Theoretical modeling and experiments on a DBR waveguide laser fabricated by the femtosecond laser direct-write technique

We present a model for a Yb-doped distributed Bragg reflector (DBR) waveguide laser fabricated in phosphate glass using the femtosecond laser direct-write technique. The model gives emphasis to transverse integrals to investigate the energy distribution in a homogenously doped glass, which is an important feature of femtosecond laser inscribed waveguide lasers (WGLs). The model was validated with experiments comparing a DBR WGL and a fiber laser, and then used to study the influence of distributed rare earth dopants on the performance of such lasers. Approximately 15% of the pump power was absorbed by the doped “cladding” in the femtosecond laser inscribed Yb doped WGL case with the length of 9.8 mm. Finally, we used the model to determine the parameters that optimize the laser output such as the waveguide length, output coupler reflectivity and refractive index contrast. ©2013 Optical Society of America OCIS codes: (140.0140) Lasers and laser optics; (130.2755) Glass waveguides; (140.3430) Laser theory; (140.3615) Lasers, ytterbium; (140.7090) Ultrafast lasers. References and links 1. K. M. Davis, K. Miura, N. Sugimoto, and K. Hirao, “Writing waveguides in glass with a femtosecond laser,” Opt. Lett. 21(21), 1729–1731 (1996), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-21-21-1729. 2. G. Della Valle, S. Taccheo, R. Osellame, A. Festa, G. Cerullo, and P. Laporta, “1.5 mum single longitudinal mode waveguide laser fabricated by femtosecond laser writing,” Opt. Express 15(6), 3190–3194 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-6-3190. 3. R. Mary, S. J. Beecher, G. Brown, R. R. Thomson, D. Jaque, S. Ohara, and A. K. 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Introduction
The advent of the femtosecond laser direct-write technique offers promising opportunities for realizing all-optical devices [1].It enables simple creation of compact 3D structures in transparent materials with greater flexibility than traditional fabrication methods involving ion exchange or photolithography.This technique has already been successfully utilized to create active devices such as amplifiers and lasers inside various host materials including many glasses and crystals.For example, both femtosecond laser written waveguide lasers with external optical feedback [2,3] and monolithic distributed feedback (DFB) WGL [4] have been demonstrated.
As more femtosecond laser direct-written active devices become experimentally realized, theoretical studies for these devices become increasingly desirable in order to predict device behavior, and optimize their design.In the literature, several models of femtosecond laser written waveguides in rare earth doped glass have been reported.Notably, a model was developed by Valles et al. to calculate the transmission loss and coupling loss of the ultrafast laser written waveguides in Er/Yb codoped glass [5].
The laser model that has been widely investigated to date is the same as that applied to conventional fiber lasers.In that specific case, because the rare earth dopants are only distributed throughout the core, an effective overlap coefficient can be adopted, which defines the fraction of the power that is overlapped with the doped core [6,7].However, these models are not suitable for femtosecond laser written waveguide lasers in which the dopants are distributed not just in the waveguide core but throughout the glass, including the surrounding unmodified glass, referred to as a doped "cladding" below.In contrast, more sophisticated models are required to analyze the behavior of WGLs fabricated using ion exchange or indiffusion technique [8,9].In particular, due to the existence of dopants in both the core and the "cladding", rate equation modeling that spatially integrates the guided mode as it propagates along the waveguide need to be employed.In these reports the model was used to predict the laser properties such as the threshold, the slope efficiency and the output power.However, the influence of the doped cladding on the WGL performances has not been studied quantitatively.The study about the doped cladding effects can help us to determine whether the doped cladding has a serious detrimental influence on laser performance.
In this paper, we present a detailed characterization of a DBR waveguide laser fabricated by a femtosecond laser and a numerical model to analyze the laser performance.This model is based on rate equations, combined with the transverse integration of the propagating mode field and was validated with an experimental comparison of a DBR fiber laser and WGL.It enables us to predict the laser performance and optimize the laser design by changing the waveguide length or increasing the refractive index contrast.Moreover, it enables us to get a better understanding of the influence of the doped cladding on the laser performance.The doped cladding region of the DBR waveguide laser absorbed approximately 15% of the pump power.

Waveguide fabrication
Waveguides were written in a "QX" phosphate glass (Kigre Inc.) doped with 9 wt% Yb ( 26 8.2 10 × ions/m 3 ) using the femtosecond laser direct-write technique.The regeneratively amplified Ti:sapphire laser had a 1 kHz repetition rate, 120 femtosecond pulse length, and operated at 800 nm.The laser beam was focused at a depth of 170 µm below the surface of the glass sample by a 20 × (NA 0.46) microscope objective.The pulse energy was 1 µJ corresponding to intensity of the order of 10 14 W/cm 2 .The beam was circularly polarized and focused into the glass through a slit with width of 500 µm in order to fabricate circular cross sectional waveguides [10].A detailed description of the writing technique can be found elsewhere [11].
The glass sample was translated using a computer-controlled XYZ stage at a speed of 25 µm/s.Multiple-writing-passes were used for waveguide fabrication.After fabrication, the glass sample was polished with a Logitech PM4 lapping and polishing machine at each end to expose the waveguide.The post-polishing length of the glass sample was 9.8 mm.
The morphology of the waveguides was first checked with a differential interference contrast (DIC) microscope (Olympus IX81).Figure 1(a) shows a top view of a waveguide written with 1 µJ pulse energy and four writing passes.Figure 1(b) shows that the crosssectional profile of the same waveguide is symmetric.The physical diameter of the waveguides is ~7 μm.The propagation loss of the waveguide was measured to be 0.43 dB/cm.The waveguides were then characterized in terms of guided mode profile using a CCD camera imaging system.The measured mode field diameter (MFD) was ~14.2 µm at 976 nm.The refractive index distribution has a Gaussian profile which was measured by a refractive index profilometer (Rinck Elektronik).However, the absolute refractive index contrast obtained from the instrument is incorrect due to the color-center-related absorption of the profilometer's probe beam at 635 nm.In order to determine a more accurate refractive index contrast, we matched the experimentally measured MFD to a numerically simulated MFD calculated using the beam propagation software BeamPROP (RSoft) using the measured profile but a variable refractive index contrast.Thus, the induced refractive index modification was estimated to be 1.6 × 10 −3 .

Experimental setup
A schematic of the DBR waveguide laser setup is shown in Fig. 2. Two fiber Bragg gratings centered around 1032 nm were used as the high reflector (HR) and output coupler (OC) mirrors and offered 99% and 45% reflectivity respectively.Pump light at 976 nm was launched into the Yb-doped waveguide through the HR grating.The signal output was monitored through WDMs with a power meter and optical spectrum analyzer with a resolution of 0.01 nm.Short sections of graded index optical fiber (GIF625) were spliced to the fiber tips to better match the MFD of the waveguide to that of the coupling fiber.The coupling loss between the fiber and waveguide was calculated by integrating the overlap between experimental intensity profiles of the waveguide and the fiber mode [12].The coupling loss between the waveguide and the fiber is approximately 6%.A DBR fiber laser was also assembled in order to validate the modeling results.The experimental setup of the DBR fiber laser is the same as that shown in Fig. 2 except that the glass sample was replaced by a spliced Yb doped fiber (Nufern Inc.) with the MFD of ~6.5 µm at 1060 nm.The length of the fiber was 11.2 cm.The Yb concentration of the fiber is 26 1.48 10 × ions/m 3 .

Rate equations
The DBR waveguide laser model is shown schematically in Fig. 3, consisting of a waveguide in a homogeneously doped glass and two fiber Bragg gratings on both sides serving as narrow bandwidth wavelength selective mirrors.The theoretical model considers a homogeneously broadened two-level system.All other sub-levels are assumed to be unpopulated because Yb ions emission from these levels takes place by a fast non-radiative transition.The rate equations describe the effects of absorption, stimulated emission and propagation losses of the laser system.Instead of using a confinement factor, widely applied in fiber modeling, a specific emphasis is made herein on the spatial integration of the cross sectional region that carries the guided mode.The integral calculation describes the transverse energy distribution of the pump and signal along the waveguide and accounts for the interaction between the pump and signal power within the doped cladding.At steady state the rate equations are given by [13,14] # where n t is the Yb ion density which is spatially constant as the dopants are uniformly distributed; n 2 is the upper lasing level population density with fluorescence lifetime τ. n 2 varies radially along the waveguide.By defining n t = n 1 + n 2 , the lower level population n 1 has been eliminated from the equations.The pump and total signal power at position z along the waveguide is represented by P p,s .Each mode is travelling either in the forward or backward directions represented by the plus and minus superscripts respectively.The emission and absorption cross sections σ e and σ a vary with wavelength, but here they can be assumed to be constant since the pump and signal bandwidths are very narrow.The scattering loss of the pump and signal are denoted by α p and α s .v p and v s are the pump and signal frequencies respectively.h is Planck's constant and c the speed of light in vacuum.The normalized optical intensity of the pump and signal are denoted by i s and i p which are both radially and azimuthally dependent.The waveguide supports single mode operation at both the pump and signal frequency which was confirmed in the experiment.A Gaussian approximation to the normalized optical mode is used with R are the reflectivity of the pump and signal at z = 0 and z = L for respectively.Coupling losses between the waveguide and fibers were taken into account by multiplying a transmission factor T 2 (since the junction is encountered twice for each reflection).The fiber gratings used in the experiment have a bandwidth of 0.5 nm which is much narrower than the gain bandwidth of Yb.The linewidth of the lasing signal measured in experiments was approximately 10 pm, limited by the resolution of OSA.Thus we can take the absorption and emission cross sections as being constant, and assume that the grating reflectivity and penetration depth does not vary over the signal bandwidth.Thus we use the standard boundary conditions for non-distributed reflectors as given in Eq. ( 2), with zero reflectivity near the pump wavelength and a fixed reflectivity across the signal band.The spontaneous emission into the propagating laser mode is used to initiate the lasing process [15].Equations (1) were solved by integrating them from z = 0 to z = L using the fourth-order Runge-Kutta method.The integration process was iterated until convergence was obtained.

Comparison between the model and experimental results
In order to first validate the model, the simulated results were compared with the experimental results of both a DBR WGL and conventional fiber laser for which the cladding is free of dopants.
Rate equations were solved with the parameters listed in Table 1.Most parameters used in the modeling, namely absorption and emission cross sections σ, and life time τ, were set according to the glass manufacturer's data.The output signal is centred at 1031.7 nm corresponding to the peak in reflectivity of the OC grating.
The output power with respect to the pump power obtained from experiments (circles) and modeling (red line) is illustrated in Fig. 4. The output power increased linearly as the pump power increased.The experimental data shows that the slope efficiency of the waveguide laser and fiber lasers were 31% and 61% respectively.The threshold of the waveguide laser and fiber laser is approximately 112.5 mW and 12 mW respectively.Figure 4 shows that the modeling results (red line) agree with the experimental data (black dots) up to a pump power of 250 mW, above which the modeling and experimental results deviate from one another.The mechanism underpinning this type of deviation is still the subject of much debate.Lifetime quenching has been proposed as a possible mechanism [16], whereby Yb ions can transfer their excitation energy to color centers and impurities in the glass.As a result, the lifetime of the Yb upper energy level is reduced.Cooperative luminescence has also been proposed as an alternative mechanism [17].In this case, the transmission coefficient is affected by the presence of Yb 3+ -Yb 3+ ion-pairs, especially in heavily-doped substrates.The de-excitation of the Yb ion-pairs channels a portion of the stored energy into separate emission at half the wavelength of the laser signal.This effect becomes more prominent at high pump power with a resultant roll over of laser output power.A third possible mechanism is energy transfer between Yb and trace amounts of other rare earth ions in the glass.For example, Tm and Er are reported to exist in trace amounts in many Yb doped materials.Indeed, mass spectrometric probing confirmed the presence of Tm and Er in our samples.Furthermore, fluorescence spectra of our samples revealed emission at wavelength of 500 nm and 650 nm which corresponds to the de-excitation of Yb and Tm respectively [18][19][20].Clearly, the gradual degradation of laser performance at high pump powers is a complex issue and further studies are required to resolve this effect.The agreement between the modeled and experimental results at moderate pump powers for a WGL and conventional fiber laser validates our approach utilizing the transverse integration of the propagating profile to predict the laser output.

Doped cladding influences on the waveguide laser performance
The model is used to study the influence of the doped cladding on the output power of the waveguide laser.Firstly, the model was used to simulate the experimental case where the Yb dopant is evenly distributed in the bulk glass.Secondly, for comparative purpose, the model was used to simulate the virtual case, similar to that for a conventional fiber laser, where the dopant was restricted to the waveguide core.Figure 5(a) shows the modeled output power with respect to the incident pump power of the DBR WGL for both the fully doped and inhomogeneously doped case, the latter with a doped radius of 10 μm to a maximum in modeled output power.The refractive index contrast and the length were set to 1.6 × 10 −3 and 9.8 mm corresponding to the earlier experiment.When the glass is inhomogeneously doped with radius of 10 μm, the threshold reduced to 85 mW with maximum output power of 117 mW.It can be calculated that approximately 15% pump power was lost in the doped cladding.Figure 5(b) shows the slope efficiency and threshold as a function of the waveguide length in fully doped glass and inhomogeneously doped glass with doped radius of 10 μm.The optimum length for the waveguide laser in fully doped glass and inhomogeneously doped glass is 8 mm and 10 mm respectively.When the waveguide length is shorter than 4 mm, the absorbed pump is not sufficient to overcome the intrinsic losses.For waveguide lengths up to 6 mm, the homogeneously and inhomogeneously doped WGLs have comparable thresholds and slope efficiencies.However as the waveguide lengths increases the pump light deposited into the doped cladding also increases, this results in extra losses and a higher threshold but consequently also provides gain to the wings of the laser mode which lies outside the core region.This additional gain (in comparison to undoped cladding case) leads to optimal laser efficiency at shorter waveguide lengths.At the optimal length the slope efficiency of the laser is predicted to reach 37.8% and 40% for the two cases with threshold values of 110 mW and 85 mW respectively.

Optimization of the laser output
The model also enabled us to determine parameters for optimizing the output power.The first aspect to investigate is the influence of the output coupler on the laser performance.Figure 6 displays the slope efficiency for various waveguide lengths as a function of different reflectivities of the OC grating.It is clear to see that the maximum slope efficiency corresponds to the waveguide laser that has the lowest reflectivity output coupler, that is, the least amount of the feedback.As expected, it can also be seen that the waveguide length at which the peak in slope efficiency occur becomes shorter as the reflectivity of the OC increases.
#  Another important parameter for laser design is the refractive index contrast of the waveguide, which varies with different writing parameters, such as pulse energy, translation speed and focusing optics in the writing process.The slope efficiency and output power versus waveguide length is shown in Fig. 7 for refractive index contrasts ranging from 1.4 × 10 −3 to 2 × 10 −3 .Over this refractive index contrast range, the WGL oscillates on a single transverse mode.The OC reflectivity is assumed to be 40% with the optimal length of 9 mm.As the refractive index contrast increases, the output power and slope efficiency of the laser increases correspondingly.The increased refractive index contrast decreases the mode field diameter of the pump and signal.Thus, more energy is distributed in the doped waveguide rather than the doped cladding, with an associated improvement in propagated energy.

Conclusion
In conclusion we have developed a predictive, rate equation model for DBR waveguide lasers fabricated using the femtosecond laser direct-write technique.The model accounts for the presence of a homogeneously distributed rare earth dopant, and was validated by analyzing a conventional fiber laser and an ultrafast laser inscribed waveguide laser.The model revealed that approximately 15% of the pump power was lost in the doped cladding of the Yb doped DBR WGL.The model can also enable us to determine parameters to optimize the performance of an ultrafast laser inscribed waveguide lasers, such as the waveguide length, reflectivity of the OC and induced refractive index contrast.Moreover, the model is not limited.It can be expanded to DBR WGLs of other kinds of rare earth dopants or symmetric structures like the W geometry [21] in order to provide reasonable predictions and further design optimizations.

Fig. 1 .
Fig. 1.(a) DIC top-view image of a waveguide (b) respective end-on cross section image.
.(1) is to be solved subject to the boundary conditions:

Fig. 4 .
Fig. 4. Experimental (black dots) and theoretical (red line) output power versus pump power of the DBR fiber and waveguide laser.

Fig. 5 .
Fig. 5. (a) Output power versus pump power of the DBR waveguide laser (L = 9.8 mm) and (b) slope efficiency and threshold of the WGL as a function of waveguide length in fully doped glass and inhomogeneously doped glass (r = 10 μm).

Fig. 7 .
Fig. 7. Slope efficiency and output power (with incident power of 300 mW) as a function of refractive index contrasts.