ISSN 10637850, Technical Physics Letters, 2011, Vol. 37, No. 9, pp. 870–872. © Pleiades Publishing, Ltd., 2011. Original Russian Text © Yu.N. Kul’chin, Yu.A. Zinin, I.G. Nagorny, 2011, published in Pis’ma v Zhurnal Tekhnicheskoі Fiziki, 2011, Vol. 37, No. 18, pp. 58–64.
Bragg Fibers with Matching Layer Yu. N. Kul’chin, Yu. A. Zinin, and I. G. Nagorny* Institute of Automation and Control Processes, Far East Branch, Russian Academy of Sciences, Vladivostok, 690041 Russia *email:
[email protected] Received April 12, 2011
Abstract—The optical properties of Bragg fibers with an intermediate (matching) layer between the core and periodic microstructured cladding have been studied. It is established that the optical losses exhibit a periodic dependence on the thickness of the matching layer, which is related to the presence of resonances and anti resonances in this layer. It is demonstrated that the transmission spectra of Bragg fibers can be controlled by varying the parameters of the internal (matching) layer of the cladding. DOI: 10.1134/S1063785011090240
The properties of Bragg optic fibers have been extensively studied, but they still receive the increasing attention of researchers [1–6]. Almost all investiga tions were devoted to fibers comprising a core and periodic layered cladding. Mizrahi and Schächter [7] considered a waveguide with an additional matching layer between the core and a periodic cladding and studied the influence of this layer on the distribution of electric and magnetic fields in the cross section. Ouy ang et al. [8] studied the case of an optically dense core and an additional airfilled layer and showed the pos sibility of single TM mode propagation with zero dis persion. Optical losses in the fibers considered in [7, 8] were not studied. This Letter presents the results of an investigation of the influence of an additional intermediate layer on the optical characteristics of Bragg fibers. Following the terminology used in [7], this layer will be referred to as the matching layer.
⎛ l E z = [ A l J m ( k l r ) + B l H m ( k l r ) ] ⎜ cos mφ ⎝ sin mφ
⎞ iβz ⎟e , ⎠ (1)
⎛ ⎞ iβz l H z = [ C l J m ( k l r ) + D l H m ( k l k ) ] ⎜ sin mφ ⎟ e , ⎝ cos mφ ⎠ where kl is the transverse component of the wave vec 2
2
2
2
tor in the lth layer; k l = k 0 ( n l – β2/ k 0 ); nl is the refractive index in the lth layer; k0 is the wave vector in vacuum, β is the propagation constant of the waveguide mode; β/k0 = neff is the effective refractive of the waveguide mode; m is the azimuthal mode num ber; and Al, Bl, Cl, and Dl are the coefficients for the lth layer. The electromagnetic field components that are tangential to the surface (Ez, Eφ, Hx, Hφ) must be con tinuous on the boundaries of layers. The boundary n
Figure 1 shows a radial profile of the refractive index in the Bragg fiber with a matching layer. The optical characteristics of this fiber are conveniently described using the transfermatrix method. The mathematical apparatus for this method is presented in detail elsewhere [1–5]. Below, we will use the trans fermatrix method as developed in [5].
n2
For an arbitrary homogeneous cylindrical layer, the axial components of the electromagnetic field (Ez, Hx) can be expressed using a linear combination of two cylindrical functions, namely, the Bessel and Hankel
n1 nco
1
functions of the first kind, i.e., Im(kr) and H m (kr). For the lth layer, these components can be written as fol lows: 870
d1 nh
d2
rco
rco + h
r
Fig. 1. Radial profile of refractive index in Bragg fiber with matching layer.
BRAGG FIBERS WITH MATCHING LAYER
conditions can be written in matrix form and each boundary features the corresponding transformation of the field components. Upon N consequent transfor mations (where N is the number of boundaries), one can obtain a matrix relation between AN + 1, BN + 1, CN + 1, and DN + 1 and the Al, Bl, Cl, and Dl values of the core. The condition of finiteness of the field compo nents on the fiber axis implies that Bl and Dl must be zero. The condition of absence of the reflected wave outside the fiber implies that AN + 1 and CN + 1 vanish as well. The obtained system of equations has a nontrivial solution only under certain conditions determined by the dispersion relation. This relation is used to deter mine the propagation constant β of the waveguide mode. Using the real part of β, we find the real part of effective refractive index for the mode, while the imag inary part of β can be used to calculate the optical losses [5]. For the present study, we used the wellknown Bragg fiber [3, 5, 9] and varied the thickness of the first layer of cladding. At a fixed structure of the remaining periodic cladding, the fundamental mode in the given fiber is TE0K, where K = 1 or 2. The HE11 mode exhib its much (several orders of magnitude) greater optical losses, while those of TM01 mode are even higher. The optical properties were calculated for a fiber with core refractive index nco = 1.0, matching layer refrac tive index nh = 1.49, and 31layer periodic cladding structure with d1 = 0.346 μm (n1 = 1.17) and d2 = 0.2146 μm (n1 = 1.49). Figure 2 shows the theoretical dependence of the optical losses γ (for TE0K mode) on the matching layer thickness h for λ0 = 1 μm and various values of the core radius rco. As can be seen, the optical losses exhibit a periodic dependence on the thickness of the matching layer and the minimum losses for each period are somewhat smaller than those for the preceding period. On the contrary, the maximum optical losses for TE0K mode in each subsequent period are significantly lower than those for the preceding period (curve 1). As the matching layer thickness h increases, the amplitude of variation of the optical losses within a period decreases. In manufacturing a Bragg fiber designed to guide a definite wavelength, the first cladding layer (adjacent to the core) may possess a significant thickness with out increasing optical losses. This circumstance can be used to obtain lowloss Bragg fibers, in particular those with a hollow core. The resonance condition for the matching layer can be written as follows [5]: k 2 h S = πs, (2) where k2 is the transverse component of the wave vec tor in the matching layer, s is an integer, and hs is the layer thickness corresponding to the sth resonance. For a fiber corresponding to curve 3 in Fig. 2, the values of h corresponding to maximum optical losses are 0.48, 0.92, 1.36, and 1.80 μm. The real part of the TECHNICAL PHYSICS LETTERS
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γ, dB/m 10
1 2 3 4
1 10−1 10−2 10−3 10−4 0
0.5
1.0 h, μm
1.5
2.0
Fig. 2. Dependence of the minimum optical losses γ for TE0K mode on the matching layer thickness h for radiation with wavelength λ = 1 μm in Bragg fiber with nco = 1.0, nh = 1.49, 31layer periodic cladding structure with d1 = 0.346 μm (n1 = 1.17) and d2 = 0.2133 μm (n2 = 1.49), and various core radii rco = 1.3278 (1), 1.525 (2), 1.8278 (3), and 9.0 μm (4).
effective refractive index of the mode is 0.9840, 0.9809, 0.9788, and 0.9771, respectively. The change in Reneff leads to small variations in the transverse component of the wave vector: k2 = 7029994.32, 7047076.22, 7058594.69, and 7067886.95 m–1, respectively. The corresponding s values calculated using formula (2) are 1.088, 2.074, 3.062, and 4.052. As the matching layer thickness grows, the maxima of optical losses are determined from the resonance con dition with increasing accuracy. For the minima of optical losses, the s values are as follows: 1.597, 2.596, 3.603, and 4.602. For curve 4 in Fig. 2, the s values corresponding to the sequential extrema of optical losses are as follows: 1.079, 1.587, 2.079, 2.587, 3.079, 3.588, and 4.072. With neglect of the insignificant variations in k2 for the adjacent resonances, the distance between these peaks is Δh = π/k2. For the average transverse component of the wave vector in the matching curve according to curve 3, the calculated distance between the resonances is Δh = 446.1 nm. The average distance between the maxima of optical losses for this curve in Fig. 2 is 437.7 nm. The obtained estimations suggest that the periodic dependence of optical losses on the thickness of the matching layer is related to the presence of resonances and antiresonances in this layer. Figure 3a shows the transmission spectrum of a Bragg fiber with rco = 1.8278 μm for three variants of the matching layer thickness. In order to construct these plots, TE0K mode with minimum optical losses was taken for each wavelength. For curve 1, the matching layer thickness is h = 0.2133 μm, which coincides with the thickness of periodic highindex layers (fiber B in [9]). The corresponding values for
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d2 = 1.9973 μm) exhibits several transmission bands in the interval of λ = 0.75–1.15 μm, which are related to the presence of antiresonances [5, 10]. For Bragg fibers with the matching layer, the trans mission spectra significantly differ from those of a standard Bragg fiber (Fig. 3, curve 1) only in some intervals of wavelengths, namely, λ = 0.7–0.85 μm for curve 2 and λ = 0.85–0.90 μm for curve 3. As can be seen, the curves exhibit repeated mutual intersections, but for the radiation with λ = 1 μm the minimum opti cal losses decrease with every period (Fig. 3b). Thus, by changing the parameters of the matching layer, it is possible to modify the optical losses in Bragg fibers. At an insignificant variation of the matching layer thickness along the fiber (for constant rco, d1, and d2), the optical characteristics of the fiber exhibit peri odic changes.
(a)
100 10 1 1 2 3
0.1 0.01 0.75
0.85
λ, dB/m 0.07
0.95 λ, μm
1.05
1.15
(b)
0.06 0.05
0.03 0.98
REFERENCES
1 2 3
0.04 0.99
1.00 λ, μm
1.01
1.02
Fig. 3. Calculated transmission spectrum of a Bragg fiber with rco = 1.8278 μm for three values of the matching layer thickness h = 0.2133 (1), 0.6936 (2) and 1.9973 μm (3).
curves 2 and 3 are h = 0.6936 and 1.9973 μm, respec tively. Figure 3b shows the region of spectrum in the vicinity of λ = 1 μm on a greater scale. An analysis of these spectra shows that a change in thickness of the first layer in the periodic cladding influences the properties of the Bragg fiber, which is manifested by (i) broadening of the transmission band (curve 2) and (ii) shift of the region of minimum opti cal losses (curve 3). The minimum optical losses for curve 1 amount to 0.027 dB/m at λ = 0.95 μm, while those for curve 3 amount to 0.018 dB/m at λ = 0.88 μm. The transmission spectrum of a Bragg fiber with large thickness of the matching layer (e.g., h =
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TECHNICAL PHYSICS LETTERS
Translated by P. Pozdeev
Vol. 37
No. 9
2011
