Numerical Simulation of the Multilayer W Type Optical Fibers

ISSN 10642269, Journal of Communications Technology and Electronics, 2014, Vol. 59, No. 5, pp. 407–413. © Pleiades Publishing, Inc., 2014. Original Russian Text © A.E. Ulanov, V.E. Ustimchik, Yu.K. Chamorovskii, S.A. Nikitov, 2014, published in Radiotekhnika i Elektronika, 2014, Vol. 59, No. 5, pp. 445–451.

ELECTRODYNAMICS AND WAVE PROPAGATION

Numerical Simulation of the Multilayer WType Optical Fibers Radiation Mode Structure A. E. Ulanova, b, V. E. Ustimchika, b, Yu. K. Chamorovskiia, and S. A. Nikitova, b a

Kotel’nikov Institute of Radio Engineering and Electronics, Russian Academy of Sciences, ul. Mokhovaya 11, str. 7, Moscow, 125009 Russia b Moscow Institute of Physics and Technology (State University), Institutskii per. 9, Dolgoprudnyi, 141700 Russia email: [email protected] Received November 27, 2013

Abstract—Numerical simulation of the properties of guided and leaky modes in Wtype multilayer cylindri cal optical fibers is performed. The dependences of the losses of leaky modes on the parameters of the Wtype optical fiber are studied. It is shown that, for certain structures of the Wfiber, the leaky modes have substan tial effect on the behavior of propagating radiation. It is shown that, when the mode structure of the radiation in Wtype optical fibers is studied, both guided and leaky modes should be taken into account. DOI: 10.1134/S1064226914050088

INTRODUCTION A significant increase in the power of fiberoptic lasers [1–3] in recent years has led to replacement of traditionally used lasers with fiber lasers in many prac tical areas of science and engineering. In modern world, fiber lasers are used in the industry for laser material processing, engraving, marking, cutting, and welding, as well as in medicine and science. Fiber optic lasers have compact designs, and, in addition, they are characterized by a high efficiency and high quality of the output beam. At present, the output power of fiberoptic lasers reaches several kilowatts. When intensity of the output radiation is high, the threshold of excitation of nonlinear effects and even the destruction threshold are reached in standard opti cal fibers, making them unsuitable for applications in the areas, where high power of the output optical radi ation is required. In connection to this, the possibility of design and study of fibers with large mode areas is now investi gated for applications related to fiberoptic lasers and highpower amplifiers. An increase in the effective area of the mode leads to a decrease in the power den sity of the optical radiation propagating in the fiber. As a result, nonlinear effects exert smaller influence on these fibers and the fibers have higher destruction threshold. Important parameters of these fibers are high quality of the output beam and insensitivity to bending. Different methods for the design of optical fibers with large efficient areas of the mode and high quality of the output beam are described in the pub lished literature. They include filtering of higher

modes in multimode optical fibers by bending the fiber [4]; chirally coupled core fibers [5, 6]; leakage channel and photoniccrystal fibers [7, 8]. This study presents an alternative approach to the design of almost single mode optical fibers with a large mode area: Wtype optical fibers with two and three claddings and a step wise profile of the refractive index [9]). These fibers are easiertofabricate than fibers with leakage chan nels and chirally coupled core fibers. In addition, the Wfibers are stable to bending due to strong field local ization in the core [10]. In this case, the freedom of selection of fiber parameters (such as the number of claddings and their radii) and the values of refractive indices allow one to vary properties of the fiber in a wide range. A great number of studies devoted to the analysis of optical properties of Wfibers with two claddings [11, 12] were published. Properties of the radiation propagating in these fibers and dependences of the guided modes cutoff frequencies on the parameters of the optical fiber and the wavelength of the propagating radiation were studied in detail. It was demonstrated that the fundamental mode in the Wtype optical fibers has nonzero cutoff frequency and extended sin glemode propagation range [11, 12]. Introduction of the third cladding is intended to expand the range of optical properties of these fibers. The purpose of this paper is to study the mode struc ture of the radiation of Wtype optical fibers with three claddings and compare it with that of Wtype optical fibers with two claddings.

407

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ULANOV et al. Profile of the refractive index of the Wlight guide n1

Refractive index

1.445

n4

n3

1.440

1.435

1.430 n2 a b

1.425 –80

–60

–40

–20

0 20 Radius, μm

c 40

60

80

Fig. 1. Profile of the refractive index of a Wtype optical fiber with three claddings.

1. THEORETICAL DESCRIPTION The radiation propagating in optical fibers can be classified as follows: (1) guided modes (have real propagation constant β), (2) leaky modes (have com plex propagation constant β), and (3) the radiation field [13]. The radiation field is not of any practical interest, since it is completely radiated within a finite length of the optical fiber. In the published studies, when the mode structure of optical fibers was studied, no proper attention was paid to the properties of leaky modes and the properties of guided modes were only studied. In the process of development of fiberoptic technolo gies, it became possible to create structures that meet modern quality requirements for the output radiation, with a large effective size of the fundamental mode. In this case, in some of these structures, leaky modes can propagate over significant distances with small losses and, thereby, affect the behavior of the radiation prop agating in the fiber and the transverse distribution of field intensity at the output. Both guided and weakly leaking modes are consid ered in this paper. The study is based on the numerical simulation of the radiation propagating in multilayer cylindrical Wtype optical fibers. Earlier, planar light guides were analyzed in detail [14]. The computer model for studying the modal structure of the radia tion in Wtype optical fibers is based on the matrix transfer method [15]. During the numerical simula tion, the Wtype optical fiber mode properties were determined as a function of the refractive indices and radii of claddings. The refractive index profile of this optical fiber (Fig. 1) enables one to vary a great num

ber of fiber parameters and efficiently filter higher order modes by varying refractive indices n3 and n4. As a result of adding the third cladding with n3 = const < n1, three fundamentally different cases of the refractive index profile are possible: Case 1: n4 < n3. Both guided and leaky modes are present in this case, but this structure has a clear mul timode character of radiation propogating therefore it is out of practical interest. With such refractive index profile, leaky modes have significant losses as com pared to guided modes. Therefore, this case is not con sidered in this paper. Case 2: n1 > n4 ≥ n3. Again, both guided and leaky modes are present in this structure. Here, the condi tion n1 – n4 = Δn Ⰶ 1 corresponds to the weakly lead ing approximation, in which the number of guided modes significantly decreases. Leaky modes have again significant losses, as in case 1. As the thickness of the second cladding (n3) significantly increases, the influence of the third cladding on propagation of radi ation vanishes and cases 1 and 2 become indistinguish able. In this case, one can efficiently filter higher order modes by varying n4 from n3 to n1. Case 3: n4 ≥ n1. Only leaky modes are present in such structures. This case is the most interesting, because nowdays there is no proper analysis of leaky modes. In the weakleading approximation, the modes of the light guide are obtained from the scalar wave equation: 2 2 2 ∂⎞ ⎛ ∇ 2 –  2 ψ + [ k 0 n ( x, y ) – β ]ψ = 0, ⎝ ⎠ ∂z

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where ψ are the components of the electric or mag netic field, k0 = 2π/λ is the wave number in vacuum, and β is the propagation constant of the mode. Let us consider the refractive index profile that meet case 2. In the analysis of guided modes proper ties, for simplicity, we select n3 = n4. The cases of n3 ≠ n4 are considered similarly. This profile corre sponds to the maximum number of guided modes that can be filtered by increasing n4. We find ψ ( r, ϕ, z, t ) in the following form: ψ ( r, ϕ, z, t ) = AR(r)exp(ilϕ). Then, in the scalar approximation, the equation for the radial component of the field can be written as fol lows: 2

2 2 2d R 2 2 2 r 2 + r dR  + [ ( k 0 n i – β )r – l ]R = 0. dr dr

(2)

Equation (2) is the wellknown Bessel equation. The solution to Eq. (2) can be found in the form ⎧ A 1 J l ( ur ) + B 1 N l ( ur ), r < a; ⎪ R ( r ) = ⎨ A 2 I l ( wr ) + B 2 K l ( wr ), a < r < b; ⎪ ⎩ A 3 I l ( vr ) + B 3 K l ( vr ), r > b, 2

2

2

2

u = kn 1 – β ,

where 2

2

2

(3)

2 2

w = β – k n2 ,

2 2

v = β – k n 3 are the transverse wave numbers. Boundary conditions at r = a and r = b require that R(r) and dR/dr must be continuous. Expressions for columns R(r) and dR/dr can be written in each layer in the form of the following matrices: ⎛ ⎜ ⎜ ⎝

R(r) 1dR  βdr

⎛ ⎞ J ( ur ) ⎟ = ⎜⎜ l ⎟ ⎜ uJ 'l ( ur ) ⎠ ⎝ β

⎞ N l ( ur ) ⎟ ⎛ A1 ⎞ ⎜ ⎟, ⎟ uN 'l ( ur ) ⎟ ⎝ B 1 ⎠ β ⎠

(4)

for r < a; ⎛ ⎜ ⎜ ⎝

R(r) 1dR  βdr

⎛ ⎞ I ( wr ) ⎟ = ⎜⎜ l ⎟ ⎜ w J l' ( wr ) ⎠ ⎝ β

⎞ K l ( wr ) ⎟ ⎛ A2 ⎞ ⎜ ⎟, ⎟ w K l' ( wr ) ⎟ ⎝ B 2 ⎠ β ⎠

(5)

for a < r < b; ⎛ ⎜ ⎜ ⎝

R(r) 1dR  βdr

⎛ ⎞ I ( vr ) ⎟ = ⎜⎜ l ⎟ ⎜ v  I 'l ( vr ) ⎠ ⎝ β

⎞ K l ( vr ) ⎟ ⎛ A ⎞ ⎟⎜ 3 ⎟, v  K 'l ( vr ) ⎟ ⎝ B 3 ⎠ β ⎠

for r > b.

(6)

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Let us substitute r = a and r = b into expressions (4), (5), and (6) and designate the obtained matrices as ⎛ ⎜ P = ⎜ ⎜ ⎝

⎞ N l ( ua ) ⎟ ⎟, u uJ 'l ( ua ) N 'l ( ua ) ⎟ β β ⎠

(7)

⎛ ⎜ Q = ⎜ ⎜ ⎝

⎞ K l ( wa ) ⎟ ⎟, w w I 'l ( wa ) K 'l ( wa ) ⎟ β β ⎠

(8)

⎛ ⎜ R = ⎜ ⎜ ⎝

⎞ K l ( wb ) ⎟ ⎟, w w ' I l ( wb ) K 'l ( wb ) ⎟ β β ⎠

(9)

⎛ ⎜ T = ⎜ ⎜ ⎝

⎞ K l ( vb ) ⎟ ⎟. v v  I 'l ( vb )  K 'l ( vb ) ⎟ β β ⎠

(10)

J l ( ua )

I l ( wa )

I l ( wb )

I l ( vb )

Then, writing the continuity conditions for R(r) and dR/dr on the boundaries at r = a and r = b, we obtain the following matrix expression: ⎛ A3 ⎞ ⎛ M M ⎞⎛ A ⎞ –1 –1 ⎛ A ⎞ ⎜ ⎟ = T RQ P ⎜ 1 ⎟ = ⎜ 11 12 ⎟ ⎜ 1 ⎟ . (11) ⎝ B3 ⎠ ⎝ B 1 ⎠ ⎝ M 21 M 22 ⎠ ⎝ B 1 ⎠ Using boundary conditions at the zero and at infinity and knowing forms of the Bessel, Neumann, Infeld, and Macdonald functions, we infer that B1 = 0 and A3 = 0, i.e., the following ratio can be obtained: ⎛ 0 ⎞ ⎛ M M ⎞⎛ A ⎞ ⎜ ⎟ = ⎜ 11 12 ⎟ ⎜ 1 ⎟ . ⎝ B3 ⎠ ⎝ M 21 M 22 ⎠ ⎝ 0 ⎠

(12)

Hence, we can deduce that M11 = 0. This condition is an eiganvalue problem. Further, studying M11(β) as a function of β, we can find propagation constants of guided modes. Here, these modes must obey the con dition k 0 n 1 > β > k 0 n 3 . A Wtype optical fiber with two claddings and parameters n1 = 1.44692, n2 = 1.42692, n4 = n3 = 1.44452, a = 20 μm, b = 21.5 μm, and the optical radiation wavelength λ = 1 μm was selected as a tested sample. The results are shown in Fig. 2. The positions of the minima in this curve give us propaga tion constants β of the guided modes. At the point where function M11(β) is zero, function log M 11 ( β ) tends to minus infinity. As can be seen from the plots, 13 modes propagate along the fiber having this config uration. Using the obtained data, we can easily find

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0 LP61 log |M11|

–5 LP13 LP32

LP51 LP03

–10

LP22

LP41 LP31 LP12

LP01 LP21 LP11 LP02

–15

–20 9.0775

9.0800

9.0825

9.0850 β, μm–1

9.0875

9.0900

9.0925

Fig. 2. Dependence of function log|M11(β)| on parameter β.

the effective refractive index for each mode from the formula neff = β/k 0 . Let us now pass to case 3, when n4 ≥ n1. For sim plicity, we select the case of n4 = n1, since it is simpler and of the most practical interest. This structure sup ports only leaky modes and there are no guided modes. Now, the solutions to Eq. (2) can be found in the form of the following system: ⎧ A 1 J l ( ur ) + B 1 N l ( ur ), r < a; ⎪ ⎪ A 2 I l ( wr ) + B 2 K l ( wr ), a < r < b; R(r) = ⎨ ⎪ A 3 I l ( vr ) + B 3 K l ( vr ), b < r < c; ⎪ ⎩ A 4 J l ( qr ) + B 4 N l ( qr ), r > c,

(13)

where q is the external cladding transverse wave number, 2 2 2 which is defined by the ratio q = kn 4 – β . Thus, two matrices ⎛ ⎜ F = ⎜ ⎜ ⎝

⎞ K l ( vc ) ⎟ ⎟ v v  I 'l ( vc )  K 'l ( vc ) ⎟ β β ⎠ ⎞ N l ( qc ) ⎟ ⎟ u uJ 'l ( qc ) N 'l ( qc ) ⎟ β β ⎠ J l ( qc )

⎛ A1⎞ ⎛ S S ⎞⎛ A ⎞ –1 –1 –1 ⎛ A ⎞ ⎜ ⎟ = P QR TF G ⎜ 4 ⎟ = ⎜ 11 12 ⎟ ⎜ 4 ⎟ .(14) ⎝ B 1⎠ ⎝ B4 ⎠ ⎝ S 21 S 22 ⎠ ⎝ B 4 ⎠ Since the leaky mode in the external cladding corre sponds to the radiated wave, we should select coeffi cients A4 and B4 as follows: B4 = iA4. Then, we substi tute them in Eq. (14) and obtain A 1 = A 4 ( S 11 + iS 12 ),

(15)

B 1 = A 4 ( S 21 + iS 22 ).

As before, B1 = 0, i.e., we obtain the equation B 1 ( β ) = 0, where β < k 0 n 4 . In this case, the propagation constant is a complex quantity. Assume now that β 0 = β 0r – iβ 0i

(16)

meets the equation B 1 ( β 0 ) = 0. Thus,

I l ( vc )

⎛ ⎜ G = ⎜ ⎜ ⎝

are added to matrices P, Q, R, and T defined in (7–10). Then, the following equation is valid:

and

B1 ( βr ) = B1 [ β0 + ( βr – β0 ) ] (17)

dB ≈ B 1 ( β 0 ) + ⎛ 1⎞ ( β r – β 0 ), ⎝ dβ ⎠ β0

where β0 is specified by Eq. (16), and, since B1(β0) = 0, then B 1 ( β r ) = C ( β r – β 0r + iβ 0i ),

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400 350 300

|1/B1|2

250 200

β0i

150 100 50 β0r 0 6.9850 × 106

6.9851 × 106

6.9852 × 106 β, μm–1

6.9853 × 106

6.9854 × 106

1 2 on parameter β . Fig. 3. Dependence of function   r B1

where C = ( dB 1 /dβ ) β0 is a constant. Thus, we obtain 1. 1 2  =  2 2 2 B1 ( βr ) C { ( β r – β 0r ) + β 0i }

(19)

1 2 as a function of real β , we Now, considering   r B1 ( βr ) obtain the Lorentzian. The position of the peak of this distribution is determined by β0r and the halfheight at halfwidth specifies the value of β0i (see Fig. 3). As is well known, the loss of the leaky mode is determined from the formula: α = 20 log ( e )β 0i dB/m.

(20)

2. RESULTS OF NUMERICAL SIMULATION In this section, we consider properties of the first three leaky modes as functions of the fiber parameters. Figure 4 shows the dependences of the leakage loss on the differences in the refractive indices of the core and the claddings and on the radii of the claddings in a cylindrical Wtype optical fiber with three claddings. As can be seen from the plots, all the dependences are expo nential. Fig. 4a shows the dependence of the leakage loss on the difference in the refractive indices of the core and the second cladding. When n3 – n1 ≈ 1.2 × 10–3, the loss of the first leaky mode α ≈ 0.05 dB/m, while the losses of the next modes are substantially higher. Figure 4b shows the dependence of the leakage loss on the differ

ence in the refractive indices of the core and the first cladding. When n2 – n1 ≈ 0.02, the loss of the first mode is compared with gray losses of guided modes (the losses caused by different irregularities of the structure and defects in the fiber), the loss of the sec ond mode α ≈ 0.25 dB/m, and the loss of the third mode α ≈ 0.9 dB/m. As (n1 – n2) increases, the losses of all modes exponentially increase. The dependence of the leaky modes losses on the radius of the third cladding c is shown in Fig. 4d, and Fig. 4c shows the dependence on δ = b – a. As can be seen from Fig. 4c, when radius c of the second clad ding substantially increases, the losses of leaky modes exponentially decrease, pointing to the fact that, in the extreme case, leaky modes are transformed into guided modes. This means that, for large c, the influ ence of the third cladding on the propagation proper ties of radiation in such optical fiber vanishes. Now let us consider the dependence of the loss level on δ. As can be seen from Fig. 4c, when δ tends to the value of wavelength of propagating radiation λ (in this case, λ = 1 μm), losses of leaky modes expo nentially increase, pointing to strong tunneling of radiation from the core to the claddings. At the same time, as δ increases, the losses of leaky modes tend to zero, i.e., leaky modes are transformed into propagat ing modes. Therefore, for large δ, external claddings do not exert a substantial effect on the properties of the radiation propagating in the fiber. As can be seen from Figs. 4a–4d, there are ranges of the fiber parameter at which the loss of the first mode is small and losses of

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(b)

(a) LP02

LP02

1.4 1.2

2.0

1.0 α, dB/m

α, dB/m

LP11

1.5

LP11

LP01

1.0

LP01

0.8 0.6 0.4

0.5

0.2

0

0

–0.0025 –0.0020 –0.0015 –0.0010 –0.0005

0

–0.028 –0.024 –0.020 –0.016 –0.012 –0.008

1.0 LP01 LP11 0.9 0.8 LP02 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 δ, μm

LP11

2.5

n1 – n2 (d) LP02

2.0 α, dB/m

α, dB/m

n1 – n3 (c)

1.5

LP01

1.0 0.5 0 22

23

24

25

26 27 c, μm

28

29

30

Fig. 4. Dependence of the leakage loss on (a) the difference in the refractive indices of the core and the second cladding, (d) the radius of the second cladding, (b) the difference in the refractive indices of the core and the first cladding, and (c) parameter δ = b – a.

the next modes are substantially higher. For some parameters of the fiber, the loss of the first leaky mode can be comparable in its order of magnitude even with the gray losses of guided modes. This suggests that the leaky mode propagates over substantial distances with out significant losses. In such structures, the wave front is specified by both the guided and leaky modes. Hence, when the behavior of the propagating radia tion is studied in such structures, it is necessary to con sider both the guided and leaky modes.

of multilayer Wtype optical fibers at which the losses of leaky modes take small values and are compared with the gray losses of guided modes. Hence, in these structures, leaky modes propagate over large distances without substantial distortions. This indicates that, in the studies of the modal structure of radiation in mul tilayer Wtype optical fibers and during their fabrica tion and design, it is necessary to consider both the guided and leaky modes and their interaction. REFERENCES

CONCLUSIONS A mathematical model that allows one to study properties of guided and leaky modes in multilayer cylindrical Wtype optical fibers has been developed. This model is a simple tool for the design of optical Wtype waveguides with specified properties. From the results of modal structure computer simulation of the radiation in multilayer Wtype optical fibers, it is possible to draw several conclusions. As the calcula tions show, there are such profiles of refractive indices

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