Midinfrared supercontinuum generation via As2Se3 chalcogenide photonic crystal fibers Hamed Saghaei,1,* Majid Ebnali-Heidari,2 and Mohammad Kazem Moravvej-Farshi3,4 1
Department of Electrical Engineering, Science and Research Branch, Islamic Azad University, Tehran, 1477893855, Iran 2
Faculty of Engineering, Shahrekord University, Shahrekord, 8818634141, Iran
3
Faculty of Electrical and Computer Engineering, Advanced Devices Simulation Lab, Tarbiat Modares University, P.O. Box 14115-194, Tehran, 1411713116, Iran 4
e-mail: [email protected]
*Corresponding author: [email protected] Received 9 October 2014; revised 9 January 2015; accepted 20 January 2015; posted 30 January 2015 (Doc. ID 224739); published 9 March 2015
Using numerical analysis, we compare the results of optofluidic and rod filling techniques for the broadening of supercontinuum spectra generated by As2 Se3 chalcogenide photonic crystal fibers (PCFs). The numerical results show that when air-holes constituting the innermost ring in a PCF made of As2 Se3 based chalcogenide glass are filled with rods of As2 S3 -based chalcogenide glass, over a wide range of midIR wavelengths, an ultra-flattened near-zero dispersion can be obtained, while the total loss is negligible and the PCF nonlinearity is very high. The simulations also show that when a 50 fs input optical pulse of 10 kW peak power and center wavelength of 4.6 μm is launched into a 50 mm long rod-filled chalcogenide PCF, a ripple-free spectral broadening as wide as 3.86 μm can be obtained. © 2015 Optical Society of America OCIS codes: (060.4370) Nonlinear optics, fibers; (060.5295) Photonic crystal fibers; (160.4330) Nonlinear optical materials; (260.2030) Dispersion; (320.6629) Supercontinuum generation. http://dx.doi.org/10.1364/AO.54.002072
1. Introduction
A very short and coherent optical pulse with relatively high power passing through a nonlinear material would generate a supercontinuum spectrum [1]. Supercontinuum generation has attracted much attention due to its numerous applications in various fields, including spread-spectrum communication based on dense wavelength-division multiplexing [2,3], noninvasive imaging of sensitive surfaces such as eye and skin based on optical coherence tomography [4], designing tunable ultrafast femtosecond laser sources [5], and precise measurement of optical frequencies [6]. The last application was deemed
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sufficiently important in the measurement science to merit part of the 2005 Nobel Prize in Physics. The advent of a new class of optical waveguides, in the late 1990 s, known as photonic crystal fibers (PCFs), has also attracted widespread interest throughout the scientific community, revolutionizing the generation of ultrabroadband high-brightness spectra [7,8]. The presence of the air-holes in the cladding of a solid-core PCF, whose core is basically made of the same material, makes the cladding effective refractive index to be smaller than that of the core. Therefore, the guiding mechanism is provided by total internal reflection along the solid core [9]. An advantage of a solid-core PCF over a conventional optical fiber is its capability of being engineered for a target dispersion [10]. Nonetheless, the possibility of modifying the cladding air-hole periodicity and dimensions offers additional degrees of freedom in
engineering the guiding properties and dispersion profile of a PCF, in a way that is not possible for a conventional optical fiber [11]. Typical PCFs due to their relatively small core diameters can guide single optical modes over a wide wavelength range. In recent years, a few research groups have demonstrated the possibility of generating supercontinuum spectra using silica-based PCFs [12–14]. They have been employed various means to broaden the generated supercontinuum spectra. One group used dispersion engineering by means of varying PCF structural parameters, like air-hole diameter, lattice constant, and the number of the air-hole rings, to achieve the intended supercontinuum [12,15,16]. Another group achieved the planned spectral broadening by decreasing the PCF effective area via a tapering technique [13]. In the most recent report, the present group has demonstrated the possibility of generating 1.2 μm wide supercontinuum via dispersion engineering of a 250 mm long silica PCF, by means of selective optofluidic infiltration of the PCFs’ air-holes as an alternative and reconfigurable post-fabrication approach that avoids any alteration in the PCF geometrical and structural parameters [14]. Unlike the first two approaches, the precise implementation of the optofluidic approach does not depend very much on the fabrication accuracy. In fact, any small deviation from the target design due to inaccuracies in devising air-holes during the fabrication processes that can lead to a significant deviation from the desired dispersion profile can be compensated by an appropriate choice of the optical fluid refractive index. Moreover, the possibility of using selective optofluidic infiltration of PCFs’ air-holes in obtaining an almost near-zero and flat dispersion profile has been demonstrated elsewhere by the same group [17]. Of course, the practical implementation of the optofluidic approach depends on the availability of optical fluids with the desired refractive indices that must be nontoxic, highly nonlinear, and transparent to the wavelengths of interest. Experimental techniques for filling one or more specific air-holes in a PCF have already been developed by the authors of [18,19]. All studies on supercontinuum generation via PCFs, reported so far, have used silica as their PCFs’ background material. Since material loss in silica increases exponentially for wavelengths greater than 2.5 μm, any intention of generating a supercontinuum centered about and beyond 2.5 μm becomes futile. Nevertheless, As2 Se3 chalcogenide with a relatively wider transmission window that cuts off at about 10 μm on the long-wavelength side, a nonlinear Kerr index more than 200 times larger than that of silica, and a relatively constant material loss could be used as an alternative background material for generating a supercontinuum in the mid-IR region. With this anticipation, we have used the same approach as in [14] to investigate the possibility of supercontinuum generation via un-infiltrated and selectively infiltrated PCFs made of As2 Se3
Table 1.
Material As2 Se3 As2 S3
Constants Ai and B i (μm2 ) (i 1, 2, and 3) for As2 Se3 and As2 S3 , used in Thompson’s Sellmeier Equation [20]
A1
A2
A3
4.994 0.120 1.710 1.8983 1.9222 0.8765
B1 (μm2 ) B2 (μm2 ) B3 (μm2 ) 0.0583 0.0225
361 0.0625
0.2332 0.1225
chalcogenide. Numerical results demonstrate the possibility of generating supercontinua with spectral widths greater than 3 μm with center wavelengths in the mid-IR range of 3–6 μm. The rest of this paper is organized as follows: in Section 2, the profiles of the wavelength dependencies of linear parameters like dispersion, total loss, and effective area, and the nonlinear parameter, of each of the un-infiltrated and infiltrated PCFs under study are plotted and compared. The nonlinear Schrödinger equation (NLSE) governing the optical pulse propagation across the As2 Se3 chalcogenide PCF is presented in Section 3. Then, using the numerical values for the linear and nonlinear parameters obtained in Section 2, the governing equation is solved numerically employing the symmetrized splitstep Fourier method (S-SSFM) across the PCFs. In Section 4, the spectral distributions of the PCFs outputs verses the center wavelengths of the input optical pulses in the mid-IR range are shown and compared. Finally, the paper is closed with a conclusion in Section 5. 2. PCFs’ Linear and Nonlinear Parameters
The real part of the linear refractive index of the chalcogenide materials based on both As2 Se3 and As2 S3 as a function of wavelength can be calculated using Thompson’s Sellmeier equation [20]: nλ f1 A1 λ2 λ2 − B1 −1 A2 λ2 λ2 − B2 −1 A3 λ2 λ2 − B3 −1 g1∕2 ;
(1)
where λ is the signal wavelength in free space and Ai and Bi (μm2 ) (i 1, 2, and 3) are material constants whose values for the As2 Se3 - and As2 S3 -based chalcogenide materials are given in Table 1. In this section, we consider four single-mode PCFs with various configurations whose background materials are all assumed to be made of As2 Se3 -based chalcogenide, as depicted in Fig. 1. The first uninfiltrated PCF, Fiber A, represented by Fig. 1(a), is assumed to have five rings of air-holes of the same diameter (dh 3.6 μm) arranged in a triangular lattice of constant Λ 5 μm, encompassing the PCF solid core of dc 2Λ − dh 6.4 μm. The number of rings of air-holes in this PCF is large enough that a change in the PCF’s dispersion would be negligible if another ring is added on the outer side of the cladding. The second un-infiltrated PCF, Fiber B, represented by Fig. 1(b), is assumed to be the same as Fiber A, except for the diameter of its solid core and also the diameters of the air-holes constituting its innermost ring, which are chosen to be dc 10 March 2015 / Vol. 54, No. 8 / APPLIED OPTICS
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makes it the most favorable fiber among the four proposed fibers under study. We are speculating that Fiber D can hopefully be fabricated, similar to that proposed for silica PCFs by Knight [11], by stacking 86 capillary As2 Se3 tubes (equal to the number of air-holes) and an As2 Se3 rod (for the core) surrounded by six As2 S3 rods (forming the innermost ring). The preform speculated can be drawn in a high-temperature drawing tower to form the desired PCF and can at last be coated by a standard protective jacket like that in [23]. Moreover, the chromatic dispersion of a PCF, also known as the group velocity dispersion, including both the waveguide and material dispersions, experienced by the fundamental mode of wavelength λ, is determined by [24] λ d2 Reneff λ c dλ2 2πc − 2 β2 ps · nm · km−1 ; λ
Dλ −
Fig. 1. Cross-sectional view of the As2 Se3 chalcogenide PCF with a triangular lattice of constant Λ 5 μm, consisting of five hexagonal rings: (a) fiber A with un-infiltrated air-holes of the same diameter (d 3.6 μm); (b) Fiber B with smaller air-holes in its innermost ring (d1 3 μm); (c) fiber C, which is the same as Fiber A, except the infiltrated air-holes in its innermost ring; (d) fiber D, which is the same as Fiber A, except the air-holes in its innermost ring, which are filled with As2 S3 rods.
7 μm and dh1 3 μm, respectively. As shown already in [12], a decrease in the diameters of the air-holes constituting the innermost ring of the PCF is expected to increase its effective mode area, resulting in reduced confinement loss as well as reduced dispersion. Practical implementation of this scheme, which is solely considered for the sake of comparison, as mentioned earlier in Section 1, is very much dependent on the precision of the fabrication processes in accurate positioning and devising of the smaller air-holes. Fiber C shown in Fig. 1(c) depicts a crosssectional view of a PCF with the same geometrical dimensions as Fiber A, whose innermost air-hole rings are infiltrated selectively with an appropriate choice of optical fluid. As demonstrated already in [14], appropriate selective infiltration prevents leakage of light into the cladding region by confining the guided optical mode within the core region, as desired. Nevertheless, the real part of the linear refractive indices of the optical microfluids, available in the market, is almost wavelength independent over a wide range of wavelengths [21,22], unlike those of As2 Se3 - and As2 S3 -based chalcogenide materials. An alternative approach to overcome this deficiency while reducing and flattening its dispersion via increasing the PCF effective mode area is to refill the air-holes of the innermost ring of Fiber A with rods made of As2 S3, having a similar dispersive behavior to that of the PCF background material, As2 Se3 . This property enabling Fiber D to control and engineer the dispersion and loss profiles, simultaneously, 2074
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(2)
where c, β2 , and neff λ are the light velocity in free space, the second-order dispersion, and the PCFs effective index, respectively. Higher order dispersions become important when the input pulse center wavelength approaches the zero-dispersion wavelength— i.e., when β2 0—and are given by βm
dm β ; dωm ωω0
(3)
where ω0 is the light center frequency. In order to calculate the neff λ for each of the four proposed PCFs, we employ a full-vector modal solver based on the finite-difference time-domain numerical method. Then using Eq. (2), dispersion profile as a function of wavelength can be calculated. Moreover, in order to reduce the simulation window and to evaluate the confinement losses of the proposed fibers, we assume the anisotropic perfectly matched layers (absorbing boundaries) to be positioned outside the outermost ring of the air-holes. Figure 2 shows the dispersion profiles of all four PCFs in terms of the input signal wavelengths. For Fiber C, due to fluid availability over the wide range of refractive indices, three different refractive indices have been studied. It should be noted that the maximum value of refractive index that can be infiltrated by capillary force through the PCF is equal to 1.8 [25]. As can be observed in the figure, going from Fiber A to Fiber D, the slope of the dispersion profile and the dispersion value both decrease. Fiber C with an nF of 1.8 has both a lower and flatter dispersion profile than the other infiltrated fluids mentioned (i.e., nF 1.32 and 1.6). Therefore, this refractive index of fluid will be considered in the rest of paper. Fibers C and D enjoy nearly flattened near-zero dispersion in the vast wavelength ranges of 3 μm and 5 μm, respectively. Nevertheless, as anticipated,
Fig. 2. Dispersion profiles versus wavelength for all proposed fibers shown in Fig. 1.
the dispersion profile of Fiber D has been engineered most favorably. Figure 3 shows the refractive indices of all materials used in this paper as a function of wavelength. We have assumed that the fluids are almost wavelength independent and have constant values over a wide wavelength range [21,25]. However, using Eq. (1), one can easily conclude that for either As2 S3 or As2 Se3, nλ decreases as λ increases, and also, that both have the same behavior. The failure of achieving a near-zero flattened dispersion profile for the As2 Se3 -based chalcogenide PCF, via selective optofluidic infiltration, is the consequence of the unavailability of the desired optical fluid having a linear refractive index with similar dispersion behavior to that of the PCF background material. Another linear parameter of interest that needs to be examined is the PCF’s total loss, L Lm Lc , with Lm and Lc being the material and confinement losses, respectively, all in dB/m. The wavelength dependencies of the material losses of As2 Se3 - and As2 S3 -based chalcogenide materials are well known [26–29]. The background material for all proposed
Fig. 4. Total loss versus wavelength for all four PCFs under study.
fibers is the same in this study. Nevertheless, confinement loss, which causes the main difference observed among the total losses of the different fibers, depends on the PCF structural parameters, like lattice types and constants, air-hole diameters, and the refractive index of the infiltrated fluid for Fiber C and that of the As2 S3 rods for Fiber D. In our calculations, confinement loss can be determined as LC λ 8.686k0 Im neff λ[12], with k0 2π∕λ being the free-space wavenumber. Figure 4 shows the wavelength dependencies of the total loss of each of the four fibers under study. As can be observed in the figure, the total loss for Fiber D is less than those obtained for the other three fibers, over the vast wavelength range of 6–10 μm. This is due to its larger effective cross-sectional area caused by refilling of the air-holes of its innermost ring with rods of As2 S3 glass having a refractive index very close to but smaller than that of the PCF background material. An increase in the PCF effective cross-sectional area, for a signal of a given wavelength, prevents the guided light to leak into the cladding region, and hence, lessens the PCF confinement loss. Having obtained the wavelength dependencies of the linear parameters of the four fibers under study, we now look into the profile of their nonlinear parameter, defined as [20] γW · m−1 n2 ω0 ∕cAeff λ;
(4)
where c and ω0 are the speed of light and its radian frequency in free space, n2 is the nonlinear Kerr index of the PCF constituents, and Aeff λ represents the wavelength dependency of the PCF effective cross-sectional area, defined as [20] RR ∞ Aeff λ
−∞
RR ∞ −∞
Fig. 3. Refractive index versus wavelength for all materials used in this paper.
2 jFx; yj2 dxdy jFx; yj4 dxdy
;
(5)
where Fx; y is the optical field distribution across the PCF cross section. The numerical values of the 10 March 2015 / Vol. 54, No. 8 / APPLIED OPTICS
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Fig. 5. Effective area versus wavelength for all four PCFs under study.
PCF effective cross section calculated for all four fibers are depicted in Fig. 5. As anticipated and as discussed earlier in this section, among the four given fibers, Fiber D has the largest effective cross-sectional area. The nonlinear parameter related to all fibers is plotted in Fig. 6. As can be observed in this figure, to our expectations, going from Fiber A to Fiber D, the nonlinear parameter decreases over the entire wavelength range of 2–10 μm. Up to λ 3.5 μm, this parameter is nearly the same for Fibers C and D. However, for wavelengths λ > 3.5 μm, the nonlinear parameter for Fiber D becomes smaller than that for Fiber A. When the nonlinear parameter of the mentioned fibers has almost the same values, the dispersion profiles would determine which scheme yields a wider spectrum at the end of the fiber; this is discussed in the next section. In order to increase the bandwidth of the supercontinuum spectrum, it is important to have a single-mode PCF in a wide wavelength range. If several modes exist in the PCF, the fundamental mode will couple with higher order modes that have a larger effective area.
It reduces the nonlinearity interactions along the fiber, which are needed for supercontinuum generation. So, the fiber should be designed to operate in a single-mode regime [26]. Note that for a fiber to be single-mode, it needs to be designed with a combination of a small core size-to-wavelength ratio and a small difference in refractive indices between the core and the cladding. We examine and calculate the mode distribution for all proposed fibers. Our results confirm that all fibers are single-mode in the wavelength range of interest. Further, a 2D schematic view of the fundamental mode distribution for Fiber D is shown in the inset in Fig. 5. This fundamental mode is similar in each length of the fiber. Our numerical results, reported elsewhere, demonstrated that if the second and third inner rings of air-holes are infiltrated by a fluid, then the PCF operates as multimode and its nonlinearity is decreased [17]. So, we may not see supercontinuum generation at the end of the fiber. 3. Mathematical Background
In this section, by extracting both linear parameters, including dispersion (from second to eighth order, as shown in Table 2) and loss, and the nonlinear parameter for the proposed fibers shown in Fig. 1, the general NLSE (GNLSE) will be solved using the symmetrized S-SSFM, to describe the propagation of femtosecond pulses inside the fiber and to broaden the generated supercontinuum. The GNLSE is described as follows [20]: 10 X βq ∂q A ∂A α i ∂ q−1 A × Az; t i iγ 1 q! ∂tq ∂z 2 ω0 ∂t q1 Z∞ Rt0 jAz; t − t0 j2 dt0 ; × −∞
(6) where z and t are the spatial coordinate along the fiber and the time variable, α is the total fiber loss, βq represents the qth-order dispersion parameter, and γ is the nonlinear parameter. Moreover, Rt in the integrand on the right-hand side of the equation is the response function including the Kerr and Raman contributions. Assuming that the electronic contribution is nearly instantaneous, the functional form of Rt can be written as Table 2.
Values of the Dispersion Orders, β2 –β8 , of all Four Proposed PCFs at the Desired Pump Wavelengths, λ
Fiber A βn (sn ∕m) (λ 3.6 μm)
Fig. 6. Nonlinear parameter versus wavelength for all proposed fibers shown in Fig. 1. 2076
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β2 β3 β4 β5 β6 β7 β8
9 × 10−26 2.33 × 10−39 −1.07 × 10−53 9.95 × 10−68 −1.06 × 10−81 1.37 × 10−95 −1.77 × 10−109
Fiber B (λ 4 μm)
Fiber C (λ 4 μm)
Fiber D (λ 4.6 μm)
1.99 × 10−26 2.64 × 10−39 −1.44 × 10−53 1.41 × 10−67 −1.64 × 10−81 2.57 × 10−95 −5 × 10−109
1.16 × 10−26 2.57 × 10−39 −1.28 × 10−53 1.17 × 10−67 −1.2 × 10−81 1.67 × 10−95 −3.2 × 10−109
1.14 × 10−26 1.96 × 10−39 −4.55 × 10−54 −1.52 × 10−68 1.09 × 10−81 −2.66 × 10−95 4.56 × 10−109
Rt 1 − f R δt f R hR t;
(7)
where δt is the Dirac delta function and f R represents the fractional contribution of the delayed Raman response function hR t, which in turn takes the approximate analytic form of τ21 τ22 t t exp − sin ; hR t 2 τ τ τ1 τ2 2 2
(8)
with f R 0.148, τ1 0.23 fs, and τ2 164.5 fs, for As2 Se3 [26,30]. The response time for the As2 Se3 chalcogenide fiber is longer than the response time for the silica fiber; so the Raman response must be taken into our account. We will use the nonlinear response function throughout the remainder of this paper. Using the S-SSFM, one can obtain the numerical solution to Eq. (6) [20]. The evolution of an input pulse, having a slowly varying electric field amplitude represented by the envelope function Az; t, propagating along the fibers in Fig. 1 is given by p t Az 0; t P0 sech ≡ T0
s jβ2 j N t sech ; γ T0 T0 (9)
where P0 , T 0 , and N are the peak power, time width, and soliton order of the input source pump, respectively. The soliton order, N, is usually determined as p N T 0 γP0 ∕jβ2 j:
(10)
A small perturbation, such as a higher order dispersion, self-steepening, and stimulated Raman scattering, can break up a given soliton of order N into N fundamental solitons [20]. As a result of the self-frequency shift induced by Raman scattering, the center wavelengths of the resulting fundamental solitons continuously shift toward the longer wavelengths of the spectrum (red side). On the other hand, nonsolitonic radiations result in a broadening on the blue side of the spectrum [31]. These nonlinear processes are dominant, in a PCF of length L, when the nonlinear length (LNL ) and the dispersion lengths (LD ) satisfy the conditions LNL < LD and LNL < L. Under such conditions, initially, self-phase modulation leads to a symmetric spectral broadening of the optical pulse, after which the requirement for phase matching conditions is fulfilled, and then soliton fission occurs, and finally, the self-frequency shift and nonsolitonic radiations processes broaden the spectral components on both sides of the spectrum, as a consequence of which a supercontinuum is generated. 4. Numerical Simulation
In this section, to have the benefit of a lower average power and a wider bandwidth, an optical pulse as that described by Eq. (9) of 50 fs and 10 kW peak
Fig. 7. Spectral broadening versus wavelength for a 50 fs input pulse of peak power P0 10 kW launched into all four PCFs under study, of the same length of 50 mm.
power is considered. The supercontinuum generation for each of the proposed fibers in a wide wavelength range of input pulses has been calculated numerically over a length of 50 mm of the optical fibers as depicted in Fig. 7. In the figure, we can observe that the maximum spectral bandwidth for Fiber A has been obtained to be around 1.9 μm for an input wavelength of 3.6 μm. These values for Fibers B and C are 2.84 μm and 2.91 μm, respectively for a 4 μm input pulse. For Fiber D, maximum spectral bandwidth of 3.86 μm for a 4.6 μm input pulse has been achieved. As can be observed in the figure, going from Fiber A to Fiber D, the value of the spectral bandwidth increases more than 100%. This result can be expected from the ultraflattened near-zero dispersion profile that is calculated for Fiber D in Section 2. In order to justify the ignoring of the nonlinear Kerr refractive index in As2 S3 , for an input pulse with peak power of 10 kW at 4.6 μm traveling along Fiber D, we have taken three snapshots of the pulse power distribution in x–y (top) and x–z and y–z (sides) cross-sectional views. These snapshots are shown in Figs. 8(a), 8(b), and 8(c) (Media 1, Media 2, and Media 3), respectively. As can be observed in the figure, the power leakage into the As2 S3 rods surrounding the As2 Se3 core of Fiber D is negligible. Figure 9 shows the propagation of optical pulses with input wavelengths corresponding to the maximum spectral bandwidth along the proposed schemes shown in Fig. 1. It can be observed that Fiber D has a flat and ripple-free spectral broadening along the fiber, which is suitable for supercontinuum applications. In this fiber, the pulse initially starts to broaden symmetrically, due to the self-phase modulation effect. Then, after the input pulse traverses a distance of about 10 mm along the fiber, due to higher order dispersions and the self-steepening nonlinear effect, it becomes unstable and soliton fission occurs, as a consequence of which fundamental solitons of various center wavelengths are formed. These 10 March 2015 / Vol. 54, No. 8 / APPLIED OPTICS
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Fig. 8. Snapshots of power distributions when an input pulse with 10 kW peak power and center wavelength of 4.6 μm is propagating along Fiber D, in (a) x–y (top) view (Media 1), (b) x–z (side) view (Media 2), and (c) y–z (side) view (Media 3). The color bar to the right indicates the normalized intensity.
Fig. 9. Evolution of the spectral distribution of a 50 fs optical pulse of P0 10 kW peak power and center wavelength of (a) λ 3.6 μm along Fiber A, (b) λ 4 μm along Fiber B, (c) λ 4 μm along Fiber C for a fluid of refractive index nF 1.8, and (d) λ 4.6 μm along Fiber D. All PCFs are 50 mm long.
isolated optical pulses are temporally narrower and hence spectrally broader than the pulse from which they originate, and hence, a broader bandwidth is created. Then, after traveling another 20 mm (i.e., Table 3. Bandwidths of the Supercontinua Generated by the Optical Signal at Various Lengths of all Four Proposed Fibers Measured at the −30 dB Level, as Depicted in Fig. 9
Supercontinuum Bandwidth (μm) Measured at −30 dB Fiber Length (mm) 10 20 30 40 50
2078
Fiber A (3.6 μm)
Fiber B (4 μm)
Fiber C (4 μm)
Fiber D (4.6 μm)
2.2 1.85 1.75 1.71 1.9
2.01 2.23 2.61 2.75 2.83
2.1 2.55 2.67 2.86 2.91
2.23 3.25 3.67 3.76 3.86
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at z 30 mm), the spectrum is spread toward wavelengths smaller than the pump wavelength, where the resonance condition prevails and due to transfer of energy to a higher frequency, dispersive radiation occurs. Finally, at about z 40 mm, stimulated Raman scattering comes into the picture and the most intense soliton slowly shifts toward longer wavelengths. All expressed phenomena result in supercontinuum generation at the end of the fiber. For more details on the proposed schemes, the bandwidths calculated at the level of −30 dB for the various cases illustrated in Fig. 9 are tabulated in Table 3. 5. Conclusion
We have studied different types of As2 Se3 chalcogenide PCFs for supercontinuum generation, using optofluidic and rod filling techniques. Linear profiles,
including the PCFs’ losses and dispersion, and the nonlinear parameter, have been calculated and have been discussed for each of the PCFs under study. The simulations show that when a 50 fs input optical pulse of peak power 10 kW and center wavelength of 4.6 μm is launched into a 50 mm long rod-filled chalcogenide PCF, a ripple-free spectral broadening as wide as 3.86 μm can be obtained, whereas the maximum spectral bandwidth of the supercontinuum of the As2 Se3 -based chalcogenide PCF whose innermost ring air-holes are infiltrated with an optical fluid is 2.93 μm. The advantage of chalcogenidebased PCFs in supercontinuum generation lies in their relatively constant material loss over a wide transmission window that cuts off at about 10 μm, as compared with an exponentially increasing material loss in silica-based PCFs for wavelengths beyond 2.5 μm. This together with their nonlinear Kerr index, which is more than 200 times larger than that of silica, makes As2 Se3 chalcogenide PCFs promising alternative media for supercontinuum applications in the mid-IR region. References 1. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006). 2. H. Saghaei, B. Seyfe, H. Bakhshi, and R. Bayat, “Novel approach to adjust the step size for closed-loop power control in wireless cellular code division multiple access systems under flat fading,” IET Commun. 5, 1469–1483 (2011). 3. C. A. Brackett, “Dense wavelength division multiplexing networks: principles and applications,” IEEE J. Sel. Areas Commun. 8, 948–964 (1990). 4. G. Humbert, W. Wadsworth, S. Leon-Saval, J. Knight, T. Birks, P. St. J. Russell, M. Lederer, D. Kopf, K. Wiesauer, and E. Breuer, “Supercontinuum generation system for optical coherence tomography based on tapered photonic crystal fibre,” Opt. Express 14, 1596–1603 (2006). 5. B. W. Liu, M. L. Hu, X. H. Fang, Y. Z. Wu, Y. J. Song, L. Chai, C. Y. Wang, and A. Zheltikov, “High-power wavelength-tunable photonic-crystal-fiber-based oscillator-amplifier-frequencyshifter femtosecond laser system and its applications for material microprocessing,” Laser Phys. Lett. 6, 44–48 (2009). 6. T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416, 233–237 (2002). 7. J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Visible continuum generation in air–silica microstructure optical fibers with anomalous dispersion at 800 nm,” Opt. Lett. 25, 25–27 (2000). 8. W. J. Wadsworth, A. Ortigosa-Blanch, J. C. Knight, T. A. Birks, T. P. M. Man, and P. S. J. Russell, “Supercontinuum generation in photonic crystal fibers and optical fiber tapers: a novel light source,” J. Opt. Soc. Am. B 19, 2148–2155 (2002). 9. B. Eggleton, C. Kerbage, P. Westbrook, R. Windeler, and A. Hale, “Microstructured optical fiber devices,” Opt. Express 9, 698–713 (2001). 10. W. Reeves, D. Skryabin, F. Biancalana, J. Knight, P. S. J. Russell, F. Omenetto, A. Efimov, and A. Taylor, “Transformation and control of ultra-short pulses in dispersion-engineered photonic crystal fibres,” Nature 424, 511–515 (2003). 11. J. C. Knight, “Photonic crystal fibres,” Nature 424, 847–851 (2003).
12. F. Begum, Y. Namihira, T. Kinjo, and S. Kaijage, “Supercontinuum generation in square photonic crystal fiber with nearly zero ultra-flattened chromatic dispersion and fabrication tolerance analysis,” Opt. Commun. 284, 965–970 (2011). 13. U. Møller, S. T. Sørensen, C. Larsen, P. M. Moselund, C. Jakobsen, J. Johansen, C. L. Thomsen, and O. Bang, “Optimum PCF tapers for blue-enhanced supercontinuum sources,” Opt. Fiber Technol. 18, 304–314 (2012). 14. M. Ebnali-Heidari, H. Saghaei, F. Koohi-Kamali, M. NaserMoghadasi, and M. K. Moravvej-Farshi, “Proposal for supercontinuum generation by optofluidic infiltrated photonic crystal fibers,” IEEE J. Sel. Top. Quantum Electron. 20, 582 (2014). 15. K. Saitoh, M. Koshiba, T. Hasegawa, and E. Sasaoka, “Chromatic dispersion control in photonic crystal fibers: application to ultra-flattened dispersion,” Opt. Express 11, 843–852 (2003). 16. M. Tiwari and V. Janyani, “Two-octave spanning supercontinuum in a soft glass photonic crystal fiber suitable for 1.55 m pumping,” J. Lightwave Technol. 29, 3560–3565 (2011). 17. M. Ebnali-Heidari, F. Dehghan, H. Saghaei, F. Koohi-Kamali, and M. Moravvej-Farshi, “Dispersion engineering of photonic crystal fibers by means of fluidic infiltration,” J. Mod. Opt. 59, 1384–1390 (2012). 18. F. Wang, S. W. Yuan, O. Hansen, and O. Bang, “Selective filling of photonic crystal fibers using focused ion beam milled microchannels,” Opt. Express 19, 17585–17590 (2011). 19. K. Nielsen, D. Noordegraaf, T. Sørensen, A. Bjarklev, and T. P. Hansen, “Selective filling of photonic crystal fibres,” J. Opt. A 7, L13–L20 (2005). 20. G. P. Agrawal, Nonlinear Fiber Optics (Springer, 2000). 21. H. P. Zappe, Fundamentals of Micro-optics (Cambridge University, 2010), Vol. 90. 22. R. Synowicki, G. K. Pribil, G. Cooney, C. M. Herzinger, S. E. Green, R. H. French, M. K. Yang, J. H. Burnett, and S. Kaplan, “Fluid refractive index measurements using rough surface and prism minimum deviation techniques,” J. Vac. Sci. Technol. B 22, 3450–3453 (2004). 23. NKTPhotonics, “Crystal fibre technology tutorial,” http://www .nktphotonics.com/side5302.html. 24. G. P. Agrawal, Fiber-optic Communication Systems (Wiley, 2011), Vol. 222. 25. M. N. Polyanskiy, “Refractive index database,” retrieved http://refractiveindex.info. 26. J. Hu, C. R. Menyuk, L. B. Shaw, J. S. Sanghera, and I. D. Aggarwal, “Maximizing the bandwidth of supercontinuum generation in As2Se3 chalcogenide fibers,” Opt. Express 18, 6722–6739 (2010). 27. V. Q. Nguyen, J. S. Sanghera, F. H. Kung, I. D. Aggarwal, and I. K. Lloyd, “Effect of temperature on the absorption loss of chalcogenide glass fibers,” Appl. Opt. 38, 3206–3213 (1999). 28. Q. Coulombier, L. Brilland, P. Houizot, T. Chartier, T. N. N’Guyen, F. Smektala, G. Renversez, A. Monteville, D. Méchin, and T. Pain, “Casting method for producing low-loss chalcogenide microstructured optical fibers,” Opt. Express 18, 9107–9112 (2010). 29. S. Madden, D.-Y. Choi, D. Bulla, A. V. Rode, B. Luther-Davies, V. G. Ta’eed, M. Pelusi, and B. J. Eggleton, “Long, low loss etched As2S3 chalcogenide waveguides for all-optical signal regeneration,” Opt. Express 15, 14414–14421 (2007). 30. R. Cherif and M. Zghal, “Ultrabroadband, midinfrared supercontinuum generation in dispersion engineered As2Se3-based chalcogenide photonic crystal fibers,” Int. J. Opt. 2013, 876474 (2013). 31. A. V. Husakou and J. Herrmann, “Supercontinuum generation, four-wave mixing, and fission of higher-order solitons in photonic-crystal fibers,” J. Opt. Soc. Am. B 19, 2171– 2182 (2002).
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Department of Electrical Engineering, Science and Research Branch, Islamic Azad University, Tehran, 1477893855, Iran 2
Faculty of Engineering, Shahrekord University, Shahrekord, 8818634141, Iran
3
Faculty of Electrical and Computer Engineering, Advanced Devices Simulation Lab, Tarbiat Modares University, P.O. Box 14115-194, Tehran, 1411713116, Iran 4
e-mail: [email protected]
*Corresponding author: [email protected] Received 9 October 2014; revised 9 January 2015; accepted 20 January 2015; posted 30 January 2015 (Doc. ID 224739); published 9 March 2015
Using numerical analysis, we compare the results of optofluidic and rod filling techniques for the broadening of supercontinuum spectra generated by As2 Se3 chalcogenide photonic crystal fibers (PCFs). The numerical results show that when air-holes constituting the innermost ring in a PCF made of As2 Se3 based chalcogenide glass are filled with rods of As2 S3 -based chalcogenide glass, over a wide range of midIR wavelengths, an ultra-flattened near-zero dispersion can be obtained, while the total loss is negligible and the PCF nonlinearity is very high. The simulations also show that when a 50 fs input optical pulse of 10 kW peak power and center wavelength of 4.6 μm is launched into a 50 mm long rod-filled chalcogenide PCF, a ripple-free spectral broadening as wide as 3.86 μm can be obtained. © 2015 Optical Society of America OCIS codes: (060.4370) Nonlinear optics, fibers; (060.5295) Photonic crystal fibers; (160.4330) Nonlinear optical materials; (260.2030) Dispersion; (320.6629) Supercontinuum generation. http://dx.doi.org/10.1364/AO.54.002072
1. Introduction
A very short and coherent optical pulse with relatively high power passing through a nonlinear material would generate a supercontinuum spectrum [1]. Supercontinuum generation has attracted much attention due to its numerous applications in various fields, including spread-spectrum communication based on dense wavelength-division multiplexing [2,3], noninvasive imaging of sensitive surfaces such as eye and skin based on optical coherence tomography [4], designing tunable ultrafast femtosecond laser sources [5], and precise measurement of optical frequencies [6]. The last application was deemed
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sufficiently important in the measurement science to merit part of the 2005 Nobel Prize in Physics. The advent of a new class of optical waveguides, in the late 1990 s, known as photonic crystal fibers (PCFs), has also attracted widespread interest throughout the scientific community, revolutionizing the generation of ultrabroadband high-brightness spectra [7,8]. The presence of the air-holes in the cladding of a solid-core PCF, whose core is basically made of the same material, makes the cladding effective refractive index to be smaller than that of the core. Therefore, the guiding mechanism is provided by total internal reflection along the solid core [9]. An advantage of a solid-core PCF over a conventional optical fiber is its capability of being engineered for a target dispersion [10]. Nonetheless, the possibility of modifying the cladding air-hole periodicity and dimensions offers additional degrees of freedom in
engineering the guiding properties and dispersion profile of a PCF, in a way that is not possible for a conventional optical fiber [11]. Typical PCFs due to their relatively small core diameters can guide single optical modes over a wide wavelength range. In recent years, a few research groups have demonstrated the possibility of generating supercontinuum spectra using silica-based PCFs [12–14]. They have been employed various means to broaden the generated supercontinuum spectra. One group used dispersion engineering by means of varying PCF structural parameters, like air-hole diameter, lattice constant, and the number of the air-hole rings, to achieve the intended supercontinuum [12,15,16]. Another group achieved the planned spectral broadening by decreasing the PCF effective area via a tapering technique [13]. In the most recent report, the present group has demonstrated the possibility of generating 1.2 μm wide supercontinuum via dispersion engineering of a 250 mm long silica PCF, by means of selective optofluidic infiltration of the PCFs’ air-holes as an alternative and reconfigurable post-fabrication approach that avoids any alteration in the PCF geometrical and structural parameters [14]. Unlike the first two approaches, the precise implementation of the optofluidic approach does not depend very much on the fabrication accuracy. In fact, any small deviation from the target design due to inaccuracies in devising air-holes during the fabrication processes that can lead to a significant deviation from the desired dispersion profile can be compensated by an appropriate choice of the optical fluid refractive index. Moreover, the possibility of using selective optofluidic infiltration of PCFs’ air-holes in obtaining an almost near-zero and flat dispersion profile has been demonstrated elsewhere by the same group [17]. Of course, the practical implementation of the optofluidic approach depends on the availability of optical fluids with the desired refractive indices that must be nontoxic, highly nonlinear, and transparent to the wavelengths of interest. Experimental techniques for filling one or more specific air-holes in a PCF have already been developed by the authors of [18,19]. All studies on supercontinuum generation via PCFs, reported so far, have used silica as their PCFs’ background material. Since material loss in silica increases exponentially for wavelengths greater than 2.5 μm, any intention of generating a supercontinuum centered about and beyond 2.5 μm becomes futile. Nevertheless, As2 Se3 chalcogenide with a relatively wider transmission window that cuts off at about 10 μm on the long-wavelength side, a nonlinear Kerr index more than 200 times larger than that of silica, and a relatively constant material loss could be used as an alternative background material for generating a supercontinuum in the mid-IR region. With this anticipation, we have used the same approach as in [14] to investigate the possibility of supercontinuum generation via un-infiltrated and selectively infiltrated PCFs made of As2 Se3
Table 1.
Material As2 Se3 As2 S3
Constants Ai and B i (μm2 ) (i 1, 2, and 3) for As2 Se3 and As2 S3 , used in Thompson’s Sellmeier Equation [20]
A1
A2
A3
4.994 0.120 1.710 1.8983 1.9222 0.8765
B1 (μm2 ) B2 (μm2 ) B3 (μm2 ) 0.0583 0.0225
361 0.0625
0.2332 0.1225
chalcogenide. Numerical results demonstrate the possibility of generating supercontinua with spectral widths greater than 3 μm with center wavelengths in the mid-IR range of 3–6 μm. The rest of this paper is organized as follows: in Section 2, the profiles of the wavelength dependencies of linear parameters like dispersion, total loss, and effective area, and the nonlinear parameter, of each of the un-infiltrated and infiltrated PCFs under study are plotted and compared. The nonlinear Schrödinger equation (NLSE) governing the optical pulse propagation across the As2 Se3 chalcogenide PCF is presented in Section 3. Then, using the numerical values for the linear and nonlinear parameters obtained in Section 2, the governing equation is solved numerically employing the symmetrized splitstep Fourier method (S-SSFM) across the PCFs. In Section 4, the spectral distributions of the PCFs outputs verses the center wavelengths of the input optical pulses in the mid-IR range are shown and compared. Finally, the paper is closed with a conclusion in Section 5. 2. PCFs’ Linear and Nonlinear Parameters
The real part of the linear refractive index of the chalcogenide materials based on both As2 Se3 and As2 S3 as a function of wavelength can be calculated using Thompson’s Sellmeier equation [20]: nλ f1 A1 λ2 λ2 − B1 −1 A2 λ2 λ2 − B2 −1 A3 λ2 λ2 − B3 −1 g1∕2 ;
(1)
where λ is the signal wavelength in free space and Ai and Bi (μm2 ) (i 1, 2, and 3) are material constants whose values for the As2 Se3 - and As2 S3 -based chalcogenide materials are given in Table 1. In this section, we consider four single-mode PCFs with various configurations whose background materials are all assumed to be made of As2 Se3 -based chalcogenide, as depicted in Fig. 1. The first uninfiltrated PCF, Fiber A, represented by Fig. 1(a), is assumed to have five rings of air-holes of the same diameter (dh 3.6 μm) arranged in a triangular lattice of constant Λ 5 μm, encompassing the PCF solid core of dc 2Λ − dh 6.4 μm. The number of rings of air-holes in this PCF is large enough that a change in the PCF’s dispersion would be negligible if another ring is added on the outer side of the cladding. The second un-infiltrated PCF, Fiber B, represented by Fig. 1(b), is assumed to be the same as Fiber A, except for the diameter of its solid core and also the diameters of the air-holes constituting its innermost ring, which are chosen to be dc 10 March 2015 / Vol. 54, No. 8 / APPLIED OPTICS
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makes it the most favorable fiber among the four proposed fibers under study. We are speculating that Fiber D can hopefully be fabricated, similar to that proposed for silica PCFs by Knight [11], by stacking 86 capillary As2 Se3 tubes (equal to the number of air-holes) and an As2 Se3 rod (for the core) surrounded by six As2 S3 rods (forming the innermost ring). The preform speculated can be drawn in a high-temperature drawing tower to form the desired PCF and can at last be coated by a standard protective jacket like that in [23]. Moreover, the chromatic dispersion of a PCF, also known as the group velocity dispersion, including both the waveguide and material dispersions, experienced by the fundamental mode of wavelength λ, is determined by [24] λ d2 Reneff λ c dλ2 2πc − 2 β2 ps · nm · km−1 ; λ
Dλ −
Fig. 1. Cross-sectional view of the As2 Se3 chalcogenide PCF with a triangular lattice of constant Λ 5 μm, consisting of five hexagonal rings: (a) fiber A with un-infiltrated air-holes of the same diameter (d 3.6 μm); (b) Fiber B with smaller air-holes in its innermost ring (d1 3 μm); (c) fiber C, which is the same as Fiber A, except the infiltrated air-holes in its innermost ring; (d) fiber D, which is the same as Fiber A, except the air-holes in its innermost ring, which are filled with As2 S3 rods.
7 μm and dh1 3 μm, respectively. As shown already in [12], a decrease in the diameters of the air-holes constituting the innermost ring of the PCF is expected to increase its effective mode area, resulting in reduced confinement loss as well as reduced dispersion. Practical implementation of this scheme, which is solely considered for the sake of comparison, as mentioned earlier in Section 1, is very much dependent on the precision of the fabrication processes in accurate positioning and devising of the smaller air-holes. Fiber C shown in Fig. 1(c) depicts a crosssectional view of a PCF with the same geometrical dimensions as Fiber A, whose innermost air-hole rings are infiltrated selectively with an appropriate choice of optical fluid. As demonstrated already in [14], appropriate selective infiltration prevents leakage of light into the cladding region by confining the guided optical mode within the core region, as desired. Nevertheless, the real part of the linear refractive indices of the optical microfluids, available in the market, is almost wavelength independent over a wide range of wavelengths [21,22], unlike those of As2 Se3 - and As2 S3 -based chalcogenide materials. An alternative approach to overcome this deficiency while reducing and flattening its dispersion via increasing the PCF effective mode area is to refill the air-holes of the innermost ring of Fiber A with rods made of As2 S3, having a similar dispersive behavior to that of the PCF background material, As2 Se3 . This property enabling Fiber D to control and engineer the dispersion and loss profiles, simultaneously, 2074
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(2)
where c, β2 , and neff λ are the light velocity in free space, the second-order dispersion, and the PCFs effective index, respectively. Higher order dispersions become important when the input pulse center wavelength approaches the zero-dispersion wavelength— i.e., when β2 0—and are given by βm
dm β ; dωm ωω0
(3)
where ω0 is the light center frequency. In order to calculate the neff λ for each of the four proposed PCFs, we employ a full-vector modal solver based on the finite-difference time-domain numerical method. Then using Eq. (2), dispersion profile as a function of wavelength can be calculated. Moreover, in order to reduce the simulation window and to evaluate the confinement losses of the proposed fibers, we assume the anisotropic perfectly matched layers (absorbing boundaries) to be positioned outside the outermost ring of the air-holes. Figure 2 shows the dispersion profiles of all four PCFs in terms of the input signal wavelengths. For Fiber C, due to fluid availability over the wide range of refractive indices, three different refractive indices have been studied. It should be noted that the maximum value of refractive index that can be infiltrated by capillary force through the PCF is equal to 1.8 [25]. As can be observed in the figure, going from Fiber A to Fiber D, the slope of the dispersion profile and the dispersion value both decrease. Fiber C with an nF of 1.8 has both a lower and flatter dispersion profile than the other infiltrated fluids mentioned (i.e., nF 1.32 and 1.6). Therefore, this refractive index of fluid will be considered in the rest of paper. Fibers C and D enjoy nearly flattened near-zero dispersion in the vast wavelength ranges of 3 μm and 5 μm, respectively. Nevertheless, as anticipated,
Fig. 2. Dispersion profiles versus wavelength for all proposed fibers shown in Fig. 1.
the dispersion profile of Fiber D has been engineered most favorably. Figure 3 shows the refractive indices of all materials used in this paper as a function of wavelength. We have assumed that the fluids are almost wavelength independent and have constant values over a wide wavelength range [21,25]. However, using Eq. (1), one can easily conclude that for either As2 S3 or As2 Se3, nλ decreases as λ increases, and also, that both have the same behavior. The failure of achieving a near-zero flattened dispersion profile for the As2 Se3 -based chalcogenide PCF, via selective optofluidic infiltration, is the consequence of the unavailability of the desired optical fluid having a linear refractive index with similar dispersion behavior to that of the PCF background material. Another linear parameter of interest that needs to be examined is the PCF’s total loss, L Lm Lc , with Lm and Lc being the material and confinement losses, respectively, all in dB/m. The wavelength dependencies of the material losses of As2 Se3 - and As2 S3 -based chalcogenide materials are well known [26–29]. The background material for all proposed
Fig. 4. Total loss versus wavelength for all four PCFs under study.
fibers is the same in this study. Nevertheless, confinement loss, which causes the main difference observed among the total losses of the different fibers, depends on the PCF structural parameters, like lattice types and constants, air-hole diameters, and the refractive index of the infiltrated fluid for Fiber C and that of the As2 S3 rods for Fiber D. In our calculations, confinement loss can be determined as LC λ 8.686k0 Im neff λ[12], with k0 2π∕λ being the free-space wavenumber. Figure 4 shows the wavelength dependencies of the total loss of each of the four fibers under study. As can be observed in the figure, the total loss for Fiber D is less than those obtained for the other three fibers, over the vast wavelength range of 6–10 μm. This is due to its larger effective cross-sectional area caused by refilling of the air-holes of its innermost ring with rods of As2 S3 glass having a refractive index very close to but smaller than that of the PCF background material. An increase in the PCF effective cross-sectional area, for a signal of a given wavelength, prevents the guided light to leak into the cladding region, and hence, lessens the PCF confinement loss. Having obtained the wavelength dependencies of the linear parameters of the four fibers under study, we now look into the profile of their nonlinear parameter, defined as [20] γW · m−1 n2 ω0 ∕cAeff λ;
(4)
where c and ω0 are the speed of light and its radian frequency in free space, n2 is the nonlinear Kerr index of the PCF constituents, and Aeff λ represents the wavelength dependency of the PCF effective cross-sectional area, defined as [20] RR ∞ Aeff λ
−∞
RR ∞ −∞
Fig. 3. Refractive index versus wavelength for all materials used in this paper.
2 jFx; yj2 dxdy jFx; yj4 dxdy
;
(5)
where Fx; y is the optical field distribution across the PCF cross section. The numerical values of the 10 March 2015 / Vol. 54, No. 8 / APPLIED OPTICS
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Fig. 5. Effective area versus wavelength for all four PCFs under study.
PCF effective cross section calculated for all four fibers are depicted in Fig. 5. As anticipated and as discussed earlier in this section, among the four given fibers, Fiber D has the largest effective cross-sectional area. The nonlinear parameter related to all fibers is plotted in Fig. 6. As can be observed in this figure, to our expectations, going from Fiber A to Fiber D, the nonlinear parameter decreases over the entire wavelength range of 2–10 μm. Up to λ 3.5 μm, this parameter is nearly the same for Fibers C and D. However, for wavelengths λ > 3.5 μm, the nonlinear parameter for Fiber D becomes smaller than that for Fiber A. When the nonlinear parameter of the mentioned fibers has almost the same values, the dispersion profiles would determine which scheme yields a wider spectrum at the end of the fiber; this is discussed in the next section. In order to increase the bandwidth of the supercontinuum spectrum, it is important to have a single-mode PCF in a wide wavelength range. If several modes exist in the PCF, the fundamental mode will couple with higher order modes that have a larger effective area.
It reduces the nonlinearity interactions along the fiber, which are needed for supercontinuum generation. So, the fiber should be designed to operate in a single-mode regime [26]. Note that for a fiber to be single-mode, it needs to be designed with a combination of a small core size-to-wavelength ratio and a small difference in refractive indices between the core and the cladding. We examine and calculate the mode distribution for all proposed fibers. Our results confirm that all fibers are single-mode in the wavelength range of interest. Further, a 2D schematic view of the fundamental mode distribution for Fiber D is shown in the inset in Fig. 5. This fundamental mode is similar in each length of the fiber. Our numerical results, reported elsewhere, demonstrated that if the second and third inner rings of air-holes are infiltrated by a fluid, then the PCF operates as multimode and its nonlinearity is decreased [17]. So, we may not see supercontinuum generation at the end of the fiber. 3. Mathematical Background
In this section, by extracting both linear parameters, including dispersion (from second to eighth order, as shown in Table 2) and loss, and the nonlinear parameter for the proposed fibers shown in Fig. 1, the general NLSE (GNLSE) will be solved using the symmetrized S-SSFM, to describe the propagation of femtosecond pulses inside the fiber and to broaden the generated supercontinuum. The GNLSE is described as follows [20]: 10 X βq ∂q A ∂A α i ∂ q−1 A × Az; t i iγ 1 q! ∂tq ∂z 2 ω0 ∂t q1 Z∞ Rt0 jAz; t − t0 j2 dt0 ; × −∞
(6) where z and t are the spatial coordinate along the fiber and the time variable, α is the total fiber loss, βq represents the qth-order dispersion parameter, and γ is the nonlinear parameter. Moreover, Rt in the integrand on the right-hand side of the equation is the response function including the Kerr and Raman contributions. Assuming that the electronic contribution is nearly instantaneous, the functional form of Rt can be written as Table 2.
Values of the Dispersion Orders, β2 –β8 , of all Four Proposed PCFs at the Desired Pump Wavelengths, λ
Fiber A βn (sn ∕m) (λ 3.6 μm)
Fig. 6. Nonlinear parameter versus wavelength for all proposed fibers shown in Fig. 1. 2076
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β2 β3 β4 β5 β6 β7 β8
9 × 10−26 2.33 × 10−39 −1.07 × 10−53 9.95 × 10−68 −1.06 × 10−81 1.37 × 10−95 −1.77 × 10−109
Fiber B (λ 4 μm)
Fiber C (λ 4 μm)
Fiber D (λ 4.6 μm)
1.99 × 10−26 2.64 × 10−39 −1.44 × 10−53 1.41 × 10−67 −1.64 × 10−81 2.57 × 10−95 −5 × 10−109
1.16 × 10−26 2.57 × 10−39 −1.28 × 10−53 1.17 × 10−67 −1.2 × 10−81 1.67 × 10−95 −3.2 × 10−109
1.14 × 10−26 1.96 × 10−39 −4.55 × 10−54 −1.52 × 10−68 1.09 × 10−81 −2.66 × 10−95 4.56 × 10−109
Rt 1 − f R δt f R hR t;
(7)
where δt is the Dirac delta function and f R represents the fractional contribution of the delayed Raman response function hR t, which in turn takes the approximate analytic form of τ21 τ22 t t exp − sin ; hR t 2 τ τ τ1 τ2 2 2
(8)
with f R 0.148, τ1 0.23 fs, and τ2 164.5 fs, for As2 Se3 [26,30]. The response time for the As2 Se3 chalcogenide fiber is longer than the response time for the silica fiber; so the Raman response must be taken into our account. We will use the nonlinear response function throughout the remainder of this paper. Using the S-SSFM, one can obtain the numerical solution to Eq. (6) [20]. The evolution of an input pulse, having a slowly varying electric field amplitude represented by the envelope function Az; t, propagating along the fibers in Fig. 1 is given by p t Az 0; t P0 sech ≡ T0
s jβ2 j N t sech ; γ T0 T0 (9)
where P0 , T 0 , and N are the peak power, time width, and soliton order of the input source pump, respectively. The soliton order, N, is usually determined as p N T 0 γP0 ∕jβ2 j:
(10)
A small perturbation, such as a higher order dispersion, self-steepening, and stimulated Raman scattering, can break up a given soliton of order N into N fundamental solitons [20]. As a result of the self-frequency shift induced by Raman scattering, the center wavelengths of the resulting fundamental solitons continuously shift toward the longer wavelengths of the spectrum (red side). On the other hand, nonsolitonic radiations result in a broadening on the blue side of the spectrum [31]. These nonlinear processes are dominant, in a PCF of length L, when the nonlinear length (LNL ) and the dispersion lengths (LD ) satisfy the conditions LNL < LD and LNL < L. Under such conditions, initially, self-phase modulation leads to a symmetric spectral broadening of the optical pulse, after which the requirement for phase matching conditions is fulfilled, and then soliton fission occurs, and finally, the self-frequency shift and nonsolitonic radiations processes broaden the spectral components on both sides of the spectrum, as a consequence of which a supercontinuum is generated. 4. Numerical Simulation
In this section, to have the benefit of a lower average power and a wider bandwidth, an optical pulse as that described by Eq. (9) of 50 fs and 10 kW peak
Fig. 7. Spectral broadening versus wavelength for a 50 fs input pulse of peak power P0 10 kW launched into all four PCFs under study, of the same length of 50 mm.
power is considered. The supercontinuum generation for each of the proposed fibers in a wide wavelength range of input pulses has been calculated numerically over a length of 50 mm of the optical fibers as depicted in Fig. 7. In the figure, we can observe that the maximum spectral bandwidth for Fiber A has been obtained to be around 1.9 μm for an input wavelength of 3.6 μm. These values for Fibers B and C are 2.84 μm and 2.91 μm, respectively for a 4 μm input pulse. For Fiber D, maximum spectral bandwidth of 3.86 μm for a 4.6 μm input pulse has been achieved. As can be observed in the figure, going from Fiber A to Fiber D, the value of the spectral bandwidth increases more than 100%. This result can be expected from the ultraflattened near-zero dispersion profile that is calculated for Fiber D in Section 2. In order to justify the ignoring of the nonlinear Kerr refractive index in As2 S3 , for an input pulse with peak power of 10 kW at 4.6 μm traveling along Fiber D, we have taken three snapshots of the pulse power distribution in x–y (top) and x–z and y–z (sides) cross-sectional views. These snapshots are shown in Figs. 8(a), 8(b), and 8(c) (Media 1, Media 2, and Media 3), respectively. As can be observed in the figure, the power leakage into the As2 S3 rods surrounding the As2 Se3 core of Fiber D is negligible. Figure 9 shows the propagation of optical pulses with input wavelengths corresponding to the maximum spectral bandwidth along the proposed schemes shown in Fig. 1. It can be observed that Fiber D has a flat and ripple-free spectral broadening along the fiber, which is suitable for supercontinuum applications. In this fiber, the pulse initially starts to broaden symmetrically, due to the self-phase modulation effect. Then, after the input pulse traverses a distance of about 10 mm along the fiber, due to higher order dispersions and the self-steepening nonlinear effect, it becomes unstable and soliton fission occurs, as a consequence of which fundamental solitons of various center wavelengths are formed. These 10 March 2015 / Vol. 54, No. 8 / APPLIED OPTICS
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Fig. 8. Snapshots of power distributions when an input pulse with 10 kW peak power and center wavelength of 4.6 μm is propagating along Fiber D, in (a) x–y (top) view (Media 1), (b) x–z (side) view (Media 2), and (c) y–z (side) view (Media 3). The color bar to the right indicates the normalized intensity.
Fig. 9. Evolution of the spectral distribution of a 50 fs optical pulse of P0 10 kW peak power and center wavelength of (a) λ 3.6 μm along Fiber A, (b) λ 4 μm along Fiber B, (c) λ 4 μm along Fiber C for a fluid of refractive index nF 1.8, and (d) λ 4.6 μm along Fiber D. All PCFs are 50 mm long.
isolated optical pulses are temporally narrower and hence spectrally broader than the pulse from which they originate, and hence, a broader bandwidth is created. Then, after traveling another 20 mm (i.e., Table 3. Bandwidths of the Supercontinua Generated by the Optical Signal at Various Lengths of all Four Proposed Fibers Measured at the −30 dB Level, as Depicted in Fig. 9
Supercontinuum Bandwidth (μm) Measured at −30 dB Fiber Length (mm) 10 20 30 40 50
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Fiber A (3.6 μm)
Fiber B (4 μm)
Fiber C (4 μm)
Fiber D (4.6 μm)
2.2 1.85 1.75 1.71 1.9
2.01 2.23 2.61 2.75 2.83
2.1 2.55 2.67 2.86 2.91
2.23 3.25 3.67 3.76 3.86
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at z 30 mm), the spectrum is spread toward wavelengths smaller than the pump wavelength, where the resonance condition prevails and due to transfer of energy to a higher frequency, dispersive radiation occurs. Finally, at about z 40 mm, stimulated Raman scattering comes into the picture and the most intense soliton slowly shifts toward longer wavelengths. All expressed phenomena result in supercontinuum generation at the end of the fiber. For more details on the proposed schemes, the bandwidths calculated at the level of −30 dB for the various cases illustrated in Fig. 9 are tabulated in Table 3. 5. Conclusion
We have studied different types of As2 Se3 chalcogenide PCFs for supercontinuum generation, using optofluidic and rod filling techniques. Linear profiles,
including the PCFs’ losses and dispersion, and the nonlinear parameter, have been calculated and have been discussed for each of the PCFs under study. The simulations show that when a 50 fs input optical pulse of peak power 10 kW and center wavelength of 4.6 μm is launched into a 50 mm long rod-filled chalcogenide PCF, a ripple-free spectral broadening as wide as 3.86 μm can be obtained, whereas the maximum spectral bandwidth of the supercontinuum of the As2 Se3 -based chalcogenide PCF whose innermost ring air-holes are infiltrated with an optical fluid is 2.93 μm. The advantage of chalcogenidebased PCFs in supercontinuum generation lies in their relatively constant material loss over a wide transmission window that cuts off at about 10 μm, as compared with an exponentially increasing material loss in silica-based PCFs for wavelengths beyond 2.5 μm. This together with their nonlinear Kerr index, which is more than 200 times larger than that of silica, makes As2 Se3 chalcogenide PCFs promising alternative media for supercontinuum applications in the mid-IR region. References 1. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006). 2. H. Saghaei, B. Seyfe, H. Bakhshi, and R. Bayat, “Novel approach to adjust the step size for closed-loop power control in wireless cellular code division multiple access systems under flat fading,” IET Commun. 5, 1469–1483 (2011). 3. C. A. Brackett, “Dense wavelength division multiplexing networks: principles and applications,” IEEE J. Sel. Areas Commun. 8, 948–964 (1990). 4. G. Humbert, W. Wadsworth, S. Leon-Saval, J. Knight, T. Birks, P. St. J. Russell, M. Lederer, D. Kopf, K. Wiesauer, and E. Breuer, “Supercontinuum generation system for optical coherence tomography based on tapered photonic crystal fibre,” Opt. Express 14, 1596–1603 (2006). 5. B. W. Liu, M. L. Hu, X. H. Fang, Y. Z. Wu, Y. J. Song, L. Chai, C. Y. Wang, and A. Zheltikov, “High-power wavelength-tunable photonic-crystal-fiber-based oscillator-amplifier-frequencyshifter femtosecond laser system and its applications for material microprocessing,” Laser Phys. Lett. 6, 44–48 (2009). 6. T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416, 233–237 (2002). 7. J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Visible continuum generation in air–silica microstructure optical fibers with anomalous dispersion at 800 nm,” Opt. Lett. 25, 25–27 (2000). 8. W. J. Wadsworth, A. Ortigosa-Blanch, J. C. Knight, T. A. Birks, T. P. M. Man, and P. S. J. Russell, “Supercontinuum generation in photonic crystal fibers and optical fiber tapers: a novel light source,” J. Opt. Soc. Am. B 19, 2148–2155 (2002). 9. B. Eggleton, C. Kerbage, P. Westbrook, R. Windeler, and A. Hale, “Microstructured optical fiber devices,” Opt. Express 9, 698–713 (2001). 10. W. Reeves, D. Skryabin, F. Biancalana, J. Knight, P. S. J. Russell, F. Omenetto, A. Efimov, and A. Taylor, “Transformation and control of ultra-short pulses in dispersion-engineered photonic crystal fibres,” Nature 424, 511–515 (2003). 11. J. C. Knight, “Photonic crystal fibres,” Nature 424, 847–851 (2003).
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