## Abstract

Photonic crystal fibers with three and four zero-dispersion wavelengths are presented through special design of the structural parameters, in which the closing to zero and ultra-flattened dispersion can be obtained. The unique phase-matching properties of the fibers with three and four zero-dispersion wavelengths are analyzed. Variation of the phase-matching wavelengths with the pump wavelengths, pump powers, dispersion properties, and fiber structural parameters is analyzed. The presence of three and four zero-dispersion wavelengths can realize wavelength conversion of optical soliton between two anomalous dispersion regions, generate six phase-matching sidebands through four-wave mixing and create more new photon pairs, which can be used for the study of supercontinuum generation, optical switches and quantum optics.

© 2015 Optical Society of America

## 1. Introduction

Photonic crystal fiber (PCF) can generate particular dispersion properties and interesting nonlinear properties, due to the special guiding mechanism and adjustable structure parameters. Compared to traditional fiber with one zero dispersion wavelength (ZDW), dispersion curves with two ZDWs can be obtained in PCFs, which have great influence on the nonlinear effects [1–4]. There is phase-matching blue shift dispersive wave radiation from optical soliton in the anomalous dispersion region of the fiber with one ZDW [5]. Two sides phase-matching resonance scattering can be generated in PCF with two ZDWs. Phase-matching blue-shift and red-shift dispersive waves respectively in two normal dispersion regions can be emitted from optical soliton [6–8]. On the other hand, in the four-wave mixing (FWM), two phase-matching sidebands can be generated in the fiber with one ZDW [9–11], and four phase-matching sidebands can be generated in PCF with two ZDWs [12, 13]. These have opened new opportunities for multi-wavelength FWM, optical switches, optical parametric amplification and supercontinuum generation [14–17].

However, PCFs with three and four ZDWs can be obtained through special design of the structural parameters of the fibers, in which new physical phenomena can be generated, providing a new way to study particular nonlinear effects of PCF [18–20]. Properties of PCFs with different ZDW number are shown in Table 1. Compared to one anomalous dispersion region of PCFs with one and two ZDWs, there can be more anomalous dispersion regions of PCFs with three and four ZDWs, in which wavelength conversion of optical soliton can be obtained from an anomalous dispersion region to another. The presence of three and four ZDWs can create rich phase-matching topology, generate more new wavelengths through FWM and emit more stokes photon pairs, enabling enhanced control over the spectral locations of FWM and resonant-radiation bands emitted by solitons and short pulses.

The theoretical study of FWM is described in the paper, and the high-gain phase-matching conditions are presented. The dispersion curves with three and four ZDWs have been obtained through special design of the structural parameters of PCFs. The corresponding phase-matching properties of different dispersion curves are analyzed. Variation of the phase-matching wavelengths with the pump wavelengths and pump powers is obtained. The resulting phase-matching curves have a complicated algebraic topology.

## 2. Four-wave mixing

FWM is a very important kind of nonlinear effect in PCF. Two pumping photons can be converted into Stokes and anti-Stokes photons due to the third order susceptibility. Wavelength conversion efficiency of FWM mainly depends on the phase-matching properties. The gain of phase-matching wavelength can be higher than Raman scattering. High-efficiency FWM can be used for up-conversion and down-conversion pair-photon sources for quantum optics.

For the equations of degenerate FWM, assuming un-depleted pump, it is possible to obtain an analytical expression [21,22]:

where*P*

_{i},

*P*

_{s}, and

*P*are the powers of the idler, signal and pump respectively.

*γ*is the fiber nonlinear coefficient.

*g*is the gain:and

*κ*is given bywhere ∆

*β*is the difference in the propagation constants given by ∆

*β*=

*β*

_{s}+

*β*

_{i}−2

*β*

_{p}. Maximum gain of

*g*=

*γP*is obtained when

*κ*= 0, which occurs when the nonlinear phase shift 2

*γP*is compensated by a negative wave vector mismatch (∆

*β*). In degenerate FWM, energy conservation determines the frequencies of the signal and idler waves generated from the pump light. The process is most intense when wave momentum is conserved. Both requirements can translate into phase-matching conditions:

*ω*is the angular frequency,

*β*(

*ω*

_{p}),

*β*(

*ω*

_{s}),

*β*(

*ω*

_{i}) and

*γ*can be given by the dispersion and nonlinear properties of PCF respectively. The phase-matching properties can be obtained by solving Eqs. (4) and (5).

The propagation constant *β* is given by

*D*is given by

*n*

_{eff}is the effective index of fundamental mode of PCF which can be computed by the effective index method [23, 24], finite element method [25, 26] or multi-pole method [27–29].

The effective mode area of the fiber is given by

*γ*is defined as follows:

*I*(

*x*,

*y*) is the transverse modal field distribution usually simulated by the multi-pole method or finite element method. ${n}_{2}\approx 3\times {10}^{-20}\text{\hspace{0.17em}}{\text{m}}^{2}\cdot {\text{W}}^{-1}$ is the nonlinear refractive index of the silica glass.

*c*is the vacuum speed of light.

## 3. Phase-matching properties of PCF with three ZDWs

#### 3.1 PCF with three ZDWs

Figure 1 shows the schematic of the cross-section of common solid-core PCF. The gray region is the pure silica, and the white round spots are the air holes. *Λ* is the lattice constant (pitch), *d* is the diameter of air holes in the cladding, and *d*/*Λ* is the hole-to-pitch ratio. PCF is assumed to have 60 air holes in 4-layer hexagonal structure. The guiding properties of PCF are simulated by the multi-pole method. Here, the maximum tolerance of the effective index is less than 1.0 × 10^{−7}, so the calculated results are accurate and reliable [27–29].

The dispersion curves with three ZDWs are obtained through proper design of the structural parameters of PCFs in Fig. 1. The dispersion curves of PCFs with different *d*/*Λ* and *Λ* values are shown in Fig. 2. The structural parameters of PCFs and three ZDWs are shown Table 2. Variation of the ZDWs with the structure parameters can be found. The location and spacing of the ZDWs can be adjusted flexibly in large wavelength range through careful design of *Λ* and *d*/*Λ* values.

The absolute values of dispersion parameter *D* are less than 0.3 ps/(nm·km) in the broad spectral range 1.4-1.9 μm, when *Λ* = 2.356 μm and *d*/*Λ* = 0.26. Therefore, the closing to zero and ultra-flattened dispersion is obtained in the fiber with three ZDWs, in which the phase-matching conditions can be controlled by low pump power due to the small dispersion values, and rich nonlinear effects can be obtained.

The values of the effective mode area *S*_{eff} and nonlinear coefficient *γ* of PCFs with different *Λ*, *d*/*Λ*, and λ values are shown in Fig. 3. It can be seen from Figs. 3(c) and 3(d), the variation of *S*_{eff} and *γ* is small (<1%) due to the little change of the values of *Λ* and *d*/*Λ*. Therefore, the variation of the phase-matching curves simulated in the paper is mainly attributed to the dispersion properties of PCFs.

PCFs are designed to be single mode over the wavelength range of interest, due to the low air-filling fraction (*d*/*Λ* = 0.26<0.43) [30–32]. The core area of these fibers is small, and corresponding nonlinear coefficient is high. The dispersion values of PCFs with three ZDWs are very small. The presence of three ZDWs has great influence on the phase properties, and the related nonlinear effects will be changed. Therefore, PCFs with three ZDWs can be exploited for new types of nonlinear effects.

#### 3.2 Phase-matching properties

According to the dispersion and nonlinear properties of PCFs with three ZDWs in Figs. 2(a) and 3(c), the phase-matching properties can be obtained by solving Eqs. (4) and (5). For PCF with *Λ* = 2.35 μm and *d*/*Λ* = 0.261, the phase-matching curves are shown in Fig. 4(a), when the pump peak power *P* = 0, 10 and 20 W respectively. The phase-matching curves appear in the full spectral range, including both the normal and anomalous dispersion regions. Six phase-matching wavelengths can be obtained for one pump wavelength in the first anomalous dispersion region between the first and second ZDWs. Therefore more excitation wavelengths and new photon pairs can be generated through FWM, which can be used for the study of multi-wavelength conversion, supercontinuum generation, and pair-photon sources for quantum optics.

For PCF with *Λ* = 2.35 μm and *d*/*Λ* = 0.262, the phase-matching curves are shown in Fig. 4(b), when *P* = 0, 50 and 100 W respectively. Compared to the phase-matching curves in Fig. 4(a), the first anomalous dispersion region extends, in which the pump photons can generate six sidebands. For the pump wavelength *λ* = 1.55 μm and *P* = 50 W, the phase-mismatching curve (the phase difference between emission and pump waves as a function of wavelength) of PCF with *Λ* = 2.35 μm and *d*/*Λ* = 0.262 is shown in Fig. 4(d). The values of phase-mismatching *κ* are very small in the wavelength range. Six phase-matching wavelengths of *κ* = 0 are found.

For PCF with *Λ* = 2.35 μm and *d*/*Λ* = 0.263, the phase-matching curves are shown in Fig. 4(c), when *P* = 0, 100 and 200 W respectively. Six phase-matching wavelengths are also obtained for one pump wavelength in the short wavelengths of the first anomalous dispersion region. Compared to the curves in Fig. 4(b), it can be seen slight variation of the dispersion properties can result in significant change in phase-matching curves.

For PCFs with *Λ* = 2.35 μm and *d*/*Λ* = 0.261, 0.262 and 0.263, the phase-matching curves are shown in Fig. 5(a), when *P* = 20 W. For PCFs with *d*/*Λ* = 0.26 and *Λ* = 2.355, 2.356 and 2.357 μm, the phase-matching curves are shown in Fig. 5(b), when *P* = 50 W. The phase-matching wavelengths are significantly changed with slight variation (≈1‰) of fiber structure parameters, which can be implemented by heat-expansion, cold-contraction, piezoelectric effect, or drawing force. Similar phase-matching topology also can be obtained by detailed variation of the refractive index of silica glass, which can be controlled by elastic-optical, thermo-optical, electro-optical, or magneto-optical Kerr effect. The wavelength conversion through FWM can be obtained in the phase-matching wavelengths. Therefore, the nonlinear spectrum can be controlled by several physical parameters, which can be applied to the study of fiber sensor, optical switches, and pair-photon sources for quantum optics.

## 4. Phase-matching properties of PCF with four ZDWs

#### 4.1 PCF with four ZDWs

Figure 6 shows the cross section of PCF with a small air hole in the core. *d*_{c} is the diameter of the air hole in the core. The dispersion curves with four ZDWs have been obtained through accuracy design of the structural parameters of PCFs.

The dispersion properties of PCFs with four ZDWs are shown in Fig. 7 and Table 3. The variation of the ZDWs with the structure parameters can be found. The changes of the dispersion values and the spacing of the ZDWs are approximately proportional to the change of the structure parameters of PCFs. The location and spacing of the ZDWs can be adjusted flexibly in large wavelength range by proper design of *Λ*, *d*/*Λ* and *d _{c}* values.

The absolute values of dispersion parameter *D* is less than 0.04 ps/(nm·km) in the broad spectral range 1.2-1.6 μm, when *Λ* = 2.035 μm, *d*/*Λ* = 0.723 and *d _{c}* = 0.566 μm. Therefore, nearly zero and ultra-flattened dispersion is obtained in the fiber with four ZDWs, in which the phase-matching conditions can be controlled by low pump power due to the small dispersion values, enabling enhanced control over the spectral locations of FWM and resonant-radiation bands emitted by solitons and short pulse. The presence of four ZDWs has great influence on the phase properties, so the related nonlinear effects will be changed.

The values of *S*_{eff} and *γ* of PCFs with different *Λ*, *d*/*Λ*, and λ values are shown in Fig. 8. It can be seen from Figs. 8(c) and 8(d), the variation of *S*_{eff} and *γ* is small (<5%) due to the little change of the values of *Λ* and *d*/*Λ*. Therefore, the variation of the phase-matching curves simulated in the paper is mainly attributed to the dispersion properties of PCFs.

#### 4.2 Phase-matching properties

The dispersion curve of PCF with *Λ* = 2.078 μm, *d*/*Λ* = 0.751 and *d*_{c} = 0.586 μm is shown in Fig. 9(a). The corresponding phase-matching curves are shown in Fig. 9(b), when *P* = 0, 10 and 30 W respectively. The phase-matching curves are obtained only in two anomalous dispersion regions. There isn’t phase-matching curve in the three normal dispersion regions. Therefore, pump wavelength should be in the anomalous dispersion regions to get multi-wavelength conversion through FWM.

The dispersion curve of PCF with *Λ* = 2.08 μm, *d*/*Λ* = 0.751 and *d _{c}* = 0.586 μm is shown in Fig. 10(a). The corresponding phase-matching curves are shown in Fig. 10(b), when

*P*= 0, 10 and 100 W respectively. Due to the extension of two anomalous dispersion regions and the reducing of the normal dispersion region in the middle, the range of the phase-matching curves increases. The large span of the phase-matching wavelengths may prove useful for broadband amplification and we therefore suggest implement this PCF in fiber optic parametric oscillators as well.

The dispersion curve of PCF with *Λ* = 2.082 μm, *d*/*Λ* = 0.751 and *d*_{c} = 0.586 μm is shown in Fig. 11(a). The corresponding phase-matching curves are shown in Fig. 11(b), when *P* = 0, 10 and 100 W respectively. The phase-matching curves are obtained in the full range between the first and fourth ZDWs including the middle normal dispersion region due to the small dispersion values. Six phase-matching wavelengths can be obtained for one pump wavelength in the first anomalous dispersion region near the first ZDW.

There is only one anomalous dispersion region in PCFs with one or two ZDWs, and the dispersive wave can be only generated in the normal dispersion region. However, there are three normal dispersion regions and two anomalous dispersion regions close to each other in the dispersion curve with four ZDWs in Fig. 11(a). Solitons in an anomalous dispersion region can emit dispersive wave to another across the forbidden region of normal dispersion, most of the energy can be transferred, and the dispersive wave in the anomalous dispersion region also can be in the form of soliton. The optical soliton can tunnel between two regions of anomalous dispersion across the forbidden normal dispersion region [33]. It provides a new physical condition for the study of dispersive waves, pulse compression, wavelength conversion of solitons and soliton-pairs sources.

It can be seen there are very small dispersion values and phase difference between the signal and pump waves in PCFs with three and four ZDWs. Therefore, small changes of the pump powers, pump wavelengths, ZDWs, and structural parameters of PCFs can result in very different phase-matching curves, enabling enhanced control over the spectral locations of FWM and resonant-radiation bands emitted by solitons and short pulses.

To compare the phase-matching curves of PCFs with one, two, three and four ZDWs, it can be seen that the shapes of the phase-matching curves of PCFs with the same number of ZDWs are similar, but the shapes of the phase-matching curves of PCFs with different number of ZDWs are very different. For pump wavelength located in the anomalous dispersion region of the fiber, the number of solutions to phase-matching condition of resonant radiation can equal the number of ZDWs of the fiber [7]. The presence of the third and fourth ZDWs has a strong impact on the propagation of ultrashort pulses and solitons. In particular, the phase-matching curves for resonant radiation are different from all previously reported cases. For several pump wavelengths, solitons can phase-match to three or four resonances, instead of the traditional one or two [19]. Therefore, the study of dispersion and phase-matching properties of PCFs with three and four ZDWs can be exploited for new types of nonlinear effects.

## 5. Conclusion

The FWM principle in the fiber is presented, and the high-gain phase-matching conditions are shown. PCFs with three and four ZDWs are obtained through special design of the structural parameters of the fibers. The absolute values of *D* can be less than 0.04 ps/(nm·km) in the broad spectral range 1.2-1.6 μm. The phase-matching curves of PCFs with three and four ZDWs are shown. The variation of phase-matching wavelengths with the pump wavelengths, pump powers, dispersion properties, and structural parameters of the fibers is obtained. Compared to the fibers with one or two ZDWs, there are new kinds of phase-matching curves in the PCFs with three and four ZDWs, which can generate different nonlinear physics effects.

Wavelength conversion of solitons between two anomalous dispersion regions can be obtained, which is suitable for the observation of soliton spectral tunnelling effect in nonlinear pulse compression. We find a complex phase-matching landscape that allows multiple frequencies to be generated by both continuous waves and solitons, and there are six phase-matching sidebands through FWM in PCFs with three and four ZDWs. Slight variation of the pump powers, pump wavelengths, ZDWs, and structural parameters of PCFs can result in significant change in phase-matching wavelengths, enabling enhanced control over the spectral locations of the FWM and resonant-radiation bands emitted by solitons and short pulses. These can be applied to broadband wavelength converters, optical switches, fiber optic parametric oscillators, and pair-photon sources for quantum optics.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 61405172), and the Natural Science Foundation of Hebei Province, China (Grant No. F2014203224)

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