.3.12) as U = U + u, where U represents average over one spatial period zp , the perturbations |u(ξ, τ )| induced by spatial beam-width variations remain small enough that they can be neglected (as long as q  1). In other words, the average dynamics of the soliton can be captured by solving the standard NLS equation i

∂ U 1 ∂ 2U 2 + N |U |2 U = 0, + ∂ξ 2 ∂τ 2

(14.3.14)

where N is the effective soliton order defined as 2

N = N 2 f −1 (ξ ) = N 2 c0 = N 2 /Cg ,

(14.3.15)

where we evaluated the integral in Eq. (14.3.5) to find c0 = 1/Cg . As discussed before, Cg represents the fraction by which the beam width shrinks at zp /2 before recovering its original width at z = zp . Clearly, Eq. (14.3.12) has a solution in the form of a fun damental soliton when we choose N = 1 or N = Cg . This solution exists for a wide range of input pulse widths as long as the peak power is adjusted to satisfy this soliton condition. Because of an oscillating beam width, the input peak P0 must be adjusted to make sure that N = 1 on average along the GRIN fiber. Fig. 14.9 compares the temporal and spectral profiles of a GRIN soliton (N = 1), after it has propagated a distance of 100LD inside a GRIN fiber using Cg2 = 0.2 and q = 100. These results were obtained by solving Eq. (14.3.12) numerically with the initial field U (0, τ ) = U0 sech(τ ), where U0 was chosen to ensure N = 1. On the logarithmic scale used here, perturbations induced by spatial oscillations are evident but their magnitude remains below the 50 dB level even after 100 dispersion lengths. Higherorder solitons for which both the spatial and temporal widths evolve periodically can also form inside GRIN fibers. As seen in Section 5.2.3 in the context of single-mode

Multimode fibers

Figure 14.9 Temporal and spectral profiles of a multimode soliton (N = 1) before (dashed line) and after it has propagated a distance of 100LD inside a GRIN fiber with Cg2 = 0.2 and q = 100.

fibers, the period of temporal oscillations is set by the dispersion length (soliton period z0 = π LD /2) and is much longer than the period zp of spatial-width oscillations. Numerical results indicate that higher-order solitons are less stable compared to the fundamental GRIN soliton as they remain intact only over distances ∼ 10LD [65]. They also undergo fission and breakup into multiple fundamental solitons if third-order dispersion is included in Eq. (14.3.12).

14.3.3 Solitons in specific fiber modes The single-mode solitons considered in Chapter 5 may also exist inside a multimode fiber when the input pulse is launched into a specific mode of the fiber that does not couple to any other mode linearly. An interesting question is whether two solitons (at the same or different wavelengths) can propagate stably inside a multimode fiber if they are initially launched into two distinct modes of a multimode fiber. This scenario first attracted attention during the 1980s and has remained of continued interest [70–73]. The simplest example of such a soliton pair is the vector soliton discussed in Section 6.5 in the context of birefringent single-mode fibers supporting two orthogonally polarized modes. As was discussed there, because of the XPM-induced coupling between the two polarization components, the two pulses shift their spectra in the opposite directions such that both of them propagate at a common speed, in spite of the DGD related to linear birefringence of the fiber. Another example is provided by the XPMpaired solitons of Section 7.3, where two pulses at different wavelengths were found to propagate inside a single-mode fiber at a common speed through the XPM-induced spectral shifts. These examples clearly suggest that two pulses launched into two different modes of a multimode fiber may also be able to overcome their modal DGD through the nonlinear intermodal coupling between them. To study the formation of

649

650

Nonlinear Fiber Optics

Figure 14.10 (Evolution over 100 LD of two solitons launched into two lowest-order modes of a fiber with a DGD of 10 ps/km. (After Ref. [60]; ©2015 OSA.)

such coupled solitons, one must solve Eqs. (14.2.13) and (14.2.14) numerically. The general M-mode problem requires a solution of M coupled NLS equations, given in Eq. (14.2.8) and containing a large number of nonlinear intermodal terms. If orthogonally polarized modes are also involved together with linear coupling, the problem becomes quite complicated and requires the use of the set given in Eq. (14.2.20). Consider the simplest case of two pulses at the same central wavelength launched into two lowest-order modes of a step-index fiber. If both pulses are linearly polarized along the same direction and the fiber is short enough that linear mode coupling can be ignored, we can use Eqs. (14.2.13) and (14.2.14) to study whether they can propagate as two mutually supporting solitons. These equations differ from Eqs. (7.4.1) and (7.4.2) in two ways: the presence of a FWM-like term and different coefficients appearing in the XPM terms. If the FWM term can be neglected because of a relatively large phase-mismatch, we may expect each pulse to propagate as a soliton if the DGD is not too large and the peak powers are adjusted properly to satisfy the soliton condition N = 1. Numerical simulations have been used to verify this expectation [60]. Fig. 14.10 shows the evolution of two fundamental solitons in the LP01 and LPe11 modes, assuming a DGD of 10 ps/km between the two modes. The trajectory of the LP01 soliton (left side) would be vertical in the absence of second soliton. Because of nonlinear coupling, trajectories of both solitons tilt in such a way that they overlap temporally. This can only happen if the two solitons shift their spectra in the opposite directions and move

Multimode fibers

at a common speed. More specifically, mode 1 slows down while mode 2 accelerates just enough that the two solitons trap each other and propagate as one unit. However, this situation is not expected to remain stable over long distances. As seen in Fig. 14.10, each soliton sheds some energy in the form of dispersive waves. Moreover, energy is transferred continuously from the LP11 mode to the LP01 mode such that the former has lost more than half of its energy after 100 dispersion lengths. If the DGD between the two modes is larger, only a fraction of energy in mode 2 is trapped by the mode 1. For example, less than 10% of pulse energy is trapped when DGD exceeds 50 ps/km, and no trapping occurs for a DGD of 100 ps/km.

14.4. Intermodal nonlinear phenomena The existence of multiple modes within the same optical fiber adds a new dimension to the three nonlinear effects discussed in Chapters 8 to 10. In this section we consider four-wave mixing (FWM), stimulated Raman scattering (SRS), and stimulated Brillouin scattering (SBS) in multimode fibers. In each case, two or three pulses at different wavelengths propagate in different modes of the fiber, while interacting with each other through the intermodal nonlinear coupling induced by the Kerr nonlinearity of the fiber.

14.4.1 Intermodal FWM As we discussed in Section 10.3, the phase-matching condition in Eq. (10.3.1), kM + kW + kNL = 0,

(14.4.1)

plays a critical role in any FWM process. This condition shows that the impact of phase mismatch resulting from material dispersion (kM ) can be reduced through the waveguide (kW ) and nonlinear contributions (kNL ) to the phase-matching condition. It is much easier to satisfy Eq. (14.4.1) for multimode fibers because kW can be made negative when four waves participating in FWM are allowed to propagate in different fiber modes. Indeed, several early experiments during the 1970s employed multimode fibers for this purpose [74–80]. Although single-mode fibers were used mostly after the year 1980, FWM in multimode fibers has continued to attract attention [81–90]. The magnitude of kW depends on the choice of fiber modes in which four waves participating in the FWM process propagate. The eigenvalue equation (14.1.5) can be used for step-index fibers to calculate kW for the four LP modes participating in the FWM process. Fig. 14.11 shows the resulting |kW | as a function of the pump–idler frequency shift s for a few-mode fiber with a = 5 µm and V ≈ 6 at the pump wavelength of 532 nm. A dashed line shows the quadratic variation of kM with s using Eq. (10.3.6). Its intersection with the solid lines determines the frequency shift in a

651

652

Nonlinear Fiber Optics

Figure 14.11 Phase-matching diagrams for (A) mixed-mode and (B) single-mode pump propagation. Solid and dashed lines show variations of |kW | and kM with frequency shift. Dotted lines illustrate the effect of increasing core radius by 10%. The four modes of the FWM process are indicated in the LPlm notation. (After Ref. [74]; ©1975 IEEE.)

Figure 14.12 Idler power as a function of wavelength generated by tuning the signal wavelength (upper scale). Far-field patterns corresponding to the two dominant peaks are shown as insets. (After Ref. [1]; ©1974 AIP.)

FWM experiment, assuming kNL is negligible. Two cases in Fig. 14.11 correspond to whether the pump power (A) is divided between two different fiber modes or (B) is launched only in the LP01 mode. As seen in part (B), frequency shifts are much larger in the latter case. The exact value of frequency shift is quite sensitive to various fiber parameters. The dotted lines show how much s changes with a 10% increase in the core radius of the fiber. In general, the phase-matching condition can be satisfied for several combinations of the fiber modes. In the 1974 demonstration of multimode FWM in silica fibers, pump pulses at 532 nm with peak powers ∼100 W were launched in a 9-cm-long fiber, together with a CW signal (power ∼10 mW) that was tunable from 565 to 640 nm [1]. FWM generated an idler wave in the blue region (ω4 = 2ω1 − ω3 ). Fig. 14.12 shows the observed idler spectrum obtained by varying ω3 . The five peaks correspond to different combinations of fiber modes for which phase matching was realized. The far-field patterns of the two dominant peaks clearly indicate that the idler was generated in different fiber modes

Multimode fibers

Figure 14.13 Observed spectra at the output of a 150-m-long multimode fiber as the peak power of 25-ps pump pulses is increased from (A) to (D). Supercontinuum formed (bottom trace) when peak intensity was increased to 1.5 GW/cm2 . (After Ref. [80]; ©1987 IEEE.)

for two different signal frequencies. In this experiment, the entire pump power was launched in the LP01 mode. As expected from Fig. 14.11, phase-matching occurred for relatively large frequency shifts (50 to 60 THz). In another experiment, the frequency shift was as large as 130 THz, a value that corresponds to 23% change in the pump frequency [78]. FWM with much smaller frequency shifts (νs < 10 THz) can occur when the pump power is divided between two different fiber modes (see Fig. 14.11). This configuration is also relatively insensitive to variations in the core diameter [74]. When s is close to 10 THz, the SRS process can interfere with the FWM process as this value is near the location of the Raman-gain peak (see Fig. 8.2). In a 1975 experiment in which 532-nm pump pulses with peak powers ∼100 W were sent through a multimode fiber, the Stokes band was always more intense than the anti-Stokes band because of its Raman amplification [74]. When the pump beam is in the form of a train of short optical pulses, the FWM process is affected not only by SRS but also by SPM, XPM, and GVD. In a 1987 experiment, 25-ps pump pulses were transmitted through a 15-m-long fiber, supporting four modes at the pump wavelength of 532 nm [80]. Fig. 14.13 shows the optical

653

654

Nonlinear Fiber Optics

spectra observed at the fiber’s output as the pump’s peak intensity was increased from 0.1 to 1.5 GW/cm2 . As nothing else was launched in this experiment, the signal and idler waves buildup from noise through spontaneous FWM occurring at high pump powers. Indeed, only the pump spectrum is seen at the lowest pump power (trace A). Three pairs of FWM sidebands with frequency shifts in the range 1–8 THz appear spontaneously just above the FWM threshold of 0.5 GW/cm2 (trace B). As all sidebands have nearly the same amplitude as SRS does not play a significant role at this pump power. As pump power is increased further, the red-shifted lines become more intense than the blue-shifted lines as a result of Raman amplification (trace C). With a further increase in the pump power, the Stokes line closest to the Raman-gain peak becomes as intense as the pump itself, whereas the anti-Stokes lines are nearly depleted (trace D). At the same time, the pump and the dominant Stokes line exhibit spectral broadening and splitting that are characteristic of SPM and XPM (see Section 7.4). As the pump power is increased further, higher-order Stokes bands are generated through cascaded SRS. At a pump intensity of 1.5 GW/cm2 , the broadened multiple Stokes bands merge and form a supercontinuum extending from 530 to 580 nm (bottom trace). Note that solitons played no role here as GVD was normal (β2 > 0) in the entire spectral range. Supercontinuum generation is discussed further in Section 14.5. With the advent of photonic crystal fibers (PCFs) during the late 1990s (see Section 11.4), it was realized that such fibers could be used to realize FWM with large frequency shifts if they supported multiple modes. In a 2009 experiment, a spectral shift of 140 THz was realized by launching femtosecond pump pulses at 808 nm into the fundamental mode of a 20-cm-long PCF with a core diameter of 10-µm [81]. Both the signal and idler pulses appeared in higher-order modes of the fiber, and their spectra were located at wavelengths near 590 nm and 1250 nm. Moreover, intermodal FWM with such a large spectral shift (about 135 THz) was realized with the pump experiencing normal GVD inside the PCF. The spectral shift was even larger (175 THz) in a 2015 experiment in which the signal and idler wavelengths were close to 550 nm and 1590 nm, respectively, when 800-nm pulses were launched into a 28-cm-long PCF [83]. Also, the FWM efficiency was close to 20%. In a later experiment, cascaded intermodal FWM led to the generation of ultraviolet light at 376 nm when the signal at 538 nm became intense enough that it acted as a pump and created its own FWM sidebands in another mode of a multimode PCF [88]. A GRIN fiber was first used in 1981 for FWM [77]. The phase-matching condition (14.4.1) can be used to find the sideband frequencies in an analytical form for GRIN fibers using the relatively simple expression for the effective mode indices given in Eq. (14.1.14). In the degenerate case in which both pump photons come from the same mode group gp , the waveguide contribution is found to be [86] kW = −mg (π/zp ),

mg = gs + gi − 2gp ,

(14.4.2)

Multimode fibers

where gs and gi are the mode groups for the signal and idler waves, respectively. It follows that kW is negative if mg is a positive integer. The material contribution kM is given in Eq. (10.3.6). If we neglect the relatively small nonlinear contribution in Eq. (14.4.1), the frequency shift 2s satisfies a quadratic equation whose solution is given by [89] 2s

=

6 β4

 −β2 +

 β22

+ mg

πβ4

3zp

 .

(14.4.3)

When β4 is negligible, the shift from the pump frequency is given by a remarkably  simple expression: s = ± mg π/(β2 zp ) for the signal and idler sidebands. Here the integer mg > 0 is restricted to take only even values because of a constraint set by the conservation of angular momentum [86]. Even with this restriction, FWM can occur for several mode combinations when the pump propagates in the LP01 mode (gp = 1). For example, mg = 2 for the two (gs , gi ) combinations (1,3) and (3,1). Thus, one of the FWM sidebands must be generated in the LP02 mode belonging to the third group. The sideband frequencies in Eq. (14.3.7) found in the context of modulation instability coincide with the preceding s values because the two have the same origin (periodic self-imaging). However, the analysis of Section 14.3.1 assumed that the GRIN fiber supported a large number of modes and that the pump beam excited several of these modes simultaneously. The intermodal FWM requires that the pump propagate in a single mode or at most in two distinct modes. In a 2017 experiment, a short, 1-m-long, GRIN fiber was used to avoid random mode coupling. Its core diameter of 22 µm was chosen to ensure that it supported only a few mode groups [87]. The pump was at 1064 nm in the form of a 400-ps pulse train, and its spatial size was controlled to ensure that it excited mostly the LP01 mode. Fig. 14.14 shows the observed output spectra at four peak power levels ranging from 44 to 80 kW. As seen there, spontaneous intermodal FWM generated many sidebands, covering a wide spectral range from 400 to 1600 nm. The SRS also played a role at such high power levels. The four sidebands close to the pump result from non-degenerate FWM (pump in two different modes) and are affected by SRS. The most interesting feature of Fig. 14.14 is the narrow spectral line at 625 nm at the 44-kW pump level. This line is due to intermodal FWM, and its frequency shift of 181 THz corresponds to the choice mg = 1 in Eq. (14.4.2); its partner sideband lies in the mid-infrared region beyond 2.5 µm. As the pump power is increased, the 625-nm feature broadens, becomes more intense, and eventually generates its own set of FWM sidebands acting as a pump. Their frequencies can also be predicted from Eq. (14.4.2) by using gp = 1 because this wave was generated in the LP01 mode. Even though the 625-nm line can also be generated in the second mode group, the efficiency of that FWM process, governed by the extent of mode overlaps, is relatively small [86]. The beam profiles recorded at 433, 473, and 625 nm agree with this interpretation [87].

655

656

Nonlinear Fiber Optics

Figure 14.14 Measured spectra at the end of a 1-m-long GRIN fiber at four peak power levels of 1064-nm input pulses. The 625-nm peak acts as a pump at high power levels and creates its own FWM sidebands that extend in the UV region. (After Ref. [87]; ©2017 OSA.)

In a 2018 experiment, a 1-m-long, GRIN fiber, whose 50-µm core supported a large number of modes, was used to observe intermodal FWM [89]. The spatial extent of the 1064-nm pump in the form of a 400-ps pulse train was controlled to ensure that its power was coupled mostly into the LP01 mode. The observed output spectra at peak powers ranging from 20 to 70 kW were qualitatively similar to those in Fig. 14.14, with multiple sidebands covering a spectral range from 400 to 2400 nm. The 625-nm peak seen in this figure moved to near 809 m because of a larger core size of the fiber. Since its idler sideband was located near 1555 nm, the FWM process could be seeded by injecting light at this wavelength from a CW diode laser. Seeding not only amplified the 1555-nm band (as expected) but also created new spectral peaks in the visible and infrared regions. In essence, the GRIN fiber was used as a parametric amplifier in the seeded mode of operation. Multimode parametric amplifiers require long fibers, and lengths ∼ 1 km were used in several studies [91–94]. A 5-km-long fiber supporting three spatial modes was used in a 2013 experiment to observe intermodal FWM [37]. The two pumps and the signal were launched into distinct fiber modes, and an idler wave was observed in the spectrum in two different pumping configurations. The FWM efficiency was relatively low (< 1%) in this experiment. The results could be explained using the coupled set in Eq. (14.2.20) that includes random linear mode coupling along the fiber length [92]. In a later experiment, the idler power was found to fluctuate considerably with the signal wavelength when two pumps at 1541 and 1542 nm wavelengths were launched into the LP11 mode [93]. A specific mode-coupling model based on fiber’s random birefringence was employed in another study to understand the process of parametric amplification [94]. The FWM efficiency depends on the relative magnitudes of the coherence length LC (related to random rotations of the fast and slow axes) and the beat length LB related to the magnitude of the birefringence. The so-called Manakov region

Multimode fibers

can be approached when LC is smaller than the shortest LB , resulting in better FWM efficiency. The idler power at the fiber output can exceed the injected signal power by up to a factor of 100 for fiber lengths > 1.5 km and pump power levels ∼ 0.2 W.

14.4.2 Intermodal SRS As discussed in Chapter 8, a Stokes pulse is generated though SRS when the pump pulse is intense enough to exceed the Raman threshold. Both pulses belong to the same mode in single-mode fibers, but they travel at different speeds because their spectra are shifted by 13.2 THz (as dictated by the Raman-gain spectrum). SRS in multimode fibers has also been studied extensively [95–108]. Such fibers offer the possibility that the Stokes pulse may appear in a mode different from that of the pump pulse but the two pulses propagate at nearly the same speeds inside the fiber. SRS in a higher-order mode of a step-index fiber (core diameter 100 µm, length 7.5 m) was observed as early as 1987 by launching 25-ps pump pulses into the fiber at the 532-nm wavelength such that pump energy was distributed among several modes of the fiber [97]. At energy levels below the SRS threshold (< 1 nJ), the output beam covered the entire fiber core and exhibited a speckle pattern resulting from multimode interference. However, at the energy level of 10 nJ, a narrow ring appeared within the speckle pattern. The 11-µm-diameter ring represented the spatial pattern of a higherorder mode in which the Stokes pulse was created through SRS at the Raman-shifted wavelength of 550 nm. This Raman process was relatively efficient because the ring contained about 50% of the input energy. It was argued that self-focusing induced by the Kerr nonlinearity played a role by creating a spatially varying index profile, but other explanations are also possible [100]. When an optical filter is used to block the pump, improved quality of the output beam realized through SRS is referred to as Ramanbeam cleanup [109]. Indeed, the use of GRIN fibers for this purpose has attracted considerable attention [105]. For sufficiently intense pump pulses, the Stokes pulse (red-shifted by 13.2 THz in silica fibers) may become intense enough that it acts as a pump and creates its own Stokes pulse that is red-shifted again by the same amount. This process can occur multiple times and is referred to as cascaded SRS; it results in the formation of new pulses at longer and longer wavelengths. In a 1988 experiment, 150-ps pump pulses at 1064 nm were injected into a 50-m-long GRIN fiber with a core diameter of 38 µm [98]. The output spectrum exhibited multiple spectral bands extending up to 1700 nm, each belonging to a different Stokes pulse and red-shifted by about 13.2 THz from the preceding band. The pulses whose wavelengths exceeded 1300 nm were narrower than the pump pulse because they experienced anomalous GVD and formed solitons. An investigation of cascaded SRS was carried out in 1992 by launching 3-ns pulses (at 585 nm) into a 30-m-long GRIN fiber with a core diameter of 50 µm [99]. A grating was employed to separate the pump and Stokes pulses and to record their spatial

657

658

Nonlinear Fiber Optics

Figure 14.15 (A) Cascaded SRS peaks observed at the end of a 1-km-long GRIN fiber by launching 8-ns pulses at 523 nm with 22 kW peak power; (B) spectrum in the infrared region at higher pulse energies. Spectral power of the data in part (A) plotted on a log scale (right) as a function of the frequency shift. (After Ref. [107]; ©2013 AIP.)

Figure 14.16 Measured spatial profiles using a CCD-based beam profiler. Image (A) was obtained without any filter. The other 4 images were obtained using color filters centered at (B) 610 nm, (C) 700 nm, (D) 770 nm, and (E) 890 nm. (After Ref. [107]; ©2013 AIP.)

patterns at the fiber output. With careful adjustment of the launch conditions, four different Stokes pulses were found to propagate in different low-order modes of the fiber. Cascaded SRS over a much wider wavelength range was observed in a 2013 experiment by launching 8-ns pulses at 523 nm into a 1-km-long GRIN fiber with a 50-µm core diameter [107]. As seen in Fig. 14.15, the output covered a large wavelength range with many cascaded Stokes peaks that were separated by about 13 THz when plotted on a frequency scale. The spectrum also extended into the infrared region up to 1750 nm with a spectral dip around 1300 nm. The intermodal nature of SRS was verified through spatial profiles of different SRS peaks. The pattern (A) in Fig. 14.16 was obtained without any optical filter. Remaining four patterns were obtained by using color filters centered at wavelengths of the four specific peaks in Fig. 14.15; they correspond to different modes of the GRIN fiber used in the experiment. Two important questions are: (i) why a pump whose power may be distributed over many fiber modes ends up creating a Stokes pulse that propagates in a specific fiber mode, and (ii) how that mode is selected as the SRS process builds up from noise. In

Multimode fibers

general, one must consider amplification of noise in each fiber mode through SRS by all fiber modes in which pump power resides. In a simple approach, we can generalize Eq. (8.2.3) obtained for Raman amplification in single-mode fibers to the multimode case as [99]

Gm = exp gR (P0 /Aeff )L





f mn pn ,

(14.4.4)

n

where Gm is the amplification factor for the mth fiber mode, gR is the Raman gain coefficient at the pump wavelength, L is the fiber length, pn is the fraction of total pump power P0 in the nth mode, and f mn is the overlap factor obtained using Eq. (14.2.10):



f mn = fmmnn = Aeff

|Fm (x, y)|2 |Fn (x, y)|2 dx dy.

(14.4.5)

As before, Aeff is the effective mode area of the fundamental mode. The overlap factors depend on the fiber design and can be easily calculated for any fiber. Even though spontaneous Raman noise is amplified in all fiber modes, the mode for which Gm is the largest wins the race because of the exponential nature of the amplification process. Eq. (14.4.4) has been used to compare relative magnitudes of the modal Raman gain for different fibers [104]. It has also been used to explain why the Raman-beam cleanup occurs in GRIN fibers but not in step-index fibers [105]. When the pump is in the form of a short optical pulse, the effects of fiber dispersion begin to play an important role. As the group velocity in a fiber depends on the wavelength of a pulse, the Raman and Stokes pulses do not travel at the same speed even inside a single-mode fiber. As discussed in Section 8.3.3, energy transfer between the pump and Stokes pulses is reduced considerably because they separate from each other within a short distance when pump pulses are shorter than 10 ps. Indeed, no Stokes pulse is generated through SRS for femtosecond pump pulses propagating inside singlemode fibers. In its place, the spectrum of pump pulse undergoes a continuous red-shift through intrapulse Raman scattering (see Section 12.2). The situation is different in a multimode fiber because the speed of a pulse also depends on the mode in which it is traveling. As a result, a femtosecond pump pulse, traveling in one mode of the fiber, may undergo intermodal SRS and generate the Stokes pulse in another mode whose group velocity happens to nearly coincide with that of the pump, even if the modal Raman gain Gm is smaller for this mode. Indeed, this phenomenon was observed in an experiment [110] in which a 100-fs pump pulse was launched into the LP0,19 mode of a 12-m-long step-index fiber (core diameter 87 µm), and its energy was transferred to several lower-order LP0,m modes in a cascaded fashion such that m = 15 at the end of the fiber. The dynamical details of this process require multimode numerical simulations because one must consider many dispersive and nonlinear effects, including intrapulse Raman scattering that red-shifts the spectrum

659

660

Nonlinear Fiber Optics

Figure 14.17 Numerically simulated spectra at the end of a 20-m-long step-index fiber for pulses of different widths, launched initially into the LP0,19 mode at the 1100-nm wavelength. Pulse width varies from 60 to 100 fs in steps of 5 fs. (Courtesy of Aku Antikainen)

of the pulse before it jumps to the next mode [111]. As an example, Fig. 14.17 shows the numerically simulated spectra in the same step-index fiber using pump-pulse widths in the range 60 to 100 fs. Input pulses at 1100 nm excite the LP0,19 mode and propagate a fundamental soliton inside the fiber. For the 100-fs pump pulse, three spectral peaks belong to three different fiber modes. They originate from an intermodal SRS process in which energy of a pulse is transferred to both the LP1,18 and LP0,18 modes depending on how closely their group velocities match with that of the LP0,19 mode. For pump pulses shorter than 80 fs, which red-shift faster through intrapulse Raman scattering, most of the energy is transferred directly to the LP0,18 mode. This behavior can be understood from Fig. 14.18 where the group index (ng = c /vg ) is shown as a function of wavelength for the three modes involved in this intermodal process. As we saw in Section 8.2.3, SRS can be used for Raman amplification if a Stokes pulse is launched into a single-mode fiber together with the pump. The use of multimode fibers for Raman amplification has attracted attention in recent years [112–116]. In the multimode case, pump and Stokes pulses may propagate in the same mode or may be launched into two different modes. Both cases can be studied by solving Eqs. (14.2.15) and (14.2.16), after modifying them to include the Raman gain. One way to do so is by making the Kerr nonlinearity n2 a complex quantity such that its imaginary part is related to the Raman susceptibility [101]. A more accurate approach should follow the analysis in Section 8.3 and modify Eqs. (8.3.1) and (8.3.2) with the

Multimode fibers

Figure 14.18 Effective group index as a function of wavelength for the three modes involved in Fig. 14.17; their shapes are shown in the insets.

appropriate overlap factors to obtain ∂ Ap iβ2p ∂ 2 Ap gp + = iγp [f pp |Ap |2 + (2 − fR )f ss |As |2 ]Ap − f ps |As |2 Ap , (14.4.6) 2 ∂z 2 ∂T 2 ∂ As ∂ As iβ2s ∂ 2 As g s −d + = iγs [f ss |As |2 + (2 − fR )f pp |Ap |2 ]As + f ps |Ap |2 As , (14.4.7) ∂z ∂T 2 ∂T 2 2

where T = t − z/vgp and d = β1p − β1s is the DGD between the modes p and s in which the pump and Stokes pulses are propagating. The parameter d accounts for the group-velocity mismatch between the pump and Stokes pulses. The Raman-gain coefficients are different for the two pulses because of their different carrier frequencies. The preceding equations can be used to study Raman amplification for any mode pairs by employing the corresponding overlap factors. They still need to be modified for femtosecond optical pulses by including higher-order dispersion terms and intrapulse Raman scattering through the Raman response function. An important conclusion can be drawn by comparing Eqs. (14.4.6) and (14.4.7) with Eqs. (8.3.1) and (8.3.2) of Section 8.3 for single-mode fibers. In particular, we can understand the detrimental role played by the DGD from Fig. 8.11, where only a fraction of pump energy is transferred to the Stokes pulse because of the walk-off effects that separate it from the pump pulse within a short distance. In the case of multimode fibers, the parameter d can be close to zero if the GVD of the fiber helps in canceling the DGD between the two fiber modes in which the pump and Stokes pulses are propagating (see Fig. 14.17). As a result, entire pump energy can be transferred to the Stokes pulse for this specific mode pair, resulting in an efficient amplification process.

661

662

Nonlinear Fiber Optics

Eqs. (14.4.6) and (14.4.7) are not suitable for fibers long enough that linear mode coupling cannot be ignored. Theory of Raman amplification in multimode fibers with random mode coupling was developed in 2013 by considering the Raman gain for both co-polarized and cross-polarized pump and Stokes pulses [112]. It can be simplified considerably by recalling that fiber modes can be divided into multiple groups of nearly degenerate modes. One can then assume that modes within each group are strongly coupled but modes in two different groups do not couple linearly, and use it to obtain a set of equations describing how the pump and Stokes powers evolve in different mode groups. Such an approach has been used to optimize the gain of a multimode Raman amplifier [113]. Raman amplification is often used to compensate for fiber losses in fiber-optic communication systems [69]. In this case, the same fiber that is used to transmit information is also used for distributed Raman amplification by pumping it periodically, resulting in long amplifier lengths (60 to 100 km) over which random linear mode coupling cannot be ignored. In a 2016 experiment, this scheme was used to transmit data over 1050 km of a six-mode fiber by pumping it bidirectionally every 70 km [114]. The pump power was launched into the LP11 mode group (four-fold degenerate) because the same Raman gain could be realized in all six fiber modes in this configuration. The experiment showed that fiber losses could be fully compensated for all modes through Raman amplification inside a six-mode fiber.

14.4.3 Intermodal SBS As discussed in Chapter 9, a backward propagating Stokes wave is generated though SBS in single-mode fibers when the pump is intense enough to exceed the Brillouin threshold. Its frequency is down-shifted from that of the pump by about 11 GHz when pump wavelength is near 1550 nm. This Brillouin shift corresponds to the frequency of an acoustic wave that accompanies the SBS process. The gain spectrum of SBS can exhibit multiple narrow-bandwidth peaks that correspond to different acoustic modes of the fiber [117]. In the case of a multimode fiber, launched pump power is generally divided into multiple modes, all of which can participate in generating the Stokes through excitation of multiple acoustic modes of the fiber [118–128]. This affects the pump power level at which the SBS threshold is reached. A theoretical model was developed in 2001; it predicted that the SBS threshold increases in multimode fibers by a factor that depends on the numerical aperture, but not on the core diameter of the fiber [118]. In a non-modal approach, increase in the SBS threshold was attributed to inhomogeneous broadening of the Brillouin-gain spectrum. A simple model was able to fit the experimental dependence of the gain bandwidth on fiber’s numerical aperture [119]. This model was later extended to predict the SBS threshold of GRIN fibers [121]. Similar to the case of single-mode fibers, SBS threshold depends inversely on the width of pump pulses.

Multimode fibers

In a 2003 study the threshold power increased by a factor of eight as the pulse width decreased from 80 to 20 ns [120]. In a more accurate calculation of the SBS process, the analysis of Section 9.4 is extended to the case of multimode fibers [126–128]. In this approach, Eq. (9.4.1) is solved by using modal expansion for the pump and Stokes fields in the form E(r, t) = xˆ Re



Fk (x, y, ωp )Ak (z, t) exp[iβk (ωp )z − iωp t]

k

+





Fk (x, y, ωs )Bk (z, t) exp[−iβk (ωs )z − iωs t] ,

(14.4.8)

k

where the sum extends over all fiber modes with the propagation constants βk (ω). A similar expansion of the material density in terms of the acoustic modes of the fiber takes the form [127] kl 



N

ρ  (r, t) = Re

Wklu (x, y)Qklu (z, t) exp(ikA z − it) ,

(14.4.9)

k,l u=1

where Wklu (x, y) is the spatial profile of the acoustic mode with the amplitude Qklu , frequency  = ωp − ωs , and propagation constant kA . For each pair of the pump and Stokes modes, several acoustic modes may contribute. This is indicated by the sum over u whose upper limit Nkl denotes the number of acoustic modes involved in that specific pair. Clearly, such a modal analysis of SBS is cumbersome. However, if we assume CW pumping, it eventually leads to the coupled power equations in the form [127] kl d |Ak |2 = Gklu |Ak |2 |Bl |2 , dz l u=1

(14.4.10)

kl d|Bl |2 − = Gklu |Ak |2 |Bl |2 , dz k u=1

(14.4.11)

N

N

where the intermodal Brillouin gain Gklu , resulting from a specific acoustic mode, is given by Gklu () =

B ( /2)2 |f |2 gklu klu klu . ( − klu )2 + (klu /2)2

(14.4.12)

Here klu is the frequency and klu is the damping rate of the acoustic mode with the B , and f peak Brillouin gain gklu klu is the overlap factor defined similar to Eq. (14.2.10) and

663

664

Nonlinear Fiber Optics

is given by



fklu = Aeff

∗ Fk (x, y, ωp )Fl∗ (x, y, ωs )Wklu (x, y) dx dy.

(14.4.13)

The physical meaning of Eqs. (14.4.10) and (14.4.11) is quite clear. They show that the pumps in all fiber modes contribute in building up the Stokes signal in any fiber mode, but the amount of SBS gain depends on the acoustic modes involved and the overlap factors fklu . If we neglect pump depletion, the net Brillouin gain for the Stokes propagating in a specific fiber mode is given by glB () =

Nkl

Gklu ()(Pk /P0 ),

(14.4.14)

k u=1

where Pk = |Ak |2 is the pump power in the kth mode and P0 is the total pump power. Since each Gklu has a Lorentzian spectral profile, glB is a superposition of several such profiles, with the fraction of power in that mode acting as a weighting factor. Even when the pump is propagating in only one mode, the Brillouin-gain spectrum may have more than one peak resulting from the sum over u in Eq. (14.4.14). When a pump beam is launched into the fiber, the Stokes signal builds up from noise and the overlap factors fklu play an important role in selecting the mode in which the Stokes wave propagates in the backward direction. It turns out that fklu is the largest when l in Eq. (14.4.14) corresponds to the LP01 mode of the fiber, irrespective of how the pump power is distributed among the fiber modes. As the Stokes power builds up exponentially from a noise seed, the mode with the largest gain wins the competition and carries the Stokes wave generated through SBS. It was noticed in several experiments that SBS occurring inside multimode fibers can be used to cleanup the spatial profile at the fiber’s output end [129,130]. SBS can also be used to combine several lowquality pump beams into one high-quality Stokes beam [131]. Indeed, SBS-induced beam cleanup and beam combining are important SBS applications. Another useful feature of SBS is that the Stokes beam is often phase-conjugated, i.e., its spatial phase is negative of that of the pump [132]. This feature has been known since 1978 for SBS occurring inside multimode waveguides [133]. The Brillouin gain given in Eq. (14.4.14) has been measured through amplification of an input Stokes signal inside fibers supporting a few modes. A 2013 experiment employed a 100-m-long fiber with an elliptic core that supported only the LP01 and LP11 modes [124]. The pump power from a 1550-nm laser was coupled selectively into one of these modes and the SBS gain spectra were recorded for all four possible mode combinations. A two-peak spectrum with 30-MHz spacing was recorded only when both the pump and Stokes were in the LPe11 mode. In a later experiment, a circular-core fiber (a = 5.09 µm,  = 1.15%) supporting four mode groups was used and gain spectra

Multimode fibers

Figure 14.19 Measured intermodal gain spectra for SBS occurring inside a step-index fiber supporting LP01 , LP11 , LP21 and LP01 mode groups. Thin lines show fitting three Lorentzian profiles used to fit the data. (After Ref. [125]; ©2013 OSA.)

were recorded for all mode combinations [125]. Fig. 14.19 shows the measured spectra in four cases. As seen there, spectra exhibited more than one peak and could be fitted by superimposing three Lorentzian profiles whose center frequencies varied from 10.25 to 10.5 GHz with bandwidths ranging from 33 to 39 MHz. Numerical estimates show that the Brillouin shift does not vary much for intermodal SBS processes involving Stokes generation into the LP01 mode from a pump in the LP0m mode, where m is an integer [123]. This is the reason why a Gaussian pump beam that couples its power into multiple LP0m modes ends up creating a single Stokes wave confined to the LP01 mode.

14.5. Spatio-temporal dynamics In this section we focus on the dynamic interplay between the spatial and temporal features when a pulsed optical beam is launched into a multimode fiber. As mentioned earlier, this problem requites solving Eq. (14.2.23), which couples the spatial and temporal evolution through the Kerr nonlinearity. This is so because the same Kerr coefficient n2 that leads to SPM of optical pulses in time also provides self-focusing in the transverse

665

666

Nonlinear Fiber Optics

Figure 14.20 Spatial beam profiles from a 20-cm-long GRIN fiber at six energy levels of 80-fs input pulses; black bar is 13 μm wide. Two images on the right are for 0.9 ns pulses inside a 12-m-long GRIN fiber; black bar is 10 μm wide. (After Ref. [135] ©2016 OSA.)

spatial dimensions. Such spatio-temporal dynamics, studied for bulk Kerr media in the past [58], has also been observed in the context of multimode GRIN fibers [134–137].

14.5.1 Spatial beam cleanup As we saw in the preceding section, both SRS and SBS can be used to cleanup the spatial profile of a pump beam at the output of a multimode fiber. It is thus natural to ask whether the Kerr nonlinearity itself can be used for this purpose, resulting in Kerr-induced self-cleaning. The term ‘beam cleanup’ generally implies that the optical beam exiting at the output end of the fiber has a smooth spatial intensity profile with a single peak. A 20-cm-long GRIN fiber with a 62.5-µm core diameter was used in 2015 for studying Kerr-induced self-cleaning [135]. When 80-fs pulses at 1030 nm were launched into this fiber such that their energy coupled into multiple modes of the fiber, near-field images of the fiber’s output indicated that the spatial pattern changed considerably as pulse energy was increased. Fig. 14.20 shows six images for energies ranging from 0.44 to 46.6 nJ. At the highest energy level, the Kerr nonlinearity leads to a smooth intensity profile with a nearly circular spot. In the time-domain, the pulse at this energy level was only 34 fs wide with a broad spectrum. In another experiment, much longer pulses (width 0.9 ns) at 1064 nm were launched into a 12-m-long GRIN fiber [136]. The 40-µm-width of the input beam was wide enough to excite nearly 200 modes of this fiber. Two images taken at the fiber’s output end are shown on the right in Fig. 14.20. At the low peak power of 3.7 W, nonlinear effects are negligible, and the beam exhibits a speckle pattern. When the peak power of input pulses is increased to 1.2 kW, the pattern becomes smoother and narrower with a single central peak. These results show that the Kerr-induced beam cleanup occurs over a wide range of pulse widths. Also, soliton effects play little role in this process because the fiber’s GVD was normal for most modes at the wavelengths used in the two experiments. The beam’s cleanup indicates intermodal energy transfer. Numerical results

Multimode fibers

Figure 14.21 Output beam diameter as a function of the peak power of 60-ps input pulses launched into a 11-m-long fiber. The images on the top and bottom show spatial patterns for the pump and Stokes beams at few input power levels. (After Ref. [139] ©2018 OSA.)

also show a nonlinear transfer of energy from higher-order modes to the fundamental mode [60]. The use of a GRIN fiber is not essential for observing Kerr self-cleaning. A Ybdoped step-index fiber with the 50-µm core diameter was used in one experiment. It was found that the beam quality improved dramatically beyond a certain peak power of 0.5-ns input pulses [137]. This fiber acted as an amplifier when it was pumped suitably. The self-cleaning threshold was near 40 kW for the unpumped fiber (because of high losses), but it was reduced to below 0.5 W when the fiber was pumped to provide 23-dB gain. A multicore Yb-doped fiber was used in another experiment to show that the spatial shape of the amplified beam can be controlled through the nonlinear effects [138]. The competition between the Kerr- and Raman-induced beam cleaning has also been studied using a microstructured fiber whose 30-µm silica core was surrounded with a holey cladding [139]. The beam diameter at the output of the 11-m-long fiber was measured as a function of the peak power of 60-ps input pulses at 1064 nm. The results are shown in Fig. 14.21 together with the observed beam patterns at several power levels. The pump’s beam diameter shrinks initially due to Kerr self-cleaning, takes a minimum value at the 18-kW peak power where the SRS threshold is reached,

667

668

Nonlinear Fiber Optics

and increases once a fraction of the pump’s energy is transferred to the Stokes beam. The diameter of the Stokes beam reaches a minimum value of about 10 µm at a peak power near 70 W and then begins to increase after this beam generates the second-order Stokes through a cascaded Raman process. The physical mechanism behind spatial beam cleanup occurring inside multimode fibers has two origins. First, self-focusing must play a role because it reduces beam size even in a bulk medium, and it leads to beam collapse if the input power is large enough. In most experiments power levels remained below this collapse but some self-focusing of the beam did occur. Second, different propagation constants of fiber modes lead to spatial beating. The resulting intensity modulations along the fiber length are converted into index modulations by the Kerr nonlinearity. This effect is the strongest in GRIN fibers, where equal spacing of modal propagation constants results in beam-width oscillations with a period of < 1 mm. The Kerr nonlinearity converts these oscillations into periodic variations of the refractive index that acts as a grating along the fiber length (period about 0.5 mm). Such a grating helps in transferring energy from higherorder modes to the fundamental mode. This is the reason why the threshold of Kerr self-cleaning is lower for GRIN fibers.

14.5.2 Supercontinuum generation Although two-mode fibers were considered earlier in the context of fiber’s birefringence [140–142], it was only after 2008 that supercontinuum generation in multimode fibers attracted attention [143–154]. As intrapulse Raman scattering plays an important role in this process (see Section 13.3), the set of multimode NLS equations in Eq. (14.2.20) must be generalized to include the delayed Raman response of optical fibers. For this purpose we extend the vector theory of Section 8.5.1 to the multimode case [30,112]. Only the nonlinear term in Eq. (14.2.20), containing the triple sum changes. ˆ it takes the form If we denote this term by N, Nˆ =

∞  iγ iβmnpq z    fmnpq e 2Aq (t) R(t )[AH p (t − t )An (t − t )]dt 3 n p q 0 + A∗p (t)



∞

0





R(t )[ATn (t − t )Aq (t − t )]e2iω0 t dt ,

(14.5.1)

where R(t) is the nonlinear response function (see Appendix B). The form of R(t) reduces to that in Eq. (2.3.36) if we ignore the cross-polarization Raman gain. One more change is often made in Eq. (14.2.20) before solving it numerically. The exponential term containing the phase-mismatch βmnpq in Eq. (14.5.1) oscillates rapidly along the fiber. It can be removed by defining Bm = Am exp[i(β0m − β0r )z],

T = t − β1r z,

(14.5.2)

Multimode fibers

where β0r and β1r serve as reference values. In practice, the fundamental mode of the fiber is used for setting these reference values. In terms of Bm , Eq. (14.2.20) takes the form ∞ ∂ Bm ∂ Bm ik βkm ∂ k Bm + (β1m − β1r ) − = i (β − β ) B + i κmn Bn ei(β0n −β0m )z 0m 0r m ∂z ∂T 2 ∂T k n k=2

∞  iγ    + fmnpq 2Bq (t) R(t )[BH p (t − t )Bn (t − t )]dt

3

n

p

q

+ B∗p (t)

0



∞

0





R(t )[BTn (t − t )Bq (t − t )]e2iω0 t dt ,

(14.5.3)

where we also included higher-order dispersion terms through a sum in the third term. This equation incorporates all important dispersive and nonlinear effects and is appropriate for studying supercontinuum generation in multimode fibers. If a fiber is short enough that linear mode coupling is not a major factor, one can set κmn = 0 in Eq. (14.5.3) to simplify it. The equation resulting from this simplification has been solved numerically to study the nonlinear intermodal effects in few-mode fibers. Typical simulations consider femtosecond pulses launched in the anomalous-GVD region of the fiber such that their energy is carried by the fundamental mode alone, or by the first two or three modes. When only the fundamental mode is excited, energy is transferred to the orthogonally polarized version of this mode but not to higher-order modes [146]. When more than one mode is exited initially, other LP0m modes may also be excited, but the fraction of energy transferred to them remains relatively small [143]. Experiments have shown that when a femtosecond pulse is launched into a fewmode optical fiber, some of the spectral bands within the supercontinuum propagate in higher-order modes of the fiber [144–147]. In all cases, the soliton effects play a dominant role, and supercontinuum generation evolves in a fashion similar to that in single-mode fibers. Starting in 2013, multimode GRIN fibers (core diameter 50 µm) were used in several experiments. The input beam diameter was kept large enough to excite multiple modes of the fiber simultaneously. In one experiment, 0.5-ps pulses at 1550 nm were launched into a 1-m-long GRIN fiber [134]. The output was in the form of a supercontinuum extending from violet to mid-infrared regions. Moreover, it exhibited multiple peaks in the visible region that were not equally spaced. The results are shown in Fig. 14.22 together with the numerical simulations [148]. The spectral peaks seen there have the same origin as those in Fig. 14.7. They originate from periodic spatial oscillations of the beam width inside GRIN fibers discussed in Section 14.2.4 and constitute an example of spatio-temporal coupling in such fibers [41]. Physically, narrow spectral peaks in Fig. 14.22 result from a Kerr-induced index grating (created by periodic oscillations of the peak intensity along the fiber) that helps

669

670

Nonlinear Fiber Optics

Figure 14.22 Numerical (top) and experimental (bottom) spectra at the end of a GRIN fiber for 0.5-ps input pulses launched at 1550 nm. (After Ref. [148] ©2015 APS.)

in phase matching of dispersive waves (see Section 12.1.2). Their frequencies can be calculated after modifying the phase-matching condition given in Eq. (12.1.6) as β2

2

2 +

β3

6

3 +

β4

24

4 = δβ1  +

c0 2π m γ Ps + , 2 zp

(14.5.4)

where the last term represents the contribution of the Kerr-induced index grating. The integer m can take any positive or negative value. As before, c0 = 1/Cg with Cg given in Eq. (14.2.26). If we keep only the β2 term and assume δβ1 = 0, the frequency shift  can be calculated from 2 =

1/Cg 4π m + , β2 zp β2 LNL

(14.5.5)

compared where LNL = (γ Ps )−1 is the nonlinear length. In practice, zp is much smaller  to LNL , and the first term dominates. In this approximation, T0 = ± 4π |m|q, where q = LD /zp . The effects of higher-order dispersion can be included through Eq. (14.5.4). The results in Fig. 14.22 were obtained using femtosecond pulses at 1550 nm. In two 2016 experiments [150,151], much longer pulses (close to 1 ns) from a Nd:YAG laser operating at 1064 nm were launched into GRIN fibers (core diameter 50 µm) of lengths up to 30 m. Fig. 14.23 shows the observed supercontinua for 12 and 30 m long fibers [151]. The supercontinuum extends from 500 to 2000 nm in the shorter fiber, with multiple peaks in the visible reason. It extends beyond 2500 nm in the longer fiber and becomes smoother as the peaks near 600 nm disappear. The peak power of Q-switched pulses exceeded 30 kW at the power levels used in this experiment. The important question is how the pulse spectrum evolves inside the fiber. This evolution can be easily simulated numerically but is difficult to observe experimentally.

Multimode fibers

Figure 14.23 Experimental spectra observed at the end of 12 m (top) and 30 m (bottom) long GRIN fibers using 0.9-ps input pulses at 1064 nm. Symbols G and G’ mark the locations of modulationinstability sidebands. (After Ref. [151] ©2016 OSA.)

One way is to employ multiple fibers of different lengths but it is difficult to maintain the same experimental conditions, including coupling to the fiber. In a better approach, a single fiber is used but its length is cut back in discrete steps, while recording output spectra each time. The results show that the spectrum broadens initially through SPM and SRS around the pump wavelength, but soon several widely separated sidebands appear on both sides of the pump in the visible and infrared regions [150,151]. Their origin lies in modulation instability discussed in Section 14.3.1 that sets in because of the relatively wide pump pulses used in the experiments. The frequencies of these sidebands agree with the predictions of Eq. (14.3.7). The sidebands nearest to the pump are shifted by about 125 THz from the pump’s central frequency (close to 285 THz) and appear at wavelengths near 730 nm and 1900 nm (marked as G and G’ in Fig. 14.23). These bands interact with each other and with the pump through XPM and FWM, and are also intense enough to create their own Stokes peaks through SRS. The net results is that more and more frequencies are generated nonlinearly and a supercontinuum begins to form if the GRIN fiber is long enough. As seen in Fig. 14.23, a broader and smoother supercontinuum is observed when the GRIN fiber is 30 m long. It should be clear from the preceding discussion that the physical mechanism behind the supercontinuum generation in GRIN fibers is quite different from those discussed in Sections 13.4 and 13.4, where anomalous GVD was required at the pump wavelengths for soliton effects to play a dominant role. The spectra in Fig. 14.23 were obtained using relatively long pulses (close to 1 ns) at a wavelength of 1064 nm, where GVD is normal for the low-order modes. In this situation, modulation instability occurring in the normal-GVD regime (because of periodic self-imaging of the beam) plays the dominant role. To quantify the spectro-temporal dynamics of the underlying physical process, the spectrogram shown in Fig. 14.24 was constructed experimentally at the end

671

672

Nonlinear Fiber Optics

Figure 14.24 Experimental spectrum (top) and its corresponding spectrogram at the end of a 6-mlong GRIN fiber. Symbols mark the location of various spectral peaks; P corresponds to the input pump pulse. (After Ref. [151] ©2016 OSA.)

of a 6-m-long fiber by dispersing its output with a grating. It shows the temporal extent of different spectral peaks marked by the letters on the top. One can understand the results in Fig. 14.24 as follows. Peaks A to G correspond to the modulation-instability sidebands in the visible region. They represent pulses whose temporal extent is in the range of 100 to 200 ps and whose positions are shifted from the center of input pulse (T = 0) because of the wavelength dependence of the group velocity. The peak P corresponds to the pump pulse itself. Clearly, pulse wings remain nearly intact because of their relatively low intensity, but energy from the central part has shifted to three Stokes bands (R, S, and L) at longer wavelengths. These peaks represent a cascaded SRS process. Energy from the central part of the pump pulse is initially transferred to the first Stokes band, which creates the second and third sidebands in a cascaded fashion. Notice also that the G peak has created its own Stokes band. One can conclude that SRS plays an important role in producing the supercontinua seen in Fig. 14.23. To identify the role played by Kerr or Raman beam cleanup, it is important to consider spatial shapes of different spectral components. This can be easily done experimentally by imaging the output, after passing it through narrow-band spectral filters centered at different wavelengths [150,151]. The results show that the mode profiles at all wavelengths are relatively smooth (bell shaped), with a central bright spot and without any speckle-like structure. The Kerr beam cleanup occurring at the pump wavelength appears to apply across the entire bandwidth of the supercontinuum. These results suggest that the GRIN fibers can be used to make broadband sources that are spatially coherent. Because of their relatively large core area, such fibers can be pumped with high pump powers, resulting in a much brighter broadband source.

Multimode fibers

Figure 14.25 Top: simulated spectra out of a 20-cm-long GRIN fiber under the on-axis (gray) and offaxis (black) conditions shown in the two insets. Bottom: experimental spectra under the same two launch conditions out of a 1-m-long GRIN fiber in the infrared (left) and visible (right) regions. Spatial profiles are shown as insets with a 20-μm scale bar. (After Ref. [152] ©2017 OSA.)

In a 2017 study, 0.5-ps-wide pump pulses were launched into a GRIN fiber at a wavelength of 1550 nm at which the fiber’s GVD was anomalous [152]. The launch conditions were varied by varying the center of the input Gaussian beam with respect to the core center. Only the radially symmetric (LP0m ) modes are exited when these two are perfectly aligned. Fig. 14.25 compares the numerically simulated spectra at the end of a 20-cm-long fiber under the on-axis and off-axis launch conditions. Both spectra extend over 500 THz, covering a 0.5–2.5 µm wavelength range, but they differ considerably in the visible region. In contrast with the spectra in Fig. 14.23, the soliton effects play an important role in shaping these spectra. In particular, they are far from being spectrally uniform and exhibit a dominant broad peak centered at 1600 nm. This peak results from soliton fission that creates an ultrashort Raman soliton (width close to 10 fs), whose spectrum shifts toward the red side through intrapulse Raman scattering. The initial alignment of the input beam also affects the output spectrum. The broad peak of the Raman soliton shifts more toward the red side in the on-axis case. Also, the modulation-instability sidebands in the visible region are well separated in the case off-axis excitation but they merge together in the on-axis case. The lower part in Fig. 14.25 shows the experimental spectra in the infrared (left) and visible (right) regions at the end of a 1-m-long GRIN fiber for both the on-axis and off-axis launch conditions. The supercontinuum is wider and more uniform in the on-axis case, features that are in qualitative agreement with the numerical results. The

673

674

Nonlinear Fiber Optics

spatial profiles of the beam (obtained without any filter) show that most of the pulse energy resides in the fundamental mode in the case of on-axis excitation. In the off-axis case, pulse energy remains distributed over multiple modes, resulting in a much wider and non-uniform spatial pattern.

14.6. Multicore fibers Multicore fibers whose cores support a single spatial mode can be thought of as an array of single-mode fibers. Nonlinear effects in such fiber arrays have been analyzed since the early 1990s [155–163]. With the advent of multicore fibers and their potential applications in space-division multiplexing, considerable work has been done since 2010 to further explore the nonlinear phenomena in such fibers [164–175]. We can use Eq. (14.2.20) for multicore fibers with some modifications. First, the Jones vector Am and the dispersion parameters β1m and β2m corresponds to the mth core. Second, the coupling coefficient κmn represents linear coupling between the modes in the cores identified by the integers m and n, a non-random quantity whose magnitude depends on the spacing between these two cores. Third, the overlap factors fmnpq are still defined by Eq. (14.2.11), but the modal distributions appearing there are centered at the location of individual cores. As a result, all overlap factors are relatively small except of the type fpppp corresponding to the SPM effect. If we assume that all cores are identical, only the x polarized modes are excited, the fiber has no birefringence, and each core is coupled only to its two nearest neighbors, Eq. (14.2.20) is reduced to   ∂ Am iβ2 ∂ 2 Am + = i κ A + A + i γ | Am |2 A m , m + 1 m − 1 ∂z 2 ∂T 2

(14.6.1)

where we introduced the reduced time as T = t − β1 z, used fmmmm = 1, and dropped the subscript m from β0m , β1m , and β2m as all cores are assumed to be identical. Eq. (14.6.1) is appropriate when all cores of a multicore fiber lie along a circle with equal spacing, as shown in Fig. 14.26(A). The parameter κ represents linear coupling between any two neighboring cores. If the cores are so far apart that κ ≈ 0, Eq. (14.6.1) reduces to the standard NLS equation and shows that each core of the fiber acts as a single-mode waveguide that supports soliton formation if a pulse experiences anomalous GVD inside it. The important question is how the linear coupling affects such solitons. To answer it, we introduce the soliton units and write Eq. (14.6.1) in the form i √

  ∂ um 1 ∂ 2 um 2 + | u | u + K u + u + m m m+1 m−1 = 0, ∂ξ 2 ∂τ 2

(14.6.2)

where um = γ LD Am and K = κ LD represents the linear coupling. The specific case of three-core fibers is interesting because the three coupled NLS equations permit analyt-

Multimode fibers

Figure 14.26 (A) A circular fiber array. (B) Discrete spatial solitons forming for three values of k when a CW beam excites five of the cores initially. (From Ref. [158]; ©1994 OSA.)

ical solution in both the CW and pulsed cases [155–157]. In particular, one can find soliton triplets that propagate without changing their shapes. Let us consider the CW case first and neglect the time-derivative term in Eq. (14.6.2). If we launch CW light into one core of a multicore fiber, some power will be transferred to the neighboring cores through linear coupling. The question we ask is how the power is distributed at the fiber output. Fig. 14.26(B) shows the power transfer along a 101-core array by solving Eq. (14.6.2) numerically for a CW beam launched initially with the amplitude [158] √

um = Ka sech[a/ 2(m − mc )] exp[−ik(m − mc )],

(14.6.3)

where mc = 51, a2 = 1.1, and five cores are exited initially at ξ = 0. The parameter k determines the initial phase difference among the excited cores. The beam remains confined to the same five cores when k = 0. However, when k = 0, the power is transferred to the neighboring cores as the CW beam propagates down the array. The remarkable feature is that the initial shape of the spatial envelope is maintained during this process. For this reason, such a structure is referred to as a discrete soliton [161]. It should be stressed that this is a kind of spatial soliton since the temporal term in Eq. (14.6.2) is not present for CW propagation. Such discrete solitons were studied extensively during the 1990s; see Chapter 11 of Ref. [58]. We have seen in Section 14.3 that modulation instability destabilizes the CW solution in a multimode fiber, and the same thing can occur in a multicore fiber. The case of two fibers was analyzed as early as 1989 in the context of a directional coupler [176]. The case of three-core fibers can also be treated analytically by using Eq. (14.6.1) with m = 1 to 3 after setting A0 = A3 and A4 = A1 . The results show that such a fiber supports three CW states with different features of modulation instability in each case [172]. The

675

676

Nonlinear Fiber Optics

symmetric CW state has equal powers in each core with the amplitudes 

A1 = A2 = A3 = P /3 exp[i(γ P /3 + 2κ)z],

(14.6.4)

where P is the total power in all three cores. It becomes unstable only when the GVD is anomalous, and the gain spectrum of modulation instability exhibits features qualitatively similar to those of a single-mode fiber (see Section 5.1). The second CW solution of a three-core fiber is A1 = 0,



A2 = −A3 = P /2 exp[i(γ P /2 − κ)z].

(14.6.5)

It becomes unstable even in the case of normal GVD, and its gain spectrum exhibits two peaks on each side of the pump frequency. The second pair of sidebands has a relatively low gain but its shift from the pump frequency can exceed 50 THz at high pump powers. The third CW solution of a three-core fiber has amplitudes 

A1 = P1 e i ψ ,



A2 = A3 = P 2 e i ψ ,

(14.6.6)

with the total power P = P1 + 2P2 and a complicated expression for ψ [172]. It also exhibits modulation instability in normal-GVD region, and its gain spectrum has three peaks on each side of the pump frequency. These results suggest that gain spectrum of modulation instability is likely to exhibit multiple peaks in fibers with more than three cores, and the number of sideband pairs can be as large as the number of cores within the fiber. The situation becomes even more complex when the fiber has a central core in addition to the multiple cores arranged in a circle [165,166]. Consider next what happens when short optical pulses are launched into different cores of the fiber simultaneously (at the same or different wavelengths). Linearly arranged fiber arrays were studied as early as 1994 and were found to support soliton-like pulses [159]. In essence, energy is transferred among the cores such that several neighboring cores eventually carry pulses of different energies in a self-sustaining fashion. One way to understand the formation of such multicore solitons is to assume that the number of cores is so large that we can take the continuum limit. It is useful to write Eq. (14.6.2) in the form i

∂ um 1 ∂ 2 um + K (um+1 − 2um + um−1 ) + 2Kum + |um |2 um = 0. + ∂ξ 2 ∂τ 2

(14.6.7)

The linear term 2Kum can be removed through the transformation um = um e2iK ξ . In the continuum limit, we obtain a (2+1)D NLS equation [159] i

2 ∂ U 1 ∂ 2U 2∂ U + Kd + |U |2 U = 0, + ∂ξ 2 ∂τ 2 ∂ x2

(14.6.8)

Multimode fibers

Figure 14.27 Nonlinear energy transfer inside a fiber whose seven cores are arranged in a circle with equal spacing. The input amplitudes are modulated (b = 0.3) such that pulse energy is slightly enhanced in the seventh core. (From Ref. [173]; ©2016 APS.)

where x = md represents the position of the mth core along the linear array, d being the spacing between any two adjacent cores, and we replaced um (t) with U (x, t). In the CW case, this equation reduces to the standard NLS equation whose spatial solitons correspond to the discrete solitons discussed earlier. Eq. (14.6.8) does not apply to fibers whose cores are arranged in a circular fashion as seen in Fig. 14.26(A). The reason is that x is replaced with the angle φ that is defined only modulo 2π . In that case, one must solve Eq. (14.6.2) numerically with the periodic boundary condition uM +1 = u1 for a fiber with M core. In one study, Gaussian pulses were launched with their amplitudes modulated such that [173] um (t) = e−τ

2 /2

[1 + b cos(2π m/M )],

(14.6.9)

where b is the modulation depth. The results for two fibers with 7 and 19 cores show that energy is transferred from all cores such that a single core for which the amplitude is the largest initially carries most of the launched energy at the fiber output. An example is shown in Fig. 14.27 where 83.5% of total energy appears in the seventh core. From a practical perspective, the Kerr nonlinearity in multicore fibers can be used to create a single, energetic, compressed pulse by combining energies of several pulses. In a 2019 experiment, 370-fs pulses could be compressed down to 53 fs using a seven-core fiber [175]. Multicore fibers are useful for enhancing the capacity of optical communication systems [177]. Another application of a multicore fiber makes use of the property that when light is launched into a single core, it spreads to all cores if the input power is relatively low but the spreading is limited to a few nearest neighbors at power levels high enough that a discrete soliton is formed. As a result, the transmission loss is high at low powers but becomes low as the input power increases. Moreover, the transmitted pulse is narrower than the input pulse because only energy in the pulse wings (less intense

677

678

Nonlinear Fiber Optics

part of the pulse) is transferred to other cores. In other words, a multicore fiber acts as a saturable absorber. This effect was quantified in a 2012 experiment by measuring the transmission loss and pulse widths as a function of input peak power when 60-fs pulses were launched into one core of a seven-core fiber [164]. It should be stressed that the study of nonlinear phenomena in multicore fibers is far from being complete. For example, one may dope some of the cores of such a fiber and pump them to provide amplification, while the remaining cores remain lossy. A multicore fiber may even be twisted along its length so that its cores follow a helical path. The parity-time (PT) symmetry in such fibers has attracted some attention [178]. One should expect further advances in the coming years.

Problems 14.1 Derive the eigenvalue equation (14.1.3) for the modes of a step-index fiber in the weakly guiding approximation. 14.2 Use the eigenvalue equation (14.1.3) to reproduce Figure (14.2) showing the effective mode index for the few low-order modes of a step-index fiber. 14.3 Derive the eigenvalue equation (14.1.13) for the modes of a graded-index fiber whose refractive index varies as indicated in Eq. (14.1.7) for all values of ρ . 14.4 Use the eigenvalue equation (14.1.13) and the concepts of mode groups to show that the effective mode index of the gth mode group of a graded-index fiber is given by Eq. (14.1.14). Identify all degenerate modes for g = 3 and 4. 14.5 Prove that only the radially symmetric LP0m modes of a multimode fiber are excited by a Gaussian beam launched such that its intensity peaks at the core center. 14.6 Derive the time-domain equation (14.2.6) starting from the frequency-domain equation (14.2.5). Hint: you need to expand βm (ω) around ω0 as a Taylor series. 14.7 Use the time-domain equation (14.2.6) to derive Eq. (14.2.8) assuming that the nonlinear response is much faster than the duration of optical pulses. 14.8 Starting from Eq. (14.2.8), derive Eqs. (14.2.13) and (14.2.14) indicating nonlinear coupling between the two specific modes of a fiber. 14.9 Use the variational technique to solve Eq. (14.2.23) and prove that a Gaussian beam evolves inside a GRIN fiber as B(r) = A0 F (r) with F (r) and w (z) given in Eqs. (14.2.24) and (14.2.25). You are allowed to consult Ref. [42]. 14.10 Solve Eq. (14.3.4) and prove that the frequencies of the modulation instability sidebands for a GRIN fiber are given by Eq. (14.3.7). Estimate the frequencies of the first two sidebands when β2 = −20 ps2 /km, zp = 5 mm, c0 = 2, and LNL = 1 cm. 14.11 Average Eq. (14.3.12) over the self-imaging period of a GRIN fiber and derive Eq. (14.3.14). Show that N is related to N as indicated in Eq. (14.3.15).

Multimode fibers

14.12 Consider FWM inside a GRIN fiber and show that the phase-matching condition is satisfied when the frequency shift s of the idler from the pump satisfies Eq. (14.4.3). You may consult Ref. [89]. 14.13 Plot the group index ng over the wavelength range 1 to 1.6 µm for the first six LP0m modes (m = 1–6) for a step-index fiber with a 50-µm core diameter and  = 5 × 10−3 . Can intermodal SRS be observed in this fiber? Explain your reasoning well. 14.14 Explain the concept of beam cleanup by a multimode fiber. What is the difference between the Kerr-induced and Raman-induced beam cleanup? How can the phenomenon of SBS be used for the beam cleanup? 14.15 Reproduce Fig. 14.27 by solving Eq. (14.6.2) numerically for a fiber with seven cores arranged in a circular fashion. Assume Gaussian pulses are launched into all cores with the initial amplitudes indicated in Eq. (14.6.9).

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]

R.H. Stolen, J.E. Bjorkholm, A. Ashkin, Appl. Phys. Lett. 24 (1974) 308. D. Marcuse, Theory of Dielectric Optical Waveguides, Academic Press, 1991. A. Ghatak, K. Thyagarajan, An Introduction to Fiber Optics, Cambridge University Press, 1999. J.A. Buck, Fundamentals of Optical Fibers, 2nd ed., Wiley, 2004. G.P. Agrawal, Lightwave Technology: Components and Devices, Wiley, 2004. N. Kishi, E. Yamashita, IEEE Trans. Microw. Theory Tech. 36 (1988) 1861. J. Hudgings, L. Molter, M. Dutta, IEEE J. Quantum Electron. 36 (2000) 1438. Y.C. Meng, Q.Z. Guo, W.H. Tan, Z.M. Huang, J. Opt. Soc. Am. A 21 (2004) 1518. C. Xia, N. Bai, I. Ozdur, X. Zhou, G. Li, Opt. Express 19 (2011) 16653. M. Koshiba, K. Saitoh, K. Takenaga, S. Matsuo, Opt. Express 19 (2011) B102. J. Zhou, Opt. Express 22 (2014) 673. G.P. Agrawal, Applications of Nonlinear Fiber Optics, 2nd ed., Academic Press, 2008. F. Dubois, Ph. Emplit, O. Hugon, Opt. Lett. 19 (1994) 433. W.Q. Thornburg, B.J. Corrado, X.D. Zhu, Opt. Lett. 19 (1994) 454. I. Giles, A. Obeysekara, R. Chen, D. Giles, F. Poletti, D. Richardson, IEEE Photonics Technol. Lett. 24 (2012) 1922. K. Aoki, A. Okamoto, Y. Wakayama, A. Tomita, S. Honma, Opt. Lett. 38 (2013) 769. D. Flamm, C. Schulze, D. Naidoo, S. Schröter, A. Forbes, M. Duparré, J. Lightwave Technol. 31 (2013) 1023. S.G. Leon-Saval, N.K. Fontaine, J.R. Salazar-Gil, B. Ercan, R. Ryf, J. Bland-Hawthorn, Opt. Express 22 (2014) 103. J. Lars, L. Rishøj, S. Ramachandran, Opt. Express 23 (2015) 28531. A.M. Yao, M.J. Padgett, Adv. Opt. Photonics 3 (2011) 161. S. Ramachandran, P. Kristensen, Nanophotonics 2 (2013) 455. S. Ramachandran, P. Gregg, P. Kristensen, S.E. Golowich, Opt. Express 23 (2015) 3721. R.D. Niederriter, M.E. Siemens, J.T. Gopinath, Opt. Lett. 41 (2016) 5736. S. Chen, J. Wang, Sci. Rep. 7 (2017) 3990. S. Wu, Y. Li, L. Feng, X. Zeng, W. Li, J. Qiu, Y. Zuo, X. Hong, H. Yu, R. Chen, I.P. Giles, J. Wu, Opt. Lett. 43 (2018) 2130. A. Wang, L. Zhu, L. Wang, J. Ai, S. Chen, J. Wang, Opt. Express 26 (2018) 10038.

679

680

Nonlinear Fiber Optics

[27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69]

B. Bendow, P.D. Gianino, N. Tzoar, M. Jain, J. Opt. Soc. Am. 70 (1980) 539. B. Crosignani, P. Di Porto, Opt. Lett. 6 (1981) 61. B. Crosignani, A. Cutolo, P. Di Porto, J. Opt. Soc. Am. 72 (1982) 1136. F. Poletti, P. Horak, J. Opt. Soc. Am. B 25 (2008) 1645. A. Mafi, J. Lightwave Technol. 30 (2012) 2803. F. Tani, J.C. Travers, P.St.J. Russell, J. Opt. Soc. Am. B 31 (2014) 311. A. Mecozzi, C. Antonelli, M. Shtaif, Opt. Express 20 (2012) 11673, Opt. Express 20 (2012) 23436. S. Mumtaz, R.-J. Essiambre, G.P. Agrawal, J. Lightwave Technol. 31 (2013) 398. C. Antonelli, M. Shtaif, A. Mecozzi, J. Lightwave Technol. 34 (2016) 36. L.G. Wright, Z.M. Ziegler, P.M. Lushnikov, Z. Zhu, M.A. Eftekhar, D.N. Christodoulides, F.W. Wise, IEEE J. Sel. Top. Quantum Electron. 24 (2018) 5100516. R.J. Essiambre, R.W. Tkach, R. Ryf, in: I. Kaminow, T. Li (Eds.), Optical Fiber Telecommunications, Vol. 6B, Elsevier, 2013. S. Ramachandran, J.W. Nicholson, S. Ghalmi, M.F. Yan, P. Wisk, E. Monberg, F.V. Dimarcello, Opt. Lett. 31 (2006) 1797. S. Ramachandran, Laser Photonics Rev. 2 (2008) 429. S. Buch, S. Mumtaz, R.-J. Essiambre, A.M. Tulino, G.P. Agrawal, Opt. Fiber Technol. 47 (2018) 123. M. Conforti, C.M. Arabi, A. Mussot, A. Kudlinski, Opt. Lett. 42 (2017) 4004. M. Karlsson, D. Anderson, M. Desaix, Opt. Lett. 17 (1992) 22. R.W. Boyd, Nonlinear Optics, 3rd ed., Elsevier, 2008. A. Mussot, T. Sylvestre, L. Provino, H. Maillotte, Opt. Lett. 28 (2003) 1820. A. Tonello, S. Pitois, S. Wabnitz, et al., Opt. Express 14 (2006) 397. M. Guasoni, Phys. Rev. A 92 (2015) 033849. R. Dupiol, A. Bendahmane, K. Krupa, et al., Opt. Lett. 42 (2017) 3419. A. Hasegawa, Opt. Lett. 5 (1980) 416. B. Crosignani, P. Di Porto, Opt. Lett. 6 (1981) 329. S. Longhi, Opt. Lett. 28 (2003) 2363. F. Matera, A. Mecozzi, M. Romagnoli, M. Settembre, Opt. Lett. 18 (1993) 1499. K. Krupa, A. Tonello, A. Barthélémy, et al., Phys. Rev. Lett. 116 (2016) 183901. L.G. Wright, Z.I. Liu, D.A. Nolan, M.-J. Li, D.N. Christodoulides, F.W. Wise, Nat. Photonics 10 (2016) 771. C.M. Arabi, A. Kudlinski, A. Mussot, M. Conforti, Phys. Rev. A 97 (2018) 023803. R. Chang, J. Wang, Opt. Lett. 18 (1993) 266. S.-S. Yu, C.-H. Chien, Y. Lai, J. Wang, Opt. Commun. 119 (1995) 167. S. Raghavan, G.P. Agrawal, Opt. Commun. 180 (2000) 377. Y.S. Kivshar, G.P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals, Academic Press, 2003, Chap. 7. W.H. Renninger, F.W. Wise, Nat. Commun. 4 (2013) 1719. S. Buch, G.P. Agrawal, Opt. Lett. 40 (2015) 225. L.G. Wright, W.H. Renninger, D.N. Christodoulides, F.W. Wise, Opt. Express 23 (2015) 3492. Z. Zhu, L.G. Wright, D.N. Christodoulides, F.W. Wise, Opt. Lett. 41 (2016) 4819. S. Buch, G.P. Agrawal, J. Opt. Soc. Am. B 33 (2017) 2217. O.V. Shtyrina, M.P. Fedoruk, Y.S. Kivshar, S.K. Turitsyn, Phys. Rev. A 97 (2018) 013841. A.H. Ahsan, G.P. Agrawal, Opt. Lett. 43 (2018) 3345. Z. Deng, Y. Chen, J. Liu, C. Zhao, D. Fan, Opt. Express 27 (2019) 483. Y. Silberberg, Opt. Lett. 15 (1990) 1282. A. Hasegawa, Y. Kodama, Opt. Lett. 15 (1990) 1443. G.P. Agrawal, Fiber-Optic Communication Systems, 4th ed., Wiley, 2010.

Multimode fibers

[70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83] [84] [85] [86] [87] [88] [89] [90] [91] [92] [93] [94] [95] [96] [97] [98] [99] [100] [101] [102] [103] [104] [105] [106] [107] [108] [109] [110] [111]

E. Caglioti, B. Crosignani, P. Di Porto, Phys. Rev. A 38 (1988) 4036. V.V. Afanasyev, Y.S. Kivshar, V.V. Konotop, V.N. Serkin, Opt. Lett. 14 (1989) 805. E.M. Wright, G.I. Stegeman, S. Wabnitz, Phys. Rev. A 40 (1989) 4455. T. Ueda, W.L. Kath, Phys. Rev. A 42 (1990) 563. R.H. Stolen, IEEE J. Quantum Electron. 11 (1975) 100. K.O. Hill, D.C. Johnson, B.S. Kawasaki, R.I. MacDonald, J. Appl. Phys. 49 (1978) 5098. A. Säisy, J. Botineau, A.A. Azéma, F. Gires, Appl. Opt. 19 (1980) 1639. K.O. Hill, D.C. Johnson, B.S. Kawasaki, Appl. Opt. 20 (1981) 1075. C. Lin, M.A. Bösch, Appl. Phys. Lett. 38 (1981) 479. R.H. Stolen, J.E. Bjorkholm, IEEE J. Quantum Electron. 18 (1982) 1062. P.L. Baldeck, R.R. Alfano, J. Lightwave Technol. 5 (1987) 1712. H. Tu, Z. Jiang, D.L. Marks, S.A. Boppart, Appl. Phys. Lett. 94 (2009) 101109. J. Cheng, M.E.V. Pedersen, K. Charan, K. Wang, C. Xu, L. Grüer-Nielsen, D. Jakobsen, Appl. Phys. Lett. 101 (2012) 161106. J. Yuan, X. Sang, Q. Wu, et al., Opt. Lett. 40 (2015) 1338. S.R. Petersen, T.T. Alkeskjold, C.B. Olausson, J. Lægsgaard, Opt. Express 23 (2015) 5954. H. Pourbeyram, E. Nazemosadat, A. Mafi, Opt. Express 23 (2015) 14487. E. Nazemosadat, H. Pourbeyram, A. Mafi, J. Opt. Soc. Am. B 33 (2016) 144. R. Dupiol, A. Bendahmane, K. Krupa, et al., Opt. Lett. 42 (2017) 1293. J. Yuan, Z. Kang, F. Li, et al., Opt. Lett. 42 (2017) 3537. A. Bendahmane, K. Krupa, A. Tonello, et al., J. Opt. Soc. Am. B 35 (2018) 295. H. Pourbeyram, A. Mafi 4 (2019) 022805. R.-J. Essiambre, M.A. Mestre, R. Ryf, et al., IEEE Photonics Technol. Lett. 25 (2013) 539. Y. Xiao, R. Essiambre, M. Desgroseilliers, A. Tulino, R. Ryf, S. Mumtaz, G.P. Agrawal, Opt. Express 22 (2014) 32039. M. Esmaeelpour, R.-J. Essiambre, N.K. Fontaine, R. Ryf, J. Toulouse, Y. Sun, R. Lingle, J. Lightwave Technol. 35 (2017) 2429. M. Guasoni, F. Parmigiani, P. Horak, J. Fatome, D.J. Richardson, J. Lightwave Technol. 35 (2017) 5296. C.-H. Lin, B.R. Marshall, M.A. Nelson, J.K. Theobald, Appl. Opt. 17 (1978) 2486. F.R. Barbosa, Appl. Opt. 22 (1983) 3859. P.L. Baldeck, F. Raccah, R.R. Alfano, Opt. Lett. 12 (1987) 588. A.B. Grudinin, E.M. Dianov, D.V. Korbkin, A.M. Prokhorov, D.V. Khaidarov, JETP Lett. 47 (1988) 356. K.S. Chiang, Opt. Lett. 17 (1992) 352. K.S. Chiang, Opt. Commun. 95 (1993) 235. S.J. Garth, R.A. Sammut, J. Opt. Soc. Am. B 10 (1993) 2040. A. Sharma, M. Dokhanian, Z. Wu, A. Williams, P. Venkateswarlu, Opt. Lett. 19 (1994) 1122. K. Rottwitt, J. Povlsen, J. Lightwave Technol. 23 (2005) 3597. A. Polley, S.E. Ralph, IEEE Photonics Technol. Lett. 19 (2007) 218. N.B. Terry, T.G. Alley, T.H. Russell, Opt. Express 15 (2007) 17509. M. Ziemienczuk, A.M. Walser, A. Abdolvand, P.S.J. Russell, J. Opt. Soc. Am. B 29 (2012) 1563. H. Pourbeyram, G.P. Agrawal, A. Mafi, Appl. Phys. Lett. 102 (2013) 201107. L. Rishøj, P. Kristensen, S. Ramachandran, K. Rottwitt, Opt. Express 21 (2013) 28836. J. Reintjes, R.H. Lehmberg, R.S.F. Chang, M.T. Duignan, G. Calame, J. Opt. Soc. Am. B 3 (1986) 1408. L. Rishøj, B. Tai, P. Kristensen, S. Ramachandran, Optica 7 (2019) 304. A. Antikainen, L. Rishøj, B. Tai, S. Ramachandran, G.P. Agrawal, Phys. Rev. Lett. 122 (2019) 023901.

681

682

Nonlinear Fiber Optics

[112] [113] [114] [115] [116] [117] [118] [119] [120] [121] [122] [123] [124] [125] [126] [127] [128] [129] [130] [131] [132] [133] [134] [135] [136] [137] [138] [139] [140] [141] [142] [143] [144] [145] [146] [147] [148] [149] [150] [151] [152] [153] [154]

C. Antonelli, A. Mecozzi, M. Shtaif, Opt. Lett. 38 (2013) 1188. J. Zhou, Opt. Express 22 (2014) 21393. M. Esmaeelpour, R. Ryf, N.K. Fontaine, et al., J. Lightwave Technol. 34 (2016) 1864. E.N. Christensen, J.G. Koefoed, S.M.M. Friis, M.A.U. Castaneda, K. Rottwitt, Sci. Rep. 6 (2016) 34693. J. Li, J. Du, L. Ma, M.-J. Li, K. Xu, Z. He, Opt. Express 25 (2017) 810. A. Kobyakov, M. Sauer, D. Chowdhury, Adv. Opt. Photonics 2 (2010) 1. K. Tei, Y. Tsuruoka, T. Uchiyama, T. Fujioka, Jpn. J. Appl. Phys. 40 (2001) 3191. V.I. Kovalev, R.G. Harrison, Opt. Lett. 27 (2002) 2022. M. Sjöberg, M.L. Quiroga-Teixeiro, S. Galt, Sverker Hård, J. Opt. Soc. Am. B 20 (2003) 434. A. Mocofanescu, L. Wang, R. Jain, K.D. Shaw, A. Gavrielides, P.R. Peterson, M. Sharma, Opt. Express 13 (2005) 2019. A.H. McCurdy, J. Lightwave Technol. 23 (2005) 3509. B. Ward, M. Mermelstein, Opt. Express 18 (2010) 1952. K.Y. Song, Y.H. Kim, B.Y. Kim, Opt. Lett. 38 (2013) 1805. K.Y. Song, Y.H. Kim, Opt. Lett. 38 (2013) 4841. A. Minardo, R. Bernini, L. Zeni, Opt. Express 22 (2014) 17480. W.-W. Ke, X.-J. Wang, X. Tang, IEEE J. Sel. Top. Quantum Electron. 20 (2014) 0901610. H. Lu, P. Zhou, X. Wang, J. Lightwave Technol. 33 (2015) 4464. B.C. Rodgers, T.H. Russell, W.B. Roh, Opt. Lett. 24 (1999) 1124. Q. Gao, Z. Lu, C. Zhu, J. Zhang, Appl. Phys. Express 8 (2015) 052501. T.H. Russell, W.B. Roh, J.R. Marciante, Opt. Express 8 (2001) 246. S.M. Massey, T.H. Russell, Opt. Express 16 (2008) 11496. R.W. Hellwarth, J. Opt. Soc. Am. 68 (1978) 1050. L.G. Wright, D.N. Christodoulides, F.W. Wise, Nat. Photonics 9 (2015) 306. Z. Liu, L.G. Wright, D.N. Christodoulides, F.W. Wise, Opt. Lett. 41 (2016) 3675. K. Krupa, A. Tonello, B.M. Shalaby, et al., Nat. Photonics 11 (2017) 237. R. Guenard, K. Krupa, R. Dupiol, et al., Opt. Express 25 (2017) 4783. R. Florentin, V. Kermene, J. Benoist, A. Desfarges-Berthelemot, D. Pagnoux, A. Barthélémy, Light Sci. Appl. 6 (2017) e16208. R. Dupiol, K. Krupa, A. Tonello, M. Fabert, D. Modotto, S. Wabnitz, G. Millot, V. Couderc, Opt. Lett. 43 (2018) 587. F. Lu, Q. Lin, W.H. Knox, G.P. Agrawal, Phys. Rev. Lett. 93 (2004) 183901. R. Cherif, M. Zghal, L. Tartara, V. Degiorgio, Opt. Express 16 (2008) 2147. C. Xiong, W.J. Wadsworth, Opt. Express 16 (2008) 2438. F. Poletti, P. Horak, Opt. Express 17 (2009) 6134. X.-H. Fang, et al., Opt. Lett. 37 (2012) 2292. J. Ramsay, S. Dupont, M. Johansen, L. Rishøj, K. Rottwitt, P.M. Moselund, S.R. Keiding, Opt. Express 21 (2013) 10764. R. Khakimov, I. Shavrin, S. Novotny, M. Kaivola, H. Ludvigsen, Opt. Express 21 (2013) 14388. Y. Chen, Z. Chen, W.J. Wadsworth, T.A. Birks, Opt. Express 21 (2013) 17886. L.G. Wright, S. Wabnitz, D.N. Christodoulides, F.W. Wise, Phys. Rev. Lett. 115 (2015) 223902. I. Kubat, O. Bang, Opt. Express 24 (2016) 2513. G. Lopez-Galmiche, Z.S. Eznaveh, M.A. Eftekhar, et al., Opt. Lett. 41 (2016) 2553. K. Krupa, C. Louot, V. Couderc, et al., Opt. Lett. 41 (2016) 5785. M.A. Eftekhar, L.G. Wright, M.S. Mills, M. Kolesik, R.A. Correa, F.W. Wise, D.N. Christodoulides, Opt. Express 25 (2017) 9078. Z. Eslami, P. Ryczkowski, C. Amiot, L. Salmela, G. Genty, J. Opt. Soc. Am. B 36 (2019) A72. S. Perret, G. Fanjoux, L. Bigot, J. Fatome, G. Millot, J.M. Dudley, T. Sylvestre, APL Photonics 4 (2019) 022905.

Multimode fibers

[155] [156] [157] [158] [159] [160] [161] [162] [163] [164] [165] [166] [167] [168] [169] [170] [171] [172] [173] [174] [175] [176] [177] [178]

N. Finlayson, G.I. Stegeman, Appl. Phys. Lett. 56 (1990) 2276. C. Schmidt-Hattenberger, U. Trutschel, F. Lederer, Opt. Lett. 16 (1991) 291. N. Akhmediev, A.V. Buryak, J. Opt. Soc. Am. B 11 (1994) 804. W. Królikowski, U. Trutschel, M. Cronin-Golomb, C. Schmidt-Hattenberger, Opt. Lett. 19 (1994) 320. A.B. Aceves, C. De Angelis, A.M. Rubenchik, S.K. Turitsyn, Opt. Lett. 19 (1994) 329. A.B. Aceves, M. Santagiustina, C. De Angelis, J. Opt. Soc. Am. B 14 (1997) 1807. H. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, J. Aitchison, Phys. Rev. Lett. 81 (1998) 3383. P.M. Ramos, C.R. Paiva, J. Opt. Soc. Am. B 17 (2000) 1125. T. Pertsch, U. Peschel, J. Kobelke, et al., Phys. Rev. Lett. 93 (2004) 053901. T.F.S. Büttner, D.D. Hudson, E.C. Mägi, A.C. Bedoya, T. Taunay, B.J. Eggleton, Opt. Lett. 37 (2012) 2469. S.K. Turitsyn, A.M. Rubenchik, M.P. Fedoruk, E. Tkachenko, Phys. Rev. A 86 (2012) 031804. A.M. Rubenchik, E.V. Tkachenko, M.P. Fedoruk, S.K. Turitsyn, Opt. Lett. 38 (2013) 4232. E. Nazemosadat, A. Mafi, J. Opt. Soc. Am. B 30 (2013) 2787. T.X. Tran, D.C. Duong, F. Biancalana, Phys. Rev. A 90 (2014) 023857. A.M. Rubenchik, I.S. Chekhovskoy, M.P. Fedoruk, O.V. Shtyrina, S.K. Turitsyn, Opt. Lett. 40 (2015) 721. A. Radosavljevic, A. Danicic, J. Petrovic, A. Maluckov, L. Hadzievski, J. Opt. Soc. Am. B 32 (2015) 2520. A.B. Aceves, O.V. Shtyrina, A.M. Rubenchik, M.P. Fedoruk, S.K. Turitsyn, Phys. Rev. A 91 (2015) 033810. J.H. Li, H. Zhou, K.S. Chiang, S.R. Xiao, J. Opt. Soc. Am. B 33 (2016) 2357. I.S. Chekhovskoy, A.M. Rubenchik, O.V. Shtyrina, M.P. Fedoruk, S.K. Turitsyn, Phys. Rev. A 94 (2016) 043848. I.S. Chekhovskoy, A.M. Rubenchik, O.V. Shtyrina, M.A. Sorokina, S. Wabnitz, M.P. Fedoruk, J. Opt. Soc. Am. B 35 (2018) 2169. A.V. Andrianov, N.A. Kalinin, M.Yu. Koptev, O.N. Egorova, A.V. Kim, A.G. Litvak, Opt. Lett. 44 (2019) 303. S. Trillo, S. Wabnitz, G.I. Stegeman, E.M. Wright, J. Opt. Soc. Am. B 6 (1989) 889. K. Saitoh, S. Matsuo, J. Lightwave Technol. 34 (2016) 55. X. Zhang, V.A. Vysloukh, Y.V. Kartashov, X. Chen, F. Ye, M.R. Beli´c, Opt. Lett. 42 (2017) 2972.

683