## Abstract

The results of theoretical and experimental studies of active tapered double-clad fibers, intending the optimization of its imperative parameters - tapering ratio, longitudinal profile, core/cladding diameters ratio, are presented. Using a refined taper geometry we have demonstrated power scaling of a ytterbium fiber laser pumped by low-brightness, cost-effective laser diodes up to 750 W, with 80% efficiency.

©2010 Optical Society of America

## 1. Introduction

Recently we proposed an active tapered double-clad fiber (T-DCF) as a gain medium for high power fiber lasers and amplifiers with nearly diffraction-limited beam quality [1–4]. Having large cladding diameters, these fibers can be pumped by high-power, low-brightness laser diode bars which establishes the basis for a cost-effective source. We have demonstrated a fundamental mode ytterbium fiber laser with 600 W of output power and 63% slope efficiency [4]. However, due to the non-optimized longitudinal profile of the T-DCF used, the efficiency observed was somewhat lower than the theoretically expected figure of 85%.

The design strategy and operation of tapered active fibers with cladding pumping have been discussed earlier in our paper [4], which presents a precise description of absorption mechanism, guiding characteristics and amplification of T-DCFs.

The goal of this paper is the detailed analysis and optimization of an active tapered fiber, with the aim of improving the performance of lasers and amplifiers using this concept.

## 2. Parameters of a T-DCF and their optimization

The parameters of tapered fibers involved in the optimization procedure include

- • Tapering ratio;
- • Shape of the longitudinal profile;
- • Core/cladding ratio;
- • Dopants concentration;
- • Cladding shape.

#### 2.1. Tapering ratio

A double-clad pumping scheme is widely used to efficiently couple the radiation from high power sources into the fiber amplifier. As a result, the low brightness of pumping sources can be effectively converted to the high brightness exhibited by single-mode fiber emitters. For a conventional double-clad fiber the brightness enhancement factor ** K_{DCF}** can be written as:

**is the slope efficiency of the laser,**

*S***,**

*A*_{clad}**,**

*A*_{core}**,**

*NA*_{clad}**are the areas and numerical apertures, respectively, of the double-clad fiber cladding and core; and**

*NA*_{core}**and**

*D*_{clad}**are the cladding and core diameters.**

*D*_{core}Typical parameters for a ytterbium double-clad fiber are S = 0.8, NA_{clad}/NA_{core}~3, A_{clad}/A_{core}~400 and K_{DCF}~3000. Modified for the tapered double-clad fiber, Eq. (1) takes the form:

**is the numerical aperture of the launched pump beam, which preferably satisfies the condition**

*NA*_{launch}**<**

*NA*_{launch}

*NA*_{clad}^{2,4},

**is the tapering ratio, and**

*T = D*_{input}/D_{output}**/**

*F = NA*_{launch}**is the cladding filling factor. Obviously, it is assumed that the light is coupled to the T-DCF through the large diameter facet. Thus, for the tapered double-clad fiber operating in a laser regime, the brightness is increased by a factor of**

*NA*_{clad}**as compared to the cylindrical (untapered) double-clad fiber. The tapering ratio beneficial for brightness improvement cannot, however, be increased above a certain value. It should be noted that in terms of ray optics the propagation of pump light through a T-DCF is accompanied by a gradual increase in the angle of the propagating ray relative to the fiber axis, which strongly affects the rate of pump absorption [1–4]. Consequently, the pump absorption for rays propagating in the T-DCF cladding at angles of α<NA**

*(F*T)*^{2}_{clad}/T and α>NA

_{clad}/T could be very different, due to their different interaction lengths with the active material.

Specifically, the absorption of pump radiation propagating at angles α<NA_{clad}/T is similar to pump absorption in untapered DCF with T = 1. Typically, α≤5°, which allows us to consider these pump rays as “near-paraxial”. In this case the unabsorbed pump power ${P}_{unabs}^{pump}$ exiting through the small diameter facet of the T-DCF is determined by the tapering ratio T and the so-called “paraxial” absorption coefficient ** γ** which takes into account in-core absorption and pump ray tracing [4]:

**– ray launching angle,**

*α*

*I(α)**–*normalized angular distribution of pump intensity, and

**– length of the T-DCF.**

*L*Pump rays propagating at angles α>NA_{clad}/T have different absorption lengths, L*(α),* in T-DCF which depend strongly on launching angle α. These pump rays are depleted during the propagation through a T-DCF partly through absorption in the core, and partly through leakage from the fiber due to violation of the total internal reflection condition, the so-called vignetting effect [4]. The unabsorbed vignetting fraction of pump, which leaks from T-DCF through the side surface, can be expressed as [4]:

_{clad}/T and α>NA

_{clad}/T could be very different. To avoid pump losses due to the vignetting effect, the aperture of T-DCF cladding should be underfilled, i.e. NA

_{launch}<NA

_{clad}[1–4]. The fraction of pump power ${P}_{parax}^{pump}$ which satisfies the condition α<NA

_{clad}/T can be written as:

Figure 1 shows that the paraxial pump power ${P}_{parax}^{pump}$ decreases strongly with an increase in tapering ratio and fill factor.

Since the rays propagating at angles α>NAclad/T exhibit lower absorption, preferential excitation of these rays could deteriorate the slope efficiency of the laser (amplifier).

Unlike the pump radiation in a tapered cladding, which enters the large diameter end and propagates in one direction, the laser light in a tapered core circulates in both directions. The single mode output is then preserved in tapered fiber by filtering out the higher-order modes to the cladding. The remarkable attribute of tapered geometry is nonreciprocal amplification. Specifically, the propagation from the large size multimode end of the tapered core to the small diameter single-mode section of the T-DCF suffers both amplification and loss due to the mode filtering effect, while the propagation in the opposite direction undergoes gain only.

Assuming that in the multimode section of the T-DCF the average number of guided modes is *N*, their power could be estimated as ${P}_{core}\approx \delta P\cdot N$, where ** δР** is the partial average power of one mode,

**is the number of guided modes in the fiber with core radius**

*N = 2π*^{2}a^{2}NA^{2}_{core}/λ**and numerical aperture**

*a***[12], and λ is the lasing wavelength. With the “worst” case assuming equal sharing of pump power over the guided modes, the estimation for the upper value of vignetting loss due to mode filtering during signal propagation towards the section with the small diameter core gives:**

*NA*_{core}*L*for

_{vgnt}= 0*T =*1 and approaches

*L*for large tapering ratios. This feature imposes a limitation on the upper value of the tapering ratio.

_{vgnt}= 1The leakage of high-order modes from the fiber core to the cladding owing to the vignetting effect can result in a high density of light propagating as cladding modes, which negatively affects the quality of the output beam. The figure of merit describing this process is the core guiding ratio or beam contrast ** C,** defined as the ratio of lasing power in the core to the total power propagating in the fiber. Therefore, although increasing the tapering ratio

*T*improves the brightness enhancement factor

**~**

*K*_{DCF}*T*, it could lead to a decrease in laser slope efficiency, an increase of the lasing threshold, and deterioration of the core guiding ratio due to reduced pump absorption and higher intracavity loss.

^{2}**for T-DCF can be normalized to the similar factor for the non-tapered cylindrical double-clad fiber with an analogous geometry which gives**

*K*_{DCF}**– slope efficiency for the laser with T-DCF,**

*S*_{T-DCF}**– slope efficiency for a laser with similar conventional DCF and**

*S*_{DCF}**– contrast of the emitted light. Here, only the radiation coming out of the core is accounted for.**

*C*In T-DCF-based amplifiers the mode filtering from the core is entirely suppressed since the signal propagates unidirectionally towards the large core fiber end, therefore the tapering ratio in the amplifiers can be varied in a wider range.

#### 2.2. Longitudinal shape of T-DCF profile

The axial taper shape should provide high output power supplied by preferential pump conversion in the large-core section, and preserve fundamental mode operation by spatial filtering in the single-mode section. The relative lengths of these sections, determined by the outer diameter variation along the fiber, affect the output beam characteristics. The multimode waveguide in the double-clad fiber should allow for efficient pump launching into the fiber, and intense absorption in the single-mode core. Moreover, in a T-DCF the longitudinal profile can be engineered to accept the radiation of low-brightness pump sources and guarantee robust fundamental mode operation of the tapered core. Figure 2 shows some feasible axial profiles of T-DCF.

As discussed above, the absorption of near-paraxial pump rays (α<NA/*T*) in T-DCF cladding is fairly insensitive to the longitudinal taper shape, while for rays with α>NA/*T* it is critically dependent on taper profile. The fraction of the latter rays with large values of α can be significant, as can be seen from Fig. 1. For example, 75% of the pump power propagates under this condition for a tapering ratio *T* = 3 and fill factor *F* = 0.68 (** NA_{launch}**/

**= 0.15/0.22), and for tapering ratios above**

*NA*_{clad}*T*~5-6 the fraction of these rays rises to 90%. The strategy for T-DCF profile optimization is to prevent the vignetting effect primarily in the large diameter section of the cladding, where the pump power density is high [4].

A “step-like” T-DCF longitudinal profile, as shown in Fig. 2 (blue curve 1), consisting of a regular cylindrical fiber and short tapered section has some attractive features. Particularly, a taper with this shape exhibits no vignetting in the regular non-tapered section of a double-clad fiber whose length and cross section are optimized primarily to ensure efficient pump absorption. Then, the residual pump light reaching the small-diameter tapered fiber section is strongly depleted, which allows for a short (few cm-long) length of this section, thus making the vignetting effect negligible [5–10]. The “step-like” tapered fiber, however, exhibits low slope efficiency as compared to a uniform fiber, 30% in [6–8], 45% in [9] and 54.1% in [10]. This is due to the strong spatial mode filtering which is required because the length of the single-mode section is much shorter than the length of the multimode section. A significant fraction of pump power consumed by higher-order non-oscillating modes in the multimode section of a “step-like” taper is filtered out in the single-mode section, resulting in losses and reduced laser efficiency. Therefore, although application of the “step-like” shape of T-DCF prevents pump loss due to vignetting, it can cause losses for signal radiation propagating in the core because of spatial mode filtering.

Fundamental mode operation is one of the most important characteristics related to the T-DCF format. Another problem associated with a “step-like” design with a short length of the tapered single-mode section is low filtering extinction, especially for the LP_{11} mode, which can reduce output beam quality to M^{2} ~1.3-1.7 [5,9,10].

Contrary to the “step-like” tapered fiber, the diameter of bowl-shaped “concave” T-DCF is gradually varied along the entire length, as shown in Fig. 2 (red curve 2), thus representing a certain compromise between output parameters and providing some advantages. An essential feature of axially non-uniform T-DCF is an efficient cladding mode mixing mechanism leading to enhanced pump absorption, which in turn leads to an improvement in the slope efficiency, as shown in [1]. Since the multimode section of bowl-shaped taper is short, the gain in the core is depleted mostly by fundamental and low-order modes, which allows for efficient and low-loss spatial mode filtering, occurring here over the long length of the taper.

Recently, we have demonstrated a 600 W single-mode fiber laser using long distributed T-DCF with a concave profile [4]. The slope efficiency and beam quality of this laser were 63% and M^{2} = 1.07, respectively, demonstrating a significant improvement over the short-length step-like tapered laser producing a power of 56 W with a slope efficiency of 54% reported earlier [10]. It should be noted, however, that the upper limit for slope efficiency of a ytterbium laser pumped at 915 nm is 84%, which suggests that the laser efficiency could be further improved by advancing the taper parameters.

Another option is a fiber with an exponential axial shape, exhibiting a strong increase of the local tapering angle at the large-diameter (pump launching) input of the fiber where pump power is high. Nearly exponential axial variation of diameter can be fabricated by direct pulling from a drawing tower [1–4,11]. Considerable vignetting of unabsorbed pump power is expected in T-DCFs with this geometry, which would impose losses, reduce the slope efficiency, and could lead to optical damage at the fiber segment with a large tapering angle.

Basically, the “step-like” and exponential bowl-shaped tapers are two extreme designs each suffering, respectively, from low fundamental mode selectivity and pump power losses. A compromising solution is a convex shape with a gradual increase of the tapering angle and a negative slope of diameter change with length, as shown in Fig. 2 (black curve 3). Specifically, linear or parabolic diameter distributions provide a reasonable trade-off between the “step-like” and the exponential profiles.

Ray optics analysis of the tapered fiber developed in [4] can be applied to determine the optimal structure of the taper. Efficient laser operation requires high absorbed pump power, ${P}_{abs}^{\text{'}}+{P}_{abs}^{\text{'}\text{'}}$ (Eq. (5)), meaning that ${P}_{unabs}$ and ${P}_{vgnt}$ are low. ${P}_{unabs}$depends on the length of the T-DCF and its paraxial absorption coefficient ** γ**, while ${P}_{vgnt}$ is influenced by the taper shape. A T-DCF with arbitrary shape can be represented as a sequence of N elementary short-length linear tapers, as shown in Fig. 3
. The propagation angle of a meridional ray at the output of the

*k*

^{th}elementary linear taper can be determined from the recurrent formula ([4], Eq. (7)):

**is the outer diameter of the T-DCF as a function of length,**

*D(z)***is the first derivative of**

*D(z)´***, and**

*D(z)***and**

*k***are the number and length of the k**

*Δz*^{th}elementary linear taper, respectively.

The meridional ray angle, after passing a T-DCF with length *L,* can be written as:

Vignetting of a ray begins when the angle of propagation reaches the numerical aperture value, θ (α_{0}) = NA. L*(α)* depends on the numerical aperture of the fiber NA, and on the shape of the taper, ** D(z)**. The effective length of fiber L

*(α)*when the vignetting starts, for given pump launch angle α, can be determined from the parametric equation

**and the paraxial absorption coefficient**

*NA*_{clad}**. For a certain acceptable level of vignetted power ${P}_{vgnt}^{pump}$ and a given angular intensity distribution**

*γ***of the pump source, the optimal longitudinal shape**

*I(α)***of the T-DCF can be defined by solving Eqs. (4) and (11). Alternatively, by setting the parameters of the pump beam and T-DCF, these equations provide an estimate of vignetted ${P}_{vgnt}^{pump}$ pump power. Obviously, the angular distribution of the pump source determines the optimal taper shape to achieve a low level of ${P}_{vgnt}^{pump}$.**

*D(z)*Assuming a parabolic shape of the taper, the diameter variation along the length ** L** can be written as

**is the parabolic shape factor,**

*b***is the average angle of the taper, and**

*b*_{0}= (D_{2}-D_{1})/L**,**

*D*_{1}**are the diameters at the wide and narrow fiber ends, respectively.**

*D*_{2}Figure 4
shows various T-DCF shapes for different parabolic factors. ** b<b_{0}**,

**, and**

*b>b*_{0}**correspond to concave, convex and linear shapes, respectively (Eq. (12). Assuming a Lorentzian angular distribution of the pump intensity, the dependence of vignetted power versus**

*b = b*_{0}**can then be derived from Eqs. (4) and (11). This dependence, calculated for different values of the paraxial absorption coefficient**

*b***is shown in Fig. 5 .**

*γ,*The concave taper shape exhibits larger vignetting losses for the pump, however, it provides better filtering of the fundamental mode and, therefore, improved output beam quality, as discussed above. In effect, an acceptable level of vignetting losses determines the optimal axial profile of the taper. For instance, Fig. 5 (inset) shows the T-DCF shape corresponding to a 0.5%-level of pump power losses caused by vignetting.

To sum up, the optimal longitudinal shape of T-DCF exhibiting the required level of pump power loss ${P}_{vgnt}^{pump}$ is defined by the angular distribution of the pump source intensity, the paraxial absorption coefficient ** γ,** and the tapering ratio

**.**

*T*#### 2.3. Shape of the cladding, core-cladding ratio and dopant concentration profile

The shape of the cladding cross sections, the core-cladding diameter ratio, and the concentration of the dopants determine the paraxial absorption coefficient ** γ**, which is an essential characteristic of T-DCF. The dependence of the vignetted pump power on the absorption coefficient

**derived from Eqs. (4) and (11), is shown in Fig. 6 for linear (b/b**

*γ*,_{0}= 1), concave (b/b

_{0}= 2) and convex (b/b

_{0}= 0) T-DCFs. As expected, the vignetted pump power ${P}_{vgnt}^{pump}$ decreases rapidly with the increase of absorption coefficient

**. For large values of absorption, γ>2 dB/m, the shape of the taper has little effect on the vignetted power, as seen from Fig. 6.**

*γ*## 3. Experimental results

Three tapers with convex, concave and near-linear shapes with different tapering ratios have been pulled from the same preform. The all-silica fiber preform was fabricated with using plasma chemical deposition. First Yb-doped alumosilicate core layer was deposited on the inner surface of a silica substrate tube. After collapsing, redrawing and jacketing the cylindrical preform was mechanically shaped into an octagon; the perform surface has been polished then by flame. The secondary outer cladding of the octagon was formed by jacketing it with another silica tube comprising a fluorine-doped silica layer deposited on its inner surface. The entire structure was collapsed again resulted in a monolithic perform.

Both Yb-doped and F-doped silica layers were formed by the SPCVD process as described in [13] and references therein. The refractive index profile of the octagon preform is shown in the inset of Fig. 7 . The essential advantage SPCVD technology as compared with conventional MCVD solution-doping technique is strong suppression of photodarkening confirmed with high-power fiber lasers. Fiber made by SPCVD technique did not reveal any measurable changes during long-term tests contrary to the lasers fabricated by conventional method which exhibit frequently decay of performance.

Tapered fiber has been formed at drawing tower during the pooling process. The axial shape of the tapered fiber depends on several factors: geometry of preform; the volume of melted part of preform in furnace determined by high temperature furnace design; drawing speed and speed of preform feed into a hot zone. Proper control of all above mentioned parameters allows to achieve a good reproducibility of longitudinal shapes of T-DCF for diameter range of 150-1000 µm and typical length 3-20 m. The accuracy of tapers shape is estimated to be better than 10% which is sufficient for control of laser parameters. The parameters of manufactured tapers are listed in Table 1 . The shapes of tapers, expressed in the form of dependences of the outer diameter on the fiber length, are shown in Fig. 7.

The fibers described in Table 1 were then incorporated into fiber laser configurations, as shown schematically in Fig. 8 . This experimental set up is similar to that report in [2,4]. The pump radiation from a 915 nm high-power source was launched via fiber cable with a core diameter of 600 µm and a NA = 0.22 into T-DCF using a beam shaping optical unit and dichroic filter. Beam shaping optics comprising lenses and a diaphragm forms the beam to a diameter of 880 µm and divergence corresponding to NA = 0.15. The laser cavity is terminated by a dielectric mirror located behind the wide end of the T-DCF, and 4% Fresnel reflection at the output small-diameter facet of the taper. The wide end of T-DCF was angle cleaved.

The dependence of the output power on the launched pump power measured for each T-DCF reveals the slope efficiency and maximum output power shown in Fig. 9 .

The contrast, defined as the relative fraction of output power emitted from the core, was determined by examining the angular distribution of output radiation in the far field. Pump power losses caused by vignetting ${P}_{vgnt}^{pump}$ have been calculated for each T-DCF, as described above, using experimental parameters - absorption coefficient, taper shape and angular intensity distribution of the pump source. The results are summarized in Table 2 .

## 4. Discussion

Next, the characteristics of lasers using different T-DCFs have been examined.

#### T-DCF#1

T-DCF#1 has a concave parabolic shape with a pump absorption coefficient of 0.8 dB/m at 915 nm. The initial 4 m section of T-DCF has a nearly flat shape, which is followed by a concave section with rapid diameter decrease along the length. The tapering ratio and the parabolic factor of the concave section are 6 and 2.05, respectively. The measured slope efficiency of the laser using this taper was only 60.2%. The poor slope efficiency is caused by high pump vignetting losses of 10.19% due to both non-optimal T-DCF shape and a relatively small value of pump absorption coefficient. A similar concave T-DCF used previously in [2] has demonstrated a better slope efficiency of 71.4% due to a 3-4 times higher pump absorption coefficient, which made the laser performance tolerant to taper shape, in agreement with results presented in Fig. 6. The measured beam contrast for this T-DCF of 64.3% is determined by the high tapering ratio of *T = 6* and the 4 m long large core diameter section. The relatively low value of contrast indicates a significant fraction of higher-order modes leaking from the core into the cladding.

#### T-DCF #2

T-DCF#2 is a convex shape taper with parabolic factor b/b_{0} = 0.56, absorption coefficient γ = 1.2 dB/m and tapering ratio of 5.6, as shown in Fig. 7 (red curve). In accordance with the above analysis, it demonstrated a high slope efficiency of 81.9%, which is close to the theoretical limit of 85%. Record slope efficiency is the result of a low level of vignetting of 0.45%, and only 0.08% unabsorbed pump power. The parabolic shape with b/b_{0} = 0.56 is actually close to the “step-like” shape shown in Fig. 2 (blue curve 1). The high-order modes efficiently deplete the population inversion and are amplified over significant lengths of the taper. The laser exhibited, however, a poor output beam contrast of only 35%. The taper section with a large core diameter of 2a = 110 µm guides a large number of modes, N~a^{2}, (Eq. (7) thus reducing the relative fraction of fundamental and lower-order modes. In the narrow part of the T-DCF higher-order modes are filtered out from the core and afterwards propagate in the cladding, degrading the output beam contrast.

#### T-DCF #3

T-DCF#3 is a convex taper with parabolic factor of ** b/b_{0}** = 0.753, i.e. taper profile is close to the linear shape, paraxial absorption coefficient of 0.61 dB/m and tapering ratio of 2.9. Vignetted pump power and unabsorbed pump power losses are 2.22% and 3.13%, respectively, resulting in a total loss of pump power of 5.35% (see Table 2). The laser with this tapered fiber demonstrates a combination of good slope efficiency, 77.9%, and high output beam contrast, 87%.

An analysis of the simulations and experimental results allows the following conclusions to be drawn:

- • The slope efficiency of a laser with T-DCF is determined by the axial profile of the taper diameter, for a given angular distribution of the pump beam. The slope efficiency of the tapered laser has been studied experimentally using a pumping source with NA = 0.15 for various longitudinal shapes of taper. We show that an optimized taper configuration could approach the theoretical limit of the efficiency.
- • The slope efficiency is strongly dependent on the pump absorption coefficient. For absorption above 2 dB/m this effect, however, is less pronounced.
- • The slope efficiency suffers from a pump-vignetting effect dependent on the tapering ratio. When the tapering ratio is low, the fraction of pump power leakage decreases and, consequently, slope efficiency increases.
- • The contrast of the output beam, defined as the relative fraction of power propagating in the core, depends critically on the tapering ratio and shape of the taper. In the T-DCF with a large tapering ratio and parabolic shape factor, the population inversion is strongly depleted by high-order modes, thus reducing the beam contrast. It was shown that the beam contrast decreases with the tapering ratio
*T*. Some discrepancy between calculations using Eq. (7) and experimental data summarized in Table 2 is due to the simplified assumption of uniform excitation of all guided modes. - • T-DCF with properly chosen parameters demonstrates superior brightness enhancement characteristics compared with regular double-clad fiber. The measured normalized brightness enhancement factors presented in Table 2 show an improvement up to a factor of 7.6. It should be noted that in T-DCF-based amplifiers there are no signal losses in the core due to mode filtering, and consequently no limitations on the tapering ratio. The brightness gain factor in tapered amplifiers can be significantly higher.

In summary, we demonstrate that T-DCF with appropriate parameters values allows considerable brightness improvements to be achieved, compared to regular non-tapered double-clad fiber. High brightness, efficiency and beam contrast from an active T-DCF are accomplished by means of high paraxial absorption, a modest tapering rate and a nearly linear (or slightly parabolic) shape.

## 5. Conclusion

In this paper, an optimization strategy for active tapered double-clad fibers has been developed based on ray optics formalism. In particular, the effect of the tapering ratio, the axial shape of the profile, and the absorption on the brightness enhancement and slope efficiency has been examined in detail. We have shown that T-DCF with an optimized design exhibits a slope efficiency close to the theoretical limit. The brightness characteristics of an active T-DCF, superior to regular non-tapered active DCF, have been demonstrated experimentally. Active T-DCF can be pumped with powerful laser diode bars, whose application is usually limited due to their low brightness. Experimentally, we have demonstrated an efficient ytterbium laser based on T-DCF as the active medium, producing an output power of 750 W with a slope efficiency of 81.9% and near diffraction-limited beam quality (M^{2} = 1.7).

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