Recently, mode instability was observed in optical fiber lasers at high powers, severely limiting power scaling for single-mode outputs. Some progress has been made towards understanding the underlying physics. A thorough understanding of the effect is critical for continued progress of this very important technology area. Mode instability in optical fibers is, in fact, a manifestation of stimulated thermal Rayleigh scattering. In this work, a quasi-closed-form solution for the nonlinear coupling coefficient is found for stimulated thermal Rayleigh scattering in optical fibers. The results help to significantly improve understanding of mode instability.
©2013 Optical Society of America
Fiber lasers are rapidly becoming very an important tool for machining, welding and materials processing in advanced manufacturing, among many other important applications in medical, science, sensing, and defense. Recently, mode instability has been observed in optical fiber lasers at high powers, severely limiting power scaling for single-mode outputs [1–3]. A fiber amplifier would operate normally at low output powers. Once the output is over a certain threshold, the output mode pattern starts to oscillate between the lowest two modes, with no apparent loss of total output power. With a further increase of pump power, more chaotic mode behavior starts to develop. The process can be repeated by turning pump off and re-start again. Many experimental [3, 4] and theoretical [5–10] studies have been conducted. Temperature grating as result of quantum defect heating and mode interference were identified fairly early on to be responsible for the phase-matched mode coupling [5–7]. A critical piece of the puzzle, the need for a traveling wave and, therefore, a frequency shift in the two coupled modes, for the stimulated nonlinear process to occur, is first identified in . A stationary grating can cause mode coupling as in the case of long period gratings. If such a grating is created by mode inference between a fundamental mode and a higher order mode, the phase of the field in the higher order mode coupled from the fundamental mode will be π/2 different from the original field in the higher order mode which created the grating in the first place, prohibiting a stimulated process to take place. A more recent numerical study based on a dynamic model  was used to study transient process of mode instability. Numerical models used in [8, 9] are very powerful for taking into considerations of a large variety of effects and very useful for precise quantitative analysis. Many aspects of the underlying physics are, however, often lost in the numerical process. A simple steady-state physics model was recently developed in . The difficulty of solving heat transportation equations in optical fibers prevents it from finding more closed-form solutions for the nonlinear mode coupling. The simple model does include all key elements of physics and provides significant insights into the problem.
In this work, a quasi-closed-form solution for nonlinear coupling coefficient is obtained for the first time for mode instability. One key enabling breakthrough is the finding of a quasi-closed-form steady-state solution for heat transportation equation. The simple quasi-analytical model developed as a consequence provides a great deal of insights in mode instability process.
2. Some historic background
Mode instability in optical fibers is a manifestation of Stimulated Thermal Rayleigh Scattering (STRS) first observed in the sixties in absorbing liquids with giant-pulse Ruby lasers [11–21]. Interference between pump and scattered lights leads to a traveling temperature wave via absorptive heating, which in turn stimulates further power coupling to the scattered light . The basic physics was quickly understood [12, 13], including the nature of the traveling wave and frequency shift in the scattered light. In the following few years, the effect was thoroughly studied and well understood [14–18]. In early eighties, there was a resurgence of interests in STRS for phase conjugation and four-wave mixing [19–21]. The similarity between stimulated Rayleigh scattering (SRS) and stimulated Brillouin scattering (SBS) was noted in . In fact, a simultaneous theoretical treatment of the two effects together is given in  considering both thermal and electrostrictive effects. The analysis in  provides most of the basic physics for understanding mode instability in optical fibers, which will be referred to as STRS in optical fibers in this paper thereafter. The simple analytical solutions in  are, however, obtained by ignoring any transverse dependence of electric field and thermal gradient. This may be appropriate for bulk media, but is not valid for optical waveguides. The analysis in  is also more appropriate for liquid, for which it was developed. As it will be noted later, there are some very important differences between liquid and solid media.
In optical fibers, co-propagating waveguide modes at slightly different optical frequencies form moving interference pattern along a fiber amplifier. Quantum defect heating as a result of the amplification process generates a traveling temperature wave which leads to further nonlinear coupling between the two waveguide modes via thermal-optics effect. Stimulated scattering can take place akin to SBS involving a traveling perturbation. It is important to point out that the traveling temperature wave is not from thermal diffusion along the fiber. The period of the interference pattern is typically much larger than fiber diameter and thermal diffusion dominates in the radial direction. The traveling temperature wave is effectively damped at the rate of thermal diffusion in the radial direction. It is worth noting that the acoustic wave in SBS is even more severely damped in optical fibers. The recently observed mode instability is essentially STRS. One variation is the means of heating, i.e. quantum defect heating in mode instability versus absorptive heating in STRS observed in the sixties. The second variation is optical waves involved, i.e. co-propagating optical waveguide modes in mode instability versus two interfering beams in early STRS. .
The analysis in  provides quantitatively thermal and electrostrictive contributions for both SRS and SBS. It is concluded that electrostrictive effect dominates in SBS and thermal effect dominates in SRS. This conclusion mainly comes from the frequency response of the two effects. SBS requires higher frequency traveling wave for phase-matching the counter-propagating optical waves, where the much faster but weaker electrostrictive effect dominates. SRS requires a much lower frequency traveling wave for phase-matching co-propagating optical waves, where a stronger traveling temperature wave dominates in absorbing or amplifying media. Consequently, electrostrictive effect is ignored in this analysis and the effort is focused on STRS.
Gain saturation effect is not considered in this work for simplicity. Gain saturation can take place in fiber amplifiers. Since not all signal power is used to generate heat through quantum defect heating in this case, a higher STRS threshold is expected than that predicated by this work. This effect can be potentially dealt with by finding the appropriate amplitude for the traveling optical intensity wave to use in the analysis. Quantum defect heating is assumed to be the only source of heat and single frequency is assumed for the optical waves. Thermal lensing is ignored. Constant temperature at the circular fiber surface is assumed, simulating the case of actively cooled fiber amplifier. The simulations were carried out for step-index fiber and LP11 modes for simplicity. The basic model can be modified for other fibers. It can also handle other higher order modes. Threshold condition is developed for amplifiers with constant gain along its length in this work. This is again for simplicity only. It can be developed for other amplifier designs.
3. Fields in optical fibers
Almost all fibers used in high power fiber lasers are operating in the weakly guided regime, where fields can be well approximated by linearly polarized (LP) modes. The conventional LP mode representations will be used in this analysis, i.e. LPmn, where m and n is azimuthal and radial mode numbers respectively. The electric field of LPmn mode can be written as:
The reason for the normalization used here will become clear later. In step index fibers,23] and determined by the eigenvalue equation. Nmn is normalization factor.
Electric field in a multimode optical fiber can be written as a summation of all propagating modes.Eq. (6), intensity in the optical fiber can be found. Ignoring interfering terms which do not include fundamental modes,
It is easy to see the optical intensity contains traveling waves with wave vector ± (βmn-β01) and angular frequency ± (ωmn-ω01).
4. Steady-state solution for traveling temperature waves
Unlike traveling acoustic wave involved in SBS, traveling temperature wave cannot exist due to heat diffusion. In typical optical fiber amplifiers, the period of the traveling wave for phase matching co-propagating optical modes is in the orders of few hundred μm to few cm and is typically much larger than fiber diameter. Heat diffusion, therefore, dominates in the radial direction. The traveling optical intensity wave as a result of interference between modes will deposit heat in the fiber core. The deposited heat will diffuse mostly in the radial direction and eventually lost to the media outside the fiber at a rate determined by heat transportation equation. This process is continuously driven by the traveling optical intensity wave as a result of modal inference. The result is a seemingly traveling temperature wave severely damped by thermal diffusion. At any fixed point in the fiber core, heat is deposited periodically in time. It is worth noting that the traveling acoustic wave in SBS in optical fibers is even more severely damped due to high material absorption at GHz regime and there is a strong analogy between SBS and STRS.
It is worth noting that, in case where thermal diffusion dominates axially, amplitude of the traveling temperature wave diminishes due to the equalization of temperature along the fiber over time. STRS in optical fiber, therefore, is stronger when the period of the traveling wave is significantly larger than fiber diameter. Unfortunately for mode instability, this is typically the case in fiber amplifiers.
In this section, a steady-state solution for the traveling temperature wave as a result of the traveling intensity wave and quantum heating is found. It is assumed that temperature is held constant at the circular outer fiber boundary. This is similar to the case where the fiber is actively cooled at the surface. Spatial temperature modes of the fiber are found first by solving the non-driven heat transportation equation. The solution of the heat transportation equation driven by the traveling intensity wave is then found by summing all the spatial temperature modes. This method is used for the first time to solve heat transportation equation to this author’s knowledge.
A traveling temperature wave driven by the traveling intensity wave as result of mode interference are, therefore, sought.
The right hand represents the heat deposited by the interfering modes through quantum defect heating. Gain coefficient profile g(r) is assumed to be only depend on r. This is typically the case in fiber amplifiers. λs and λp are signal and pump wavelength respectively. Note that q and Ω are dependent on mode numbers m and n (not expressed explicitly for clarity). Note also the ¼ reduction comparing to Eq. (7) for converting the two cosines to exponential. Also, ρ is density; C is specific heat; and κ is thermal conductivity. UsingEquation (10) can be transformed to
Considering the case of active cooling where the fiber surface r = b is held at a constant temperature, the boundary condition dictatesEq. (16) into Eq. (14), solution of each mode can be written as
Using the orthogonality equations, al can be determined by substituting Eq. (18) into Eq. (12), multiplying both side of the resulting equation by Tml and integrating over the fiber cross section, the amplitude terms in Eq. (18) can be obtained,
5. Coupled nonlinear equations
Taking a similar approach to , the nonlinear polarization which is phase-matched for the mode coupling is found first. Index change due to a change in temperature can be written as,
We can now write the relevant nonlinear polarization terms:Eq. (24) and Eq. (6) into nonlinear wave equation, where c is the speed of light,
The nonlinear gain coefficient in Eq. (28) consists of four parts. The first part at the front consists of mainly materials constants and vacuum wave number. The second part in the bracket describes the quantum defect heating. The third part immediately after the summation sign describes the frequency dependence. The last part describes the two overlap integrals involved in the three-wave interaction. The first integral in the numerator describes the process where the interference of the two modes deposits heat in the active region to create the traveling temperature wave. The second integral in the numerator describes the process where the traveling temperature wave leads to further nonlinear coupling of the two modes. The terms in the denominator in the last part are all normalization factors except the damping factor Γml.
The third part describing the frequency dependence deserves some more comments. Phase of the nonlinear coupling coefficient is determined by this part alone. The real part of the gain coefficient determines the nonlinear coupling strength. The frequency dependence of the nonlinear coupling can be clearly seen by examining the real part of this third term, which achieves a maximum of 1 when Ω = Γml/2. It is easy to see that the there is no coupling at Ω = 0 rad/s.
Equation (26) can be re-written for clarity.
6. Characteristics of the nonlinear gain coefficient
The physical constants used in this work are summarized in Table 1 , unless stated otherwise. Refractive index of silica is taken as 1.45. The data are obtained from . There are some variations in thermal optics coefficient used in the literature. For example, KT = 1.2 × 10−5 K−1 was used in .
The nonlinear gain coefficient in Eq. (28) was first bench marked for the cases in simulated in  for step index fibers. Only V value is give in , there is some uncertainty in what value was used for KT. The bench marking results are summarized in Table 2 . It can be seen that the error is within 3% for all the cases if KT = 1.2 × 10−5 K−1 was used.
Fiber design can now be studied to understand their impact on the nonlinear coupling coefficient χ. It should be noted that the χ in Eq. (28) is the total nonlinear gain normalized against amplifier gain coefficient for the fundamental mode g01. A fiber amplifier is usually designed for a target fundamental mode gain. Varying some fiber parameters such doping radius d can impact target gain for the fundamental mode. This definition of χ, unlike that in , is normalized against this change, so that fiber designs can be studied while keeping the target gain for the fundamental mode constant.
The first fiber in the study is a typical LMA 30/400 fiber, commonly found in many fiber amplifiers. This fiber has a NA of 0.06, core diameter 2a = 30μm, and cladding diameter 2b = 400μm. Core is uniformly doped, i.e. d = a. Pump wavelength of 976nm and signal wavelength of 1060nm will be used throughput this paper. Nonlinear coupling between the fundamental mode and the first higher order mode LP11 is studied thereafter in this paper. Other higher order modes are expected to have lower nonlinear gain and therefore higher STRS thresholds. The solutions for the spatial temperature modes are found first. The nonlinear coupling coefficients amplitude χmnl defined in Eq. (28) and damping factor Γml at each l are plotted in Fig. 1(a) . The largest χmnl is achieved at the mode number l where the deposited heat overlaps best with the spatial temperature mode. The damping factor Γml increases with mode number l. This is due to a reduced physical dimension of the spatial temperature mode features at large l. The simulated χ is shown in Fig. 1(b). The real part of χ is responsible for the nonlinear coupling. It becomes 0 at Ω/2π = 0 kHz and reaches a maximum at Ω/2π = ~3.4kHz. It decays slowly towards higher frequency as a Lorentzian function. The positive sign of real part of χ at Ω/2π<0kHz indicates gain for LP11 mode at the Stoke frequency. The negative sign of real part of χ at Ω/2π>0kHz indicates loss for LP11 mode at anti-Stoke frequency. This is opposite to what observed in  for STRS in liquid. This is largely due to the fact that index change in liquid and gas is dominated by thermally-induced expansion with a negative n2, while, in glass, index change is dominated by thermal optic effect with a possible n2.
The frequency separation between the LP01 and LP11 mode at the maximum nonlinear coupling is Ω/2π = ~3.4kHz, which is much smaller than spectral width of almost all seed lasers. This implied LP11 mode can be seeded by a seed laser at the required Stoke frequency.
These fibers are good representations of LMA fibers commonly found in high-power fiber amplifiers. The variations of core diameters lead to different V values at 5.3348, 4.4456, 3.5565 and 2.6674 respectively for the 4 fibers. Nonlinear coupling coefficient amplitude χmnl are plotted in Fig. 2(a) , while real part of χ and its phase is plotted in Fig. 2(b). The l for peak χmnl moves to larger number for smaller cores to account for the smaller active area. The damping factors do not change at all with a change of core diameters. The data for various core diameters essentially overlap in Fig. 2(a). This is due to the fact that the period of traveling wave is significantly larger than fiber diameter 2b. The second term in the bracket in Eq. (29), therefore, dominates, which only depends on m, l and b. The peak of the real part of χ moves towards larger frequency separation for smaller core diameter, accompanied by a broadening of the spectrum. This move of peak towards higher frequency is a reflection of movement towards larger temperature mode number l with higher damping factor shown in Fig. 2(b). It will be shown later on that the smaller nonlinear coupling coefficient at the peak for smaller core diameter is largely a reflection of smaller V value of these fibers.
For the 30μm core step index fiber, the dependence on V value is studied by varying NA while keeping the rest of parameters unchanged. The real and imaginary part of χ at the real part peak are plotted in Fig. 3(a) , along with the corresponding frequency of the peak fmax. When V is reduced from 5.5, the peak nonlinear coupling coefficient decreases initially very slowly and this decrease then accelerates near LP11 mode cut-off at around 2.405. The absolute value of the peak frequency fmax decreases at smaller V, reflecting the increasing delocalization of LP11 mode while moving towards its cut_off. A reduction of the doped area also reduces the peak nonlinear coupling coefficient (see Fig. 3(b)). This reduction is, however, small when d/a is near 1. Reducing the nonlinear coefficient by 50% requires d/a≈0.45, i.e. a doped area reduction by ~80%. This would require a significant increase of doping level to maintain the same level of gain/absorption per unit length. Considering doping levels are already near their upper limits in many fibers, this may not be possible. Absolute value of the peak frequency fmax increases with a reduction of d/a, reflecting the smaller active area.
The dependence of the nonlinear coupling coefficient at fmax is also studied for various core diameters while V is kept constant for step index fibers with 2b = 400μm and 2a = 2d. NA is varied to keep V constant. The results are shown in Fig. 4 . Absolute value of the peak frequency fmax increases with a reduction in core diameter as expected. The nonlinear coupling coefficient remains almost constant at various core diameters. The overlap integrals in Eq. (28) are essentially dependent only on V. As core diameter gets smaller, peak mode number l moves to higher mode number with a larger Γml (see Fig. 2(a)), accompanied by a broadening of the distribution (see Fig. 2(a)). These effects cancel each other out overall to keep χ at the peak frequency fmax largely constant.
7. STRS power threshold
Finding out threshold power for modal instability is very important for amplifier designs. Assuming a fiber amplifier of length L where threshold is reached at z = L, the gain described in Eq. (30b) can be integrated from 0 to L to find power in the higher order mode, noting that, in the high gain regime over most part of the fiber, P11N(z)<<P01N(z) and P01N(z)≈P01(0), assuming uniform gain along the fiber and threshold is reached at Pmn(L) = x P01(L) where x<<1.Eq. (33) is reduced to,
Equation (33) is only applicable in the high gain regime. It is interesting to see the threshold power depends only on real part of χmn and weakly on input condition P01(0)/Pmn(0) once x is known in this high gain regime and independent of any other amplifier parameters. In the low gain regime, the assumption P01N(z)≈P01(0) is no longer true. The lower P01N(z) leads to a higher threshold power. It can be seen in Eq. (33) that smaller higher-order-mode gain and larger higher-order-mode loss can also increase the threshold power. This is, however, very limited, due to the fact that the first term in the bracket in Eq. (33) dominates in most cases.
To test Eq. (34), amplifier based on a LMA fiber with 30μm core diameter and NA of 0.06 are studied numerically by solving the coupled mode equations described in Eq. (30) for threshold powers for a range of input conditions. The fiber parameters are NA = 0.06, 2b = 400μm, 2a = 2d = 30μm, V = 5.3348 and α11 = 0. Both g01 and g11 are considered with α11 = 0.The results versus total gain factor g01L (plotted in dBs) are summarized in Fig. 5 . The predicated threshold powers from Eq. (34) are plotted as solid red lines in Fig. 5(a). The threshold power are independent of gain when g01L>4, i.e. ~17dB. Below this, threshold power increases with a reduction in gain. For g01L>4, Eq. (34) fits the numerical data very well. Equation (34) can be modified slightly to account for the threshold increases at lower gains.
The modified threshold powers described in Eq. (35) are plotted as dotted black lines in Fig. 5(a) and (b). It fits very well with the numerical data through the entire gain range. The threshold power can be seen in Fig. 5 to increase with a decrease in P11(0)/P01(0) as expected from Eq. (35). The only two amplifier parameters which can be used to increase the threshold powers are lower gain and lower P11(0)/P01(0).
Using the FWHM STRS gain bandwidth of ~20kHz for the 30micron core LMA fiber (see Fig. 1(b)), the quantum noise at 1060nm is estimated to be ~2 × 10−28 W. In practice, the seed to the higher order mode are likely from the input signal when the input signal spectrum is broader than just few kHz for core diameter over ~30μm, and, can be, therefore, much higher than this quantum limit.
8. Mode coupling dynamics
The power evolution along a fiber amplifier is simulated by numerically solving Eq. (30) and illustrated in Fig. 6 . The fiber parameters are NA = 0.06, 2b = 400μm, 2a = 2d = 30μm, V = 5.3348 and α11 = 0. It can be clearly seen in Fig. 6(a) that the LP01 mode is amplified normally at the first part of the fiber, while the LP11 mode experiences significant nonlinear gain of >250dB in this case. The fraction of LP11 mode power over the total power is also plotted in Fig. 6(a), showing the rapid switch over at threshold. Once over the STRS threshold, the power continues to couple from LP01 mode to LP11 mode, because the sign of nonlinear coupling coefficient remains the same in Eq. (30). This coupling coefficient diminishes as power in LP01 mode decreases. The LP11 mode undergoes continued linear amplification in the second part of the fiber amplifier. In practice, it is expected that LP01 mode can be re-seeded when its power is below the noise level in the fiber. This can be simulated by switching the sign of the nonlinear coupling coefficient in Eq. (30). This is done in Fig. 6(b), showing the reversing of coupling from LP11 to LP01 after the first coupling cycle. This behavior is repeated with an increasing spatial frequency, driven by the increased total power. This is confirmed to some extent by the observed chaotic behavior when operating well above the threshold power .
9. Spectral considerations
Up to this point, the optical powers are assumed to be at single frequencies ω01 and ωmn. In practice, seed lasers can have much broader spectrum than that of the STRS gain. In case where the input power spectrum is broader than that of the STRS gain spectrum, if the power in the higher order mode at a given frequency is seeded by the input signal at frequency very close by (less than few kHz frequency separation for core diameter over 30μm), it reasonable to assume that phases of the fields in LP01 and LP11 modes with a small frequency separation of Ω are identical at the amplifier input. This may in fact be true across the entire input signal spectrum even for the case where the seed is an ASE source as in . In this case, input signal power at any given frequency can interfere with power in LPmn mode at an adjacent frequency Ω away to produce an intensity traveling wave described by exp(i(qz-Ωt)). In another word, power at any frequency within the input power spectrum can interfere with its corresponding power in the LPmn mode to add to the intensity of the traveling temperature wave described by exp(i(qz-Ωt)). This is only true when the phase difference of the two fields in the interfering modes is constant across the power spectrum. This is not the case, for example, for SBS, where the counter-propagating wave is seeded from quantum noise without any fixed phase relationship to the input signal. This collaborative effect can lead to the possibility that total power of the input signal contributes towards nonlinear coupling at any local frequency within the input signal spectrum in STRS, despite the fact that the power spectrum of the signal is much larger than the STRS gain spectrum. This effect can lead to threshold being independent of input signal bandwidth but more dependent on the total power of the input signal as experimentally observed in . This effect is unique only to STRS due to the fact that the interfering fields involved are originated from the same source and can, therefore, have identical phase. In case where the seed laser spectrum is significantly wider than the STRS gain spectrum, replacing power by power spectrum density represented by S and keeping the same subscripts and superscripts, the nonlinear coupled mode equations can be written as,Eq. (36) is, in fact, square root of product of total powers in the two modes. It is easy to see in Eq. (36) that the total powers in the two modes are contributing towards nonlinear coupling at any frequency within the power density spectrum. This can also lead to uniform nonlinear gain across the power spectrum for the LPmn mode.
10. Discussions and conclusions
The mode instability observed recently in high-power fiber lasers is based on the same nonlinear coupling mechanism as STRS first observed half a century ago. By finding quasi-closed-form solution for the heat transportation equation in optical fibers, it is able to find quasi-closed-form solution for the nonlinear coupling coefficient for STRS in optical fibers. The quasi-analytic solution shines a great deal of insights on the underlying physics involved as well as on critical design parameters both in optical fibers and fiber amplifiers. Threshold power can be obtained by a simple analytical formula, surprisingly depends only on the nonlinear coupling coefficient and seed condition in the high gain regime, not on amplifier designs. The evolution of power along the fibers are also studied, showing well behaved power growth when under the threshold power, rapid switch over at the threshold power, and oscillatory behavior above threshold power.
It should be pointed out that only the steady-state solutions are studied in this work. In reality, high environmental sensitivity of thermal process can lead to temporal fluctuations in the nonlinear coupling. It can take several thermal diffusion cycles to reach steady state. It is, therefore, not surprising to see oscillatory behavior of mode coupling over time when operating above threshold at a frequency corresponding to the thermal diffusion rate. It is also expected to be more chaotic when operating well above threshold, due to the higher power levels in the fiber, especially considering the increased number of higher order modes reaching thresholds. When a fiber amplifier starts from cold at above threshold powers, it is expected that there is initially a great deal of temperature fluctuations along the fiber due to different part of the fiber heating up slightly differently. Assuming the power is mostly in the fundamental mode, the deposited heat will resemble the fundamental mode spatial pattern at this early stage. This large temperature fluctuation along the fiber can easily overwhelm those from much smaller amount of heat deposition from mode interference, and, consequently, suppresses STRS at startup. STRS can only start when a thermal equilibrium is reached from a cold start.
After the initial submission of this manuscript, the author was made aware of two additional references [25, 26]. Some concepts regarding transient process are discussed in , while steady-state is left largely ignored. Mode instability is referred to as STRS in  for the first time to this author’s knowledge. Quantum noise corresponding to one photon per beat cycle at 2 × 10−16 W is suggested to seed LP11 mode in .
This material is based upon work supported in part by the U. S. Army Research Laboratory and the U. S. Army Research Office under contract/grant number W911NF-10-1-0423 through a Joint Technology Office MRI program.
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