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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 25 — Dec. 6, 2010
  • pp: 25873–25886
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Dynamical, bidirectional model for coherent beam combining in passive fiber laser arrays

Tsai-wei Wu, Wei-zung Chang, Almantas Galvanauskas, and Herbert G. Winful  »View Author Affiliations


Optics Express, Vol. 18, Issue 25, pp. 25873-25886 (2010)
http://dx.doi.org/10.1364/OE.18.025873


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Abstract

We generalize the recently proposed model for coherent beam combining in passive fiber laser arrays [Opt. Express 17, 19509 (2009)] to include the transient gain dynamics and the complication of counterpropagating waves, two important features characterizing actual experimental conditions. The extended model reveals that beam combining is not affected by the population relaxation process or the presence of backward propagating waves, which only serve to co-saturate the gain. The presence of nonresonant nonlinearity is found to reduce the coherent combining efficiency at high power levels. We show that the array lases at the frequencies with minimum overall losses when multiple loss mechanisms are present.

© 2010 OSA

1. Introduction

The possibility of multi-kW power scaling through passive coherent phasing of fiber laser arrays has spurred intense research into the physics and technology of beam combining. Several groups have successfully demonstrated highly efficient coherent combining of up to eight discretely coupled fiber lasers by 50:50 directional couplers [1

1. H. Bruesselbach, D. C. Jones, M. S. Mangir, M. Minden, and J. L. Rogers, “Self-organized coherence in fiber laser arrays,” Opt. Lett. 30(11), 1339–1341 (2005). [CrossRef] [PubMed]

5

5. A. Shirakawa, T. Saitou, T. Sekiguchi, and K. Ueda, “Coherent addition of fiber lasers by use of a fiber coupler,” Opt. Express 10(21), 1167–1172 (2002). [PubMed]

]. Theoretical work has addressed various steady state aspects of beam combining, such as the scaling of combining efficiency with array size [6

6. J. Q. Cao, J. Hou, Q. S. Lu, and X. J. Xu, “Numerical research on self-organized coherent fiber laser arrays with circulating field theory,” J. Opt. Soc. Am. B 25(7), 1187–1192 (2008). [CrossRef]

10

10. K. Wiesenfeld, S. Peles, and J. L. Rogers, “Effect of Gain-Dependent Phase Shift on Fiber Laser Synchronization,” IEEE J. Sel. Top. Quantum Electron. 15(2), 312–319 (2009). [CrossRef]

]. However there have been few detailed dynamical studies that probe the process in which the independent fiber amplifiers organize themselves to produce a mutually coherent output. We recently presented a theoretical model of fiber laser coherent beam combining that showed clearly that the mechanism of beam combining is the selection of the composite cavity mode that satisfies the condition of minimum loss [11

11. 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]

]. In that work a ring cavity was assumed in order to avoid the computational complexities of counter-propagating waves while still gaining insight into the nature of the beam combining process. Although there have been some demonstrations of passive coherent phasing in ring cavity geometries, all the attempts to scale up beyond two coupled lasers have involved standing wave cavities. It would be desirable to have a simulation tool that permits detailed studies of beam combining with Fabry-Perot cavities.

In this paper we present a full bidirectional model of passive coherent phasing based on mutually-coupled nonlinear Schrödinger equations for counterpropagating waves. These equations are coupled to dynamic rate equations that include the effects of cross saturation by the forward and backward waves. The model allows a detailed look at how the composite cavity modes evolve from noise and how the phase-difference between coupled amplifiers locks to a fixed value. It is evident from the model that the coherent phasing results from the selection of the composite cavity mode with the lowest loss. We show how the spacing between array modes is determined and examine the controversial role of the nonlinear refractive index. Our results yield insight into the mechanisms that limit beam combining efficiency.

2. Model

A two-channel fiber laser array is depicted in Fig. 1
Fig. 1 A two-channel fiber laser array.
with two independent single mode fibers coupled discretely by a directional coupler. The continuous-wave pump beams, launched into each fiber by a wavelength division multiplexer (WDM) at z = 0, excite active ions and give rise to gain at longer wavelengths. The fiber Bragg gratings (FBG) provide ~100% feedback at the right ends of the fibers, while differential power reflectivities, R1 and R2 in Fig. 1, are applied to the output ports of the 50:50 coupler at the left-hand sides. Assuming single polarization, the coherent waves propagating in + z and −z directions in each fiber laser are governed by the nonlinear Schrödinger equation together with the rate equation [12

12. G. P. Agrawal, Nonlinear fiber optics, third edition (Academic Press, 2001), pp. 263–264.

, 13

13. A. E. Siegman, Lasers (University Science Books (January 1986)), pp. 485–487.

]:
Ejfz=12(gjα)Ejfβ1Ejft+12(biβ2)2Ejft2+iγ(|Ejf|2+2|Ejb|2)Ejf
(1)
Ejbz=12(gjα)Ejb+β1Ejbt12(biβ2)2Ejbt2iγ(2|Ejf|2+|Ejb|2)Ejb
(2)
τdgj(z,t)dt=g0jgj(z,t)|Ejf|2+|Ejb|2Psatgj(z,t)
(3)
Ejf and Ejb refer to the slowly varying envelopes of the forward and backward electromagnetic waves in the first (j = 1) and second (j = 2) fiber respectively. Various effects including linear gain gj, fiber losses α, the inverse of the group velocity β1, the frequency-dependent losses b, the group velocity dispersion β2, and the nonresonant Kerr nonlinearity γ are all incorporated in Eqs. (1) and (2). In Eq. (3) for gain dynamics, g0j specifies the unsaturated gain while the second and third terms respectively describe the process of excited population relaxation with upper-state lifetime τ and laser gain saturation at high intensity fields. We normalize the electric field amplitudes so |Ej|2 corresponds to the power distribution. Based on the coupled mode theory [14

14. A. Yariv, Photonics: Optical Electronics in Modern Communications, 6th Edition (Oxford University Press, USA, 2007), pp. 563–565.

], the 50:50 directional coupler connecting the inputs E1b,E2b to the outputs A1,A2 is represented by a linear matrix.

(A1A2)=12(1jj1)(E1bE2b)
(4)

In the dynamical aspects of the array, rather than turning on gj abruptly as assumed in many cases [15

15. V. Roy, M. Piché, F. Babin, and G. W. Schinn, “Nonlinear wave mixing in a multilongitudinal-mode erbium-doped fiber laser,” Opt. Express 13(18), 6791–6797 (2005). [CrossRef] [PubMed]

, 17

17. J. Kanka, “Numerical simulation of subpicosecond soliton formation in a nonlinear coupler laser,” Opt. Lett. 19(22), 1873–1875 (1994). [CrossRef] [PubMed]

, 18

18. E. Martí-Panameño, L. C. Gomez-Pavon, A. Luis-Ramos, M. M. Méndez-Otero, and M. D. I. Castillo, “Self-mode-locking action in a dual-core ring fiber laser,” Opt. Commun. 194, 409–414 (2001). [CrossRef]

], we allow the gain to build up gradually and retain its dependence on both time and position. Due to the discrete nature of numerical computation, the fiber length is partitioned into segments and we will deal with an integrated gain variable g˜j=z0z0+gjdz over each segment instead of gj(z,t) in Eq. (3). The product term involving the gain variable and the intensity field on the right hand side of Eq. (3) is, however, difficult to integrate without losing accuracy; we thus follow Ref [9

9. J. L. Rogers, S. Peles, and K. Wiesenfeld, “Model for high-gain fiber laser arrays,” IEEE J. Quantum Electron. 41(6), 767–773 (2005). [CrossRef]

]. in replacing that term in the manner described below.

3. Simulation results

The simulation results are shown in Fig. 2
Fig. 2 The spatial distributions of one of the fiber laser (L1 = 24.3 m) are plotted as an example for (a) both propagating waves and (b) the gain field along the z axis. The three curves consisting of red circles present the self-consistent steady-state solutions obtained from our model, while that of solid black lines are calculated from Matlab with its built-in BVP solver. As for array dynamics, the time evolution of the output power and the averaged gain variable (over z) of each fiber are displayed in (c) and (d). The output power refers to the combined power coming out of the partially-reflected, R1, port as seen in Fig. 1.
to Fig. 4
Fig. 4 Evolution diagram of the output power spectrum for (a) the array modes, (b) the zoom-in longitudinal modes and (c) the relative phase difference Δϕ (π) between two incident (backward) waves at z = 0. All of them start from random and noisy spontaneous emissions. The free spectral range in (b) is 4.1MHz.
. In addition to the standard SSFM outputs (temporal and spectral domain profiles), the spatial distributions and dynamical evolutions of the array can also be retrieved from our model. The spatial distributions refer to the self-consistent solutions of the coupled equations and they are plotted with red circles in Fig. 2 for (a) both forward and backward waves and (b) the gain field within one of the fiber lasers (L1 = 24.3 m) at steady state. It is clear that the backward signal dominates in this efficient backward pumping configuration [19

19. B. N. Upadhyaya, U. Chakravarty, A. Kuruvilla, A. K. Nath, M. R. Shenoy, and K. Thyagarajan, “Effect of steady-state conditions on self-pulsing characteristics of Yb-doped cw fiber lasers,” Opt. Commun. 281(1), 146–153 (2008). [CrossRef]

] and the resulting gain exhibits stronger saturation near the front end of the fiber. To verify the numerical solutions of the dynamic equations, the time derivatives and the nonlinear index terms of Eqs. (1-3) are set to zero since they do not affect the field distributions at steady states. We then solve the simplified ODEs together with the boundary conditions using the built-in BVP (boundary value problems) solver of Matlab. The solid black line represents such solution in Fig. 2(a), 2(b) and its agreement with the red circles supports our simulation results. As for array dynamics, the time evolutions of (c) the output power, coming out of the partially-reflected port, and (d) the gain variable (averaged over z) in both fiber lasers are clearly observed in Fig. 2. The array exhibits transient oscillations in the beginning of the excitation and settles eventually after a few milliseconds. At steady state, the averaged gain variables g¯1,2 amount to 0.1244 and 0.1251 m−1 and these equal the roundtrip losses of αln(R)/2/L1,2 as expected from fundamental laser theory.We now turn to the beam combining properties of the array. The temporal (left) and spectral (right) domain outputs are shown in Fig. 3
Fig. 3 An Er-doped fiber laser array in Fig. 1 with L1 24.3 and L2 24.0 m. The output powers from (a) upper port with partial reflectivity and (b) lower, angle-cleaved, port are plotted for time (left) and frequency (right) domains respectively. The separation between spikes in the frequency domain is 0.333 GHz.
. As expected, almost all the power comes out of the upper, partially reflected (R1 = 0.04) port, while a negligible amount leaks through the lower, angle-cleaved (R2 = 0) one. Taking Pout as the output of the straight-cleaved end and Pi as the power from the ith single laser if uncoupled, we define the power combining efficiency for an N - channel array as

Nchannel combined power efficiency=Pout/1NPi

To fully characterize the array dynamics, the model is used to study the formation process of the coincident modes and also the associated phase-locked states. Figure 4 illustrates the evolutionary spectrum for (a) the array modes, (b) the longitudinal modes and (c) the relative phase difference Δϕ between two incident (backward) waves at z = 0 before the 50:50 coupler, where Δϕ is defined as ( ϕ1ϕ2 ) modulo 2π. The initial noise in the frequency domain is modeled as uniformly distributed complex numbers with both signs of the real and imaginary parts of the field assigned stochastically to be positive or negative. The serial snapshots show that the array output grows out of noisy spontaneous emission and is continually filtered due to the interferometric nature of the composite cavity (Fig. 4(a)). After a few milliseconds the initial random spectrum is transformed into a set of discrete mode clusters spaced by Δv=c/(2n1ΔL)=0.333GHz . (We show in the Appendix that for larger arrays the spacing between the mode clusters is given by Δv=c0/(2n1ΔLgcd) , where gcd stands for Greatest Common Divisor.) As time increases, the width of these clusters shrinks considerably owing to gain competition. Zooming in the spectrum further (Fig. 4(b)) shows that each mode cluster consists of the Fabry-Perot resonances of a single cavity with a free spectral range of 4.1 MHz, i.e. c/(2n1L). The result is particularly appealing since the fundamental characteristic of the laser cavity manifests itself naturally out of the model as anticipated. The phase difference between the two lasers also evolves from an initial random distribution to a linear function of frequency centered at the peak of the mode cluster. This indicates that those modes are phase locked. Note these oblique lines center around 1.5π on the vertical axes of Fig. 4(c) because this particular phase difference yields constructive interferences in the upper output port of the directional coupler and destructive ones in the other. Finally, cross-referencing the time orders in Fig. 2(c) and Fig. 4 reveals that both mode formation and the phase-locking behavior are established very early before the transient oscillation begins. The transient relaxation oscillations are thus a collective phenomenon of the coupled lasers.

4. Nonlinearity

The role of an intensity-dependent nonlinear phase shift in beam combining or in the phase locking process remains controversial. Some groups have claimed finding support for the enforcement of self locking through a non-resonant nonlinear index n2 [21

21. E. J. Bochove, “Effect of nonlinear phase on the passive phase locking of an array of fiber lasers of random lengths,” in Integrated Photonics and Nanophotonics Research and Applications (IPNRA) (Honolulu, Hawaii, 2009).

23

23. B. S. 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]

], while at the same time opposing evidence is demonstrated experimentally with high power, > 50W, fiber laser arrays [23

23. B. S. 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]

, 24

24. H. Bruesselbach, M. Minden, J. L. Rogers, D. C. Jones, and M. S. Mangir, “200 W self-organized coherent fiber arrays,” in 2005 Conference on Lasers & Electro-Optics (2005), pp. 532–534.

]. This confusing state of affairs needs to be clarified and so we utilize the bidirectional model to study the role of nonlinear phases in coherent combining.

To completely account for the decrease of the combining efficiency owing to nonlinearity, we calculate the output powers using Eq. (4) assuming equal amplitudes of the incident waves before the coupler. Their power ratio is expressed as a function of their phase difference Δϕ by [14

14. A. Yariv, Photonics: Optical Electronics in Modern Communications, 6th Edition (Oxford University Press, USA, 2007), pp. 563–565.

]
P1P2=1sin(Δϕ)1+sin(Δϕ)
(9)
A logarithmic plot of Eq. (9) is shown in Fig. 6
Fig. 6 The logarithmic plot of the output power ratio in terms of relative phase Δϕ.
to illustrate how rapidly the powers transfer between one port to the other when Δϕ changes. It is clear that the singularities occur at 1.5π and 0.5π representing the two extremes of power combining. Observe when Δϕ is confined within 1.4π and 1.6π (in the case of the linear arrays), the power ratio is high and most of the power resides in P1. On the other hand, when Δϕ deviates far from 1.5π and approaches π or 2π, P1 decreases and more power emerges out of the lower, angle-cleaved port as is evident in Fig. 5. The drop in combining efficiency is thus seen to be a result of the increasing bandwidth of the power spectrum, in particular, the broadening of each spectral packet under modulation.

5. Array lasing frequencies - the minimum loss

As mentioned in Ref [11

11. 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]

], the nonzero loss dispersion coefficient b may give rise to reduced combining efficiency as well as shifted lasing frequencies in a two-channel fiber laser array. It is essential to understand how the resonant frequencies are determined and why the frequency shift occurs in the presence of additional loss sources. A reasonable expectation is that the array chooses to lase at the frequency that experiences the least overall losses. In order to verify this point, we present a simple loss analysis based on the unidirectional two-channel fiber laser array. The ring cavity configuration is adopted here as seen in Fig. 7
Fig. 7 A unidirectional two-channel fiber laser array.
since it is simpler and we have shown earlier that the coherent beam combining is not affected by the backward propagating waves of standing-wave cavities.

To derive the frequency-dependent array loss, the circulating power within each fiber laser before the coupler is assumed to be P. The coupler output powers are calculated, according to Eq. (9), as P(1+sin(Δϕ)) and P(1sin(Δϕ)) depending on the accumulated phase difference of Δϕ=ω/cn1(L1L2) between the two incident fields. Only one of the output powers is reflected and fed back into the other end of the two fibers. Take P(1+sin(Δϕ)) for example; the steady-state laser oscillation requires that the power be restored to P at the reference plane just before the coupler, so we can write
P(1+sin(Δϕ))×R×12exp[(gαbω2)L]=P
(10)
where R is the power reflectivity, g is the saturated gain, α is the linear loss and b is the loss dispersion coefficient. We take the logarithm of Eq. (10) and the expression for the loss is given as the right hand side of Eq. (11).
g=α+bω2log(R(1+sin(Δϕ))/2)L
(11)
The frequency dependent loss profile can thus be readily plotted by plugging in Δϕ=ω/cn1(L1L2).

Consider an example of a two-channel fiber laser array of lengths 24.0005 m and 24.0 m, with the simulated power spectra shown in Fig. 8
Fig. 8 Power spectra of a two channel fiber laser array with fiber lengths 24.0005m and 24.0m for (a) b = 0 ps2m−1 and (b) b = 0.13 ps2m−1.
for (a) b = 0 ps2m−1 and (b) b = 0.13 ps2m−1 respectively. (The very small length difference is chosen to ensure the spike separation and the frequency shifts are large enough for clear visualization.) It is clear the combining efficiency drops considerably and the lasing frequency shifts from 126.5 GHz to 45.79 GHz in the presence of nonzero loss dispersion. Utilizing Eq. (11), we plot the frequency-dependent loss on a logarithmic scale with blue solid lines and further overlap them with the output lasing frequencies (red solid lines) in Fig. 9
Fig. 9 The frequency dependent losses (m−1), plotted in the log scale with blue lines, are overlapped with the lasing spectrum of the output fields (red spikes) for (a) zero and (b) nonzero b coefficients respectively.
for better visualization. The loss curves exhibit minimum values near −300 GHz and 100 GHz in Fig. 9(a) and around 50 GHz in Fig. 9(b). In both cases, the good agreement between array resonant modes and the location of the minimum losses validates the hypothesis that the coupled array finds the mode with minimum overall losses.

6. Conclusion

To conclude, we have extended our dynamic model of passive beam combining in fiber lasers to include transient gain dynamics and the interaction of counterpropagating waves. The model allows us to study the process in which composite cavity modes are selected and the establishment of a fixed phase relationship between the coupled amplifiers. We find that the phase locked state is established relatively soon within several hundred roundtrips and that the amplifiers exhibit collective transient relaxation oscillations upon turn-on. The unsettled issue of nonlinearity is also studied. Our simulation suggests the nonresonant n2 induces spectral broadening and reduces the combining efficiency at high power levels. We explore the working principle of the array and demonstrate that it is based on the selection of composite cavity modes with the minimum overall losses.

Appendix

Array mode spacing – the greatest common divisor

It is well known that when two lasers of length L1 and L2 are combined in a composite cavity, the individual Fabry-Perot frequency combs become modulated with an envelope whose peaks are separated by Δv=c/(2n1ΔL) , where ΔL=L2L1 . For N coupled lasers the separation between the maxima of the modulation envelope can be found by examining the condition for constructive interference at all the 50:50 couplers. Since lasing of the composite cavity should occur near these maxima (corresponding to frequencies of minimum loss) this analysis helps to make sense of the complicated spectra observed in multi-element arrays.

Fig. A1 A four-channel fiber laser array. The figure is taken and modified from Ref [25].

Consider a four-channel fiber laser array in Fig. A1, where the coherent combining is governed by the 50:50 directional couplers and the linear coupling matrix of Eq. (4). For any frequency f, constructive interferences can occur at the lower output port of the couplers M1 and the upper output ports of M2 and M3 respectively when

2πn1fc02(L2L1)=3π2+m12π2πn1fc02(L3L4)=3π2+m22π2πn1fc02(L3L2)=3π2+m32π
(A1)

Here n1 is refractive index of the fiber and m1,m2,m3 are integers. Note the third equation of Eq. (A1) describes the power addition criterion in M3 and it depends only on the fiber lengths L2 and L3. This can be understood by calculating the phase of the output fields from the coupler M1 by

12[1jj1][ej2kL1ej2kL2]=12[ej2kL1jej2kL2jej2kL1+ej2kL2]
(A2)

Assuming the input waves interfere destructively at the upper output port of M1 such that ej2kL1jej2kL2=0, the emerging field of the lower port jej2kL1+ej2kL2 can then be written as 2ej2kL2 if we replace ej2kL1 by jej2kL2. Similar calculation can be applied to M2 with the result of 2ej2kL3 and so the phases of the two input fields into M3 are merely characterized by fiber lengths L2 and L3 in Eq. (A2).

Given random combinations of lengths L1 to L4, the exact solution f generally does not exist for all three equations in Eq. (A1) even with the degrees of freedom provided by m1,m2andm3. In most cases only an optimal frequency f¯ can be obtained. Let us assume f¯ satisfies the following conditions:

2πn1f¯c02(L2L1)=3π2+m12π+Δϕ12πn1f¯c02(L3L4)=3π2+m22π+Δϕ22πn1f¯c02(L3L2)=3π2+m32π+Δϕ3
(A3)

where Δϕk, k = 1⋯3, indicates the deviations of f¯ away from the exact solution of each equation in Eq. (A1) and is responsible for the imperfect power combining due to the residual phase mismatch. The optimal solution f¯ is recognized as the frequency of minimum coupling loss in the four-channel array simulation as shown in Ref [11

11. 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]

]. We can then calculate the period of these modes Δv by substituting f¯+Δv into f¯ in Eq. (A3);

2πn1(f¯+Δv)c02(L2L1)=3π2+(m1+p1)2π+Δϕ12πn1(f¯+Δv)c02(L3L4)=3π2+(m2+p2)2π+Δϕ22πn1(f¯+Δv)c02(L2L3)=3π2+(m3+p3)2π+Δϕ3
(A4)

where pk,k=13 are new integers. Upon eliminating f¯ from the above two sets of equations we find

2πn1Δvc02(L2L1)=p12π2πn1Δvc02(L3L4)=p22π2πn1Δvc02(L2L3)=p32π
(A5)

The maxima of the modulation envelope are thus spaced by a frequency given by

Δν=c02n1ΔLc=LCM[c02n1(L1L2),c02n1(L2L3),c02n1(L3L4)]
(A6)

where LCM represents the least common multiple of the arguments and ΔLc represents some equivalent path length difference. For this solution p1,p2,p3 are integers with no common factor. From this result it can also be shown that Δν is determined by the greatest common divisor of the length differences. Thus the frequencies of minimum loss are spaced by

Δv=c02n1ΔLgcd
(A7)

where ΔLgcd=GCD[L1L2,L2L3,L3L4]=ΔLc. This result can be generalized to any number of lasers in the tree architecture described here.

It is important to note that the composite cavity modes have to satisfy the condition that the field at any point reproduces itself after a round trip, within a phase shift of an integral multiple of 2π. These modes form a very dense comb structure which is modulated by an envelope representing the transfer function of the multiple-coupler interferometer. Maximum combining efficiency occurs where these modes coincide with maxima of the transfer function.

To support the analysis, a numerical example is given for a unidirectional four-channel fiber laser array of lengths 24.0, 24.3, 23.733 and 24.633 m. The greatest common divisor of their length differences is 3 mm and the mode periodicity is calculated to be 66.7 GHz according to c0/(n1ΔLgcd) where the factor of two differences from Eq. (A7) is due to the single pass nature of the laser cavity in this case. The simulation result is shown in Fig. A2 with Δv seen to be 66.7 GHz and is consistent with the theoretical prediction.

Fig. A2 Coherent combining of a four-channel fiber laser array with lengths 24.0, 24.3, 23.733 and 24.633 m. The period of the power spectrum pattern indicated by the red arrow is measured to be 66.7 GHz.

Acknowledgments

References and links

1.

H. Bruesselbach, D. C. Jones, M. S. Mangir, M. Minden, and J. L. Rogers, “Self-organized coherence in fiber laser arrays,” Opt. Lett. 30(11), 1339–1341 (2005). [CrossRef] [PubMed]

2.

W. Z. Chang, T. W. Wu, H. G. Winful, and A. Galvanauskas, “Array size scalability of passively coherently phased fiber laser arrays,” Opt. Express 18(9), 9634–9642 (2010). [CrossRef] [PubMed]

3.

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]

4.

A. Shirakawa, K. Matsuo, and K. Ueda, “Fiber laser coherent array for power scaling, bandwidth narrowing, and coherent beam direction control,” in Conference on Fiber Lasers II, L. N. Durvasula, A. J. W. Brown, and J. Nilsson, eds. (San Jose, CA, 2005), pp. 165–174.

5.

A. Shirakawa, T. Saitou, T. Sekiguchi, and K. Ueda, “Coherent addition of fiber lasers by use of a fiber coupler,” Opt. Express 10(21), 1167–1172 (2002). [PubMed]

6.

J. Q. Cao, J. Hou, Q. S. Lu, and X. J. Xu, “Numerical research on self-organized coherent fiber laser arrays with circulating field theory,” J. Opt. Soc. Am. B 25(7), 1187–1192 (2008). [CrossRef]

7.

D. Kouznetsov, J. F. Bisson, A. Shirakawa, and K. Ueda, “Limits of coherent addition of lasers: Simple estimate,” Opt. Rev. 12(6), 445–447 (2005). [CrossRef]

8.

W. Ray, J. L. Rogers, and K. Wiesenfeld, “Coherence between two coupled lasers from a dynamics perspective,” Opt. Express 17(11), 9357–9368 (2009). [CrossRef] [PubMed]

9.

J. L. Rogers, S. Peles, and K. Wiesenfeld, “Model for high-gain fiber laser arrays,” IEEE J. Quantum Electron. 41(6), 767–773 (2005). [CrossRef]

10.

K. Wiesenfeld, S. Peles, and J. L. Rogers, “Effect of Gain-Dependent Phase Shift on Fiber Laser Synchronization,” IEEE J. Sel. Top. Quantum Electron. 15(2), 312–319 (2009). [CrossRef]

11.

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]

12.

G. P. Agrawal, Nonlinear fiber optics, third edition (Academic Press, 2001), pp. 263–264.

13.

A. E. Siegman, Lasers (University Science Books (January 1986)), pp. 485–487.

14.

A. Yariv, Photonics: Optical Electronics in Modern Communications, 6th Edition (Oxford University Press, USA, 2007), pp. 563–565.

15.

V. Roy, M. Piché, F. Babin, and G. W. Schinn, “Nonlinear wave mixing in a multilongitudinal-mode erbium-doped fiber laser,” Opt. Express 13(18), 6791–6797 (2005). [CrossRef] [PubMed]

16.

D. Hollenbeck and C. D. Cantrell, “Parallelizable, bidirectional method for simulating optical-signal propagation,” J. Lightwave Technol. 27(12), 2140–2149 (2009). [CrossRef]

17.

J. Kanka, “Numerical simulation of subpicosecond soliton formation in a nonlinear coupler laser,” Opt. Lett. 19(22), 1873–1875 (1994). [CrossRef] [PubMed]

18.

E. Martí-Panameño, L. C. Gomez-Pavon, A. Luis-Ramos, M. M. Méndez-Otero, and M. D. I. Castillo, “Self-mode-locking action in a dual-core ring fiber laser,” Opt. Commun. 194, 409–414 (2001). [CrossRef]

19.

B. N. Upadhyaya, U. Chakravarty, A. Kuruvilla, A. K. Nath, M. R. Shenoy, and K. Thyagarajan, “Effect of steady-state conditions on self-pulsing characteristics of Yb-doped cw fiber lasers,” Opt. Commun. 281(1), 146–153 (2008). [CrossRef]

20.

D. Sabourdy, V. Kermene, A. Desfarges-Berthelemot, M. Vampouille, and A. Barthelemy, ““Coherent combining of two Nd: YAG lasers in a Vernier-Michelson-type cavity,” Appl. Phys. B-Lasers. 75, 503–507 (2002). [CrossRef]

21.

E. J. Bochove, “Effect of nonlinear phase on the passive phase locking of an array of fiber lasers of random lengths,” in Integrated Photonics and Nanophotonics Research and Applications (IPNRA) (Honolulu, Hawaii, 2009).

22.

C. J. Corcoran and K. A. Pasch, “Output phase characteristics of a nonlinear regenerative fiber amplifier,” IEEE J. Quantum Electron. 43(6), 437–439 (2007). [CrossRef]

23.

B. S. 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]

24.

H. Bruesselbach, M. Minden, J. L. Rogers, D. C. Jones, and M. S. Mangir, “200 W self-organized coherent fiber arrays,” in 2005 Conference on Lasers & Electro-Optics (2005), pp. 532–534.

25.

A. E. Siegman, “Resonant modes of linearly coupled multiple fiber laser structures,” unpublished at http://www.stanford.edu/~siegman/Coupled%20Fiber%20Lasers/coupled_fiber_modes.pdf (2004).

OCIS Codes
(060.2320) Fiber optics and optical communications : Fiber optics amplifiers and oscillators
(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers
(140.3290) Lasers and laser optics : Laser arrays
(140.3510) Lasers and laser optics : Lasers, fiber
(140.3298) Lasers and laser optics : Laser beam combining

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: September 29, 2010
Revised Manuscript: November 8, 2010
Manuscript Accepted: November 11, 2010
Published: November 25, 2010

Citation
Tsai-wei Wu, Wei-zung Chang, Almantas Galvanauskas, and Herbert G. Winful, "Dynamical, bidirectional model for coherent beam combining in passive fiber laser arrays," Opt. Express 18, 25873-25886 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-25-25873


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References

  1. H. Bruesselbach, D. C. Jones, M. S. Mangir, M. Minden, and J. L. Rogers, “Self-organized coherence in fiber laser arrays,” Opt. Lett. 30(11), 1339–1341 (2005). [CrossRef] [PubMed]
  2. W. Z. Chang, T. W. Wu, H. G. Winful, and A. Galvanauskas, “Array size scalability of passively coherently phased fiber laser arrays,” Opt. Express 18(9), 9634–9642 (2010). [CrossRef] [PubMed]
  3. 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]
  4. A. Shirakawa, K. Matsuo, and K. Ueda, “Fiber laser coherent array for power scaling, bandwidth narrowing, and coherent beam direction control,” in Conference on Fiber Lasers II, L. N. Durvasula, A. J. W. Brown, and J. Nilsson, eds. (San Jose, CA, 2005), pp. 165–174.
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  10. K. Wiesenfeld, S. Peles, and J. L. Rogers, “Effect of Gain-Dependent Phase Shift on Fiber Laser Synchronization,” IEEE J. Sel. Top. Quantum Electron. 15(2), 312–319 (2009). [CrossRef]
  11. 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]
  12. G. P. Agrawal, Nonlinear fiber optics, third edition (Academic Press, 2001), pp. 263–264.
  13. A. E. Siegman, Lasers (University Science Books (January 1986)), pp. 485–487.
  14. A. Yariv, Photonics: Optical Electronics in Modern Communications, 6th Edition (Oxford University Press, USA, 2007), pp. 563–565.
  15. V. Roy, M. Piché, F. Babin, and G. W. Schinn, “Nonlinear wave mixing in a multilongitudinal-mode erbium-doped fiber laser,” Opt. Express 13(18), 6791–6797 (2005). [CrossRef] [PubMed]
  16. D. Hollenbeck and C. D. Cantrell, “Parallelizable, bidirectional method for simulating optical-signal propagation,” J. Lightwave Technol. 27(12), 2140–2149 (2009). [CrossRef]
  17. J. Kanka, “Numerical simulation of subpicosecond soliton formation in a nonlinear coupler laser,” Opt. Lett. 19(22), 1873–1875 (1994). [CrossRef] [PubMed]
  18. E. Martı́-Panameño, L. C. Gomez-Pavon, A. Luis-Ramos, M. M. Méndez-Otero, and M. D. I. Castillo, “Self-mode-locking action in a dual-core ring fiber laser,” Opt. Commun. 194, 409–414 (2001). [CrossRef]
  19. B. N. Upadhyaya, U. Chakravarty, A. Kuruvilla, A. K. Nath, M. R. Shenoy, and K. Thyagarajan, “Effect of steady-state conditions on self-pulsing characteristics of Yb-doped cw fiber lasers,” Opt. Commun. 281(1), 146–153 (2008). [CrossRef]
  20. D. Sabourdy, V. Kermene, A. Desfarges-Berthelemot, M. Vampouille, and A. Barthelemy, ““Coherent combining of two Nd: YAG lasers in a Vernier-Michelson-type cavity,” Appl. Phys. B-Lasers. 75, 503–507 (2002). [CrossRef]
  21. E. J. Bochove, “Effect of nonlinear phase on the passive phase locking of an array of fiber lasers of random lengths,” in Integrated Photonics and Nanophotonics Research and Applications (IPNRA) (Honolulu, Hawaii, 2009).
  22. C. J. Corcoran and K. A. Pasch, “Output phase characteristics of a nonlinear regenerative fiber amplifier,” IEEE J. Quantum Electron. 43(6), 437–439 (2007). [CrossRef]
  23. B. S. 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]
  24. H. Bruesselbach, M. Minden, J. L. Rogers, D. C. Jones, and M. S. Mangir, “200 W self-organized coherent fiber arrays,” in 2005 Conference on Lasers & Electro-Optics (2005), pp. 532–534.
  25. A. E. Siegman, “Resonant modes of linearly coupled multiple fiber laser structures,” unpublished at http://www.stanford.edu/~siegman/Coupled%20Fiber%20Lasers/coupled_fiber_modes.pdf (2004).

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