## Abstract

Fiber optical parametric oscillators (FOPOs) are coherent sources that can provide ultra-broadband tunability and high output power levels and are been considered for applications such as medical imaging and sensing. While most recent literature has focused on advancing the performance of these devices experimentally, theoretical studies are still scarce. In contrast, ordinary laser theory is very mature, has been thoroughly studied and is now well understood from the point of view of fundamental physics. In this work, we present a theoretical study of OPOs and in particular we theoretically discuss the process of gain saturation in optical parametric amplifiers. In order to emphasize the significant difference between the two coherent sources, we compare the optimized coupling ratios for maximum output powers of the ordinary laser and the optical parametric oscillator and demonstrate that in contrast to ordinary lasers, highest output powers in optical parametric oscillators are achieved with output coupling ratios close to 1. We confirm experimentally our theoretical studies by building a narrowband fiber optical parametric oscillator at 1450nm with multi-watt output power. We show that the device is robust to intracavity losses and achieve peak power as high as 2.4W.

© 2013 OSA

## 1. Introduction

The requirements of coherent light sources for applications in medical imaging, biological and chemical sensing, are continuously increasing. Several kinds of coherent sources are been proposed and applied for these applications, including diode-based and fiber-based lasers. Among those candidates, fiber optical parametric oscillators (FOPOs) offer many competitive advantages such as broad tunability and high output powers. The operation of FOPOs derives from the third order nonlinearity of the fiber by the process of four-wave mixing (FWM). Because of this, its parametric gain is decided by the dispersion relation between the pump and sidebands, and thus the FOPOs can provide ultra-broad bandwidth and large tunable area. For example, B. Kuo et al. demonstrated 329nm wide tunable bandwidth using a FOPO composed of ordinary components for fiber communication systems [1]. Several reports have also demonstrated generation at the range of 1000nm and 2000nm by pumping the FOPO at wavelength around 1550nm [2, 3], largely extending the application field of FOPOs. Several schemes have been proposed to increase the sweeping rate for FOPOs. Using a pump sweeping scheme, a sweeping speed as high as 77760nm/s has been realized [1], and the dispersion tuning technique [4], developed for fiber lasers, can be also applied to FOPOs, to achieve tuning speeds exceeding 4,000,000nm/s [5]. The majority of the recent literature is focused on advancing the performance of FOPOs experimentally but theoretical studies about FOPOs are still scarce. We can gain some insight from previous works in the field of conventional optical parametric oscillators (OPOs) based on the materials with second-order nonlinearities [6]. The studies about how to optimize an OPO with more efficient conversion efficiency and higher output power are presented [7, 8]. However, for a kind of devices with third-order nonlinearities, to have a deeper understanding of FOPOs, it is essential to include the cavity properties, such as the saturation gain, loss, output coupling ratio and conversion efficiency, in the discussion. In a recent work [9], a precise nonlinear model to predict the saturation gain and the output power of a FOPO in the steady state was proposed. It considers the solutions of FOPO, but does not seem to give a recipe for getting maximum conversion efficiency or output signal power. Y. Q. Xu et al. numerically studied the maximum conversion efficiency from the pump to each sideband in the FOPO [10]. They considered the effects of dispersion mismatch, the number of times above oscillation threshold, and the sideband feedback fraction in the lossless FOPO. A conversion efficiency of 93% is achieved [11, 12]. But compared to the mature laser theory, these efforts are still incomplete. In order to optimize the performance and implementation of FOPOs, it is necessary to study and understand them from the point of view of fundamental physics.

In this work, we discuss the influence of output coupling ratio for FOPOs without the lossless approximation. Theoretical and experimental study is presented. At first, we review the laser theory for the lasers with an ordinary gain medium and present the effects of output coupling ratio and cavity losses. The readers who are familiar with the laser theory can go to section 2 directly. In section 2, we discuss the parametric gain saturation in a fiber with third order nonlinearity. The discussion focuses on the single oscillation for one sideband (signal/anti-Stokes) in the phase insensitive FOPO with one pump. We compare the optimized coupling ratios for maximum output powers of the laser using an ordinary gain medium and the FOPO. Through the theoretical analysis and numerical calculation, we show that there is an optimized output coupling ratio for a FOPO to get a maximum output power. The optimized output coupling ratio is not much less than 1, this differs from the case of lasers using ordinary gain medium that require coupling ratios close to 0. We discuss a particular example of a narrowband FOPO [13], which has an optimized ratio in the range of 85%-95%. Finally, we present experimental results that confirm our theoretical and numerical analysis. We show that the FOPO is robust to cavity losses, and capable of multi-Watts output powers even with cavity losses as high as 70%-80%.

## 2. Gain saturation in ordinary lasers versus gain saturation in FOPOs

Lasers and parametric oscillators are similar in many aspects. However, the optical processes involved are fundamentally different and this leads to different behaviors. While laser physics and in particular its saturation theory is nowadays very mature, an improved understanding of the gain saturation characteristics of FOPO is necessary to further optimize their performance. In this section, we describe and compare the well-known gain saturation characteristics of ordinary lasers and compared them to our study of gain saturation in FOPOs. While the physical processes involve in FOPOs are closer to conventional OPOs than to fiber lasers, in this paper we choose to emphasize the differences of output coupling ratio between FOPOs and fiber lasers. This is a problem that is easily overlooked in literatures, where experimental demonstrations of FOPO choose the output coupling ratio as they would for a fiber laser coupling ratio.

#### 2.1 Gain saturation in ordinary lasers

As we mentioned above, the saturation theory in rare-earth doped fiber lasers is already mature, and nowadays their gain properties can be fully described using physical and mathematical models. Lasers using ordinary gain medium are considered as energy level systems. The amplification coefficient for a signal wave passing through a laser amplifier is proportional to the population difference on the amplification transition. However, when the signal power begins to approach the saturation power for the laser medium, the population difference and hence the gain coefficient in the laser medium begins to saturate. When this happens, the rate of signal growth with distance begins to decrease and thus, the signal power grows more slowly with distance. Such saturation effects begin first at the output end of the amplifier. The input signal is amplified as it propagates through the laser media. If that input signal power is high enough, as it approaches the output end, it reaches the saturation power of the laser medium. This saturation at the output end causes the growth rate to decrease near the output end, and this in turn reduces the total saturated gain from input to output compared to the small signal gain of the amplifier. If the input signal is increased further, the saturation region will gradually move toward the input end. In the case of rare-earth doped fiber based lasers, gain saturation can be considered as homogeneous. This means that it saturates at each transverse plane according the power at that plane. If we limit the discussion to laser cavities with a low output coupling ration (i.e. most of the power is sent back into the cavity as feedback), the power along the length of the cavity for a steady state can be considered as a constant *P*_{circ}. Using basic laser theory, the gain saturation can be derived. Equation (1) gives the gain, *G* as a function of the power, *P*_{circ} for single pass amplification [14]:

*P*

_{sat}is the saturated power, which means the value of signal power passing through the laser medium that will saturate the gain coefficient down to half its small-signal or unsaturated value

*G*

_{unsat}. In Fig. 1(a), we illustrate the single pass saturation properties based on Eq. (1), where we assume the unsaturated gain

*G*

_{unsat}= 100. This means that if we want to extract the maximum power from a laser cavity with such rare-earth doped fiber as gain medium, a work point with low single pass gain is preferable. Assuming that the laser is working in a steady state, the gain G, the output coupling

*T*, and the cavity losses

*δ*, should be balanced so they should follow the following relationship

*G*= 1/(1−

*T*) (1−

*δ*). From Eq. (1), the laser with higher output power should operate at a work point with low gain, which corresponds to a lower output coupling and a lower loss in the cavity. The optimized coupling relation is shown in Fig. 1(b), which can also be calculated by the theory in [14].

#### 2.2 Gain saturation in FOPOs

The discussion in the previous section is well-known and agrees well with experiments, not only for rare-earth doped fiber lasers but also for many laser systems even with larger output coupling ratios [14]. Unlike doped fiber lasers, the origin of the gain in a FOPO lies in the nonlinear process of four wave mixing (FWM). From the viewpoint of quantum optics, FWM could be considered as an energy level transition process between several pseudo levels, but unlike ordinary lasers, the process of parametric amplification is not governed by population inversion. FWM is governed by the phase matching condition between the pump and sidebands. Thus, the approximations discussed above are not valid in the operation of FOPOs.

In an erbium doped fiber, the physical meaning of the saturated power, *P*_{sat} is the power at which there is one photon incident on each atom, within its cross section*σ*, per recovery time *τ*_{eff}. If the incident signal power (i.e. the number of incident signal photons) is high enough, the gain monotonically decreased and saturates toward the limiting *G* = 0dB. In fiber lasers the power cannot be transferred from the amplified signal back to the gain medium or pump. However, the saturation in a parametric amplifier is fundamentally different and thus far, there has not been a fully physical explanation or a clear definition of the term, saturated power, for FOPOs.

The saturation caused by the degenerate FWM in a fiber optical parametric amplifier comes from the energy transfer between the pump and the signal/idler which is periodically determined by the phase matching condition. The fiber for parametric amplification is not a homogeneously saturable gain medium, where the saturation does not take place at each transverse plane along the fiber. Usually, in the field of fiber communication systems, we say that a fiber optical parametric amplifier (FOPA) is in its “saturated state” when it can work as an amplitude limiter for the signal regeneration. Usually, the three coupled-mode equations for the degenerate four-wave mixing can be expressed as [15]

*E*_{0}, *E*_{1}, and *E*_{2} are the electric field complex amplitudes of the pump power, and two sideband amplitudes, respectively. Δ*k* = *k*_{1} + *k*_{2}−2*k*_{0}, *γ*, and *z* are the propagation constant mismatch, third order nonlinear parameter, and propagated distance in the fiber, where *k*_{0}, *k*_{1}, and *k*_{2} are the electric field complex amplitudes of the pump power, and two sideband amplitudes, respectively. From theoretical and numerical analysis [16, 17], we know that the saturation caused by FWM is periodic. The normalized power of one sideband can be expressed as:

*P*

_{sig}= |

*E*

_{1}|

^{2},

*P*

_{idl}= |

*E*

_{2}|

^{2}, and

*P*

_{pump}= |

*E*

_{0}|

^{2}, are the power of signal and pump, respectively. The idler is zero at the input,

*P*

_{idl}(0) = 0. α = (

*P*

_{sig}−

*P*

_{idl})/(

*P*

_{sig}+

*P*

_{idl}+

*P*

_{pump}) is the normalized power difference between the signal and idler.

*ξ*=

*zγ*(

*P*

_{sig}+

*P*

_{idl}+

*P*

_{pump}) is the normalized propagation length. The parameters

*a*,

*b*,

*c*, and

*d*, are the roots of equation d

*η*/d

*ξ*= 0 (Eq. (2a) in [16]), and the order is that

*a*>

*b*>

*η*(

*z*) ≥

*c*>

*d*. To illustrate the results in a concrete form, we calculated the propagation along a lossless fiber using the following parameters: nonlinear parameter

*γ*= 0.001/W/m, input pump power

*P*

_{pump}= 44.8dBm (30W), input signal power

*P*

_{sig}= 26.2dBm (0.421W). We assume a perfect linear phase matching term Δ

*β*= −2

*γ P*

_{pump}. As shown in Fig. 2, if the fiber is long enough, the power transfers from pump to signal and then from pump back to signal periodically as it propagates through the fiber. The energy transfer between the idler and the pump has the same periodicity.

For a fiber length *L* = 100 m, which is corresponding to the first peak of the signal in Fig. 2, the transmission property of the fiber working as a FOPA is presented in Fig. 3. The output signal power is not a monotone increasing function of input signal power but a periodic increasing function. Under these assumptions and parameters, saturation occurs when the input signal is increased to a power around 26.2dBm, see Fig. 3. At this input signal power level, the output power is limited to 39dBm, marked with a circle in Fig. 3. The FOPA at this work point can be regarded as entering the “saturated state”, and can be used as an amplitude limiter in the field of fiber communication systems. Notice that in Fig. 3, there is a higher order saturation state with 45dBm output at the high input signal power. For a FOPA, in principle, if we increase the input signal power in experiments, this state can be reached. However, in this work, the signal of the FOPO originates from amplified spontaneous emission (ASE). By the optimized coupling ratio design, the FOPA can only work steadily at the lower order saturation state.

From the discussion above, it can be concluded that the saturated state, in mathematics, is corresponding to a critical point that ∂*P*_{sig}/∂*z* = ∂*P*_{sig}/∂*P*_{sig}(0) = 0. In physics, this means that the phase matching condition is at a critical point. If the input signal power becomes even higher or the fiber length is even longer, the phase matching condition will change, and the energy will transfer back from the signal and idler to the pump.

We can define the gain of the FOPA as *G* = *P*_{sig}(*L*)/*P*_{sig}(0). The gain medium in FOPAs is therefore inhomogeneous and the gain of signal should not decrease as a strictly monotonic function of the input signal power. The mathematical solution of the gain in a FOPA can be expressed as a combination of elliptic functions, which can be found in [9]. By using the split-step Fourier method to numerically solve the coupled equations shown in Eq. (1), we can get the signal gain as a function of the output signal power as shown in Fig. 4, where the parameters are the same with those in Fig. 3. Here, we just consider the simplest FWM process that only three waves are involved. The higher order or cascaded FWM processes are ignored. It is reasonable that for the narrowband parametric amplification in fibers, usually we can only generate the three waves FWM in experiments [13]. The readers can find the basic Matlab code of using split-step Fourier method in Appendix B of [15].

From Fig. 4, we can see that in the FOPA the limitation of the gain, similar to ordinary laser amplifiers, is 0dB. However, in the FOPA, the point with 0dB gain does not correspond to the highest output power. This means that at the end section of the fiber, energy is no longer transferred from the signal to the pump, the energy transfer has completed a cycle and in the end section of the fiber it starts to transfer energy to the signal again. There is a work point (the red box shown in Fig. 4), which corresponds to the saturated state (the red circle in Fig. 3). Unlike ordinary laser amplifiers, a lower single pass gain does not necessarily lead to a higher output power. This saturated gain is much larger than 0dB. When the FOPO is working at the stable state, the following relation between gain *G*, cavity losses *δ*, and output coupling *T*, is satisfied: *G* = 1/(1−*T*) (1−*δ*). This relation indicates that the highest output powers are achieved with larger output coupling ratios. This is inverse to the previous discussion with ordinary lasers that showed that the optimize output power require most of the power to be fed back into the cavity and thus much lower output coupling ratios.

To verify this conclusion, we firstly numerically simulate light propagation in FOPOs. The FOPO consisted of a cavity with a fiber, a filter, and a coupler. The parametric gain medium is a 100m fiber with the same parameters in Fig. 3 and Fig. 4. The results are shown in Fig. 5, where the cases of FOPOs with different output coupling ratios and cavity losses are illustrated. All the simulations are following the assumption that the linear phase matching term Δ*β* = −2*γP*_{pump} is automatically satisfied.

From Fig. 5, it can be found that an output coupling larger than 0.9 is suitable for getting a higher output power when the loss *δ* = 0. Another interesting point is that the FOPO is robust to the loss in the cavity. Even the cavity losses as high as 70%-80%, by choosing a suitable coupling ratio, multi-watt outputs can also be realized.

## 4. Experiment

In order to confirm our theory, we constructed an FOPO as shown in Fig. 6. The FOPO consists of a cavity and a pump source. In the cavity there were a dispersion shift fiber (DSF), a polarization controller, a bend-loss low pass filter, and a 2 × 2 coupler. The connection of the 2 × 2 coupler is illustrated. The DSF (about 100m, made by Fujikura, DSM⋅8/125⋅026) was the parametric gain medium with zero-dispersion wavelength, *λ*_{ZDW} of 1570 nm, a dispersion slope, d*D*/d*λ* of 0.072 ps/nm^{2}/km, and a nonlinear coefficient *γ* of 0.001/W/km. A low pass filter was made by rotating a SMF-28 fiber or a DSF, where the bend loss was wavelength dependent. This filter ensured only the anti-Stokes light (signal) returns to the DSF through the coupler, and the Stokes light (idler) and the pump in the last round trip were filtered out. The pump source consisted of an external cavity laser which is tunable between 1520 and 1630 with an output power 5mW. The CW output of the laser was modulated by a MZ modulator with the order of nanoseconds Gaussian pulses and the duty ratio of about 1:40. Then the pump pulses were amplified by two EDFAs, and a band pass filter was placed between the two EDFAs in order to suppress the ASE noise. The repetition frequency was finely tuned to match the cavity fundamental frequency (around 1772.8KHz). A 90/10 coupler was inserted before the cavity to monitor the pump power to confirm that the pump pulses into the cavity were amplified to a peak power about 30W. Figure 8 shows the output spectrum of the FOPO in log and linear scale respectively, where the pump wavelength was fixed at 1547nm. The wavelength of the generated anti-Stokes light was about 1450nm.

In fact, as the simulation results shown obviously in section 3, the FOPO with low losses is more efficient for us to get higher power out of the cavity. In the experiment, because the bend loss at 1450nm is not low enough, we only studied the cases with high losses. Three bend low-pass filters Filter 1, Filter 2, and Filter 3 were fabricated that the total cavity losses could be controlled at about 70%, 80%, and 90%, respectively. Filter 1 and Filter 2 were made by DSF, and Filter 3 was made by SMF-28. Filter 1 has a larger wound diameter and a longer length compared with Filter 2 and Filter 3. The loss properties the filters are illustrated in Fig. 7. With these filters, no residual Stokes light (idler) > 1600nm can be feedback. For measuring the effects of the output coupling ratio on the output power, the filters with coupling ratio 99/1, 90/10, 80/20, 70/30, and 50/50 were used. In our experiments, the power (30W peak power) we used to pump the signal can produce an about 20dB unsaturated gain in the cavity, which was higher than the threshold [10]. From the conventional OPO theory, the times above threshold is a very important factor for the stability of OPO [18, 19]. Recently, the problem of stability was also discussed for the novel OPO using chirped quasi-phasematching gratings [20]. Though the stability of FOPO is not the main topic of this work, we present the temporal profile of the signal pulses compared with the pump pulses, as shown in Fig. 9. The temporal profile of the signal pulses is corresponding to the output from the cavity with a 50/50 coupler and a 70% intracavity loss, for which the times above threshold is largest in our experiments. It can be found that the signal output is stable enough, and the duration of the pump pulse can give the signal enough time to reach a steady state.

In Fig. 10, the experimental and simulation results are shown in lines and dots respectively. It can be found that the experimental results agree well with the prediction in the last section. The maximum output power corresponds to the output coupling ratio near 1. As illustrated in the figure, an output peak power of about 2.4W was measured at 0.9 output coupling with a 70% internal cavity loss. Even the loss in the cavity is as high as 90%, a 0.5W peak power was measured, which shows good robustness against the cavity loss.

## 5. Conclusion

We discussed the effects of output coupling ratio and cavity loss on the output power in FOPOs. The differences between the gain saturation process in lasers using ordinary gain medium and FOPOs were analyzed. For a phase insensitive FOPO with one pump, there is an optimized output coupling ratio to get a maximum output power. The optimized output coupling ratio is close to 1, this is significantly different from the case of lasers using an ordinary gain medium. This can help us to build a FOPO with maximum output power. By choosing a suitable output coupling ratio of the fiber optical parametric oscillator cavity, a narrowband FOPO at 1450nm based on the commercial dispersion shift fiber with multi-watt output power was proposed. The optimized ratio is in the range about 80%-90% for different cavity losses. Because the FOPO is robust to the cavity losses, a 2.4W peak-power was measured for the cavity with 70% internal loss. We observed good agreement between the theoretical and experimental results.

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