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We propose a modified gain-guided index-antiguided (GGIAG) fiber structure for large mode area laser amplifiers, in which a thin dielectric layer is placed between the low-index core and the high-index cladding. The introduced dielectric layer functions as a Fabry-Perot etalon. By letting the resonant wavelength of the Fabry-Perot layer coincide with the signal wavelength, the signal is gain-guided in the fiber core. Moreover, the pump is confined in the low-index core owing to the antiresonant reflection originated from the Fabry-Perot layer. Numerical results indicate that the leakage loss of the pump can be minified over two orders of magnitude in the proposed structure, and thus the end-pumping efficiency could be enhanced significantly.
© 2015 Optical Society of America
1. Introduction
Output powers of fiber lasers and amplifiers have been increasing in recent years [1,2]. To mitigate the nonlinear effect and catastrophic optical damage resulting from the high power, the most effective way is to increase the mode area within the fiber. However, the number of transverse modes often grows with core diameter, and the beam quality will deteriorate consequently. Hence, it is important for high power fiber lasers and amplifiers to have a fiber structure which achieves large mode area (LMA) and single-transverse-mode operation simultaneously. Several LMA fibers were reported to meet both requirements, which include: the photonic crystal fibers [3,4] the leakage channel fibers [5,6], and the chirally coupled core fibers [7,8]. Although the effective mode areas of these reported fibers are larger than that of the conventional single-mode fiber, their core diameter are all less than 100 μm. Recently, the large pitch fiber was reported having a core diameter of 135 μm [9].
The gain-guided index-antiguided (GGIAG) fiber [10,11] is another LMA fiber proposed for high power fiber lasers. From the waveguide theory, if the refractive index of the fiber core is lower than that of the cladding, leakage loss will occur and such a structure is said to be index-antiguided (IAG). It has been shown that, the leaky modes of an IAG fiber can be converted to guided ones when there is a sufficient gain present in the core [10]. This phenomenon is referred to as gain-guided (GG) and the structure is referred to as the GGIAG fiber. Single-mode laser oscillation has been demonstrated in the GGIAG fiber with a core diameter up to 400 μm [12].
Unfortunately, efficiencies of end pumping (or core pumping) for GGIAG fiber lasers and amplifiers reported so far are poor. One of the reasons is attributed to the index antiguiding of the pump [13], because the GGIAG structure for the signal is actually an IAG structure for the pump. Therefore, the pump power launched to the fiber core will leak to the cladding rapidly when propagating along the fiber. To overcome the problem, several modified GGIAG fiber structures were proposed, including the photonic bandgap fiber [14] and the dispersion-engineered fiber [15]. However, these proposed structures are complex and thus difficult to fabricate.
It has been recognized that antiresonant reflection provides a means to confine light in the waveguide core having a lower refractive index than that of the cladding. Such a technique has been applied to the antiresonant reflecting optical waveguides (ARROWs) [16,17], the photonic crystal fibers [18–20], and the pipe waveguides [21,22]. In this work, by employing the similar technique, we propose an improved GGIAG fiber to enhance the end-pumping efficiency. The proposed fiber is simple in structure and it merely introduces an additional dielectric layer between the core and the cladding of the GGIAG fiber. It will be shown later that, with the introduced dielectric layer, the pump can be successfully confined in the low-index core by antiresonant reflection, and the leakage loss of the pump can be reduced significantly. At the same time, by letting the resonant wavelength of the dielectric layer coincide with the signal wavelength, the gain-guiding operation of the signal remains unaffected.
2. Fiber structure
Figure 1 plots the cross-section and index profile of the proposed fiber structure. Refractive index of the core nco is almost equal to but slightly lower than that of the cladding ncl. If the dielectric layer between the core and the cladding is omitted, the structure is an IAG fiber and it becomes a GGIAG fiber when gain is present in the core. The function of the introduced thin dielectric layer is to behave like a Fabry-Perot etalon, therefore this layer is referred to as the FP layer, and the proposed structure is referred to as the FP-GGIAG fiber. Refractive index and thickness of the FP layer are denoted as nFP and tFP, respectively. In addition, wavelengths of the signal and the pump are denoted as λsignal and λpump.
The purposes of the FP layer are twofold: to maintain the gain-guiding operation for the signal and to minimize the leakage loss of the pump. The first purpose can be achieved by requiring the resonant wavelength of the FP layer to coincide with the signal wavelength. It iswell known that, at the resonant wavelength, the transmission of a Fabry-Perot etalon is unity, i.e., the etalon is transparent [23]. Since the FP layer functions as a Fabry-Perot etalon, when the resonant wavelength of the FP layer and the signal wavelength coincide, the FP layer seems disappear for the signal because of the transparency. In other words, the signal still sees an IAG fiber (or a GGIAG fiber if there is a gain in the core) even though the FP layer is introduced. Hence, it is expected that the gain-guiding operation of the signal is ensured for the FP-GGIAG fiber. The resonant wavelength λr of the FP layer can be expressed as [22]
where m is an integer. If the refractive index of the FP layer nFP is known, the required thickness tFP leading to λsignal = λr can be derived from Eq. (1) and it is given byTo reduce the leakage loss of the pump, the antiresonant reflection resulting from the FP layer is utilized. According to the spectral behavior of a Fabry-Perot etalon, the transmission of the etalon is very low when the operation wavelength is far from the resonant wavelengths [23]. Therefore, strong reflection occurs under the out-of-resonant (or antiresonant) condition. This antiresonant reflection has been employed in various waveguides to achieve waveguiding in a low-index core [16–22]. Here, the same technique is utilized to minimize the leakage loss of the pump. By letting the pump stay in the antiresonant condition of the FP layer, instead of leaking its power to the cladding due to the aforementioned IAG nature, the pump launched to the core of the FP-GGIAG fiber can then be well confined in the low-index core region by the antiresonant reflection originated from the FP layer. However, note that not every tFP value obtained from Eq. (2) allows the pump to meet the antiresonant condition. We will describe how to determine the desired thickness of the FP layer later.
3. Numerical results
3.1 General characteristics of the FP-GGIAG fiber
In the following, the proposed FP-GGIAG fiber is investigated numerically. A full-vectorial finite-difference mode solver especially proposed for waveguides with cylindrical symmetry [24] is used to perform numerical simulations. The perfectly matched layer (PML) absorbing boundary condition based on the complex coordinate stretching [25] is also incorporated. Essentially, the mode solver is to solve an eigenvalue equation in the matrix form:
where Er and Eφ are transverse (radial and azimuthal) components of the electric field and β is the complex propagation constant. Once Eq. (3) is solved, modal loss can be obtained from the imaginary part of the propagation constant as −2Im(β).We assume the wavelengths of the signal and the pump are λsignal = 1.055 μm (the emission wavelength of Nd-doped fibers) and λpump = 0.803 μm (a typical wavelength of laser diodes), respectively. Core diameter of the FP-GGIAG fiber is D = 100 μm, and refractive indices of the core and the cladding are nco = 1.56 and ncl = 1.561, respectively. These parameter values are taken from literature [13,14], and these values correspond to the dimensionless index parameter ΔN = –276 for the signal [11].
We first consider the case where there is no gain in the low-index core of the FP-GGIAG fiber. In addition, the refractive index of the FP layer is assumed to be nFP = 2.2. With this assumption, the required FP layer thickness for the signal wavelength and the resonant wavelength to be coincident can be obtained from Eq. (2), and tFP = 0.34, 0.68, and 1.02 μm for m = 1, 2, and 3, respectively.
Figure 2 plots the loss spectra of the fundamental mode (the LP01 mode) of the FP-GGIAG fiber with the three thicknesses of the FP layer. Result for the IAG case, i.e., the FP layer is absent (tFP = 0), is also shown for comparison. Obviously, like a Fabry-Perot etalon, the loss spectrum of the FP-GGIAG fiber also exhibits a periodic feature. The wavelengths at which local maximum loss occurs correspond to the resonant wavelengths of the FP layer. Because the FP layer is transparent at the resonant wavelengths, the (local maximum) losses of the FP-GGIAG fiber occurring at the resonant wavelengths and the losses of the IAG fiber in which the FP layer does not exist are coincident. Moreover, at the wavelength far from the resonant ones, the loss of the FP-GGIAG fiber decreases owing to the antiresonant reflection provided by the FP layer.
Fig. 2 Loss spectra of the LP01 mode of the FP-GGIAG fiber for different thicknesses of the FP layer. IAG represents the case where the FP layer is absent. The signal wavelength is 1.055 μm and the pump wavelength is 0.803 μm.
In Fig. 2, it is clear for the signal that the losses with the three different tFP values are all the same as that for the IAG case. They are all 0.296 cm–1. This provides a strong evidence that the introduced FP layer indeed will not affect the signal behavior if the resonant wavelength and the signal wavelength agree. On the other hand, for the pump, losses are much different. Without the FP layer, i.e., the IAG case, the pump loss is 0.172 cm–1; when the layer is introduced, it becomes 8.08 × 10−4, 6.58 × 10−4, and 1.43 × 10−2 cm–1 for tFP = 0.34, 0.68, and 1.02 μm, respectively. From Fig. 2, it is observed that, the pump wavelength is close to the resonant wavelength when tFP = 1.02 μm, whereas it is out of resonant when tFP = 0.34 and 0.68 μm. That's why the losses for the three thickness are so different. If we choose tFP = 0.68 μm for the FP layer, the pump loss drops significantly from 0.172 cm–1 for the IAG case to 6.58 × 10−4 cm–1 for the proposed FP-GGIAG fiber. That is to say, with the introduced FP layer, the leakage loss of the pump is improved over two orders of magnitude.
Figure 3 plots normalized field distributions of the signal and the pump for the IAG fiber and the FP-GGIAG fiber with tFP = 0.68 μm. In Fig. 3(a), except that there is an oscillation inside the FP layer, it is clear the field profiles of the signal for the IAG and FP-GGIAG fibers are the same in the core and the cladding. (The oscillation regime is zoomed in the inset.) This result again confirms that, even though the FP layer is added, the behavior of the signal remains unaffected. On the other hand, from Fig. 3(b), obviously, behavior of the pump is changed. The field in the cladding is strongly suppressed in the FP-GGIAG fiber, indicating that the pump is well confined in the low-index core region.
Fig. 3 Radial field distributions for the IAG fiber and the FP-GGIAG fiber (with tFP = 0.68 μm). (a) Signal. (b) Pump. Inset in (a) shows the field distribution inside the FP layer for the signal.
3.2 Influence of the FP layer on pump loss
Figure 4 plots the losses of the LP01 mode of the FP-GGIAG fiber with different tFP values, which are all obtained according to Eq. (2). Results with tFP = 0 correspond to the IAG case, and as stated previously, the IAG losses are 0.296 and 0.172 cm–1 for the signal and the pump, respectively. For the signal, it is clear that the loss is independent of tFP. The signal loss maintains the IAG level for all tFP values due to the transparency of the FP layer at these thicknesses. However, for the pump, the loss fluctuates significantly with tFP. In Fig. 4, the lowest pump loss is 5.60 × 10−4 cm–1 occurring at tFP = 2.72 μm, and the highest is 1.43 × 10−2 cm–1 at tFP = 1.02 μm (the loss at tFP = 0 is not counted).
Fig. 4 Losses of the LP01 mode for the FP-GGIAG fiber with different tFP values. Results with tFP = 0 correspond to the IAG case.
The loss behavior of the pump can be understood by examining the transverse phase within the FP layer. Because the layer in the FP-GGIAG fiber behaves like a Fabry-Perot etalon, the resonance occurs when the phase ϕ in the FP layer is an integer multiple of π, i.e., ϕ = mπ, where m is an integer. Under the resonant condition, no reflection is provided by the FP layer and this condition will yield a local maximum loss. Nevertheless, as long as the phase leaves the resonant condition, reflection happens and the resultant propagation loss will decrease accordingly. The strongest reflection appears when ϕ = (m – 0.5)π, and which is the condition for the local minimum loss to occur.
The phase within the FP layer of the FP-GGIAG fiber can be expressed as [26]
where λ is the wavelength. Remember that the signal is resonant with the tFP values obtained from Eq. (2), hence Eq. (4) can be validated by checking the phase of the signal ϕsignal. Replacing λ with λsignal and substituting tFP according to Eq. (2), one has ϕsignal = mπ, which is exactly the resonant condition for the signal. (In Fig. 3(a), the field distribution of the signal for the FP-GGIAG fiber shown in the inset displays a phase of 2π in the FP layer.)According to Eq. (4), the phase of the pump ϕpump in the FP layer is given by
The phases ϕpump are shown in Fig. 5(a) with tFP values derived from Eq. (2). Note that with these tFP values, neither ϕpump = mπ nor ϕpump = (m – 0.5)π is precisely satisfied. Here we define the phase range within mπ ± 0.1π as the near-resonant region. So the pump is considered to be near resonant if its phase meets the following condition:Further, we also define the antiresonant region as the phase range within (m – 0.5)π ± 0.1π. Thus the pump is antiresonant ifThe phases of the pump shown in Fig. 5(a) are plotted again in Fig. 5(b). Nevertheless, here, the integer part of ϕpump /π is neglected and only the decimal part is extracted. As a result, in Fig. 5(b), the range below 0.1 and above 0.9 corresponds to the near-resonant region, and the one between 0.4 and 0.6 is the antiresonant region. For the convenience of discussion, results in the near-resonant region are marked with the red color, while in the antiresonant region aremarked with the blue color. Figure 5(c) re-plots the loss of the pump as a function of tFP. These results have been shown in Fig. 4, but now, they are marked with color respectively according to the phase condition. For example, when tFP = 1.02 and 4.42 μm, the pump phases ϕpump locate in the near-resonant region (refer to Fig. 5(b)), so the corresponding losses shown in Fig. 5(c) are marked with the red color. In addition, when tFP = 1.70, 2.72, and 3.74 μm, the phases of the pump are in the antiresonant region, thus the losses are marked with the blue color. Clearly, in Fig. 5(c), when the pump is under the near-resonant condition, its losses (the red ones) are the highest. Hence, these tFP values which make the pump near resonant should be avoided. On the contrary, when the pump is under the antiresonant condition, the losses (the blue ones) are the lowest, and they are 5.86 × 10−4, 5.60 × 10−4, and 5.73 × 10−4 cm–1 for tFP = 1.70, 2.72, and 3.74 μm, respectively. Because the pump phase with tFP = 2.72 μm is more close to the midpoint of the antiresonant region (refer to Fig. 5(b)) at which ϕpump meets (m – 0.5)π exactly, the corresponding pump loss is lower than the other two, but the difference is very small and thus is insignificant. Therefore, the performances of these thicknesses are considered to be equal since the losses of the pump are reduced to the same level. These thicknesses, i.e., the thicknesses which make the pump antiresonant, are the optimal thicknesses for the FP layer.Fig. 5 (a) Phases of the pump ϕpump in the FP layer. (b) Decimal part of ϕpump /π. The range below 0.1 and above 0.9 is the near-resonant region, and the range between 0.4 and 0.6 is the antiresonant region. (c) Losses of the pump for the FP-GGIAG fiber. The dashed line represents the pump loss for the IAG case (tFP = 0).
In brief, the FP layer of the FP-GGIAG fiber serves two purposes: keeping the signal resonant to maintain the GGIAG operation for the signal and making the pump antiresonant to minimize the leakage loss of the pump. The optimal thickness of the FP layer can be determined by the following two steps. First, acquiring the tFP values which make the signal resonant from Eq. (2), and then, screening these tFP values according to Eq. (7) to further make the pump antiresonant.
We also examine the influence of the refractive index of the FP layer nFP on the loss of the pump. Figure 6 plots the pump loss as a function of nFP ranging from 1.57 to 2.8. For each nFP value, the loss is calculated with the corresponding optimal thickness tFP as described above. The pump loss for the IAG case is also shown as the dashed line in Fig. 6 for reference. Compared with the IAG case, it is found that introducing the FP layer reduces the leakage loss of the pump and the loss reduction is enhanced as nFP increases. However, when nFP exceeds some threshold value, say, nFP = 2.4 in this case, the pump loss stops decreasing and becomes saturated. From the figure, it is clear high index materials are desired for the FP layer. Materials such as chalcogenide glass and tantalum oxide have refractive indices ranging from 2 to 3. For example, the index of chalcogenide glass can be 2.25 [27] or 2.8 [28] depending on the ingredient. Some techniques to realize the high-index-contrast fiber structures are described in [27–30]. Nevertheless, in Fig. 6, it is also noted that when the nFP value is as lowas 1.57, which is only slightly higher than the core index nco = 1.56, the pump loss drops from 0.172 cm–1 for the IAG case to 0.0182 cm–1 for the FP-GGIAG fiber. The leakage loss of the pump is minified nearly by 10 even though such a low refractive index nFP is adopted.
Fig. 6 Loss of the pump for the FP-GGIAG fiber as a function of refractive index of the FP layer. The dashed line represents the pump loss for the IAG case.
3.3 Gain-guiding operation of the signal
For the GGIAG operation, the signal is gain-guided in the low-index fiber core. Hence we further investigate the behavior of the signal by assuming a uniform gain present in the core of the FP-GGIAG fiber. Consequently, the corresponding core index for simulation now contains an imaginary part, and it becomes nco = 1.56 + j(g/2k0), where g is the material gain coefficient and k0 is the free-space wavenumber. Modal gain of the signal is obtained from the imaginary part of the calculated propagation constant as 2Im(β). Note that if the value of the modal gain is negative, it represents a loss for the signal. Figure 7(a) plots the calculated modal gain coefficient of the LP01 mode for the signal as a function of material gain coefficient. Parameters for the FP layer are nFP = 2.2 and tFP = 2.72 μm. If the FP layer is absent, the structure reduces to a standard GGIAG fiber. Results for the GGIAG case are also shown in Fig. 7(a). It is observed that, no matter whether there is an FP layer, the modal gain coefficients agree with each other. Such an agreement confirms that the gain-guiding operation of the signal is not affected by the introduction of the FP layer. Figure 7(b) plots the field distributions of the signal with a material gain coefficient of 0.5 cm–1 for both GGIAG and FP-GGIAG fibers. Clearly, except the oscillation appearing in the FP layer, field distributions are the same for both fibers elsewhere.
Fig. 7 (a) Modal gain coefficients and (b) radial field distributions of the LP01 mode of the signal for the GGIAG fiber (without the FP layer) and the FP-GGIAG fiber. Inset: the field distribution inside the FP layer.
One advantage of the GGIAG fiber structure is its robustness of the single-transverse-mode operation, in that the threshold gain for onset of the LP11 mode is about 2.54 times larger than that for the LP01 mode [11]. Here we demonstrate that the robustness is still maintained in the FP-GGIAG fiber. Figure 8 plots the modal losses of the signal for the LP01 and LP11 modes. The losses are calculated with tFP values satisfying Eq. (2), i.e., the FP layer thicknesses with which the signal is resonant. Without the FP layer (tFP = 0), the losses of LP11 and LP01 modes are 0.749 and 0.296 cm–1, respectively, and the loss ratio of LP11 mode to LP01 mode is 2.53. When the FP layer is introduced, the signal losses remain unchanged for either the LP01 or the LP11 modes. Thus, no matter what thickness the FP layer has, the loss ratio of the two modes is always 2.53, indicating that the FP-GGIAG fiber also ensures the robust single-transverse-mode operation.
Fig. 8 Losses of the LP01 and LP11 modes of the signal for the FP-GGIAG fiber. Results with tFP = 0 correspond to the GGIAG case.
4. Conclusion
In conclusion, we have proposed a modified GGIAG fiber structure, the FP-GGIAG fiber, by introducing a thin dielectric layer between the core and the cladding of a GGIAG fiber. This layer behaves like a Fabry-Perot etalon. With proper determination of the thickness of the FP layer, the signal behavior is the same as in the GGIAG fiber whereas the pump is well confined in the low-index core by antiresonant reflection. It is shown that the pump loss can be minified over two orders of magnitude, and thus the end-pumping efficiency could be enhanced significantly. The robustness of single-transverse-mode operation is also demonstrated. The FP-GGIAG fiber is simple in structure and efficient for end pumping, which makes it a promising candidate for high power fiber laser amplifiers and oscillators.
Acknowledgment
This work was supported in part by the Ministry of Science and Technology of the Republic of China under grant MOST102-2221-E-224-069-MY3.
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