Open Access
Add to BibSonomyAbstract
Passive phase-locking of laser arrays is usually less efficient when the operating point is far above the laser threshold. We investigated the contribution due to the spectral filtering effect induced by the coupled cavities. Experiments carried out on a basic dual-arm laser as well as modeling show that some laser combining losses arise from the spectral filtering whose transmission windows have a non-uniform profile. A simple analytical model of the laser confirms this interpretation.
© 2015 Optical Society of America
1. Introduction
Coherent summation of laser radiation to get laser sources of high power and high brightness is a long-standing research field started in the eighties. A book covering the latest advances on that topic has been recently published [1]. One of the simplest systems of passive coherent beam combining consists in the association of an array of parallel amplifier with a compound cavity. Phase-dependent coupling and phase dependent losses force the laser array to operate in synchrony [2]. Various coupling techniques such as diffraction coupling [3–5], polarization coupling [6], multi-arms interferometers [7], etc., have been exploited to obtain the so called “self-organized” phase-locking. Passive coherent operation of laser arrays has been demonstrated also with different laser media: gas lasers, solid-state lasers, semi-conductor lasers and fiber lasers. However, it has been often observed (but not always discussed) that the combining efficiency, or equivalently said the quality of the phase-locking, drops whenever the system is operated far from the laser threshold. For arrays with tiled output beams, the loss of coherence when observed in the far field pattern, takes the form of a broad incoherent background that grows larger with the laser power. The main consequence is a decrease of the Strehl ratio of the beam array [8]. For laser arrays with intracavity beam combining and delivering a single output beam (filled aperture array) instead, the loss of coherence can be noticed by an increase of power leaking from the unused port of beam splitters [9] or from the loss diffraction orders of grating combiners. The magnitude of this effect strongly varies from case to case, but its presence has been observed in widespread circumstances, whatever the gain medium, whatever the coupling technique and whatever the number of lasers in the array. Coupled laser networks are complex nonlinear systems so that their operation can be altered by practical imperfections or can be influenced by more fundamental mechanisms. For instance in fiber lasers, Kerr nonlinearity was invoked as one of the mechanisms leading to the reduction of the combining efficiency [9]. This phenomenon was also attributed to the dynamical Kramers-Kronig induced phase change close to threshold, since a weakly saturated laser can suffer of high gain variation [10]. The loss of coherence is an issue that remains unexplained in its general form, leaving many open-ended questions.
In the present paper we report an experimental and theoretical study that highlights the role played by the spectral filtering whose contribution apparently has been overlooked in the past. It results from the fact that in many types of laser arrays, and in particular in fiber laser arrays, each elementary laser unit has a different physical length. The coupling among these different cavities leads to the selection of a common set of longitudinal modes causing an envelope spectral filtering. We show that the transmission profile of such filter may partly explain the drop in combining efficiency observed in laser arrays operated far above threshold.
2. Experiments with a basic two-element laser array
Experiments have been carried out on a basic two fiber laser array. The schematic drawing of the set-up is given in Fig. 1. It was based on a unidirectional ring fiber laser built up around double-clad ytterbium doped fiber amplifiers delivering up to a few Watts of power. The resonator included a section with a duplicated path, by means of a fiber coupler (50:50) and a 50/50 beam splitter (BS), forming a kind of Mach-Zehnder interferometer. The cavity consisted of a guided-wave part and of a free-space part. In the free-space section, we implemented a delay line for adjustment of the path length difference between the two branches of the laser. The intracavity beam recombination was also performed in free space for a better control of polarization and to avoid potential nonlinear effects in a fiber coupler. That choice made it necessary to carefully adjust the two beam wavefront and their overlap at the beam splitter (BS). The free-space section was completed by an output coupler of the laser (Roc = 10%) and by an isolator to ensure a unidirectional laser operation. As no spectral bandpass filter was employed, wavelengths across the full gain-bandwidth of the dual-arm laser were available for lasing.
Fig. 1 Schematic drawing of the two-element ring laser set-up including two polarization controlers (PC), two ytterbium doped fiber amplifiers (YDFA) pumped by laser diodes (LD) through fiber combiners (Co), a free-space delay line, a beam splitter (BS) for beam combining, an output coupler (Roc) and an isolator (ISO).
It is well known that such a compound laser resonator with an interferometric set-up exhibits phase-dependent losses [11,12]. The laser works on the eigen-frequencies arising from the least losses, which are associated here to common resonances for the two sub-cavities. Consequently, the fields coming from the two laser arms are expected to combine coherently at the beam splitter BS and to maximize the output power (Output in Fig. 1). This was indeed what we observed with our laser, at least when it was driven close to threshold, where almost all the power was detected at the main output and where only a faint contribution was detected at the loss port of the beam combiner (called Leaks in Fig. 1). However, we observed that the situation evolved depending on the gain setting. For a better characterization, we measured the output power Po of the compound cavity exiting from the output mirror Roc as well as the power leaks Pleak coming from the loss port of the beam splitter BS, for various values of the amplifiers’ pump diode current. An example of our results is shown in Fig. 2(a), where we plot the ratio Po/Pleak. The combining efficiency is thus simply given by η = (Po/Pleak)/(Po/Pleak + 1). We observed that when increasing the small signal gain, and so the laser power, the ratio Po/ Pleak reduced significantly. Our observation agrees with other works. See for example, the data extracted from [9] (large core fibers), which are plotted in the same way in Fig. 2(b). Although the magnitudes of the values are significantly different, connected with an all-fiber beam combining, the shapes of the curves are similar.
Fig. 2 laser power on the main output normalized to the power leaks (a) measured at the loss port of the beam splitter BS for increasing pumping of the amplifiers in the laser of Fig. 1, (b) on the loss port of couplers for increasing pumping of the amplifiers in a two-element fiber laser with a Michelson type cavity (data from [9]).
There are various possible origins for such observed leaks of power: non-phase matched summation on the combiner, incoherent contribution of amplified spontaneous emission from the two amplifiers, spectral broadening through dispersive nonlinear propagation, etc.
To have a further insight into the problem, we recorded the fiber laser spectrum under various laser gain values. The center of the laser emission was around 1085 nm and its spectral envelope broadened quickly above threshold. This fact could be partly explained by the specific configuration of the laser, whose bandwidth was only limited by the gain profile. We used a delay line to adjust the path lengths difference between the two arms. The delay line was tuned so that the differential path was of the order of 0.5 mm: with this choice, a spectral modulation was clearly visible on the optical spectrum analyzer (OSA). The laser longitudinal modes instead, related to the ring cavity of about 20 m length, werenot resolved by the OSA (resolution ~0.07nm), and the observed periodic spectra corresponded to the envelope of the laser modes. For a differential path length of about 0.4 mm and at 18 W pump level (G/Gth = 34), we recorded the laser output spectrum shown in Fig. 3. As expected, it consisted of a set of periodic peaks with a period Λ~λ2/ΔL that in turn is a function of the path-length difference ΔL between the two amplifying arms. Simultaneous measurement of the radiation leaving the cavity from the beam combiner loss port gave the spectrum shown by the red curve in the same graph. The experimental results revealed that the peaks’ spectral positions in the leak spectrum perfectly coincided with those of the coherently combined laser. However, the shape of the peaks was different. In the leak spectrum, some dips clearly appeared in the peaks profile, close to their center wavelengths. It is the first time to our knowledge that such a result is reported for phase-locked laser arrays. The comparison of the laser spectrum at the combined and uncombined output ports respectively, seems to indicate that they are closely related to each other. It proves also that the leaks did not come from the YDFA’s amplified spontaneous emission (ASE): ASE from both arms should have led to a continuous broad background as a result of their incoherent summation at the beam splitter. The following theoretical analysis will clarify the observed leaks spectrum profile as well as the laser phase locking evolution.
Fig. 3 Spectra recorded on the main laser output (blue) and on the loss port of the beam splitter BS (red).
3. Rigrod compound cavity analysis
For comparison with the theory we have used a model derived from Rigrod approach [13] applied to the case of a ring laser with a duplicate amplifier section. If we assume that the fiber coupler and the beam splitter used for combining have perfect 50:50 coupling coefficient and if we consider, for simplicity, that there are no other losses than those of the cavity output mirror with reflectivity Roc, we obtain the following expression for the laser output intensity (normalized to the saturation intensity):
G denotes the small signal power gain of the amplifiers (assumed to be identical) and φ = k.ΔL stands for the phase mismatch between the two arms of the ring with differential length ΔL (k being the free-space wave-vector). Details of the derivation have been omitted but the expression is consistent with the one derived by T.B. Simpson et al. in [11]. The intensity of the leaks Ileak from the loss port of the combiner is given by:The connection between the two above equations and the spectral response comes from the dependence of I0 and Ileak in cos2(φ/2). We now assume that the small signal gain profile G(λ) has a parabolic spectral profile with a full width of 20nm (at G = 1). The laser intensity spectrum S(λ) and the leak intensity spectrum Sleak(λ), calculated from Eqs. (1) and (2), gave the profiles that are shown in Fig. 4 with blue and red curves respectively. The parameters were as follow: ΔL = 0.4 mm, G = 2xGth where Gth denotes the small-signal gain at threshold, Roc = 0.1. The shapes of the spectra are in good agreement with the experiments although the model leading to Eqs. (1) and (2) was of purely energetic type. A small difference is found in the leaks spectrum (red curve) for each laser band. The double bump was slightly asymmetric in the experimental data and the central dip did not go down to zero. This fact can be explained by a weak imperfection of the beams superposition at the beam combiner.Fig. 4 Spectra of the laser output (blue) and of the leaks (red) computed from the Rigrod model. ΔL = 0.4 mm, Roc = 0.1, G equals twice the value at threshold.
At this point, we have identified that the leaks, which have been so far attributed to an imperfect coherent summation, can be instead related to a standard behavior of a laser that contains a device acting as a spectral filter. Because a two wave interferometer is characterized by a spectral transmission which is a smooth continuous function of the wavelength, the laser lines are broad (far above threshold) and some fractions of the laser field are rejected by the spectral filter. They appear in the present set-up on the loss port of the beam splitter used for the beam re-combination. The leaks spectrum Sleak(λ) is simply related to the laser spectrum S(λ) by the relationship:
We then computed the ratio of the laser output power to the laser leaks power upon the small signal gain G. To do that, we simply integrated the computed intensities along thewavelength domain under consideration. The predictions of the simple model (see Fig. 5) agree qualitatively well with the general trend observed in experiments. A sudden drop of the ratio Pout/Pleaks occurs immediately above threshold when increasing G, meaning that a stronger fraction of the power was lost through the leaks. The rate of such a drop gradually decreases when the laser works farther above threshold. Our calculation also indicated that the actual values of the leaks slightly depend on the cavity out-coupling. The question is now whether the same behavior can be found for laser arrays with more than just two lasers: this is the relevant situation of most applications. To find an answer for this question we extended the above Rigrod analysis to the case of a ring laser with N parallel arms including identical amplifiers but characterized by different optical lengths. The effective intensity transmission of the N ways interferometer can be described by [14]:The laser output intensity is therefore given by:And the leaks total intensity by:The evolution of the intensity upon wavelength of the multi-wave interference, T, strongly depends on the difference in length between the amplifier arms. In our computations we have chosen a set of uniformly distributed random lengths of 10 m +/− 1cm. When increasing the number of lasers in the array, the laser spectrum become more complex, with sparse peaks, in agreement with a filtering which becomes more irregular both for the position and for the value of the transmission peaks. An example of laser output (Io) is shown in Fig. 6 by the blue curve for a 4-lasers array together with the spectrum of the leaks (Ileak in red color) and the transmission T of the filter (black-dashed curve and with halved amplitude for ease of reading). The laser lines appear in those spectral regions where the filter transmission is higher, as expected. However leaks still occur, again with spectral components in the same wavelengths as those of the main output although with a different spectral shape. In this case, the dips in the peaks of the leak spectrum are less pronounced than in those of the previous two-arm configuration, because of the smaller transmission peaks at the main output.Fig. 5 Computed ratio of the two-element laser main output power to the power lost in the combining for different values of the small signal gain. Calculations are based on the Rigrod model for an output coupling of 10%.
Fig. 6 Example of spectral filtering connected with a cavity including four amplifiers in parallel of different lengths (dashed line). The transmission has been halved for clarity. The blue color curve show the laser output spectrum derived from a generalized Rigrod analysis. The spectrum associated with the combining losses is plotted in red color.
Our numerical simulations for arrays of size greater than two confirmed the same tendency that the power of the leaks grows faster than the laser output, in a way similar to our basic two-arm laser system. Figure 7 presents a comparison between laser arrays of 2, 4 and 6 lasers (from upper to lower curves). They were computed by averaging over 40 random choices of amplifier lengths. The drop in the ratio laser power/laser leaks is clearly observed in all cases. The drop in the ratio Pout/Pleak upon gain looks even steeper and stronger for larger numbers of units of the laser array.
Fig. 7 Power ratio of laser array main output power Pout to the power lost in the combining Pleaks for different values of the small signal gain and for an array of 2, 4 and 6 lasers (from upper curve to lower curve). The numerical results are based on the generalized Rigrod model for an output coupling of 4%.
5. Conclusion
Our experimental and numerical works show that the drop of coherence in a laser array is an intrinsic feature of a compound cavity driven far above threshold. It is related to the spectral filtering resulting from the presence of multi-arm cavity of different lengths. Our results emphasize, for the first time, that the leaked fraction of the laser array radiation, which was previously interpreted as an incoherent contribution, might instead be a fraction of the coherent intracavity laser field. A simple Rigrod analysis of the compound cavity confirmed this second interpretation and permitted to predict a loss of coherence for an array operated above threshold. The numerically calculated spectra were in agreement (at least in a qualitative manner) with the experimental observations carried out with a basic two-arm laser, as well as with other published data. It was also checked experimentally that when the frequency filtering becomes negligible, setting an almost perfectly balanced length for the two amplifier sections, the leakage no longer evolved with gain. The reported study does not imply that the spectral filtering is the sole explanation for the combining performances in passively phase locked laser array. In fiber lasers and at high power, the nonlinear propagation effects give an additional contribution to the degradation of coherent beam combining. It occurs through frequency couplings and generation of non-resonant components.
All along our experiments, we observed some situations where the laser outputs cannot be explained without taking into account the Kerr effect. Additionally, a signature of optical turbulence has been identified in our experiment at large length mismatch between the two lasers of the array. The laser spectrum had a power spectrum that fitted the hyperbolic secant profile predicted by theory [15] on four decades of magnitude. Clearly four-wave mixing contributes also to the loss of coherence, as it has been already demonstrated by several groups in the past, and should not be ignored in laser arrays. Our approach neglects also the potential impact of resonant nonlinearity resulting from gain-dependent refractive index changes. The deleterious spectral filtering effects described in this paper may be reduced significantly by the Kramers-Kronig self-phasing of the amplifying channels [16]. This work may help the design of new resonator configuration for a more efficient phase-locking of laser array.
Acknowledgments
The authors acknowledge CILAS Company for their financial support.
References and links
1. A. Brignon Ed, Coherent Laser Beam Combining (Wiley-VCH, 2013)
2. V. Eckhouse, M. Fridman, N. Davidson, and A. A. Friesem, “Loss enhanced phase locking in coupled oscillators,” Phys. Rev. Lett. 100(2), 024102 (2008). [CrossRef] [PubMed]
3. J. Morel, A. Woodtli, and R. Dändliker, “Coherent coupling of an array of Nd3+-doped single-mode fiber lasers by use of an intracavity phase grating,” Opt. Lett. 18(18), 1520–1522 (1993). [CrossRef] [PubMed]
4. C. J. Corcoran and F. Durville, “Experimental demonstration of a phase-locked laser array using a self-Fourier cavity,” Appl. Phys. Lett. 86(20), 201118 (2005). [CrossRef]
5. D. Paboeuf, G. Lucas-Leclin, P. Georges, N. Michel, M. Krakowski, J. J. Lim, S. Sujecki, and E. C. Larkins, “Narrow-line coherently combined tapered laser diodes in a Talbot cavity with a volume Bragg grating,” Appl. Phys. Lett. 93(21), 211102 (2008). [CrossRef]
6. S. P. Ng and P. B. Phua, “Coherent polarization locking of a diode emitter array,” Opt. Lett. 34(13), 2042–2044 (2009). [CrossRef] [PubMed]
7. L. Shimshi, A. Ishaaya, V. Eckhouse, N. Davidson, and A. Friesem, “Passive intra-cavity phase locking of laser channels,” Opt. Commun. 263(1), 60–64 (2006). [CrossRef]
8. Y.-H. Xue, B. He, J. Zhou, Z. Li, Y.-Y. Fan, Y.-F. Qi, C. Liu, Z.-J. Yuan, H.-B. Zhang, and Q.-H. Lou, “High power passive phase locking of four Yb-doped fiber amplifiers by an all-optical feedback loop,” Chin. Phys. Lett. 28(5), 054212 (2011). [CrossRef]
9. B. Wang, E. Mies, M. Minden, and A. Sanchez, “All-fiber 50 W coherently combined passive laser array,” Opt. Lett. 34(7), 863–865 (2009). [CrossRef] [PubMed]
10. B. Wang, E. Mies, M. Minden, and A. Sanchez ;“All fiber coherent arrays combining high power lasers,” SPIE Photonics West’09, Fiber Lasers VI: Technology Systems and Applications, Proc.7195.
11. T. B. Simpson, A. Gavrielides, and P. Peterson, “Extraction characteristics of a dual fiber compound cavity,” Opt. Express 10(20), 1060–1073 (2002). [CrossRef] [PubMed]
12. T. W. Wu, W. Z. Chang, A. Galvanauskas, and H. G. Winful, “Model for passive coherent beam combining in fiber laser arrays,” Opt. Express 17(22), 19509–19518 (2009). [CrossRef] [PubMed]
13. A. E. Siegman, Lasers (University Science Books, 1986), Chap.12.4.
14. D. Sabourdy, V. Kermene, A. Desfarges-Berthelemot, L. Lefort, A. Barthelemy, P. Even, and D. Pureur, “Efficient coherent combining of widely tunable fiber lasers,” Opt. Express 11(2), 87–97 (2003). [CrossRef] [PubMed]
15. S. I. Kablukov, E. A. Zlobina, E. V. Podivilov, and S. A. Babin, “Output spectrum of Yb-doped fiber lasers,” Opt. Lett. 37(13), 2508–2510 (2012). [CrossRef] [PubMed]
16. H.-S. Chiang, J. R. Leger, J. Nilsson, and J. Sahu, “Direct observation of Kramers-Kronig self-phasing in coherently combined fiber lasers,” Opt. Lett. 38(20), 4104–4107 (2013). [CrossRef] [PubMed]
