Modal interference can lead to intensity modulations in optical fibers, which can produce refractive index gratings under the influence of quantum defect heating in a fiber laser. These gratings are perfectly phased matched for mode couplings, which can lead to transverse mode instabilities at high average powers in fiber lasers. A detailed understanding of this process is critical for further power scaling of fiber lasers. We have directly observed and characterized this quantum-defect-assisted mode coupling for the first time using polarization modes in a PM fiber amplifier, providing solid experimental evidence for this key mechanism for transverse mode instability in fiber lasers.
© 2016 Optical Society of America
Modal interference causes periodic intensity modulations in an optical fiber which supports more than one mode. In an active fiber such as a fiber laser or amplifier, a higher local intensity can lead to higher quantum defect heating, i.e., heat generation due to the amplification process. This can lead to temperature gratings, resulting in a refractive index grating in the fiber that is perfectly phase matched for further power coupling of the two modes involved. This mode coupling can become very significant at high average powers, leading to mode degradation and limiting the average powers for single-mode operation. This effect has recently been theoretically studied in a number of reports [1–4].
Transverse mode instability (TMI) has indeed been observed in recent years in many fiber types [5–8], manifesting itself as severe mode quality degradation at high average powers in an otherwise well-behaved single-mode fiber laser at lower average powers. Quantum-defect-assisted mode coupling is currently a leading candidate for its main mechanism. More detailed studies are, however, required to establish a concrete link.
Quantum-defect-assisted mode coupling is in fact a manifestation of stimulated thermal Rayleigh scattering (STRS), which was first observed in the sixties . When a pump and a probe beam from a -switched Ruby laser interfered in an absorbing liquid, a power transfer between the two beams was observed. A theory was developed at that time to explain this based on the formation of a refractive index grating resulting from the combination of the periodic intensity variation from interference and heat generation due to absorption [10,11].
STRS is very similar to stimulated Brillouin scattering (SBS), except that the third wave is a temperature wave instead of an acoustic wave and the two optical waves can co-propagate. The temperature wave needs to be a traveling wave to satisfy the required phase conditions, as it does in SBS. It arises from the actions of local heat generation from a traveling interference pattern and mostly transverse heat diffusion. When the heat diffusion rate is faster than the velocity of the traveling interference pattern, a virtual traveling temperature wave develops. The coupled power, therefore, is frequency shifted. Like SBS, STRS can start from one optical wave and some noise. It is very robust and can become very significant at high powers.
In the late sixties and early seventies, there were some more studies of STRS [12–16]. The lack of suitable sources prevented further in-depth experimental studies. In the early eighties, there was a resurgence of interest in STRS for phase conjugation and four-wave mixing [17–19]. The similarity between stimulated Rayleigh scattering (SRS) and SBS was noted in . In fact, a simultaneous theoretical treatment of the two effects is given in  considering both thermal and electrostrictive effects. STRS has rarely been studied since the eighties, possibly due to a lack of applications.
In this work, we have observed and studied mode coupling, i.e., frequency-dependent STRS gain or loss, for the first time using two polarization modes in a polarization-maintaining (PM) fiber amplifier and a pump-probe technique. The choice of the polarization modes allows us to monitor the input and output of a mode easily using polarization beam splitters. The sensitive pump-probe technique allows us to perform this experiment at low powers of sub-watt levels instead of hundreds of watts, making it significantly easier. Since the interesting range of frequency differences between the two optical waves is over several kHz (determined by transverse thermal diffusion rate), the pump and probe waves are generated from the output of a single-frequency laser with a kHz linewidth by two acousto-optic modulators (AOMs) driven by two RF wave phase locked to a master oscillator. We can control the relative frequencies of the two RF waves down to Hz-level accuracy. To our knowledge, this is the first direct measurement of frequency-dependent STRS gain. Our work firmly establishes that STRS plays a significant role in mode coupling in fiber lasers at high powers.
Assuming the powers of two modes are represented, respectively, by and , and the fiber amplifier gains are independent of propagation distance , represented, respectively, by and , the following coupled non-linear equations can be written for STRS power coupling between the two modes if the background loss is ignored :4]). The normalized powers are defined as
The power in mode 2 can be obtained analytically for a fiber amplifier with a length of in this case,
The first exponential on the right-hand side of the equation is gain due to the amplification, and the second exponential term is gain due to mode coupling by STRS. If we can ignore power depletion in mode 1 due to mode coupling and the amplifier gain is sufficiently large, i.e., , we can simplify Eq. (3) to
It is interesting to see that the STRS gain is only dependent on the output power of the first mode and the STRS non-linear coupling coefficient in this case.
The choice of polarization modes was due to the ease of mode mux/demux at the input and output of a fiber amplifier using simple polarization beam splitters. This, however, brought some complications due to the nature of the polarization modes. Assuming orthogonally linearly polarized modes and a linearly polarized input, the polarization state in a PM fiber generally goes from linear to elliptical, then back to linear twice over a beat length without any intensity modulation. No temperature wave is expected in this case. It is, however, well known that the gain of a fiber amplifier has a weak dependence on the state of polarization . With a combination of the periodical variation of the local polarization state in a PM fiber amplifier and polarization-dependent gain, a temperature wave can be generated. To enable polarization mode coupling, the temperature wave also needs to perturb the local birefringence axis periodically, as in a rocking rotator filter . In a practical fiber, any asymmetry can lead to a perturbation of the local birefringence axis by a temperature wave. All the elements are therefore in place for polarization mode coupling in a PM fiber amplifier due to STRS.
In addition, it has also been known for some time that polarization modes in highly birefringent fibers are not strictly linearly polarized, with fields in both orthogonal polarizations . The minor field component of the modes is a consequence of the stress variation across the core [24–26]. This can also lead to polarization mode coupling without a rocking rotator filter.
The fiber used is PM-YDF-5/130, a double-clad PM ytterbium-doped fiber from Nufern. The core has a NA of 0.12, a diameter of 5 μm, a mode field diameter (MFD) of 6.5 μm at 1060 nm, and a birefringence of . The cladding diameter is 130 μm with a pump NA of 0.46. The pump absorption is 0.6 dB/m at . The experimental setup is shown in Fig. 1. The laser used is a non-planar ring oscillator single-frequency laser capable of providing with a linewidth less than 5 kHz. The laser beam is split into two beams and each goes through an AOM. The two AOMs are driven by two phase-locked RF waves generated from the same master oscillator, with one AOM fixed at 40 MHz. The relative frequency can be tuned with Hz-level accuracy up to few hundred kHz. The two beams are launched as the seed beam into the slow birefringence axis (see inset in Fig. 1) and the probe beam into the fast birefringence axis. A fiber length of 4 m was used, pumped by a multimode diode at . The output powers of the probe and seed beams are measured after being separated by a polarization beam splitter.
There is always a very small amount of power leaked through to the wrong detector at the polarization beam splitters. This leaked power has a different frequency and leads to a sinusoidal amplitude fluctuation at the frequency difference between the two polarization modes. It can be easily taken out from the total measured power by analyzing the measured amplitude fluctuation and average power. This setup has an extremely high stability in the frequency difference (probe frequency—seed frequency) enabling measurement down to kHz frequency separation, limited by the laser linewidth.
Since there is no mode coupling when the frequency difference is zero, the output power can be normalized against the output at a 0 Hz frequency difference to obtain gain or loss purely from STRS. With both the input probe and seed power being 19 mW, the measured STRS gain/loss for the probe and seed beams are shown in Figs. 2(a) and 2(b), respectively, for various pump powers of 0, 2.5, 3.3, 5.6, and 6.5 W, to give output powers at 7, 250, 380, 500, and 700 mW, respectively, for both the seed and probe, as shown in the legends. The transfer of power from the seed to the probe beam at a positive frequency difference is clearly shown along with that in the opposite direction for a negative frequency difference.
It is impossible to simulate the STRS gain coefficient without knowing quantitatively the nature of local heat generation and birefringence perturbation. It is, however, possible to obtain the shape of the frequency-dependent STRS gain coefficient using the formula given in  if the mode distributions are known. This is shown in Fig. 2(c) for the coupling from seed to probe. The STRS gain is zero at a 0 Hz frequency difference, and this is dictated by the phase-matching condition. The decrease in the STRS gain at larger frequency differences is due to the increasing propagation speed of the modal interference, which washes out the temperature wave. It can be seen that the measured gain spectrum is similar to that predicted by the theory.
The peak STRS gain and loss can be obtained from Fig. 2, and these are plotted against the output powers in Fig. 3. In this low-gain regime, the STRS gain and loss are expected to be linear with regard to the output power with a slope of from Eq. (4). The peak can be estimated from the linear fits in Fig. 3 to be . This is, in fact, close to that expected for the coupling between and modes, i.e., , given in .
The mode coupling for a seed input power of 19 mW and a probe input power of 9.5 mW was also studied. The output powers of two modes are given in Fig. 4(a) and the STRS gain in Fig. 4(b). A similar study was also performed for a seed input power of 9.5 mW and a probe input power of 19 mW. The results are plotted in Fig. 5. From Figs. 4(a) and 5(a), it can be seen that the power is roughly conserved for the coupling, i.e., the power loss in one mode is gained in the other. This translates into a larger STRS gain or loss for the mode with less power [see Figs. 4(b) and 5(b)].
It is also interesting to note that, unlike the case when the input powers are equal, the maximum gain and loss at either side of the 0 Hz frequency difference are no longer roughly equal in amplitude when the two polarization modes have different input powers. Also, the power coupling to the mode with more power is stronger than the other way around.
To verify this asymmetry in coupling, further experiments were performed for a seed input of 20 mW and a probe input of 2 mW, and a seed input of 19.6 mW and a probe input of 1.9 mW. These are plotted in Figs. 6(a) and 6(b), respectively. The same trend is confirmed. The power coupling to the mode with more power is significantly stronger. In this case, due to the large difference in the powers of the modes, the weaker coupling, i.e., to the mode with lower power, is hardly visible.
The reasons for the asymmetry in the coupling are not immediately clear. When the powers in the two polarization modes are different, the major axis of the elliptically polarized light in the fiber is closer to the electric field orientation of the mode with more power. This may play a part in the asymmetry, making the coupling to the mode with more power easier.
In order to better understand the polarization-dependent gain, we have conducted experiments in order to characterize this. Linearly polarized light was first passed through a half-wave plate and then launched into the fiber. This allowed the easy change of polarization states. A similar fiber amplifier arrangement as in the STRS gain/loss measurement was used. The maximum polarization-dependent gain was , similar to that measured in  for small input powers.
In order to quantitatively simulate the polarization mode coupling in this case, the extent of the perturbation of the birefringence axis also needs to be precisely known. Since this is dependent on a small asymmetry in the transverse temperature distribution relative to the birefringence axis, it is very hard to obtain. In addition, the mode coupling could potentially be influenced by other possible effects, such as a minor electric field of the polarization modes. More detailed studies are necessary before a quantitative understanding of the coupling is possible.
We also measured the temporal dynamics of the coupling to provide additional evidence for its origin. In a first experiment, the frequency difference was initially set for maximum coupling at 4 kHz. The seed input power was very low at 50 mW compared to the probe input power of 1.13 W. The probe output power was 2.33 W. The amplifier had a linear gain of only . The seed input was then turned off using the AOM. The typical dynamics of the seed output power after turning off is given in Fig. 7(a). Before turning off, there was some sinusoidal oscillation at 4 kHz in the measured seed output power due to the beating between the amplified seed and the leaked probe at the beam splitter. Turning off the seed input power caused an initial rapid change in the seed output power and the disappearance of the sinusoidal oscillation. Continued evolution in the seed output power could be clearly seen afterward before it settled down to a constant value, from the leaked power from the probe. After the turning off, the temperature wave took some time to disappear due to the finite rate of the heat diffusion. A temporary standing refractive index grating existed in the meantime and continued to couple power from the probe to the seed, contributing to the continued seed power evolution. In this regime, the amplifier dynamics had a negligible contribution to the seed output power.
We repeated this experiment several times. Very similar power evolutions were observed. The initial rapid change in the seed power, however, varied. After the seed input was turned off, the total measured seed output power was from coherently combining the leaked probe power and the power coupled from the probe by the temporary grating. It was clearly dependent on the phase between the two components. The phase of the coupled power was determined by the phase of the grating. The grating was formed at the point of turning off a traveling wave, and its phase was dependent on when the seed input was turned off. This phase was ultimately determined by the relative phase of the two modes at the time of the turning off. We could determine this phase from the coherent interference right before the turning off. The seed power difference between right after the turning off and when the seed eventually settled down, i.e., in Fig. 7(a), should be a sinusoidal function of the phase between the two modes at the point of turning off. This is shown to be largely true in Fig. 7(b).
The heat diffusion solution in an optical fiber is given in . The decay of the temperature change in the core after turning off a Gaussian-shaped heat source follows , where is the mode radius and is the thermal diffusivity of silica. A fit based on this (black line) is plotted in Fig. 7(a) along with the measured decay of the mode coupling. The expected decay from the thermal conduction theory is consistent with the measurement.
We also conducted additional experiments in which the seed input at 4 kHz frequency difference was turned on. In this case, there was a sinusoidal oscillation in the seed output power after turning on due to the coherent beating between the leaked power from the probe beam beating with the amplified seed. It could, however, be subtracted out by knowing its amplitude and phase. A similar dynamic as in Fig. 7(a) was observed. The measurement noise was, however, worse in this case due to the oscillation.
We have measured frequency-dependent STRS gain and loss for the first time and experimentally confirmed the theoretical understanding of STRS laid out in sixties [10,11,13–17,20]. This was possible due to innovations in the experiment setup which allow measurements at low powers, at extremely small frequency differences of kHz, and with easy separation of modes. This work confirms, for the first time to our knowledge, that STRS can play significant role in mode coupling in an active fiber. This is also the first direct experimental observation of some of the key features of STRS, e.g., the gain and loss at kHz frequency shifts near the pump, which have been very difficult to measure.
Army Research Laboratory (ARL)/Army Research Office (ARO), HEL-JTO MRI (W911NF-10-1-0423).
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