1 Introduction

Fiber optical parametric oscillator (FOPO) is a fiber optical resonator that relies on efficient generation of four-wave mixing (FWM) in optical fiber. The basic idea of a FOPO is to confine the gain region generated by the parametric process inside a highly nonlinear fiber (HNLF) cavity and subsequently creates laser.

When a HNLF medium is single-pumped with high power at wavelength near to its zero dispersion, symmetrical parametric gain spectra exist around the pump wavelength. For instance, if a pump wavelength is positioned in the normal dispersion region of the nonlinear fiber, a pair of symmetric narrow gain bands which is far away from the pump can be obtained provided that the fourth-order dispersion value of the HNLF is negative [1]. The interesting phenomenon here is that the further the pump wavelength is into the normal dispersion region, the narrower the gain bandwidth is. This feature allows a natural filtering mechanism and is widely applied in many FOPO structures [2–6]. On the other hand, the pump wavelength can also be positioned in the anomalous region of the nonlinear fiber to realize FOPO [7–9], but a filtering mechanism for seeding the cavity need to be included. A FOPO can also be constructed in a dual-pump configuration [10]; however, in this paper only single-pump FOPO will be discussed.

In either normally or anomalously pumped FOPOs, it is often to construct these FOPOs in singly resonant architecture, where only one side of the parametric gain band is allowed to oscillate in the fiber cavity, while the pump wave and the other gain band are removed deliberately from the cavity. It has become the norm that the gain band that resonates in a cavity is usually known as the signal band, while the other symmetrical gain band is known as the idler band. Typically in a ring FOPO, unidirectional oscillation is realized and a WDM is commonly used to ensure the pump wave and the idler wave are filtered from the cavity in each cycle of oscillation [2, 4, 5, 7]. In this case, the pump wave passes through the nonlinear fiber medium only once in each oscillation and is removed before a new cycle begins, whereas in linear cavity (Fabry–Perot) FOPOs, the easiest way to construct is by attaching fiber Bragg gratings (FBG) [8] or dichroic mirrors [6] at both ends of the nonlinear fiber medium to ensure singly resonant operation in the cavity. Note that, like in the case of ring FOPOs, the pump wave also passes through the nonlinear fiber medium once in each oscillation cycle and leaves the cavity together with idler component. In short, the pump was not fed back into the cavities mainly to avoid phase-mismatch disturbance. Since parametric gain is unidirectional [11] and reciprocal [12], a FOPO in fact can be operated in a more efficient way by propagating the pump in both forward and backward directions in the HNLF.

In this work, we propose a continuous wave wavelength-tunable FOPO by allowing the pump to propagate bi-directionally in the HNLF. However, this configuration introduces wavelength-dependent phase modulation on to the traveling light which in turn leads to wavelength-dependent gain modulation [13, 14] in the cavity. This phenomenon is due to the dispersion mismatch between the HNLF and the single-mode fiber (SMF) attached to optical components. This problem can be solved by utilizing a C/L-band wavelength division multiplexer (C/L-WDM) as an idler suppression filter.

2 Theory

Fundamentally, in degenerate FWM, a single pump acts as a reservoir to provide photons to a pair of spectrally symmetrical side-bands, which are termed as signal and idler in fiber optical parametric amplification literature. The fundamental equations that govern this process are [15, 16]

$$ \frac{{dP_{\text{p}} }}{dz} = - \alpha P_{\text{p}} - 4\gamma \left( {P_{\text{p}}^{2} P_{\text{s}} P_{\text{i}} } \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} \sin \theta , $$
(1)
$$ \frac{{dP_{\text{s}} }}{dz} = - \alpha P_{\text{s}} + 2\gamma \left( {P_{\text{p}}^{2} P_{\text{s}} P_{\text{i}} } \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} \sin \theta , $$
(2)
$$ \frac{{dP_{\text{i}} }}{dz} = - \alpha P_{\text{i}} + 2\gamma \left( {P_{\text{p}}^{2} P_{\text{s}} P_{\text{i}} } \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} \sin \theta , $$
(3)
$$ \frac{d\theta }{dz} = \Updelta \beta + \gamma \left\{ {2P_{\text{p}} - P_{\text{s}} - P_{\text{i}} + \left[ {\left( {P_{\text{p}}^{2} {{P_{\text{s}} } \mathord{\left/ {\vphantom {{P_{\text{s}} } {P_{\text{i}} }}} \right. \kern-\nulldelimiterspace} {P_{\text{i}} }}} \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} + \left( {P_{\text{p}}^{2} {{P_{\text{i}} } \mathord{\left/ {\vphantom {{P_{\text{i}} } {P_{\text{s}} }}} \right. \kern-\nulldelimiterspace} {P_{\text{s}} }}} \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} - 4\left( {P_{\text{s}} P_{\text{i}} } \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} } \right]\cos \theta } \right\}, $$
(4)

where P p is the pump power, P s is the signal power, P i is the idler power, α is the fiber attenuation coefficient, γ is the fiber nonlinear coefficient, Δβ is the linear phase-mismatch parameter defined as Δβ(ω) = β s(ω) + β i(ω) − 2β p(ω), in which β s(ω), β i(ω), β p(ω) are the propagation constant of signal, idler and pump, respectively. For ease of calculation, β is expanded around the pump wavelength using Taylor’s expansion such that Δβ now becomes [ 17 ]

$$ \Updelta \beta = \beta_{2} (\omega_{\text{s}} - \omega_{\text{p}} )^{2} + \beta_{4} (\omega_{\text{s}} - \omega_{\text{p}} )^{4} /12, $$
(5)

The θ in Eqs. ( 1 )–( 4 ) is the phase relation that is defined as

$$ \theta \left( z \right) = \Updelta \beta z + \phi_{\text{s}} \left( z \right) + \phi_{\text{i}} \left( z \right) - 2\phi_{\text{p}} \left( z \right), $$
(6)

where \( \phi_{\text{p}} \), \( \phi_{\text{s}} \) and \( \phi_{\text{i}} \) are the phase of pump, signal and idler, respectively. Note that θ determines the direction of power flow from pump to signal and idler or vice versa since it affects the sign in Eqs. (1)–(3). The first term at the right-hand side of Eqs. (4) and (6) clearly shows that θ will be modulated if Δβ changes, i.e., when the pump, signal and idler travel in a fiber medium consisting of different sections of dispersion properties.

3 Experiment and result

Figure 1 shows the proposed FOPO setup for the generation of FWM-based tunable fiber laser in a 500-m HNLF. The HNLF used in this experiment has a nonlinear coefficient of 11.5/W/km, an average zero dispersion wavelength (ZDW) of 1,556.5 nm and a dispersion slope of 0.015 ps/nm2/km. A tunable laser source (TLS) with a full-width half maximum of 0.01 nm is used as the parametric pump. We have ensured that the TLS is spectrally wide enough to avoid stimulated Brillouin scattering in the setup. The pump polarization is controlled by a polarization controller (PC) for optimum result. The pump is then boosted up by a high-power Erbium-doped fiber amplifier (EDFA). A tunable bandpass filter (TBF) is placed after the EDFA to remove excessive amplified spontaneous emission. Small fraction (2 %) of the boosted pump power is tapped out using a 98/2 coupler and is constantly monitored by an optical power meter (OPM). A wideband mirror and a fiber loop mirror form a Fabry–Perot cavity and a 90/10 coupler is used to tap out the oscillating waves in the cavity. Even though the cavity loss that originated from 90/10 coupler is considerably high, it prevents damaging of C-band TBF by high oscillating power and allows small signal generation from the loop mirror which is suitable for later theoretical analysis. FWM effect is typically a unidirectional process. By using a wideband mirror, the pump can propagate through HNLF in both forward and backward directions, thus increasing the efficiency of the lasing process. The lasing wavelength is determined by a C-band TBF which can be tuned from 1,535 to 1,565 nm with a 3-dB bandwidth of 0.35 nm. The C/L-WDM in the dashed box is employed to remove idler spectral components (L-band) in the cavity. In order to evaluate its impact, studies are performed beforehand with the C/L-WDM excluded.

Fig. 1
figure 1

Experimental setup of Fabry–Perot cavity optical parametric fiber laser

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In the equilibrium state of oscillation, the generation of laser can be understood as follows: first, the pump at 1,560.5 nm and a seed laser (controlled by TBF) enter HNLF from the left. The seed signal is amplified and an idler is generated in the HNLF via FWM process. These three waves (amplified seed signal, residual pump and idler) are reflected back by wideband mirror and pass through HNLF for the second time in the reverse direction. The residual pump and idler are eliminated in the loop mirror by the TBF, leaving the seed signal to complete its round trip oscillation and form a laser. Figure 2a shows the superposition of multiple output spectrum of the FOPO laser at several C-band TBF central wavelength positions. It can be clearly seen that the laser peak power is not smooth across the wavelength range. Figure 2b shows greater resolution of laser peak power across the wavelength of interest and the calculated parametric gain is also included. The calculated gain is numerically simulated using Eqs. ( 1 )–( 4 ). In this model, undepleted pump approximation and linear polarization of all waves are assumed. The fiber attenuation is ignored for ease of calculation. There are three steps in the simulation similar to [ 13 ]. In the first step, the pump power, seed signal power and a small but finite idler power are inserted into Eqs. ( 1 )–( 4 ) to calculate the output after passing though 500 m HNLF. The second step caters for the phase modulation by the SMF section. We assume that the SMF has negligible nonlinearity for FWM process. Hence the output θ from HNLF is changed to θ′ = θ + Δβ SMF L SMF, where Δβ SMF is the phase mismatch of SMF and L SMF is the length of the SMF section near the reflecting wideband mirror, while the power parameters (pump power, seed signal power and idler power) are readjusted to P p,s,i(reflect) = r × P p,s,i , where r = 0.74 is the reflectivity of the wideband mirror used. Finally the pump power, seed signal power, idler power as well as the new phase relation θ′ are re-inputted into the same 500 m HNLF model and their gain profile calculated. The experiment data and the calculated gain are normalized such that they can be conveniently compared. Referring to Fig. 2b, the uneven laser peak power is indeed due to the wavelength-dependent phase modulation caused by the SMF dispersion in the cavity [ 13, 14 ].

Fig. 2
figure 2

a Superposition of multiple output spectrum of fiber laser in a Fabry–Perot cavity without using C/L-WDM. b Normalized optical parametric fiber laser peak power/theoretical gain versus wavelength in C-band. The pump wavelength is 1,560.5 nm and pump power is 0.9 W

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Qualitatively, the gain distortion of FOPO can be understood as follows: In the equilibrium state, the pump at ω p and the oscillating laser ω s are constantly passing back and forth through the HNLF, as well as the generated idler at ω i = 2ω p − ω s. When the idler is first generated in HNLF, it will have a phase ϕ i that automatically fulfills the relative phase difference θ = 2ϕ p − ϕ s − ϕ i, where ϕ p and ϕ s are the phase of the pump wave and signal wave, respectively. Then three waves enter a short section of SMF of the wideband mirror, and all phases of three waves are changed. When these waves are reflected by the wideband mirror and reenter the HNLF, their new relative phase difference would become θ′ = θ + Δβ SMF L SMF, where θ′ is the modulated relative phase difference, Δβ SMF = β s + β i − 2β p is the wavelength-dependent linear phase mismatch in SMF and L SMF is the total length of SMF where the waves had passed through. As a result of the modulation of phase, the gain in the FOPO is wavelength dependent.

In order to smoothen out the uneven laser peak power, we could either use phase-locking techniques to preserve the relative phase difference of the three waves oscillating in cavity or build the entire cavity with HNLF fiber so that the dispersion properties are consistent in the cavity. However, both of the methods are more complex and expensive. Here we propose a much simpler way by removing the idler component from re-entering the HNLF using a C/L-WDM. Consequently, a new idler with correct phase is generated in the HNLF with each new cycle of oscillation, thus preserving the phase relation of all the three waves.

This C/L-WDM has a 3-dB cutoff at 1,565 nm. It is connected near the wideband mirror such that spectral component longer than 1,565 nm is filtered out. This means that the pump and C-band seed signals are retained in the cavity and pass the HNLF twice. The new idler is also generated in this second pass but is eventually eliminated by the TBF in the fiber loop mirror, thus diminishing wavelength-dependent gain. Figure 3 shows the laser peak power comparison when a C/L-WDM is inserted/not inserted into the experimental setup. With the C/L-WDM, the laser peak fluctuation in C-band is reduced and its peak power improved. The reason for the improved peak power is the mitigation of back energy conversion through the idler filtering mechanism in the cavity [ 18, 19 ]. Back energy conversion happens at the point when the pump is fully depleted, and as the result the energy from both signal and idler start to flow back to the pump until the pump is recharged. The periodic energy transfer from pump to signal/idler and vice versa causes the FOPO to operate in a less efficient manner. However, by keeping the idler power low enough throughout the cavity, one can ensure the energy transfer always happens from pump to the signal/idler pair. Hence by introducing idler loss mechanism at the two ends of the cavity (one end is C/L-WDM and the other is C-band TBF), the FOPO performance is improved.

Fig. 3
figure 3

Comparison of C-band optical parametric fiber laser peak power versus wavelength with C/L-WDM and without C/L-WDM in the cavity

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To investigate the benefit of double-pass pumped FOPO, we first connect the L-arm of the C/L-WDM to the wideband mirror and replace the C-band TBF with an L-band TBF. By configuring the experimental setup this way, we ensure only wavelength components longer than 1,565 nm oscillate in the cavity; in other words, the pump and the C-band idler will be filtered out in each cycle of oscillation to realize single-passes of pump. The result is shown in Fig. 4 with 0.9 W 1,560.5 nm pumping condition, comparing the FOPO laser generation with C/L-WDM (single-pass) and without C/L-WDM (double-pass). As we can see, in the wavelength region dominated by parametric process (<1,585 nm), the laser peak with double-passed pump configuration is generally higher than single-passed pump configuration, and more than 10 dB peak power improvement is observed by comparing the maximum points of double-passed laser peak with that of single-passed laser peak. In the wavelength region longer than 1,585 nm, the laser peak for both cases are smooth because they are mainly supported by stimulated Raman scattering process; hence almost no gain-modulation characteristic from parametric process can be observed.

Fig. 4
figure 4

Comparison of L-band optical parametric fiber laser peak power versus wavelength with C/L-WDM and without C/L-WDM in the cavity

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4 Conclusions

In conclusion, a Fabry–Perot cavity FOPO with double-pass pump is demonstrated. Due to the difference of fiber dispersion in the Fabry–Perot cavity, the FOPO laser peak power is uneven across the wavelength of interest. This can be solved by simply filtering out the idler spectral component in the cavity. This method is expected to open up more FOPO design possibility other than just limited to FBG-based and ring-based FOPO design. Besides, this method is also expected to be useful in designing fiber optical parametric amplifier (FOPA), e.g., in multi-stage FOPA where fiber-dispersion discrepancy may become an issue.