## Abstract

We propose a new gain fiber structure with an inverse index profile for high power amplification of a Gaussian single mode beam. A large mode area (LMA) design can be fulfilled with the inverse index profile by implementing a graded index in the depressed core. We numerically show that the proposed gain fiber can guide a single mode Gaussian beam with a large beam area and amplify the beam to a kW level output power.

© 2017 Optical Society of America

## 1. Introduction

High power, high-beam-quality fiber laser systems have been attractive for scientists and industries because its structure is more compact and power scalable in comparison with other high-power solid-state laser systems. However, the output power of a single-mode fiber laser is limited by the strong nonlinear phenomena combined with thermal effects occurring inside a limited fiber core size [1]. To overcome the power limit of a single-mode fiber laser, core area scaling becomes a standard approach. Many scientists have been designing new fiber-structures that allow large mode area yet maintaining a good beam quality via higher-order mode (HOM) suppression mechanisms [2–6]. Among them, a gain guided and index anti-guided (GG + IAG) fiber was proposed by Siegman [7]. The structure allows single mode operation. However, a GG + IAG fiber suffers from an inherent leakage loss, resulting in a low lasing efficiency [8]. The inherent leakage loss for the similar fiber structure had been analyzed by E. A. J. Marcatili and R. A. Schmeltzer [9]. It was concluded that the inherent leakage loss saturates output power quickly, hence limiting power scalability [9,10]. That means that it is difficult to obtain high power from a GG + IAG fiber. Nonetheless, a GG + IAG fiber is a promising design as it provides mode selective attenuation, promoting single mode operation in a large core size. To power scale the single fundamental mode (FM) beam in the GG + IAG fiber, the inherent leakage loss of the FM should be minimized. We propose a graded-index profile to replace the step index core, and reduce the FM loss. A FM of a GG + IAG fiber has approximately a Gaussian shape. Hence a fiber structure offering good guidance of a Gaussian beam should fulfill the requirement. The graded-index fiber is known to provide a good guidance of the Gaussian beam without significant leakage loss [11]. Consequently, the inherent leakage loss of the FM of a GG + IAG fiber can be reduced with the graded-index structure. Of course, a graded index fiber can also guide HOMs depending on a signal seed beam coupling condition. Furthermore, even with the well-controlled launching condition, the HOMs can be excited during an amplification process [12,13] which can degrade output beam quality. However if the graded-index core is surrounded by a raised cladding to form the GG + IAG fiber structure, the HOMs can be selectively suppressed via the mode-dependent loss feature of an GG + IAG fiber. Thus, combining the of mode-dependent loss of a GG + IAG fiber with an excellent FM guiding property of a graded index core can serve as an alternative LMA design route to realize kW level single mode operation.

Hence, in this paper, we propose a LMA gain fiber with a hybrid-index distribution, structured by the GG + IAG index distribution with a graded-index core. The structure allows LMA for amplifying a Gaussian beam to a kW level. The entire graded-index core is depressed from the cladding, which builds a mode-dependent loss, acting as a low mode pass filter. We numerically investigate the amplification characteristics of the proposed gain fiber in several large core diameters and various graded-index parameters. And we show that the core size can reach to 100 μm suitable for kW level output power. Our paper contributes to an alternative LMA fiber route to achieve single mode high power fiber laser. Fabrication feasibility of the fiber design is also discussed.

## 2. Numerical modeling for the proposed gain fiber composed of a graded- index and an GG + IAG fiber index

The proposed gain fiber has the index structure composed of a graded-index and an inverse-step- index distribution as shown in Fig. 1.The graded-index core in Fig. 1 has a refractive index profile as follows.

where*n*is the peak refractive index,

_{1}*γ*the scale factor,

*r*the core radius, and the value

_{0}*α*is in the neighborhood region of 2. In the case of

*α*= 2, the FM of a graded-index fiber has a Gaussian shape [11]. The FM beam size depends on γ and the normalized frequency parameter V [11]. A Gaussian beam non-matching the FM beam size periodically divergence and convergence when propagating along the graded-index fiber. The converging or focusing of the HOMs is detrimental to high-power operation because it can result in optical damage inside the core. To avoid the focusing effect, the beam inside the graded-index core should be collimated. This collimation is achievable by matching diffraction angle of the Gaussian beam to a critical angle for total internal reflection in the graded-index [14]. Using the above condition and the paraxial approximation, we can derive the scale factor

*γ*for the collimated Gaussian beam inside the graded-index fiber core as follows:

*r*is the input beam radius and

_{L}*λ*the wavelength of an incident beam in vacuum. This result agrees with the approximate value numerically derived for the FM beam size of a graded-index fiber in the neighborhood of $\alpha =2$ [11]. And a GG + IAG fiber has the intensity loss (α

_{0}_{m}) depending on the fiber mode number as follows [9,10]:

*u*is

_{m}*m*-th root of Bessel’s function of zeroth order,

*J*

_{0}(

*u*) = 0,

_{m}*r*the core radius,

_{0}*n*the index of the core of a GG + IAG fiber and ${\lambda}_{0}$ is the vacuum wavelength. Equation (3) shows that the loss increases with mode number. The propagation characteristics of an inverse-step-index fiber amplifier were investigated in details in [10].

_{2}Beam propagation in our fiber is investigated using the slowly varying envelope electric field (E) at a frequency (ω) in the paraxial approximation as [15–17]:

*k*(

*r*) =

*n*(

*r*)

*ω*/

*c*, c is the velocity of light in the vacuum,

*k*

_{1}=

*n*

_{1}

*ω*/

*c*, ${\nabla}_{\perp}^{2}$ is the transverse Laplacian and the last term of Eq. (4) represents the laser gain. And

*g*is a small signal gain coefficient,

_{0}*I*is the intensity inside a fiber core, and

*I*is the saturation intensity of a gain medium. Among lanthanide ions, neodymium (Nd) and ytterbium (Yb) have been the key element for technical advances in the 1 μm laser. The

_{sat}*I*for Nd-doped glass is in the range of 1.4 × 10

_{sat}^{4}to 1.9 × 10

^{4}W/cm

^{2}[14]. The Yb offers a slightly larger

*I*at 2.0 × 10

_{sat}^{4}W/cm

^{2}[18]. We choose the value of

*I*to 1.4 × 10

_{sat}^{4}W/cm

^{2}for Nd-doped glass To solve Eq. (4), we use a scalar beam propagation method. To demonstrate our concept, we perform a numerical simulation for the fiber by setting the main parameters of our amplifier to those of the first GG + IAG fiber laser demonstrated by Y. Chen

*et al*[8]. And we choose a graded-index core with $\alpha =2$, in our numerical calculation. We assume a Gaussian beam with the vacuum wavelength (λ

_{0}) of 1052 nm and 10 mW-input power as an input beam for propagating inside the fiber core.

First, we numerically investigate the leakage losses (α_{0}) of a step index GG + IAG fiber and our proposed fiber with no gain. The leakage loss is simulated for a core diameter of 50 μm and 100 μm, with various incident laser beam radii as shown in Fig. 2. Apparently, the graded-index core significantly reduces the leakage loss as compared to the step index core (See the curves with 17.5 μm beam radius, *r _{L}*). We then check the validity of our numerical simulation against the analytical model as presented by Eq. (3). The calculated leakage loss of the step index GG + IAG fiber for the core radii of 25 μm and 50 μm are calculated as 1.116/cm and 0.1395/cm, respectively, using Eq. (3). This is nearly the same as the fitted values in Fig. 2. Hence we can confirm that our numerical model is correct. The index of cladding is assumed as 1.5734 and the index difference of cladding to the core is −0.0045.

Clearly, the results of Fig. 2 confirm our prediction that the leakage loss can be improved in our graded-index fiber. We also find that the leakage loss of our proposed fiber depends on the input beam size as shown in Fig. 2. In contrast, a leakage loss of a GG + IAG fiber doesn’t depend on the input beam size [10]. Hence we calculate the loss coefficient of the proposed fiber for several input beam radii while the loss coefficient of the GG + IAG fiber is calculated for a single input beam radius. For the optimum coupling efficiency to a GG + IAG fiber, we use the known design criterion *r _{L}*/

*r*= 0.706 [10], thus selecting the input beam radii of 17.5 μm and 35 μm for the GG + IAG fiber core radii of 25 μm and 50 μm, respectively. As presented in Fig. 2, the smaller beam size is favored for lower leakage loss in our design. For 25-μm core radius, the leakage loss is ${\alpha}_{0}$< 0.003/cm, nearly three orders of magnitude lower than the GG + IAG fiber, when the input beam radius is less than 10 μm (58% of the original beam size). The loss reduction is still significant, improved by 3 times, when comparing the same core size. A similar trend is observed in the larger core of 50-μm albeit less drastic. We can reach an order of magnitude improvement by controlling the input beam size to 70% of its GG + IAG counterpart.

_{0}We further investigate the propagation characteristics of a Gaussian beam in a pure graded-index fiber without the inverse index as in the leftmost design in Fig. 1. The design satisfies the condition of Eq. (3). We used the fiber length of 300 mm, the fiber core radius of 25 μm, and n(r) = n_{2} for r ≥ *r*_{0}. A Gaussian input-beam radius of 10 μm is assumed to realize low leakage loss as shown in Fig. 2(a). The value of γ = 0.00285 is chosen for satisfying Eq. (2). The result from the Gaussian beam propagation in the pure graded-index fiber is shown in Fig. 3. We find that the Gaussian input beam can be well collimated inside the core when the condition of Eq. (2) is satisfied. However if the Gaussian beam in the graded-index fiber core is amplified by uniformly pumped active ions across the core, the beam propagating through a core cannot keep a Gaussian profile. The Gaussian shape can be distorted by a gain saturation following the intensity profile as expressed in Eq. (4). That is, the side part of a Gaussian beam with a relative low intensity is faster growing up than the center part due to the saturation. Hence the amplified beam becomes multi-moded. Consequently, the beam is no longer collimated because the excited HOMs cannot satisfy the collimating condition. This effect is represented in Fig. 4 from our numerical result. All parameters for Fig. 0.4 are the same as that of Fig. 3 except for the included gain. Figure 4 shows that the beam is distorted by HOMs induced by an amplification process and propagates through a graded-index fiber with a periodical focusing and diverging motion. This contrasts with the undoped pure graded-index fiber. The beam focused on a fiber core axis can cause optical damage. So it is difficult to obtain a high-quality and high-power beam from a pure graded-index fiber amplifier due to the beam distorting and focusing phenomena by an amplification process.

Hence, in order to amplify a Gaussian beam without the distortion and focusing phenomena inside a pure graded-index fiber, the HOMs generated by an amplification process should be removed. Combining the graded-index core with the GG + IAG design can effectively suppress the undesired HOMs via the selective modal leakage loss. Therefore, our hybrid-index gain fiber composed of a GG + IAG fiber index and a graded index distribution can amplify only a fundamental mode beam while removing HOMs generated by an amplification process.

## 3. Results and discussion

The proposed hybrid design is tested with the same parameters used in Fig. 4, but with the inverse index profile as represented in the rightmost design in Fig. 1. Additionally, we used an index difference (*Δn*) between the cladding and core boundary of −0.0045 which is the same as in Y. Chen *et al.* [8]. The numerical result for a Gaussian input beam amplification is shown in Fig. 5. In contrast to the result in Fig. 4, the results in Fig. 5 shows that the explicit intensity modulation on a fiber disappears when the beam is amplified through our proposed gain fiber. This result proves that the HOMs excited by amplification are effectively removed by the mode-dependent loss property of the inverse index design. And the result shows that the amplification rate and the saturated power limit of the proposed gain fiber can be much improved as compared to the step index GG + IAG fiber as shown in Fig. 6. The output power (*P _{out}*) of the proposed fiber proportionally increases with fiber length without power saturation in a given fiber length of 300 mm, whereas the output power from the GG + IAG fiber is quickly saturated after short-distance propagation [10]. Thus, we conclude that the proposed gain fiber outperforms the step-index GG + IAG fiber in terms of power scalability.

The behavior of the saturated output power in Fig. 6 can be understood with the help of a simple amplification equation for a Gaussian beam [19]. The amplification of a Gaussian beam can be expressed as:

*P*(

*z*) is the total power inside a fiber core. In the calculation, the loss coefficient α

_{0}is set by the leakage loss obtained in Fig. 2. A GG + IAG fiber have a fundamental mode of a Bessel function [7]. The Bessel profile of

*J*(

_{0}*r*) inside a fiber core is approximately the same as a Gaussian profile [10]. Hence we can use the Eq. (5) to describe the beam amplification of both a GG + IAG and our proposed fiber. The numerical data obtained by Eq. (5) are expressed as circle(⬤)and triangle(▲) symbols in Fig. 6 for the GG + IAG fiber amplifier and the proposed fiber amplifier, respectively. The numerical result by Eq. (5) is consistent with Eq. (4). Hence, we again confirmed that the numerical model established to solve Eq. (4) can describe a correct propagation behavior inside our proposed fiber core. The improved power scalability in our fiber design attributes to a reduced fiber loss. In Eq. (5), the saturation condition mainly depends on an inherent saturation intensity (

*I*) and an optical fiber loss coefficient (α

_{sat}_{0}). Given the same

*I*, it is apparent that lower loss in our fiber, as presented in Fig. 2, promotes higher output power before being saturated. Consequently, the saturation output power of the proposed fiber can be much higher than that of a GG + IAG fiber under the saturation condition of

_{sat}*dP*(

*z*)/

*dz*=

*0*.

We note that the reduced loss in Fig. 2 is achieved in exchange of an input beam size. To delay the output power saturation, the beam size should be reduced to lower a loss coefficient. However, too small beam size would facilitate undesired nonlinearities and confine HOMs in the center. Subsequently, the confined HOMs generated during the amplification do not experience the gradient refractive index change in the core boundary, preventing the HOMs from refraction out from the core. This suggests a trade-off between the background loss and the HOMs contents. To investigate the trade-off and find best compromise, the dependence of the mode selective loss on a FM beam size is simulated for amplification of *r _{L}* = 19 μm, 20 μm, and 21 μm when the fiber core radius is 50 μm, with g

_{0}= 1/cm. The results are shown in Fig. 7.

Unlike the beam size of 20 and 21 μm, the intensity on the fiber axis oscillates at the 19 μm beam. As mentioned in the previous section, HOMs can be induced by the amplification process. And the HOMs should be removed by the mode selective loss property of our proposed fiber as the cases of the beam size of 20 μm and 21 μm. But if the HOMs loss introduced by the refraction at a core and cladding boundary accumulates slower than the growth rate of HOMs in a small signal amplification region, an intensity modulation is caused by a self-focusing of HOMs by the graded index. Hence we conclude that the intensity modulation at *r _{L}* = 19 μm comes from the reduced HOMs loss rate due to the smaller input beam size. It is also noted that the intensity modulation depth for

*r*= 19 μm becomes more significant along the propagation. An approximated modulation length period (

_{L}*l*) by a self-focusing in a graded index fiber can be found by the following equation [20]:

*l*) for

*r*= 19 μm is calculated as 0.531/cm. Alternatively, the period can be numerically obtained by Fourier transform of the curve in Fig. 7. The obtained period from Fig. 7 is 0.530/cm, which well agrees to the result from using Eq. (6). Thus, we conclude that the beam size should be controlled to find best compromise between the single mode operation and the low leakage loss in our design.

_{L}We further extend our investigation to dependence of the Gaussian beam amplification on the input beam size when the collimation condition is not satisfied in Eq. (2). We introduce the γ value as 0.00071 to the amplifier parameters of *r _{L}* = 20 μm and

*r*= 50 μm. The chosen parameters ensure the minimum leakage loss and also satisfy the collimating condition. For a comparative study with non-collimating condition, the input beam size is varied to

_{0}*r*= 15 μm, 20 μm, and 25 μm. The results for the different input sizes are shown in Fig. 8.

_{L}As expected, when the collimation condition is satisfied with a beam of *r _{L}* = 20 μm, the FM beam is well preserved along the propagation. Its intensity linearly increases along the fiber length. However in contrast, for the beam sizes of

*r*= 15 μm and 25 μm that do not satisfy Eq. (2), HOM components are visible as intensity modulation because the launching condition excites the HOMs. Interestingly, the excited HOMs diminish after some distance propagation and linear increase of intensity is achieved, with the same slope as the beam of

_{L}*r*= 20 μm. As one of methods to remove the intensity modulation, we can introduce an undoped version of our proposed design as a HOM filter in front of a doped gain fiber. When an unmatched input beam is coupled to this undoped fiber, the excited HOMs can be quickly suppressed instead of being amplified due to absence of an active dopant and the mode selective loss mechanism in our design. Subsequently, the remained FM can be coupled to the gain fiber via splicing, with the desired matched condition. Therefore, matching of an input beam to the designed core can be practically fulfilled by introducing a passive fiber in front of the gain fiber.

_{L}We numerically find that all the spatial intensity profiles in the linear region in Fig. 8 are a Gaussian shape with the beam radius of 20 μm as shown in Fig. 9. The inverse index efficiently suppresses the HOMs and retrieve a Gaussian profile satisfying Eq. (2) after certain distance propagation.

Hence, we conclude that the amplified signal of our fiber can keep its FM by a mode filtering effect even though an input beam mode does not match to the fundamental mode of our proposed fiber.

The power scalability of our design is investigated at several gain coefficients. We used Eq. (5) to get the results shown in Fig. 10. We used the parameters of the fiber core radius of 50 μm, the beam radius of 20 μm and the loss coefficient of α_{0} = 0.02/m obtained from Fig. 2 to investigate the power amplification. And we calculate the amplified output power for two cases of typical saturation intensities for Nd-doped and Yb-doped glass fibers.

The results show that the gain fiber with the higher saturation intensity can amplify a seed signal as expected from Eq. (5). In Eq. (5), the derivative of *P*(*z*) would approaches to zero when z becomes sufficiently large due to a power saturation. That is, the signal power approaches to the maximum available power (*P*_{max}) determined by interplay between the gain and the loss of the fibers. Figure 10 shows that the maximum available power can be over a kW level power. We intentionally extend the fiber length to show the maximum available power at each gain coefficient. We note that it is not practical for a fiber amplifier to consist of fiber length of several hundred meters as the results shown in Fig. 10.

To validate mode area scalability in our fiber, we numerically demonstrated that the proposed fiber of a 100 μm core diameter can support a single Gaussian mode with a mode field diameter (MFD) of 40 μm. Of course, a commercial step-index LMA fiber with a 40 μm core diameter is available. But the commercial fiber does not offer HOM suppression. The large core size pushes up a V-number, which implies multimode nature of such fiber. In addition, the 40 μm commercial fiber only offers 26 μm MFD of FM calculated by an equation given in [21], assuming core NA of 0.08, which is much smaller than our fiber. In addition, a LMA fiber can suffer a transverse-mode instability (TMI) due to a thermal grating effect induced by LMA fiber modes interaction [22–24]. The TMI is detrimental to output beam quality. The best approach to avoid TMI would use a single mode fiber. But in practice, LMA fibers is multi-moded, unless considering a veru low NA rod-type fiber, and a strong HOM suppression mechanism is required to ensure single mode operation. Our design provides a distributed HOM filtering that can suppress the TMI and safeguard the FM operation as compared to the conventional LMA fibers. Thus, our proposed fiber has an advantage of TMI suppression due to its mode selective loss design.

Considering a practical laser design, the fiber should be coiled for packaging. We estimate a bending effect of the proposed fiber by considering the bending loss of a graded index fiber. This is justified as the proposed fiber have the same fundamental mode property as a graded index fiber [25,26]. Using a bending loss equation in a graded index fiber [26], for a Gaussian field of 20-μm input beam radius inside the proposed fiber core with 100-μm diameter in Ref [26], we can get the loss coefficients of 0.0054/m and 0.0003/m for the bending radius of 40 cm and 45 cm, respectively, with 20-μm input beam radius in a 100-μm core diameter. The value of 0.0003/m corresponds to 1.5% additional loss to the straight proposed fiber. Therefore, we expect that the proposed fiber can be packaged in 45-cm bending radius without significant output power drop.

We extend our discussion to thermal load influence on a refractive index and a thermal focusing effect. The fiber index variation by a thermal load distribution for a high power fiber laser had been well discussed in Ref [27]. The reference reported that 180 W pump power inside a 9.2-μm core diameter can induce the index variation of about 1x10^{−6} inside the fiber core. Assuming linear relationship between the pump power and the index variation, the index change is estimated to the pump power increasing by a factor of ten for 1-kW fiber laser, we can expect the index variation of about ~10^{−5} in 1 kW output power. This is negligible, only amounting to 1/50 of our 100 μm core design. Hence, we expect that the performance of our proposed fiber would not be significantly compromised by the thermal load distribution induced index variation. Also there is the fiber index variation by an intensity dependent nonlinearity. Usually, a nonlinear refractive index coefficient (*n _{2}*

_{B}) is ~10

^{−16}cm

^{2}/W [28]. The index variation for 1-kW fiber laser is estimated of

*n*(

_{2B}⋅I*r*), where

*I*(

*r*) is an intensity distribution inside a fiber core. The calculated maximum index variation is ~10

^{−7}for a 20-μm input beam radius. This value is very smaller than the graded index variation of our 100-μm core fiber. Hence we conclude that the fiber index variation by an intensity dependent nonlinearity can be neglected in our analysis.

We consider fabrication aspects of the fiber designs. As present in Fig. 1, the fiber is structured with a uniform cladding and a core with a graded index profile. Fiber design parameters can be determined from Eq. (2) and (3) for a desired core diameter. For instance, a 50 μm core size is achievable with a depressed core refractive index difference, *n _{0} - n_{2}*, of 0.0045 with respect to the cladding index,

*n*. The core index difference should increase from −0.0045 (for

_{0}*n*) to −0.0039 (for

_{2}*n*) toward the center to realize the single mode operation. That is, with the same index depression,

_{1}*n*, the core index variation should reduce to 0.00056 to realize a 100 μm core size. Assuming that the cladding is formed by silica glass as in conventional fibers, the core index profile can be directly constructed via chemical deposition processes. Today’s modified chemical vapor deposition (MCVD) process can be concurrently carried out with a rare-earth vapor delivery system [29,30]. Thus, deposition of index varying silica layer can be accompanied with uniform Yb doping under a precise flow rate control. Refractive index reducing elements such as boron or fluorine can be incorporated to silica for index suppression while vaporized Yb is continuously present throughout the entire deposition for uniform distribution. The required Yb concentration in this design is ~2 wt% or less than 2 wt% which is within achievable range. Hence, the graded index profile with uniform Yb distribution is achievable and practical. Alternatively, a stack-and-draw technique can be employed to build a pixelated core with the suppressed gradient index profile [31]. This technique should be more suitable to satisfy the smaller index variation such as in the 100 μm core.

_{0}- n_{2}## 4. Conclusions

We propose an alternative LMA design that features in-fiber mode selective filtering to generate a high power Gaussian output beam. The graded-index core ensures FM guidance while the inverted cladding in the GG + IAG structure provides the necessary HOMs suppression. We investigate characteristics of this design for adoption in high power fiber amplifiers. The numerical results confirm pure Gaussian beam guiding and HOMs suppression in core sizes of 50 and 100 μm. A seed signal coupling condition is also investigated, indicating the launched beam size needs control to maximize the design feature in terms of single mode operation and waveguide leakage loss. Therefore, our design contributes to an alternative route to single mode high power fiber amplifiers.

## Funding

Chosun University, 2014.

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