## Abstract

We analyze the scalability of diffraction-limited fiber lasers considering thermal, non-linear, damage and pump coupling limits as well as fiber mode field diameter (MFD) restrictions. We derive new general relationships based upon practical considerations. Our analysis shows that if the fiber’s MFD could be increased arbitrarily, 36 kW of power could be obtained with diffraction-limited quality from a fiber laser or amplifier. This power limit is determined by thermal and non-linear limits that combine to prevent further power scaling, irrespective of increases in mode size. However, limits to the scaling of the MFD may restrict fiber lasers to lower output powers.

©2008 Optical Society of America

## 1. Introduction

High average power lasers are needed for materials processing and defense systems. Scientific applications such as laser-based guide stars for astronomy, gravitational wave detection, coherent remote wind sensing and laser based particle acceleration could also benefit from high average power lasers. Some of these applications also require the light to be single-frequency and have a stable polarization state. Considerable attention has been focused on fiber-based lasers and amplifiers due to their potential for high average power combined with high efficiency, compactness, and reliability [1]. During the 1970’s the loss in silica fiber was significantly reduced. In the 1980’s the development of Erbium doped fiber amplifiers (EDFA’s) at 1.5 µm enabled long-haul, broadband optical networks based on wavelength division multiplexing. By building on these material and fabrication advances, Ytterbium doped fiber lasers and amplifiers at 1 µm have recently made tremendous progress and have been scaled to the multi-kW average power level with diffraction-limited beam quality.

Beam combining of multiple fiber lasers is one approach to scale fiber laser output powers beyond the rapidly approaching single aperture diffraction-limited power limit. Coherent [2–5], incoherent [6,7], and spectral [8,9] beam combining have recently been proposed to overcome the limitations in scaling a single fiber laser.

These limitations include thermal self-focusing, onset of non-linear effects [10,11], facet damage, and/or the availability of high brightness pump diodes. Large-mode-area (LMA) step-index and photonic crystal (PC) fibers [12,13] form the two major classes of fiber designs [14] that address these problems. LMA and PC fiber laser systems have been scaled to the 1.36 kW [15] and 1.53 kW [16] levels respectively.

The typical approach to power scaling a single fiber laser or amplifier is to increase the mode size, since non-linearities and facet-damage power thresholds increase with increasing mode diameter. Several papers in the last decade have discussed the issues involved in average power scaling of a single fiber laser or amplifier [17–22]. However, the theoretical scaling limit of a single fiber laser or fiber MOPA system has not been established in these papers. We reconsider this issue and rigorously derive the average power limit of a single diffraction-limited fiber-based laser system. There is an optimum mode-size, fiber length, and a maximum output power that can be achieved with good beam quality from a single fiber amplifier. The results in this paper will help the research community to establish future research direction.

This paper is organized as follows. Sect. II reviews the physical limits of interest and focuses on establishing mathematical equations that describe the output power limits in terms of core diameter and fiber length. Sect. III examines how the limits interact as the mode field diameter and fiber length are varied assuming these parameters can be varied without bounds. Sect. IV examines how large the fiber core really can be before other effects limit the effective modal area and Sect. V briefly examines energetic issues in order to validate that the optimum laser solutions we arrive at are indeed physically possible. We emphasize that these results are for lasers with diffraction limited beam quality and that fiber lasers with less than perfect beam quality may yield higher single aperture powers.

## 2. Review of fiber laser physical limitations

In this section we review the critical limits relevant to the scaling of fiber lasers. These include limits imposed by thermal considerations, fiber non-linearities, facet damage and the brightness of the diode pump lasers. We show that the combination of the limits of rare earth doping concentration, finite brightness of the diode pump lasers and requirement for efficiency places bounds on the relative ratio of the core and cladding diameters of the optical fiber. This in turn allows construction of contour plots that simultaneously consider all the limits as a function of core diameter and fiber length. With these plots we determine the maximum output power of a fiber laser or amplifier.

#### 2.1. Thermal limitations

Brown and Hoffman [18] review the thermal limits to scaling fiber lasers in detail. We summarize their analysis here and present some of our insights. They compute the maximum heat power (P_{heat}) that can be deposited in an optical fiber core for a given length (L) when limited by: rupture, melting and thermal lensing.

They then suggest that the maximum extractable power per unit length is proportional to the amount of heat power deposited per unit length multiplied by the factor (1-η_{heat})/η_{heat}, where η_{heat} is the fraction of the absorbed pump power that is converted to heat. The best reported optical-to-optical conversion efficiencies from fiber lasers in Yb:silica is 84% [15]. However, the quantum defect from these lasers is typically less than 16%. Thus the relation η_{laser}+η_{heat}=1, where η_{laser} is the optical-optical conversion efficiency does not necessarily hold. ηheat may not equal the quantum defect [23] since non-radiative decay mechanisms may contribute to heat deposited in the core. However, the laser may have less than perfect efficiencies for reasons that do not contribute to heat absorbed in the core, such incomplete absorption of the pump. To this end, we replace (1-η_{heat})/η_{heat} from Brown with η_{laser}/η_{heat} in our analysis, where η_{heat} is bounded on the low end by the quantum defect and on the high end by 1-η_{laser}.

The longitudinal dependence of the heat deposition can be safely neglected. Practically, very high power fiber lasers and amplifiers will likely have relatively low gains. Thus the difference between an exponential and linear increase in power will be small. While the final stage for a master oscillator power amplifier system may reach 20–30 dB of gain, the 10 dB of gain at the end of the amplifier will contribute 90% of the total output power. Thus we only need to consider the case of 10 dB or less of gain and the corresponding fiber length. The rest of the final-stage amplifier’s length that contributes to the remaining gain is presumed not to be the limiting portion of the amplifier.

For an oscillator configuration, the simplest and most practical design is a linear oscillator cavity with a high reflector and a low reflectivity output coupler (~1-4%). This corresponds to a round trip linear gain in the oscillator of around 20–100. From the perspective of photons starting at the output coupler and making one round trip, they grow in number proportional to the square root of the gain in each direction. The travel of the light from the output coupler back to the high reflector does not extract appreciable energy. Most of the energy will be extracted in the region between the high reflector and the output coupler. Thus a simple oscillator can be treated as an amplifier with low (~5-10) gain from the standpoint of limits on power extraction.

*Thermal fracture*

The heat power deposited per unit length leading to thermal fracture is given by [18]

where *L* is the length of the laser or amplifier, *R _{m}* is the rupture modulus of the glass (2460W/m in silica [24]),

*a*is the core radius and

*b*is the cladding radius. The corresponding output power of the amplifier or laser is obtained by multiplying the ratio of the extracted laser power to the deposited heat power and is given as

*Melting of the core*

The heat power deposited per unit length leading to melting of the fiber core is given in a form similar to Eq. (2) as [18]

where *k* the thermal conductivity (1.38 W/(m-K) for silica [25]), *T _{m}* is the melt temperature (1983 K for silica [25]),

*T*is the coolant temperature (we will assume

_{c}*T*is 300K),

_{c}*h*is the convective film coefficient and the other parameters are as defined above. he convective film coefficient can vary significantly depending on the cooling mechanism [26]. It may be as low as 1000 W/(m

^{2}-K) for forced airflow cooling or has high as 10,000 W/(m

^{2}-K) for forced liquid flow of the coolant.

Currently almost all fiber lasers are coated with a polymer coating. These coatings are the simplest and most practical way to protect optical fibers operating at low power levels. However, polymer coatings impede heat flow and will alter the overall thermal analysis. The effect of this has been neglected. Instead it is assumed that this technical issue will be eliminated in a high power fiber laser design. For example, a high power fiber laser could have a low index glass cladding for guiding the pump light and either a thin metal or diamond-like carbon coating for the mechanical protective layer. Such coatings will not have the thermal drawbacks of polymers and will withstand higher temperatures. The analysis has also neglected radiation cooling which will be negligible at the proposed convective film coefficient cooling levels discussed above.

*Thermal lens*

The power limit due to the heat induced temperature gradient in the fiber core that creates a thermal lens that is competitive with the index guiding from the fiber core is given as

where *λ* is the wavelength of the laser signal, *dn/dT* is the change in index with the core temperature (11.8×10^{-6} K^{-1} for silica [25]). The analysis leading to this result is presented in Appendix A.

#### 2.2. Non-linear optics limitations

When considering continuous wave (CW) high power fiber lasers and amplifiers there is an interest in lasers that have broad bandwidth as well as lasers that operate with very narrow spectral lines. The two systems have different, but functionally similar non-linear limitations. In the broadband case, the relevant limit is stimulated Raman scattering (SRS). In the narrowband case, the relevant limit is stimulated Brillouin scattering (SBS). The peak power at which self-focusing occurs in silica optical fibers is 4 MW [27], and is far above most of the other CW laser limits. Therefore self-focusing will not be considered further in this work.

*Stimulated Raman Scattering (SRS)*

As the signal power propagating in a fiber amplifier increases, eventually the power length product reaches a point where the Raman gain generated by the signal is very high. At this point the amplified noise at the peak wavelength of the Raman gain (about 13.2 THz lower in frequency [28]) becomes a significant fraction of the propagating power. The signal power is then effectively clamped and further increases in pump power, resulting in conversion to progressively longer unwanted wavelengths. Further, the heat deposited in the core increases with the onset of every new Stokes-shifted line.

Smith [29] calculates the critical input power into an optical fiber at which conversion to the first Stokes line is significant. It is found that for an attenuating pump source, the threshold for forward SRS is lower than for backward SRS. Our situation consists of an amplifying pump source but due to the isotropy of the Raman gain, the calculation is the same but the thresholds are interchanged. Consequently, the threshold for backward SRS is lower and the critical output power at which backwards SRS is significant is given as

where *g _{R}* is the Raman gain coefficient (about 10

^{-13}m/W for silica [28], this is the polarized gain a factor of two reduction is needed in the case of unpolarized amplifiers),

*A*is the effective area of the mode and

_{eff}*L*is the effective length of the fiber, which is dependent upon the fiber gain or loss. For a small core, high numerical aperture fiber,

_{eff}*A*is typically slightly larger than the core area. Extremely high power fiber lasers however will more likely operate with large, slightly multi-mode cores with low numerical aperture. In this regime, the mode field diameter of the fundamental mode of the fiber is typically about 80% of the fiber core diameter. We therefore define

_{eff}*A*as

_{eff}where *Γ* is the ratio of the mode field radius and the core radius. This will vary only slightly with core radius, such that we can safely approximate it as a constant.

Smith defines *L _{eff}* in terms of the loss

*α*of the fiber. We are more interested in fiber amplifiers with saturated gain coefficient, g, at the signal wavelength that serves as the pump wavelength for SRS at longer wavelengths. We wish to use positive

*g*so we will substitute

*g*for

*α*in Smith’s derivation of

*L*and the redefined

_{eff}*L*is now given as

_{eff}Note that if the fiber amplifier power gain factor is *G* then *g*=*ln(G)/L*. By approximating *e ^{gL}*-

*1*~

*e*and using Eq. (5–7) we find that the Raman limited amplifier output power is given as

_{gL}There are fiber waveguide designs [30], which can raise the SRS limited output power by a significant factor. Our calculation of the SRS limited output power does not consider advanced designs as they are technically challenging to construct and add significantly to the overall complexity of the system.

Finally, we note that the numerical factor of 16 in Eq. (5) as first derived by Smith assumed an input optical power of around 100 mW. If one goes back to that original derivation, higher input powers will lead to higher thresholds prior to the onset of significant depletion of energy by the first Stokes line. However, the allowed output power scales slowly as the natural log of the power. Thus we estimate that while the factor of 16 refers to 100 mW, scaling the input power to 1kW increases the factor of 16 only to 25.

Variations in fiber compositions may limit accurate knowledge of *g _{R}* to this degree of accuracy. As most researchers measuring Raman gain will likely apply the commonly used 16 independent of the actual power levels used in an experimental measurement of

*g*, we leave the slightly inaccurate factor in place for our analysis. However, if a more accurate calculation is desired this factor should be scaled appropriately.

_{R}
*Stimulated Brillouin Scattering*

For optical signals whose bandwidth is narrow compared to the Brillouin linewidth (~50–100 MHz), the output amplifier power clamps when electrostriction creates an acoustic wave in the fiber, leading to back scattering of the signal power. The functional form of this limit is quite similar to the SRS limit. Applying Smith’s calculation for the critical output power at which the onset of SBS occurs, we find that it is given by

where *g _{B}(Δν)* is the Brillouin gain coefficient, which depends upon the laser signal linewidth,

*Δν*. Smith gives the explicit functional form of

*g*if it is needed. Once

_{B}(Δν)*Δν*becomes sufficiently large, SRS becomes the limit rather than SBS. For very narrow

*Δν*,

*g*(

_{B}*Δν*) attains its peak value (5×10

^{-11}m/W for silica optical fibers [29]). In the numerical calculations below, we assume narrow linewidth operation and use the Brillouin gain’s peak coefficient, but carry the linewidth dependence along in our analysis. Analogous to the SRS case, we find that the SBS-limited output power of a fiber amplifier is given by

It is possible to increase the SBS limited output power of an optical fiber via waveguide design [31] or via a longitudinal temperature gradient [32]. We account for the former by simply adjusting the Brillouin gain, g_{B}(Δν), so that it accurately describes the modified fiber’s SBS properties. It is not clear that the Brillouin gain coefficient in these fibers has the same linewidth dependence as the unaltered fibers. The effect of linewidth needs to be accurately characterized if the laser line has significant bandwidth. However, this is not relevant for the cases we consider where Δν is effectively 0. As for temperature gradients, we have already indicated that we will focus on cases where the signal power grows linearly with length and the temperature is therefore uniform along the fiber length.

Again, the factor of 17 in Eq. (9) assumes a 100 mW input optical power. At an input power level of 1 kW in Smith’s derivation, this may scale from 17 up to 26. Given variations in experimental measurements of *g _{B}* between various fibers at varying power levels, we choose to leave the slightly inaccurate 17 in place to avoid potential confusion with experimental measurements that do not accurately account for the changes in this factor.

#### 2.3. Damage limitations

There is an extensive body of work on optical damage in both optical fibers and bulk silica glass [33–35], but even the relative magnitude of the damage threshold for optical fibers and bulk silica glass has not been established. Furthermore there are bulk optical damage mechanisms unique to fiber such as the so-called fiber fuse effect [36], that one may also need to consider. However, fiber damage is typically observed at the end facet. End-cap schemes that allow the fiber mode to expand in the bulk prior to striking an air-glass interface can increase the surface damage limit. All of these mechanisms however, have one thing in common: they all are limits on the allowed peak intensity at the output of the fiber. Thus for a simple Gaussian-like fundamental mode, the maximum damage-limited output power of a fiber laser or amplifier is given by

where I_{damage} is the upper limit of the intensity allowed in the fiber. Without end-caps, I_{damage} would be given by the surface damage limit of optical fibers, (~10 W/µm^{2} for silica [37]). With end caps, the bulk damage threshold is the damage limit. To operate with a margin of safety we suggest using a damage limit no higher than the surface damage limit. Thus we take I_{damage}=10 W/µm^{2} for the examples in this paper and assume endcaps are used to provide a margin of safety at the fiber end face.

#### 2.4. Pump power limitations

As fiber lasers are typically pumped by low brightness laser diodes, we must also account for the limits on the fiber laser output power due to the total absorbed pump power. There are three factors involved in calculating this limit, the finite brightness of the pump laser diodes, the finite doping concentration achievable in an optical fiber core and the requirement of an efficient laser.

Diode lasers have finite brightness or irradiance at which light can be coupled into the fiber. We shall call this intensity I_{pump} (diode lasers with I_{pump} up to 0.021 W/(µm^{2}-steradian) are commonly available commercially [38]). This irradiance should not be considered a hard upper limit, as it has increased steadily with time as technology progresses. The laser output power limit due to the pump light is then

where *b* is the radius of the pump cladding, and *NA* is the numerical aperture of the pump cladding.

To achieve efficient fiber laser operation, nearly all of the pump light must be absorbed by the rare earth ions in the core of the fiber. The core’s small signal absorption is typically limited by either concentration quenching or photo-darkening [39]. Yb fiber lasers have been shown to be the most efficient. They can be doped to reach a core small signal absorption coefficient (α_{core}) of 1000 dB/m at 976 nm prior to the onset of detrimental effects. At 915 nm (another common pump wavelength), the absorption is about 1/3^{rd} of this value. The quantum defect is significantly degraded at this wavelength as well. Presently 1000 dB/m does not appear to be a safe absorption level to avoid photo-darkening. An absorption coefficient of 250dB/m should ensure that there are no adverse effects.

Once the core’s absorption coefficient is known, the effective small signal absorption coefficient of the pump light for a cladding pumped amplifier, α_{clad}, is given by

We assume that the outer edge of the roughly circular fiber is sufficiently non-uniform as to frustrate the propagation of skew rays that never intersect the core [40].

The small signal pump absorption, A, is then given by *A*=*α _{clad}L*. Again, for efficient operation, we suggest

*A*be at least 20dB. This will be justified in Sect. V. From this relaion and Eq. (13), we can link the core and cladding diameters. Specifically,

Eq. (14) implicitly assumes the fiber is pumped only from one end. One might pump the fiber amplifier from both ends leading to a factor of two increase in the total applied pump power. However, distributed pumping will not materially change the limit implied by Eq. (14). To see this we substitute Eq. (14) into Eq. (12) yielding

Eq. (15) is the output power from a laser of length *L* due to the combined upper limit on diode pump brightness, the upper limit on rare earth doping concentration and the requirement that an efficient laser have a minimum small signal pump absorption *A*. Dividing the fiber laser into *N* segments of length *L/N* will permit a smaller cladding diameter than what is implied in Eq. (14). However, each of these segments will have a new cladding diameter *b _{N}* defined by Eq. (14), but with

*L/N*substituted for

*L*. The new

*b*when substituted into Eq. (12) will yield a new Eq. (15) with

_{N}*L*replaced by

*L/N*for the output power per section. Thus each section in the

*N*-section distributed pumping scheme produces

*1/N*of the power of the end pumped scheme. Thus, the distributed pumping scheme results in the same diode brightness limited power as a single uninterrupted length of fiber that is end pumped.

For convenience and ease of handling it may be desirable to divide up the amplifier into several segments. This permits the use of a smaller cladding, but it will not result in more power out for a given total length of fiber with regards to the pump coupling limit. Review of the other limits discussed to this point show that only the rupture (Eq. (2)) and the melting (Eq. (3)) limits have the cladding diameter as a variable in their functional forms. We will show below that even assuming Eq. (14) holds these limits are so far above any of the other limits as to be not worth considering, providing the fiber is aggressively cooled.

Some groups are investigating phosphate glass based fiber lasers [41]. Phosphate glasses permit much higher doping concentrations than silicate based glasses. However, the thermal properties of these glasses are quite different and the technology to make these compositions into optical fibers is not as well developed as silica glass technology. Nonetheless, the analysis developed here is applicable to fiber lasers based on different glass compositions.

## 3. Interaction of the fundamental physical limits

In the previous section, we reviewed the seven physical limits to scaling the power of a fiber laser. A key result is Eq. (14), which relates the cladding radius to the fiber length and core radius. Using this result we can express all seven limits in terms of the core radius and fiber length. We contend that those are the only parameters that must be selected by the fiber laser engineer. All the other parameters are either physical constants (e.g. the rupture modulus, thermal conductivity, melt temperature, dn/dT, Raman or Brillouin gain coefficients and optical damage limit) or parameters that are not easily varied and would typically be chosen to be state-of-the-art (e.g. convective film coefficient for fiber cooling, small signal pump absorption of the laser, core absorption, coolant temperature, pump brightness and the efficiency of the laser and deposited heat fraction). These latter parameters may be optimized slightly to meet application needs but not without considerable effort and expense. We summarize the parameters in Table 1.

We must now consider two cases: the case of a fiber laser with a signal that is sufficiently broadband that SBS need not be considered and the case where the signal is relatively narrowband and SBS must be considered. In each case there are 6 relevant limits. Five of these, thermal fracture (Eq. (2)), melting (Eq. (3)), thermal lensing (Eq. (4)), optical damage (Eq. (11)) and pump power limitations (Eq. (15)) are the same in both cases. Again, Eq. (14) linking the core and cladding diameters will be used in Eqs. (2) and (3) to remove cladding diameter from consideration. Only SRS (Eq. (8)) or SBS (Eq. 10) will be different for the two cases.

*SRS Limited Lasers (Current State-of-the-Art Diode Pump Brightness)*

We consider the SRS case first and construct the contour plot in Fig. 1 via the following procedure. For any given fiber core diameter *2a* (x-axis) and length *L* (y-axis), we have computed the six relevant limits and plotted the minimum power limit in kW for each point. Since a fiber laser candidate must simultaneously meet all the limits, the lowest limit will be the actual limit at any given point (2*a*, *L*) on the plot. We have color-shaded and labeled the plot mentioning the limit which dominates in each region. The boundary lines between these regions are of particular interest.

The most interesting of the three boundary lines is the 36.6 kW ridge between the SRS and thermal lens limited regions. This boundary can be calculated by setting the thermal lens limit (Eq. (4)) equal to the SRS limit (Eq. (8)). Doing so and solving for the fiber length as a function of the core radius yields

We see that the length L at which the ridge occurs scales as the square of the core diameter. The remaining parameters in Eq. (16) are almost all physical constants or are very difficult to adjust. We can now substitute Eq. (16) into Eq. (4) or (8) and determine the power level of the boundary which is given by

What is most intriguing about Eq. (17) is that the derived power limit is independent of both the core diameter and fiber length above 90 µm and 40 m respectively. This implies that the fiber laser output power would be limited by Eq. (17), even if the core radius could be scaled arbitrarily. This equation predicts a hard upper limit on the output power. Furthermore, the parameters in Eq. (17) are either physical constants or have already been chosen to be the best value that is likely to be achievable in practice. Scaling beyond the 36.6 kW limit will thus require either an increase in the laser wavelength (Tm has shown some promise as a fiber laser medium at 2µm, which could double the theoretical output power) or a scheme to suppress SRS.

We note that Eqs. (16) and (17) are valid for all core diameters, however, below about 90 µm diameter on the plot in Fig. 1, the laser is pump power limited. Understanding the boundary between the pump power limited region and the thermal lens limited region tells us how to achieve the ultimate potential of the fiber laser (the thermal lens-SRS limited power) at smaller (more realistic) core diameters. To find the thermal lens-pump power boundary, we equate Eqs. (4) and (15) and solve for the optimum core radius which is given by

Eq. (18) gives the smallest core radius at which the thermal lens-SRS power limit is reached. It varies very slowly with its underlying parameters due to the ¼ power exponent. Operating at longer wavelengths requires larger core radii. To shrink the core radii to smaller values, the brightness of the diode laser pumps or the maximum doping level of the optical fiber needs to be increased, or both. A factor of 16 increase in the product of these parameters decreases the core radius by a factor of 2 putting the diameter about into the reasonable realm of 45µm. By substituting Eq. (18) back into Eqs. (4) or (15), we see that if one were using a fiber with a core diameter at the boundary between the thermal-lens and pump-power limited regions, the power from the laser would increase linearly with the fiber length according to the equation given by

The remaining boundary in Fig. 1 is the boundary between the SRS-limited and pump power limited regions. It is less interesting, and simply yields the minimum length at which the thermal lens-SRS limited power is achieved. As length is easily scalable in a fiber laser system this does not present a difficult practical limit. By equating Eq. (8) and (15), we find that the minimum length corresponding to the minimum core radius from Eq. (18) above is given by

As expected this decreases with increasing pump diode brightness and increasing doping concentrations. Again, in the current context this is of limited usefulness as the lengths involved are still in the tens of meters range and the fiber is still so long that it needs to be bent to achieve a reasonably sized package. The analogous result for SBS limited fiber lasers is more interesting, as we will see later. Substituting Eq. (20) into Eqs. (8) or (15), we find that the output power limit is proportional to the square of the core radius along this boundary and is given by

*SRS Limited Lasers (5X Current State-of-the-Art Diode Pump Brightness)*

Eq. (18) and (20) suggest that the ultimate output power from a single aperture fiber laser can be reached using smaller cores and shorter fiber length if brighter fiber coupled pump diode lasers and high core concentrations can be achieved. To investigate this, let us assume that fiber coupled pump diode brightness continues to increase with time and are 5 times brighter than they are today. We can then construct a new contour plot, shown in Fig. 2, using the same procedure used to construct Fig. 1. We note that a new region has emerged. We have colored this region gray and it corresponds to the optical damage limited region. In this region and at its boundaries the power is limited by the damage threshold of the fiber, thus it is simply given by Eq. (11). The blue, pink and green regions continue to represent the pump-power, thermal-lens and SRS limited regions respectively. The maximum laser output power is still limited to 36.6 kW as given by Eq. (17). The pump-power and thermal-lens boundary now occurs at a 60 µm core diameter due to the brighter pumps, but the corresponding core radius (still described by Eq. (18)) is no longer the minimum core radius needed to achieve the SRS-thermal lens limited laser power.

To find the new minimum radius we first need to find the lines that describe the three new boundaries in Fig. 2. Again, the power at these boundaries will be given by Eq. (11). The optimum laser length in the optical damage limited region is however, no longer tightly confined to a single value but exists in a range of lengths. The maximum length is defined by the boundary between the optical damage and SRS limited region. The minimum length is defined by the boundary between the optical damage limited region and the pump power limited region, for core diameters less than the 60 µm defined by Eq. (18). The minimum length for core diameters larger than 60 µm is determined by the boundary between the optical damage limited region and the thermal lens limited region This core diameter is smaller than the core diameter at which the optical damage limited power first reaches the SRS and thermal-lens limited output power. We now find those four equations of interest.

First, by equating Eqs. (11) and (8) and solving for L, we find the upper boundary of the optical damage limited region. It is given by

Substitution of Eq. (22) into Eq. (16) yields the new minimum core radius at which the SRS and thermal-lens limited output power is reached. This radius is given as

To reduce the core diameter further requires a higher than assumed damage threshold for silica fiber. The CW damage threshold of fused silica fiber is still debated in the literature.

The minimum fiber length needed to achieve the optical damage limited laser power in the region where the core radius is smaller than that given by Eq. (18) can be found by equating Eqs. (11) and (15) and solving for the fiber length. In this case we find that

Finally, for the range of core radii between the values given by Eqs. (18) and (23), the minimum length at which the damage limited optical power may be achieved is found by equating Eqs. (11) and (4), and solving for the length as a function of core radius, yielding

Here, the strong dependence of the fiber length upon the core radius drives the minimum fiber length at which the optical damage limit is reached from about 8 m to about 38 m over the range of core diameters spanning 60 µm to 85 µm.

*SBS Limited Lasers*

Following the procedure for constructing Fig. 1, we now construct a contour plot given in Fig. 3 for the SBS limited case. In this case, we use the thermal, damage and pump power limits combined with the SBS limit and the parameters in Table 1. As in Fig. 1, there are only three relevant limits: the SBS limit (shaded yellow), the thermal lens limit (shaded pink) and the pump power limit (shaded blue). The upper bound for the fiber length is only 4 meters due to the severe limit imposed by SBS upon the output power of the fiber laser.

Since the functional form of SBS is the same as the functional form of SRS a constant power ridge is formed by the boundary between the thermal-lens limited region and the SBS limited region. The boundary between the pump-power limited region and thermal-lens limited region is still well defined by Eqs. (18) and (19). The other boundary is between the SBS limited region and the pump-power limited region and this determines the minimum length at which the SBS-thermal lens limited output power can be reached. Fig. 3 shows that 1.86 kW of power can be achieved at a core diameter of 90 µm and a fiber length of 2.1 m. This core diameter can be achieved with a photonic crystal fiber rod. Commercially available fiber rods are typically about 1m in length. It is quite interesting to ask the question: how much brighter would the diode laser pumps need to be to pull the optimum fiber length down to 1 m? But first, let us find the equation for the boundary between the SBS and thermal lens limited region. To do this we equate Eqs. (4) and (10) and solve for the fiber length which is given as

Again, the length scales as the core area. Substituting Eq. (26) back into Eqs. (4) or (10) yields

The power scales linearly with the laser wavelength and is independent of the core radius and fiber length (providing the core has been made large enough to exceed the pump coupling limited region). SBS suppression fiber designs that modify the SBS gain coefficient by a factor of 10 will only yield a factor of 3 improvement in the maximum achievable output power, as can be seen from Eq. (27).

As noted above, the minimum core diameter at which the SBS and thermal-lens limited power is achieved is around 90 µm in accordance with Eq. (18). This diameter is within the demonstrated parameter space achievable by photonic crystal fiber rods. Due to the requirement that these rods be kept straight, it is desirable to understand how to shorten the fiber length. The minimum length at which the SBS-thermal lens limited power is achieved is set by the boundary between the SBS limited and pump power limited regions. To find this length we set Eq. (8) equal to Eq. (15) and solve for the length. We find that

Eq. (28) suggests that a factor of 4× increase in the pump brightness should be sufficient to bring the optimum fiber length down into the 1m range. In the SRS limited case above we ran into an optical damage limit barrier as we scaled the pump brightness. However this does not happen in the SBS limited case because the power limit due to the SBS thermal lens is quite low, and at 10 W/µm^{2} an effective core area of only 200 µm^{2} is required to push the fiber out of the damage limited regime. This core area corresponds to a core diameter of less than 20 µm, something that is easily fabricated. Brighter pump diode lasers or equivalently high dopant concentrations are thus an essential part of any scheme to scale the output power of single frequency fiber laser systems.

## 4. Mode size limits

Within certain constraints, we have shown how scaling the core diameter and the fiber length leads to power scaling. In this section, we determine how the mode field diameter (MFD) scales with the core diameter in a practical fiber laser system. We consider the MFD scaling limit by looking at two properties of optical fiber waveguides that are most relevant. These are: the increasing sensitivity of the fiber mode to scattering into an adjacent mode as the core diameter is scaled and the reduction in mode field diameter of a fiber mode due to the effect of bending the fiber.

In sub-section b below, we show that the product of the fiber effective area and the difference in the effective indices between the fundamental mode and the closest propagating mode is effectively a constant. As a consequence, fibers with larger effective areas necessarily have modes with more closely spaced effective indices.

The scattering of light from one mode to another can be described by coupled mode theory, which relates the wave vectors (or propagation constants, which are proportional to the effective indices) of the modes’ photons and the scattering perturbations. A result of the theory is that the closer the modes’ wave vectors are, the more likely it is that those modes will be coupled by scattering.

Thus we conclude that the larger modes are increasingly more likely to scatter into their neighboring mode due to perturbations of the waveguide and finite manufacturing tolerance. We note that this observation is for the lowest order mode, and higher order modes are less affected by this phenomena [44].

In sub-section c below we examine bending. Given the results of Sections II and III above suggesting that the optimum high power fiber length is on the order of 40 m, it will be necessary to bend the fiber waveguide. However, bending creates a geometric distortion of the effective refractive index profile. This in turn reduces the effective area of the mode. We show below that for reasonable bend radii, the effective area of the fiber plateaus at an area equivalent to an unbent 50µm core.

#### 4.1 Definitions

We consider conventional fibers those whose refractive index profiles do not vary with azimuth, as might be manufactured by the modified chemical vapor-deposition or outside vapor-deposition methods. In the future, we will conduct similar analyses of micro-structured fibers. For weakly guiding conventional fibers [42], the field distribution *ψ* follows the wave equation given as

where *l* is the azimuthal order number. The wave equation has been normalized in two ways. First, the radius, *r*, is normalized to a scaling factor, *a*, yielding the normalized radius, *γ*. Second, the index profile, *n*(*r*), is replaced by a normalized profile, *b*(*γ*), given as

where *n _{max}* and

*n*are the maximum and minimum refractive index of the profile. Conventionally, the normalized eigenvalue,

_{min}*b*, is defined as

_{eff}and the generalized V-number *ν* is defined as

For step-index fibers, the maximum and minimum indices are equal to the core and cladding indices, respectively, and if *a* is chosen to be the core radius, then *b _{eff}* and

*v*reduce to their common definitions.

Table 2 gives the normalized parameters of the four fiber designs considered here. Step-index and W-profiles are well known, the A-profile is a step-wise approximation of a bend-tolerant graded-index design [43].

#### 4.2 Refractive index control

It can be shown that the separation between the eigenvalues of two modes, which we generically designate 1 and 2, follows the expression given as

Note that the right hand side of this equation is fixed for a given value of *ν*. It can also be shown that the ratio of *A _{eff}*/

*πa*, where

^{2}*A*represents the mode’s effective area [45], is fixed for a given value of ν. It follows that the mode index spacing-area product is given by

_{eff}depends only on ν.

Fig. 4 shows how the spacing-area product varies with the number of allowed, azimuthally non-varying (*l*=0) modes supported by the step-index, A- and W-profile defined above. The spacing considered here is the difference between the eigenvalues of the LP_{01} and LP_{11} modes if both were supported, and the difference between the LP_{01} eigenvalue and *n ^{2}_{clad}* if only the LP

_{01}is supported.

Note that choosing the *l*=0 mode-count for the abscissa allows direct comparisons between different fiber designs and makes the abscissa essentially linear with core diameter.

We illustrate the importance of Fig. 4 with an example. Assume we wish to build a step index fiber that supports exactly two *l*=0 modes. From the chart, the spacing-area product is 0.33. Thus for any combination of fiber parameters — core diameter, numerical aperture, and wavelength — that comprise the desired *ν*-number (the one that supports two *l*=0 modes), the spacing-area product will be 0.33. Note that if the mode area is made relatively large, then the intermodal spacing will necessarily be relatively small.

Now assume that the above fiber will operate at λ_{0}=1 µm and must have A_{eff}=1,000 µm^{2} to withstand some desired average power or pulse energy. Then the eigenvalue spacing would be 3.3×10^{-4}, and for weakly-guided modes in silica (n=1.45), the index difference between the LP_{01} and LP_{11} modes would be 1.1×10^{-4}. Following [44], we allege that a manufacturer’s refractive index control most be of this order, or better, for them to economically manufacture such a fiber.

Equating larger spacing-area products with ease of manufacturing and hardiness to environmental perturbations, Fig. 4 shows that:

- Step-index, W-profile, and A-profile fibers that support more than two
*l*=0 modes (and hence also several*l*≠0 modes) offer essentially no manufacturing advantages over fibers that only support those modes, since the spacing-area product asymptotically approaches 0.338 as more modes are allowed. Fibers can be manufactured that support many more modes than this, but those modes will be easily corrupted. - A W-profile that supports only the LP
_{01}is nearly as easy to manufacture as a step-index fiber that supports two*l*=0 modes (as well as the LP_{11}mode), since the spacing-area products are nearly the same. Note, though, that we have ignored the radial tolerances, which may make the W-profile more difficult to realize.

Based on the limited number of fiber designs illustrated in Fig. 4, we observe that

We find that most other conventional fiber designs, and intriguingly, step-index fibers having rectangular and triangular cross sections, also follow this observation. This latter fact can also be derived from mode density considerations. However, high-order mode [44] and flattened-mode [46] fibers, have special behaviors that we will consider in a future paper.

The following section analyzes the bend sensitivity of the three example designs.

#### 4.3 Bend sensitiνity

Bending effectively adds a linear perturbation to the refractive index profile of the fiber [47]. In the normalized units adopted above, the perturbation to the profile is given as

where *R* is the bend radius. For an unbent fiber at a fixed *ν*-number - and hence for a fixed mode count and fixed modal shapes - the perturbation is proportional to *a*
^{3}, or as noted by Schermer [48], indirectly proportional to *NA ^{3}*=(

*n*

^{2}

_{max}-

*n*

^{2}

_{min})

^{3/2}.

We have used the Beam Propagation Method (BPM) to calculate the mode field shape and effective area using Eqs. (29–36) above for various core diameters and bend radii of the optical fiber. In this calculation we assumed a numerical aperture (NA) of 0.06, common to all cases. This value of NA is typical of large core single mode fibers, and represents a practical limit based on manufacturing tolerances and mode robustness. The field is propagated by the well known BPM technique [49], from an initial field distribution determined by a finite element mode solver applied to each bent fiber configuration. Final field distributions and bend losses are determined after any initial transients have been allowed to decay.

In Fig. 5(a) we plot the effective mode area vs. the core diameter of a bent and unbent step index optical fiber. For comparison, we also plot the effective core area. Fig. 5(b) shows the effective mode field diameter corresponding to these effective areas, assuming cylindrically symmetric modes. Nevertheless, Fig. 5(a) shows that as the core diameter increases any bending of the fiber has a severe impact on the effective area. This severe impact effectively negates any gain obtained by increasing the core diameter above 40 or 50 µm. This is also reflected in Fig. 5(b).

Reports in the literature of large core diameters with good single mode properties are rare [50,51,52] and likely not very reproducible due to the ease with which power may be scattered to other modes or out of the core as the core diameter increases. Furthermore, our analysis in Sections II and III of the interaction of the thermal, pump brightness, damage and non-linear limits strongly suggest that for broadband lasers, the effective fiber length needs to be 10s of meters. This in turns suggests that a high power fiber laser must be bent in order to be practical. However, the analysis here tells us that once we start bending the fiber we quickly achieve diminishing returns in the area of mode field diameter scaling for core diameters above 50 µm. Combined with the damage threshold limitations this suggests that single aperture circularly symmetric fiber lasers will have a maximum damage limited output power of around 10 kW assuming 10 W/µm^{2} is an applicable damage limit in the bulk fiber.

## 5. Energetics

In sections II and III above, the scalability of diffraction limited fiber lasers and amplifiers to high average power was assessed. In that analysis typical values were assumed for parameters such as the small signal pump absorption required for efficient operation (A~20dB), fraction of pump light converted to laser power (η_{laser}~0.84) and fraction of pump light converted to heat in core (η_{heat}~0.1). It is important to show that those assumptions are consistent with the core radii, pump clad radii, assumed fiber absorption and fiber length found to be optimal in Figs. 1–3. In this section we construct an energetics model for Yb^{3+} doped silica fiber and validate that indeed the optimal parameters correspond to a laser system that yields efficient performance. In particular, the SRS bend limited case of a 50µm core producing 10kW output power, the SRS arbitrarily large mode 90µm core producing 40kW and the SBS limited 90µm core 2kW lasers are specifically examined for consistency with Yb^{3+} energetics. The precise parameters corresponding to these conditions are summarized in table 3 below.

We model the performance of Yb^{3+}:Silica doped fiber laser using CW theory of quasi-three level end-pumped oscillators presented by R. Beach [53]. The model system under consideration is shown in Fig. 6. The Yb:silica fiber is cladding pumped at wavelength *λ _{p}* and pump power

*P*. The laser signal, at wavelength

_{pump}*λ*and initial power

*P*, is launched into the fiber core and propagates the length of the fiber,

_{in}*L*. Recall, the fiber cladding and core radii are,

*b*and

*a*respectively.

We calculate laser efficiency as a function of fiber length, pump and lasing wavelengths, core diameter, etc. The Yb:Silica energy level structure consists of two manifolds with measured and calculated absorption and emission cross-sections, *σ _{abs}* and

*σ*[54,55].

_{em}We assume that the fiber is single passed by the pump and include effects of saturation and population depletion. The rate of change of the population in the lower state is *R _{Ex}*, given by

where the absorption coefficient, *α _{clad}* is given by

The rate of change of the population in the upper state is *R _{DEx}* is given by

where

Here, *n _{0}* is the Yb

^{3+}doping density,

*n*is the upper state population density,

_{2}*γ*and

_{laser}*γ*are, respectively, laser and pump photon energies. In steady state, population inversion is constant in time. Hence,

_{p}*R*=-

_{Ex}*R*and we can solve for the upper state population density,

_{DEx}*n*.

_{2}We calculate the theoretical efficiency of the laser system as a function of pump and lasing wavelengths. The efficiency is defined as the ratio of the extracted signal power (difference between output and input signal power) and the pump power. As an example we choose fiber laser parameters where the fiber absorption at the peak wavelength of 976 nm is 250 dB/m, corresponding to *n _{0}*=3.7×10

^{19}cm

^{-3}, fiber diameter,

*a*=50 µm and

*b*=1185 µm, and upper state lifetime,

*τ*=1 msec. We assume input pump power,

_{0}*P*=12 kW and input lasing power

_{pump}*P*=1 kW.

_{in}The resulting laser efficiency for a 45 m long fiber length is represented as a contour plot (Fig. 7). The lasing efficiency has two peaks, one at 920 nm and another at 977 nm, in agreement with the absorption curve for Yb^{3+}:Silica (Fig. 8) [54]. The lasing efficiency is highest at 977 nm pump and 1032 nm lasing wavelengths (92%). The variation in the output efficiency plotted as a function of the lasing wavelength is in Fig. 9. Away from the peak, the lasing efficiency decreases rapidly at shorter and slowly at longer wavelengths.

Next, we examine variation in the laser efficiency with fiber length. These results are dependent on doping concentration, pump power coupled into the fiber, and fiber core diameter. For the simulation parameters previously summarized and the additional cases detailed in Table 3 below, the result is presented in Fig. 10 for fiber lengths between 1 m and 100 m. For the first two cases corresponding to SRS dominated lasers (red and blue lines), the lasing efficiency increases rapidly reaching 90% for 30 m long fibers, and then continues to increase slowly to 95% at around 100m. At these fiber lengths it is likely loss would begin to effect efficiency. However, the main point here is that 85% pump to signal conversion efficiency is a reasonable value to consider. The third curve (black) in Fig. 10, representing the SBS dominated case peaks as expected at a few meters.

Finally, we examine the magnitude of the quantum defect, defined as 1-λ_{p}/λ for the cases in corresponding to the parameters in Fig. 10. The majority but not all of the unconverted pump photons result from the quantum defect and are converted to heat. Depending on fiber length, the quantum defect at the optimum lasing efficiency varies between 3.5% and 9.5% (Fig. 11). The rest of the pump power is lost due to spontaneous fluorescence from the upper level and, for short enough fiber lengths, incomplete absorption in the fiber.

Table 3 summarizes three of the most important cases considered in section III and shows that the critical assumptions of doping density, fiber length, core and cladding radii, efficiency and quantum defect all tie together consistently with an energetics model for Yb^{3+} in silica.

We recognize that the optimum laser wavelengths predicted by this formalism do not agree with the 1080 nm laser wavelength that is typical of most high average power results. There are several reasons for this discrepancy. First, for practical experimental reasons many of the high power results include pumps at 915 nm rather than the more optimum 976 nm wavelength. Second our model has not included any losses, whose values will vary between experiments. Third, most high power results employ oscillators with <10 dB of gain per pass. In a three level system this lowers the inversion and pushes the laser wavelength to longer values. Fourth, many of the high power amplifier results have employed bulk dichroic mirrors to combine 915 nm and 976 nm pump lasers with the >1µm signal laser. These bulk mirrors likely do not have optimum reflectivity close to the wavelengths suggested as optimum in Table 3. All of these would then push the laser signal wavelength to longer values.

The three critical cases detailed in table 3 have been shown to be consistent with a rigorous energetics model. These cases examine both the SRS and SBS limited cases. The SRS limited case was further divided into an bend limited case and the infinitely scalable core case. Thus, the energetics analysis ties together sections II–IV, showing that critical assumptions about doping levels, the relationship between fiber core and cladding diameters and fiber lengths and the required overall absorption needed for an efficient laser that were made early in the analysis are indeed valid.

## 6. Conclusions

We have now examined the scaling limits of common diffraction-limited circularly symmetric step index optical fibers in detail. In Sections II and III we have shown how thermal effects, finite pump brightness, limited doping concentrations, non-linear effects and damage limit the scalability of a single aperture fiber laser. These effects interact to create hard limits on the output power of broadband fiber lasers at around 36 kW and narrowband fiber lasers at around 2 kW. Waveguide designs which suppress Raman and Brillioun effects may increase these limits somewhat, particularly in the narrowband case. However, further analysis in Sect. IV suggests that there is a real upper limit to the scalability of the core diameter largely because a long fiber must be bent to be packaged. This upper limit is likely to be somewhere in the area of 50 µm diameter. This in turn will limit the output power of a simple M^{2}=1, single aperture fiber laser to around 10 kW. In Sect. V, we reviewed the energetic analysis of an Yb^{3+} fiber laser system in order to validate that the analysis in Sections II and III was consistent with a laser that could be physically realized with the claimed efficiencies. This section also revealed some interesting theoretical results having to do with optimum pump and laser wavelengths.

We note that while we have given specific numbers here for power outputs, it is the derivation of the limits that we consider most relevant and useful. Improvement in fiber technology may quickly render a significant improvement in one or more of the values in Table 1. This in turn could lead to significantly higher output powers than we have suggested possible here. In particular, fibers that suppress SRS would be a significant improvement. Another major advance would be to remove the uncertainty of the CW damage threshold of the fiber. Similarly, enhancement of diode laser pump brightness and SBS suppression fibers could have a significant impact on the scalability of single frequency fiber lasers.

In closing, we would like to acknowledge valuable discussions with Johan Nilsson of the University of Southampton, who has independently considered, and presented in short courses [56] nearly identical results for some of the same limits we have derived in this paper.

## Appendix A. Derivation of thermal focusing

Here we derive an equation giving the output power limit of a fiber based on thermal focusing that occurs in the fiber’s core. The thermal dissipation in a fiber begins to significantly perturb the spatial mode spectrum of a fiber when the positive dn/dT of the fiber material combined with the approximately parabolic thermal gradient present in the core of the fiber start to self-guide the fiber radiation such that the thermal guiding competes with the waveguide structure that is engineered into the fiber. To estimate the critical thermal power density at which this occurs, we consider a gaussian beam characterized by a waist ω and propagating in a uniformly heated medium that is cooled on its perimeter. This is the situation in the core of a fiber and leads to an approximately parabolic thermal profile there. Self-guiding occurs when the waist of the gaussian waveform is such that the thermal lensing introduced by the parabolic temperature profile just compensates the diffraction of the beam as it propagates. Formally, we can describe the propagation of the gaussian beam using ABCD matrices appropriate for a duct having a radial index grade. The ABCD matrix for such a duct [57] is given as

where

and

The thermally generated radial index profile is described by

where P_{Th} ’’’ is the thermally dissipated power density in the volume of the gain region. The laser beam can be described by a complex radius of curvature, q, given as

where R is the radius of curvature and ω is the beam waist, leads to the following equation for the self-similar propagating mode,

The solution of this equation leads to an expression for ω and R given by

This describes a collimated beam (R=∞) propagating with a constant beam waist ω. Rearranging Eq. (47) gives

By noting that the beam waists we consider in this paper are between 0.7a to 0.8a, we further simplify Eq. (48) and write it as

Thus the power limit due to the heat in the fiber core creating a thermal lens that is competitive with the index guiding from the fiber core is

which is the same as Eq. (4) in the main text.

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