## Abstract

We investigate the one-third harmonic generation (OTHG) in optical microfibers with power attenuation considered by analytically analyzing and numerically solving the coupled mode equations (CMEs). Both the strength and effective length of signal power growing in nonlinear media, which are extremely sensitive to the relative phase between the interaction waves, contribute to the final conversion efficiency. The relative phase and its evolution along the propagating direction play crucial roles in highly efficient OTHG. In order to obtain high conversion efficiency, the general expressions of optical threshold conditions are derived and discussed for choosing proper initial parameters. Numerically simulations are performed with both partial and absolute phase matching, which are corresponding to the microfibers with uniform and non-uniform diameters, respectively. Optimizations of relative phase and phase compensation are suggested by the simulations and provide significant enhancement of conversion efficiencies.

© 2015 Optical Society of America

## 1. Introduction

One-third harmonic generation (OTHG) is the reverse process of third harmonic generation (THG) [1] and possesses potential applications in mid-IR parametric amplifying and generating three simultaneously entangled photons [2]. This generation is a third order nonlinear process, which creates triple equal energy photons with a pump photon disappearing [3, 4]. OTHG is driven by an optical field oscillating at pump frequency: it can occur spontaneously or be forced by a seeding field that oscillates at triple photons’ frequencies. The first one is called parametric fluorescence [5, 6], whereas the latter one is usually known as optical parametric amplification [7]. Both the weak magnitude of third order electric susceptibility in nonlinear media and partial knowledge of the specific corresponding process impede the OTHG research [8]. In past few years, the coupled mode theory describing THG in optical microfibers [9] overcomes the obstacles and facilitates the studying of OTHG process in optical waveguides [1, 3–5]. Grubsky and Savchenko had addressed the issue of phase matching of THG by using glass microfibers with sub-wavelength diameter and derived the general expressions of coupled mode equations (CMEs) in waveguides [9]. The air-clad silica microfiber with high step-index contrast provides a tight confinement of the optical field [10–12], which leads the effective nonlinearity to be enhanced by several orders of magnitude comparing with standard single mode fibers [13]. Moreover, OTHG and THG share the same phase matching conditions and consequently can be described by the same set of CMEs deduced in [9]. Phase mismatching originated from material dispersion and power-related phase shifts is compensated by modal dispersion for achieving phase matching condition of OTHG in optical waveguides. The diameter-dependent dispersion of microfibers [14] can be tailored to obtain phase matched OTHG. Meanwhile, different nonlinear terms can be engineered through the waveguide structure to achieve efficient OTHG in optical waveguides. The dependence of maximum OTHG efficiency on waveguide structural parameters and laser power has been investigated in [1]. In addition, as nonlinear phenomena are sensitive to the relative phase between the participating waves, the effect of initial phase on initial power to support highly efficient OTHG has been demonstrated in [3]. But the optimizations of these two researches are drawn from the CMEs without material attenuation, which always exists and depletes the total power along the interaction medium in experiments. Total optical power attenuation causes the strength diminishing of power-related phase shift and makes the situation of phase matching condition in waveguides to be complicated. In [4], the varied power-related phase shift is used to dynamically compensate the accumulated linear phase mismatch along the fiber and lengthen the effective interaction length of OTHG. However, the dependence of effective interaction length and final conversion efficiency on the relative phase between the interacting waves and its evolution along propagating has not been systemically investigated in waveguides with attenuation considered.

In this paper, the phase sensitivity of OTHG process is analyzed based on rigorous manipulation of CMEs with loss included in optical microfibers. The change rate of signal power along propagating is derived and expressed as a function of relative phase and total optical power. On the other hand, the effective propagating length, in which signal power grows positively, is limited by the evolution of relative phase caused by linear modal dispersion and nonlinear optical refractive effects. Therefore, the relative phase and its evolution along propagating dominate the growth rate of signal power and effective interaction length, respectively. Both growth rate and effective length contribute to the final signal enhancement of OTHG process. The possible longest effective interaction length at given optical parameters corresponds with the situation of perfect phase matching, which means no relative phase change existing over the whole propagating length. We demonstrate the circumstances of perfect phase matching and partial phase matching, which correspond with uniform and non-uniform diameters microfibers, respectively. The corresponding approaches to select suitable initial parameters for highly efficient OTHG are proposed at both situations.

## 2. Theoretical derivation and analysis

Grubsky and Savchenko [9] firstly derived the CMEs governing the evolutions of fundamental wave and third harmonic wave propagating in silica microfibers without power attenuation. Here, we introduce the power attenuation into the CMEs to investigate the phase characteristic features of the OTHG process. The CMEs can be written as Eqs. (1a) and (1b) when considering the attenuation of the two waves propagating in microfibers.

In order to separate the amplitude and phase from the complex amplitude expression, defining${A}_{1}={\rho}_{1}{e}^{i{\phi}_{1}}$, ${A}_{3}={\rho}_{3}{e}^{i{\phi}_{3}}$ and$\theta =\delta \beta z+{\phi}_{3}-3{\phi}_{1}$, where $\theta $ is the relative phase between the interacting modes involved in OTHG process, the CMEs can be rewritten as:

We can make a substitution with the relationship of${P}_{t}={\rho}_{1}^{2}+{\rho}_{3}^{2}$,${\rho}_{1}^{2}=b{P}_{t}$ and ${\rho}_{3}^{2}=\left(1-b\right){P}_{t}$ for further simplifying the CMEs. The parameter b stands for the proportion of signal power to total power ${P}_{t}$. The Eqs. (2a)-(2d) can be derived as:

Firstly, it is supposed that signal wave shares the same attenuation with pump wave, as${\alpha}_{1}={\alpha}_{3}=\alpha $, which is quite reasonable and common for the situation of light with the spectral from visible to infrared propagating in silica microfibers [15]. Accordingly, Eq. (3a) can be rewritten as:

This expression is exactly unrelated to the nonlinear energy transfer and can be integrated directly as${P}_{t}\left(z\right)={P}_{t}\left(0\right){e}^{-2\alpha z}$, where ${P}_{t}\left(0\right)$ is the initial total optical power. Meanwhile, Eq. (3b) can only be simplified as:Secondly, in microfibers with uniform diameters, the relative phase can be supposed to be a constant over all the propagating length, which is possible by dynamically compensating the linear and nonlinear phase mismatches along propagating. As shown in Eq. (5), the most favorable relative phase for OTHG process is $\mathrm{sin}\theta =-1$, which can be expressed as$d\theta /dz=0$. With this assumption, Eqs. (5), (3c) and (3d) can be simplified as:

The last hypothesis parallel to the second one is corresponding to microfibers with uniform diameters, which means that the relationship of $d\delta \beta /dz=0$ comes into existence. The Eq. (3c) can be reorganized as Eq. (9), and the other equations remain unchanged.

The relative phase change rate $K$ caused by nonlinear phase modulations presents an exponential decay along with the propagating length, while the modal phase match detuning $\delta \beta $ maintains as a constant in uniform microfibers. Therefore, the relationship $d\theta /dz=0$ can only be satisfied at a specific position with a suitable value of detuning. This difference of the evolutions of the relative phase between the non-uniform and uniform microfibers will be discussed later in this text by numerically solving the CMEs.In order to realize positive growth of signal wave, some threshold conditions deduced from the evolution of signal power have to be satisfied. The variation rate of signal power, shown as Eq. (10), can be obtained by combining Eqs. (4) and (5) based on the relationship ${P}_{s}\left(z\right)={P}_{t}\left(z\right)b\left(z\right)$, where ${P}_{s}\left(z\right)$ is signal power at z.

The first inequality (11) stands for the power cut-off condition for efficient OTHG process at given proportional coefficient $b$ and relative phase$\theta $, whose minimum equals to ${P}_{t-cut}=2\alpha /{\gamma}_{0}{J}_{3}$ with$b=0.5$, $\mathrm{sin}\theta =-1$. This expression indicates the power-dependent property of OTHG process and determines the range of optical parameters for efficient OTHG. In order to achieve highly efficient OTHG with low total power, feasible approaches include decreasing the medium attenuation $\alpha $at signal frequency, increasing the nonlinearity ${\gamma}_{0}$ of material and enhancing the overlapping integral ${J}_{3}$ between interaction modes. Additionally, optical parameters, including proportional coefficient $b$ and relative phase$\theta $, are also helpful to reduce the required total power when they are close to the values corresponding to the threshold total power.

The inequality (12) presents an effective range of proportional coefficient of signal power to given total power with a given relative phase and total optical power. The left hand of inequality (12) is a parabolic function of $b$ with downward opening and symmetry axis at $b=0.5$. Parameter $b$diverges more from its symmetry axis, the value of this parabolic function is smaller, which corresponds with that the total optical power needs to be larger for ensuring the efficient OTHG.

The last inequality indicates a valid relative phase range of efficient OTHG process at a given total power and a given seeding signal proportional coefficient. As we have mentioned, the range of relative phase difference corresponding to OTHG process is $\mathrm{sin}\theta <0$. However, as the attenuation of microfibers has been considered, optical power dissipation weakens the nonlinear energy growth of signal power and narrows the range of relative phase for guaranteeing the efficient OTHG.

The narrowing process of OTHG phase range can be illustrated as Fig. 1, in which we draw a curve of sine value of the relative phase$\theta $. As we have emphasized, the nonlinear processes are very sensitive to the relative phase between the interacting waves. The value of relative phase determines the direction and strength of nonlinear energy transfer. In this figure, we divide the complete phase cycle of $2\pi $ into two parts: OTHG phase range and THG phase range, which is based on the term of nonlinear energy transfer in Eq. (10) and has ignored the influence of power attenuation. For considering the complete expression of Eq. (10), the effective relative phase range for efficient OTHG corresponds with the modality of inequality (13), which is corresponding to the transverse range of grey area in the OTHG phase range of Fig. 1.

Considering a single transmission point, the optimal value of the relative phase for obtaining the maximum nonlinear energy transfer equals to $-\pi /2$, which corresponds with point B in the curve of Fig. 1. However, the nonlinear energy transfer is an accumulative process along the propagation, in which the relative phase varies with the modal phase match detuning and power-dependent phase change rate. Moreover, the evolution of relative phase along propagating strongly affects the nonlinear energy conversion. Therefore, the key issue for highly efficient OTHG is whether the relative phase and its evolution along propagating were suitable for OTHG. The strength of artificial phase compensation can be adjusted by changing the diameter of microfibers. Therefore, the detuning of phase matching in a uniform microfiber is stationary along the propagating length. Meanwhile, the power-dependent phase change rate diminishes as the exponential decay of total optical power along propagating, which means that the relative phase in uniform microfibers cannot be fixed at the optimal value over the whole interaction length. By contrast, fixing the relative phase at its optimal value along the whole propagating length is possible in a non-uniform microfiber if its diameter was carefully adjusted according to the power-dependent phase change. In next part of this text, we will discuss the OTHG process in each situation in detail based on numerical calculation.

Moreover, we always pay more attention on conversion efficiency than other parameters in a nonlinear optical process. We can define a gain coefficient $g$ to illustrate the nonlinear energy conversion along the propagation, as shown in Eq. (14). The enhancement of signal power along with the interaction length can be obtained by solving the CMEs.

## 3. Numerical calculation and discussion

In order to provide a detailed discussion, we numerically solve the reduced CMEs for the uniform and non-uniform microfibers. The medium parameters involved in calculation refer to [15]. The wavelengths of the signal wave and the pump wave are 1550 nm and 517 nm, respectively. The signal wave propagates in HE_{11} mode. The HE_{12} mode of pump wave experiences the largest modal overlap with the signal mode. Accordingly, the microfiber diameter is chosen as 767 nm in order to ensure the phase matching between the two modes. Both modes have the same effective refraction indices of ${n}_{eff}=1.081$. We set the amplitude dissipation coefficient of the two modes as the same value, i.e.${\alpha}_{1}={\alpha}_{3}=5{m}^{-1}$. ${k}_{1}=2\pi /{\lambda}_{1}$ is the signal free-space propagation constant and the silica nonlinear refractive-index coefficient is${n}^{\left(2\right)}=2.7\times {10}^{-20}{m}^{2}/W$. The corresponding overlap integrals of the two modes are:

The values of optical parameters, including initial total optical power, proportion of signal power to the total power and the initial relative phase, are carefully chosen to obtain an efficient OTHG process. The methods for selecting optical parameters are given by inequality (11)-(13). Thus, the total power threshold for the OTHG process can be calculated as 234 W by the expression of ${P}_{t-cut}=2\alpha /{\gamma}_{0}{J}_{3}$with the medium parameters introduced above. However, this power threshold is quite strict with the proportional coefficient and relative phase, which are corresponding to 0.5 and$-\pi /2$, respectively. The optical power will be depleted to lower than the threshold along propagation, which means that the efficient OTHG process can only be observed with relatively high power even with the optimal values of the other parameters of the optical field. Therefore, we set the initial total optical power as 500 W to make the OTHG process easily to be observed. A large initial power is helpful to achieve a long interaction length, which is defined as the maximum propagating length with the nonlinear energy increasing stronger than the linear attenuation with respect to signal power. The propagation length from 500 W initial total power to 234 W threshold total power is 75 mm. However, in practice, the effective interaction length, which is corresponding to the propagating length with signal power positively growing, mostly depends on the relative phase rather than the optical powers.

Using this selected total optical power, we can figure out the maximum possible range of proportion of signal power to total power based on the inequality (12). The range of $b$ at the given total power is largest only when the right hand of the inequality approaches to its minimum, which corresponds with the relative phase value of $-\pi /2$. The largest range of signal power proportion to 500 W total power is from 0.058 to 0.942 by solving the inequality (12). Considering applications, the initial signal power should be the lower the better. The lower initial signal power requires a wider range of proportion, which requires the larger initial total power. In our calculation, we choose the proportion of signal power to total power as 0.08, which is 40 W of the initial signal power in 500 W of the initial total power.

With the chosen initial total power and initial proportion of signal power to total power, we work out the effective OTHG phase range using the inequality (13). The effective relative phase ranges from $-2\pi /3$ to $-\pi /3$, in which $\theta =-\pi /2$ corresponds with the maximum nonlinear energy transfer rate as shown in Eq. (10). At the same time of keeping the maximum rate of nonlinear energy transfer, the high conversion efficiency of OTHG process also requires a long interaction length with the positive growth of signal power, which means that we need to keep the relative phase around the optimal value along propagating, i.e.$d\theta /dz=0$. In order to achieve this goal, we introduce the modal phase compensation determined by microfiber diameter to counteract the power-dependent phase change. In the following paragraphs, we will discuss the OTHG process in microfibers with single diameters and variable diameters in detail, respectively.

#### 3.1 Optimizing initial relative phase in uniform microfibers

With regard to uniform microfibers, the single diameter only provides constant phase compensation over the whole propagating length. In addition, combining the required relative phase above, an apparent solution is keeping both the conditions satisfied at the initial position, namely $\delta \beta =-K\left(b\left(0\right),{P}_{t}\left(0\right),\theta \left(0\right)\right)=-183{m}^{-1}$ and$\theta \left(0\right)=-\pi /2$. However, the strength of power-dependent phase change rate would experience an exponential decay along propagating, which is the same as total optical power does. Thus, the relative phase $\theta $ presents a negatively growing trend apart from the initial point under the condition of compensating the nonlinear phase modulations at the beginning. The influence of the initial relative phase on the nonlinear conversion is related to the effective interaction length. The enhancement of signal power can be considered as the integration of Eq. (10) along the effective interaction length. Therefore, the initial relative phase would be critical for the maximum enhancement of signal power.

In our calculation, the initial relative phases have been chosen from −100 degree to −75 degree with the interval of 5 degree to verify its influence on effective interaction length. At the same time, the varying rate of relative phase caused by nonlinear phase modulations is compensated by modal phase mismatch at the beginning of propagation. The initial conditions for this simulation can be expressed as:

where the change rate of relative phase along propagating is described as Eq. (9). The strength of nonlinear phase modulation $K$ is a function of total optical power, proportional coefficient and relative phase. Therefore, the modal phase mismatches $\delta \beta $ with different initial relative phase need to be calculated based on the initial condition of Eq. (17).By solving the simplified CMEs [Eqs. (4), (5), (9) and (3d)] for uniform microfibers with the initial parameters mentioned above, we plot the relative phase $\theta $ as functions of propagating length $z$ with different initial$\theta $, as shown in Fig. 2(a). The evolution of signal power along propagating is described by Eq. (10), which shows that the sine value of relative phase $\theta $ directly controls the direction and strength of nonlinear energy transfer. Therefore, the evolutions of $\mathrm{sin}\theta $ at different initial $\theta $ are plotted in Fig. 2(b) as the function of propagating length. In the propagating length less than 10 mm, the intervals between $\mathrm{sin}\theta $ and −1 increase with the increasing of the offsets of $\theta \left(0\right)$ from its optimal value of $-\pi /2$. Consequently, the values of the right hand of Eq. (10) at the corresponding $z$ decrease with the intervals between $\mathrm{sin}\theta $ and −1 increasing. However, the final nonlinear signal gain is the integration operation of Eq. (10) along propagating length. As shown in Fig. 2(c), the maximum gain coefficient is 0.087 dB, which is calculated by using Eq. (14), and corresponds with the effective interaction length of 22 mm with the initial relative phase $\theta \left(0\right)=-{85}^{\circ}$. More accurate simulation result is shown in Fig. 2(d) with the initial relative phase enlarging with 1 degree from −100 to −75 degree. The largest gain coefficient is 0.0875 dB at the effective interaction length of 22.89 mm with initial relative phase to be −83 degree. The effective interaction length, which corresponds the right hand of Eq. (10) decreases from the initial positive value to zero, increases with the increasing of initial relative phase, as shown in Fig. 2(d). Therefore, the positive offset from the optimal relative phase is helpful for increasing the effective interaction length. The corresponding evolution of relative phase along propagating can be illustrated as the process of $\text{A}\to \text{B}\to \text{C}$ in Fig. 1. The enhancement of signal gain is about 0.006 dB for −83 degree comparing with −90 degree, which correspond with the effective interaction length of 22.89 mm and 20.25 mm, respectively. However, the effective length extending takes the cost of effective signal power enhancing rate, especially for the beginning of propagation, in which the nonlinear effects can employ the unbated optical power.

#### 3.2 Optimizing the modal phase mismatch in uniform microfibers

An improved method for avoiding the waste of optical power in uniform microfibers is to keep the initial $\theta $ as its optimal value but to tailor the evolution of $\theta $ along propagating. We expect that the effective interaction length can be extended without expensing the nonlinear energy transfer rate in the beginning of propagating. In the first simulation, the relative phase monotonically decreases with the increment of propagating length. Here, we propose that the evolution of relative phase can be adjusted arbitrarily by tailoring the modal phase mismatch $\delta \beta $. By this way, the nonlinear process with longer effective interaction length than the first simulation is possible. However, as we known, the nonlinear energy transfer is still impossible to be maintained as the peak value with the given signal and pump optical powers over the whole propagating length, which is because the relative phase between optical fields propagating in the uniform microfibers cannot be kept as its optimal value. The initial conditions for this method can be expressed as following:

The simulation results are shown in Fig. 3 with the initial condition of expression (18) and (19). The initial change rate of relative phase caused by nonlinear phase modulations is −183 ${m}^{-1}$ with the initial optical parameters of ${P}_{t}=500W$, $b=0.08$ and$\theta =-\pi /2$. Based on the initial conditions above, the initial optical parameters are kept unchanged, while the modal phase mismatch $\delta \beta $ varies from −185 ${m}^{-1}$ to −155 ${m}^{-1}$ to find the optimal modal phase mismatch value for OTHG process.

As shown in Fig. 3(a), the decreasing trend of the evolutions of $\theta $ along propagating with $\delta \beta \text{=}-183{m}^{-1}$ gradually turn to increasing trend with the increasing of modal phase mismatches provided by microfibers. This is because that the modal phase mismatches bigger than −183${m}^{-1}$ cannot fully compensate the strength of nonlinear phase modulations at the beginning of propagation and makes the relationship $d\theta /dz>0$ exist in a period of propagating length. Therefore, the bigger modal phase mismatch corresponds with the larger enhancement of the relative phase along propagating. By choosing larger $\delta \beta $, the monotonically decreasing of $\theta $ in the first simulation turns to non-monotonic variation, which can be illustrated as the process of $\text{B}\to \text{A}\to \text{B}\to \text{C}$ in Fig. 1. The corresponding evolutions of $\mathrm{sin}\theta $ with 6 different values of $\delta \beta $ are plotted in Fig. 3(b), whose values are more close to −1 comparing with the evolutions shown in Fig. 2(b). The evolutions of the nonlinear gain coefficient along propagating at different $\delta \beta $ are calculated by substituting the solutions of uniform microfibers CMEs [Eqs. (4), (5), (9) and (3d)] in Eq. (14), which are plotted in Fig. 3(c). The largest gain coefficient in Fig. 3(c) is 0.0951 dB with effective interaction length of 27.24 mm and $\delta \beta =-165{m}^{-1}$. The enhancement of signal gain is about 0.01 dB for the modal phase mismatch −165 ${m}^{-1}$ comparing with −180${m}^{-1}$, which correspond with the effective interaction length of 27.24 mm and 21.32 mm, respectively. More accurate simulation result is shown in Fig. 3(d) with the modal phase mismatch enlarging with 1 ${m}^{-1}$ from −185 to −155 ${m}^{-1}$. The largest gain coefficient is 0.0953 dB at the effective interaction length of 26.58 mm with $\delta \beta $ to be −167 ${m}^{-1}$, which corresponds 0.0078 dB enhancement of maximum gain with the largest gain obtained in Fig. 2(d).

In the first two simulations, we have demonstrated the influence of the relative phase on the nonlinear energy transfer process in uniform microfibers. The nonlinear phase modulations are considered, which means that the phase compensation provided by microfibers has to simultaneously compensate the linear material dispersion of nonlinear medium and the nonlinear phase modulations. However, as we know, the intensity of phase modulation is proportional to the total optical power. In other words, the strength variation of nonlinear phase modulations along the propagation can be described as an exponential decay function when considering the power loss in microfibers. Consequently, the nonlinear phase modulation cannot be fully compensated over the whole propagating length by the phase compensation provided by uniform microfibers. As a result, the relative phase cannot be kept as a constant in propagation under this circumstance. We have pointed out that the nonlinear energy transfer is quite sensitive to the relative phase between the waves involved in nonlinear process, as shown in Fig. 1, which determines the direction and intensity of the nonlinear energy transfer process. Therefore, a suitable range of transmission phase difference for OTHG process is crucial to high conversion efficiency. Considering the propagating process, the effective interaction length, which is corresponding to the propagating length of the positive growth of signal power, is limited by the effective range of relative phase. The effective interaction length and the intensity of signal power growth jointly decide the final nonlinear conversion efficiency. In order to achieve long effective interaction length, we have to manipulate the relative phase along the propagation, which would inevitably affect the intensity of the signal power growth. In the first simulation, the nonlinear phase modulations are fully compensated at the beginning of propagation. Then the relative phase monotonically decreases along the propagation. Therefore, the effective interaction length can be extended by setting the initial relative phase to be bigger than the optimal value for the signal power growth, which can be illustrate as the process of $\text{A}\to \text{B}\to \text{C}$ in Fig. 1. However, the initial relative phase larger than its optimal value would reduce the growth intensity of signal power at the beginning of propagating, where the nonlinear effects can take full advantage of the undepleted optical power. In order to overcome this defect, in the second simulation, we set the initial relative phase to be its optimal value of $-\pi /2$ to ensure the largest growth intensity of signal power and change the evolution of the relative phase along the propagation to keep the long effective interaction length, which correspond with the process of $\text{B}\to \text{A}\to \text{B}\to \text{C}$ in Fig. 1. The simulations turn out that the second one is more effective than the first one for improving the OTHG conversion efficiency. However, the relative phase cannot be strictly kept as the optimal value over the whole propagating length in all these two schemes, which means the optical power is more or less wasted in the uniform microfibers.

#### 3.3 OTHG in non-uniform microfibers

In microfibers with varied diameters, it is possible to keep the relative phase as $-\pi /2$ over the whole propagating length by dynamically adjusting the modal phase mismatch to fully compensate the nonlinear phase modulation at any point along propagating. The CMEs describing the OTHG process with constant value of relative phase have been derived above, as Eqs. (4) and (6)-(8) shown. In this situation, we fix the relative phase on its optimal value for OTHG, corresponding to the point B in Fig. 1, over the whole interaction length. The initial conditions for relative phase under this circumstance can be expressed as following:

By numerically solving the CMEs for non-uniform microfibers with the initial optical parameters of ${P}_{t}\left(0\right)=500W$ and $b\left(0\right)=0.08$, the evolution of gain coefficient along propagating is obtained by substituting its solutions in Eq. (14). The results are plotted in Fig. 4, in which the evolutions corresponding with the largest conversion efficiency in the first and second simulation are appended for comparing with that of non-uniform microfibers.

Figure 4(a) shows the evolutions of relative phase for the three simulations we have performed, in which we can see that the relative phase between the interacting waves cannot be kept as its optimal value over the whole propagation length for uniform microfibers, while it is possible in non-uniform microfibers. The corresponding sine values for respective evolutions of relative phase are plotted in Fig. 4(b), in which the curves are more close to −1 the values of the right hand of Eq. (10) are bigger at the same propagating point. Consequently, the final nonlinear signal gains for the different initial conditions are shown as the curves in Fig. 4(c), in which the non-uniform microfibers with relative phase fixed as its optimal value over the whole length can obtain the largest gain coefficient of 0.0979 dB with longest effective interaction length of 28.42 mm. Comparing with the largest gain coefficients in the first two simulations, 0.0875 dB for optimizing initial relative phase and 0.0953 dB for optimizing modal phase mismatch in uniform microfibers, the enhancements of largest gain coefficient are 0.0104 dB and 0.0026 dB, respectively. The corresponding evolution of modal phase mismatch along propagating is plotted in Fig. 4(d) by solving Eq. (7), which means that the microfibers with variable diameters can completely eliminate the limit of relative phase on nonlinear processes.

## 4. Conclusions

So far, we have studied the basic properties of OTHG process in microfibers with power attenuation considered. The coupled mode equations with linear loss included are transformed to derive the threshold conditions for optical and microfibers’ parameters for efficient OTHG process. The transformed CMEs have been further simplified and analyzed with the specific circumstances of uniform and non-uniform microfibers. The final gain of signal power is the combined action of growth rate of signal power and the effective interaction length of nonlinear process. The growth rate value of signal power is sensitive to the value of relative phase and the evolution of relative phase would influence the effective interaction length. By adjusting the evolution of relative phase in the uniform and non-uniform microfibers, we have numerically demonstrated the optimization for initial conditions of efficient OTHG process. Evident enhancement for gain coefficient can be observed for OTHG process in microfibers with optimized parameters.

Although we have just discussed the OTHG process in microfibers, as [9] had mentioned that this set of CMEs is suitable for the arbitrary waveguides having third-order nonlinearity, our results of this text would also hold water for any waveguides providing modal phase matching. Furthermore, the original CMEs are derived to describe the THG process in waveguides, thus the method for optimizing the initial condition in this work would provide a feasible approach for the THG process with attenuation considered.

## Acknowledgments

This research was financially supported by the Basic Research Program of Shenzhen City (No. JCYJ20140417172417146) and Key Laboratory of Network Oriented Intelligent Computation at Shenzhen graduate school, Harbin Institute of Technology.

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