Passive Phasing in a Coherent Laser Array - IEEE Xplore

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IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 15, NO. 2, MARCH/APRIL 2009

Passive Phasing in a Coherent Laser Array Christopher J. Corcoran and Frédéric Durville, Member, IEEE

Abstract—A coherent array of fiber lasers in a self-Fourier cavity is described and analyzed. With individual regenerative feedback added to each fiber laser, the integrated gain in each individual fiber is a function of its cold cavity phase shift (fiber length). This results in a gain-dependent phase shift due to the Kramers–Kronig relations, and has been shown to partially compensate for the random differences in fiber lengths often encountered in coherent fiber arrays. A coupled cavity analysis of the active gain elements and the passive external cavity is performed and a self-consistent fundamental supermode determined. The output phase distribution of the array is determined based on a random distribution in fiber lengths. The Strehl ratio of this phase distribution is calculated and compared to experimental data. Index Terms—Coherent laser array, fiber laser, injection-locked oscillator, passive beam combination, regenerative amplifier.

I. INTRODUCTION HE PERFORMANCE of high-power fiber lasers continues to improve rapidly. Single-mode fiber lasers are currently available commercially with 5-kW output power, >25% wall plug efficiency, and are expected to ultimately achieve on the order of 10 kW per element [1]. To achieve output powers beyond this with good beam quality, coherent beam combination remains an attractive option. Coherent combination techniques include active combination [2], evanescent coupling [3], Talbot cavities [4]–[6], spectral beam combination [7], [8], in-line fiber couplers [9], [10], the tapered fused bundle [11], Fourier filtering techniques [12], single-mode fiber filtering in the far field [13], [14], and the self-Fourier (SF) cavity [15]–[17]. Previous cold cavity models have suggested that the random differences in path lengths inherent in fiber laser arrays can result in poor output phasing characteristics. These models determine the probability of finding the wavelengths that satisfy the resonant condition for all of the individual cavities in the array. The results predict that the probability of finding a common resonant frequency drops sharply when the number of cavities becomes greater than about 8, indicating that there would be a limitation in the number of fiber lasers that can be coherently combined passively to no more than ∼8 elements. We note that coherent addition of larger arrays of Nd:YAG lasers has been successfully demonstrated by careful control

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Manuscript received October 1, 2008; revised November 17, 2008 and December 4, 2008. First published March 10, 2009; current version published April 8, 2009. This work was supported by Raytheon Missile Systems (Directed Energy Weapons), by the U.S. Naval Surface Warfare Center (NSWC), and by the U.S. Army Space and Missile Defense Command (SMDC). C. J. Corcoran is with Corcoran Engineering, Inc., Waltham, MA 02453 USA (e-mail: [email protected]). F. Durville is with Optical Fiber Systems, Inc., Chelmsford, MA 01824 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JSTQE.2008.2011494

of the path length differences between the individual laser resonators [18], [19]. The technique presented here has been developed to circumvent the predicted limitations for fiber lasers and permit the passive coherent combination of a large number of fiber lasers by exploiting the phasing properties of regenerative amplifiers. In this paper, we further develop the analysis of operation of a coherent fiber laser array coupled to a passive SF cavity where each fiber in the array is given individual regenerative feedback (labeled here as facet reflectivity for visualization and simplicity). This analysis theoretically demonstrates that the gain-dependent phase shift in fiber lasers with regenerative feedback can substantially compensate for random differences in fiber lengths and lead to highly phased outputs. Specific enhancements to the previous model include clarification and quantification of the gain-dependent phase shift in the active fiber lasers, inclusion of gain saturation in the active laser elements, utilization of a coupled analysis of the active and passive portions of the complete laser resonator, comparison with experimental results, and evaluation of the predicted Strehl ratio resulting from these phase errors. We begin in Section II by analyzing the passive cavity subsection of the coherent array and define its supermodes and eigenvalues. Section III then analyzes the active gain element subsection of the array and discusses the gain-dependent phase shift in the presence of regenerative feedback (quantified in Appendix A). In Section IV, the active and passive portions of the complete resonator array are coupled together. As indicated by previous analyses [20], the complete system supermode will, in general, be different than that of either subsystem alone. We first present the fundamental supermode of a simplified ideal array in which all fiber elements have exactly the same optical path length. We then use a computer model to determine selfconsistent solutions of the coupled cavities in the presence of path length errors in the fibers. The output phase noise distribution is determined based on a statistical analysis of random differences in fiber lengths. Section V then describes the effects of the phase and intensity noise on the Strehl ratio of the laser array output if a suitable aperture filling technique is utilized. These predictions are compared with experimental results. Section VI concludes with a discussion. II. SF CAVITY ANALYSIS In this paper, we utilize the SF cavity as an example of an external cavity used to provide coupled feedback to an array of fiber lasers. The SF cavity has been described before and is presented in Fig. 1 for an array with N = 9 elements. We choose a nine-element array for this analysis to more clearly demonstrate the improvement in coherence resulting from the gain-dependent phase shift and regenerative feedback.

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CORCORAN AND DURVILLE: PASSIVE PHASING IN A COHERENT LASER ARRAY

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Fig. 3. Model of single regenerative amplifier (fiber laser with regenerative feedback). Individual regenerative feedback (r) causes coherent interference of fields E 1 and E 3 , providing variations in field E 2 , which then varies the saturable amplifier gain g and gain-dependent phase shift δG D . Fig. 1. Coherent array of nine fiber lasers in SF cavity. Coherent feedback from SF cavity provides in-phase feedback to individual elements in the array.

the gain-dependent phase shift is described by the following: E3 = exp (γL) exp (−iαγL − iδ0 ) E2

Fig. 2. Fundamental supermode of nine-element passive SF cavity. Cavity supermode consists of nine Gaussian peaks multiplied by Gaussian envelope.

Through the action of the Fourier lens, the coupled feedback provided to the array can be written as EFB = r0 KEOut

(1)

where r√ 0 is the amplitude reflectivity of the cavity output coupler (= R0 ), Eout is the electric field emitted by the fiber array, K is the passive coupling matrix of the SF cavity, and EFB is the electric field of the feedback to the array. The passive coupling matrix K is unitary with a well-defined set of supermodes described previously [16]. The fundamental supermode for an array of nine elements has an eigenvalue equal to 0.98, and all higher order modes are completely extinguished with eigenvalues equal to 0.0. This fundamental supermode is shown in Fig. 2. We note that this fundamental supermode is valid only for the passive SF cavity alone. This supermode does not reflect the oscillating supermode of the complete laser system, as this analysis does not take into account the active gain element portion of the system. III. ACTIVE GAIN ELEMENT ANALYSIS We now analyze the operation of the active gain elements in the array. In particular, the analysis presented here takes into account the presence of an added regenerative feedback provided by individual facet reflectivity at the output of each gain element. Fig. 3 illustrates the model used to describe the gain elements. In this model, E1 and E4 are the same parameters as EFB and Eout , respectively, as shown previously in Fig. 1. The dependence of the amplifier output on the input can be simply written as E4 = gr E1

(2)

where gr is the diagonal N × N matrix for the regenerative gain. The total integrated round-trip gain in the fiber including

(3)

where γ is the real amplitude gain coefficient, L is the round-trip fiber length, α is the linewidth enhancement factor (≡ χ /χ ) commonly used in diode laser analysis, and δ0 is the cold cavity phase shift (=nkL). The total round-trip phase shift in the fiber amplifier δ is equal to δ0 + δGD , where δGD is the gaindependent phase shift equal to αγL. The integrated amplitude gain g is equal to exp(γL). In order to perform this analysis, we first need to determine the gain-dependent phase shift, and thus α. A. Calculation of Gain-Dependent Phase Shift We note that previous analyses have described the nonlinear phase shift as being dependent on the intensity in the fiber lasers through the Kramers–Kronig relations and changes in the electronic populations [3], [21]–[23]. A nonlinear coefficient n2 had been assigned to this dependence and was experimentally determined to vary quite widely, being as large as 10−16 m2 /W, but also approaching 0, becoming negative under certain conditions, and saturating with signal intensity. This complicated dependence can be incorporated in the system equations in a much more straightforward manner if the phase shift is written directly in terms of the integrated gain of the fiber amplifiers. This phase shift is determined by solving the Kramers–Kronig relations and evaluating the complex atomic susceptibility of Yb, leading to a much simpler analysis and more easily tractable results [24], [25]. The two descriptions of this nonlinear phase shift reflect the same physics of the lasing medium, utilizing different partial derivatives to determine the dependence of the phase shift on the electronic populations. This gain-dependent phase shift is quantified from the gain-dependent change of the refractive index ∆n as follows. We start by calculating the gain-dependent phase shift δGD as a function of the resonant dispersion δGD = kL∆n.

(4)

The refractive index change is a function of the real component of the atomic susceptibility [26] L 1 ∆n = ΓS χ (z) dz (5) 2nL 0 where ΓS is the overlap factor between the laser beam intensity profile and the active amplifier medium, n is the refractive index, L is the round-trip length of the amplifier, and χ and χ are,

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respectively, the real and imaginary components of the complex atomic susceptibility. We also have L ω χ (z) dz (6) G = exp −ΓS nc 0 where G represents the integrated power gain throughout the fiber length. Finally, using α=

χ χ

(7)

we can combine (4)–(7) to obtain δGD =

α ln(G) = α ln(g) 2

(8)

where g is the integrated amplitude gain [=exp(γL)]. This phase shift is independent of the signal intensity in the fiber laser and depends only on the integrated gain and α. We have calculated the linewidth enhancement factor commonly used in semiconductor lasers and adapted here for Yb fiber lasers as a convenient method to estimate the gain-dependent phase shift. A particularly simple method to do this has been presented in [26] for the Er ion and adapted here for Yb. The results of this analysis indicate that the value of α evaluated at a wavelength of 1.08 µm is approximately equal to 1.0. This calculated value of α results from only the electronic transitions between these two manifolds. The total value for Yb may be significantly higher than this due to the presence of strong UV transitions and their associated change in refractive index as described in [21]. As can be seen in [22], the contributions from the UV transitions will be approximately equal to the contribution from the nearby resonances in our wavelength regions of interest. Thus, we roughly incorporate the UV contribution into the model by increasing the total effective α to 2.0. The total value of this gain-dependent phase shift is presently being explored further both theoretically and experimentally. B. Single-Gain Element Graphical Solution As we have previously presented in [23] and [27], we can graphically visualize the solution to these equations for a single laser element by simultaneously graphing (8) (gain-dependent phase shift as a function of gain in the fiber) and the resonance curve of the resonator (g from (3) as a function of the total roundtrip-accumulated phase shift in the fiber). The intersections of these two curves indicate the equilibrium states of the system. The equilibrium states that have positive derivatives of both these curves will be dynamically unstable to perturbations in the electric fields. Equilibrium states that have different sign derivatives will be dynamically stable to perturbations [23], [28]. These two curves are shown in Fig. 4. We note that the natural lifetime of the Yb upper state manifold is equal to ∼0.8 ms, and it has been conjectured that this might limit the response of these phase change to times scales of this order. It should be noted, however, that the presence of the high pump and laser intensities in the fiber core changes the response time of the electronic populations [28]. With the high pump and signal intensities typically encountered in fiber

Fig. 4. Two simultaneous equations relating the total phase shift δG D and the integrated amplitude gain g (in decibels) for a single fiber amplifier. The straight dashed line indicates the gain-dependent phase shift and the solid line indicates the resonance curve of the regenerative amplifier.

lasers, this time response is estimated to decrease on the order of tens of microseconds. We note that in addition to the gain-dependent phase shift discussed in this paper, there will be large changes in the refractive index of the fiber lasers due to thermal effects. These thermally induced index changes will take place on a much slower time scale than the gain-dependent phase shift discussed here, and thus, can be incorporated into the total lengths of the fiber amplifiers, which are given a random statistical distribution anyway. Full solution of this system requires simultaneous solution of the cavity equations along with the full N × N matrix description of the cavity. This is performed in Section IV using a numerical solver. IV. COUPLED CAVITY ANALYSIS We now combine the analysis of the passive SF cavity with that of the active gain elements. The supermodes of the cavity are determined through a self-consistent analysis of the field through the round-trip through the entire cavity by combining (1) and (2) into a single equation E4 = gr E1 = r0 Kgr E4 .

(9)

In order to illustrate the fundamental supermode clearly and simply, we begin by presenting the supermode in the absence of path length errors and assuming a highly saturated gain element with no loss. Under these assumptions, and further assuming that each fiber gain element is pumped with the same pump power, gain saturation results in approximately equal power output among the N fiber laser outputs, independent of the input to the fiber lasers, and the approximate idealized supermode of the coupled laser resonator is presented in Fig. 5. We note that reproduction of the supermode envelope in the SF cavity is not required to achieve good coupling and low-loss operation of the array, it is more important to achieve good reproduction of the individual mode profiles of the individual fiber lasers [29]. The ideal supermode described before will be perturbed by the random fiber length differences unavoidable in fiber arrays, on the order of a millimeter at best. With constant refractive index amplifiers, this would lead to serious degradation in the output phase distribution of the coherent array. However, with the gain-dependent phase shift described in Section III, and


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Fig. 5. Fundamental supermode of coupled cavity with highly saturated gain elements in absence of fiber path length differences.

with the fiber amplifiers acting as regenerative amplifiers due to the individual facet reflectivity, the change in gain in the fiber amplifiers can substantially compensate for these lengthdependent phase errors. A numerical computer model was generated to find the solutions of (9) taking into account a random variation of the fiber length as follows. The total length of the fiber lasers (L) is the sum of two components L = L0 + δL with nominal lengths (L0 ) equal to 20 m along with random length increments (δL) that range between 0 and 2 mm. These random length increments include the actual fiber length differences, refractive index variations, and variable operating conditions such as vibrations, temperature variations, and stress. The different length increments (δL) were chosen by a random number generator with a uniform probability distribution over the 2 mm range. Using a statistical sample of these fiber length distributions, we estimated the coherence of the laser output. In order to better understand the passive phasing mechanism, we use the individual electric fields presented in Fig. 3 as the parameters of interest in the implementation of the model. There are five equations utilized with five unknowns: E1 , E2 , E3 , E4 , and γ, where each of the electric field variables is represented by a nine-element complex vector and the gain coefficient γ is represented by a nine-element real vector E1 = r0 KE4

(10)

E2 = tE1 − rE3

(11)

E3 = exp (γL − iαγL − iδ0 ) E2

(12)

E4 = tE3 + rE1 .

(13)

Note that (10) and (13) are the same as (1) and (3), respectively. They are repeated here for clarity and consistency. We include one additional equation describing the fields in the laser as a function of the unsaturated gain that depends on the pump power and the saturation intensity |E3 | (I3 − I2 ) γ0 L = ln (14) + |E2 | Isat where γ0 indicates the unsaturated gain of the fiber laser and Isat is the saturation intensity in the laser that equals (hν)/(στ ), where h is the Plank’s constant, ν is the frequency of the laser oscillation, σ is the cross section at the lasing wavelength, and τ is the effective lifetime of the transition. Equations (10)–(14) were used as the basis of the computer model implemented to determine the phasing characteristics of the laser array. In this

Fig. 6. Electric fields within regenerative amplifiers as noted in Fig. 3. Variations in E 2 allow for variations in gain, and thus, the gain-dependent phase shift.

model, these equations were simultaneously solved using a nonlinear Levenberg–Marquardt solver. The operating wavelength was allowed to vary initially to include longitudinal mode selection in the complete laser resonator and to begin the search algorithm with a partially phased state. In this model, an array of nine elements was chosen with a random path length distribution assigned to the fibers as described before using an output coupler of R0 = r02 = 10% and an individual regenerative reflectivity R = r2 = 6%. We chose a nine-element array for this example to ensure that the cold cavity coincident wavelength search did not result in any well-phased modes of operation. Typical results of the electric fields (E1−4 ) for each of the laser amplifiers calculated using (10)–(14) are presented next and their amplitudes are plotted in Fig. 6     0654 − 0.107i −0.029 + 0.135i  0.788 − 0.129i   0.060 − 0.140i           0.900 − 0.148i   0.170 − 0.116i       0.976 − 0.160i   0.319 + 0.171i          E1 =  1.002 − 0.164i  E2 =  0.266 − 0.174i       0.976 − 0.160i   0.252 − 0.035i           0.900 − 0.148i   0.170 − 0.116i       0.788 − 0.129i   0.062 − 0.085i  0.654 − 0.107i 0.004 + 0.214i    2.98 − 1.07i 3.05 − 1.07i  3.16 + 0.06i   3.26 + 0.03i           3.17 − 0.12i   3.29 − 0.15i       2.82 − 1.46i   2.97 − 1.46i          E3 =  3.18 + 0.06i  E4 =  3.32 + 0.02i  .      3.12 − 0.54i   3.26 − 0.57i           3.17 − 0.13i   3.29 − 0.16i       3.16 − 0.19i   3.25 − 0.21i  

2.83 − 1.43i

2.90 − 1.41i

It can be seen in Fig. 6 that E1 has the exact Gaussian envelope dictated by the SF cavity and that all the components of

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Fig. 8. Predicted far field of the coherent laser array. Solid curve indicates predicted far field with 0.1 wave rms phase noise. Dotted curve indicates ideal far field with zero phase noise.

Fig. 7. Typical output electric field vector distribution predicted by the regenerative amplifier model for a nine-element fiber laser array with a random fiber length distribution. Radial lines indicate output electric field vectors pointing away from the origin.

E1 are in phase. The amplitude of E3 is determined by the saturated gain g in the amplifiers and is approximately equal for all elements. From (13), we can see that E4 is approximately equal to E3 due to the relatively small values of r and E1 compared to t and E3 . However, as can be seen in (11), there can be large variations in the amplitude of E2 due to the phase-dependent coherent interference of fields tE1 and rE3 , which can be engineered to be approximately equal through appropriate choice of reflectivity r. The phase distribution of the output electric field E4 of the coherent array is shown in Fig. 7, portrayed as vectors with their tails at the origin and pointing in a direction indicating their respective phases φi . The

results presented in Fig. 7 demonstrate a phasing function

σ ≡ | Eout |/ |Eout | equal to 0.984, indicating a coherence C equal to 0.968. In general, solutions to the model were consistently found with degrees of coherence that varied between 0.95 and 1.00. In the absence of the gain-dependent phase shift described here, the best phasing that was achieved through calculation of cold cavity phase shifts utilizing a wavelength search was equal to 0.86, indicating a coherence equal to 0.75. From this, the benefits of using regenerative feedback are evidenced by the predicted improvement in coherence of the array. V. STREHL RATIO OF ARRAY SUPERMODE As a result of the phase noise that results from the errors in the fiber lengths as described before, the resulting Strehl ratio will be reduced from unity [30]. The results of the small intensity variations shown in Fig. 7 on the Strehl ratio are negligible. In order to determine the effects of this phase noise, the far field of the coherent array was modeled using a coherent array with phase noise superimposed on the fundamental supermode. In order to compare these predictions with previous experimental data [17], this model was performed using an array with seven elements. In this model, a series of far fields was calculated

Fig. 9. Experimentally measured far field of coherent laser array. Nonunity visibility in the far field indicates the presence of phase noise in the individual laser emitters in the coherent array.

with differing phase increments added to each fiber laser using a random number generator, and the resulting far fields were averaged, simulating a noise frequency faster than the detector response. We note that the phase errors used in the model have a zero mean and are uncorrelated between the elements. The results of this model are presented in Fig. 8, where the dashed curve represents the predicted far field for a perfectly phased supermode and the solid curve indicates the predicted far field for the supermode with 0.1 waves of phase noise (∼±36◦ ). The presence of this noise does not broaden the widths of the individual peaks in the far field, but instead, contributes to a noise floor which the individual peaks are superimposed upon. This predicted response is in rough agreement with the experimentally measured far field of the array, which is presented in Fig. 9. It can be seen that the experimental data presented in Fig. 9 have reduced intensity side lobes compared with the theoretical data presented in Fig. 8. This can be explained by a slight difference in the Gaussian radius of the envelope of the far field, which only has to change by 9% to explain the discrepancy. We note that the far field presented in Fig. 9 is not suitable for most laser applications and that a suitable aperture filling technique will need to be employed for collecting the power in the side lobes into a single diffraction-limited on-axis beam. Techniques to accomplish this have been developed, successfully demonstrated [31], and projects are underway to implement these techniques for this coherent array.


As discussed in [30], the Strehl ratio (S) of such an array, assuming perfect aperture filling, will be equal to S = C + ff (1 − C)

(15)

where ff is the fill factor and C is the coherence of the array. This result states that a small amount of the incoherent portion of the output will remain on axis, and thus, will slightly increase the Strehl ratio of the array. For the experimental data shown in Fig. 9, the coherence was 0.84 and the fill factor was 0.24, resulting in a Strehl ratio of 0.88. The fact that the experimental coherence shown by Fig. 9 is less than the coherence predicted by the regenerative amplifier model indicates that there is phase noise in the system resulting from other mechanisms. These results are also consistent with the fact that the system has not been optimized in terms of output coupler reflectivity and regenerative feedback. VI. DISCUSSION These results indicate that it is possible for the gain-dependent phase shift to substantially compensate for the random path length distribution typically encountered in fiber laser arrays if the system is engineered properly, with an appropriate level of regenerative feedback built into the system. The presence of the regenerative feedback interferes the light injected into a given amplifier from the external cavity (E1 ) with light coupled back into the amplifier after a single round-trip through the amplifier (E3 ). Depending on the relative phase of these two signals, an increase in the cold cavity phase shift (path length) of a given amplifier results in a net increase/decrease of the total feedback to that fiber amplifier (E2 ). By appropriate choice of the output coupler reflectivity (r0 ) compared to the regenerative feedback (r), we can tailor the relative contributions of E1 and E3 to E2 , which will control the variation in the magnitude of E2 resulting from the relative phase of E3 to E1 . In particular, if the contributions to E2 from E1 and E3 are of similar amplitude, then E2 could vary from almost twice its nominal value down to near zero. Due to the fact that the output powers of the amplifier (P3 and thus E3 ) are maintained approximately constant due to the highly saturated gain, the variations in the field E2 will result in variations in the gain of the amplifier g = |E3 |/|E2 |. These changes in gain result in variations in the phase shift within the amplifiers, which thus change the output phase distribution of the array. As described previously [23], this will result in the presence of stable operating modes with output phase distributions significantly better than that obtained with cold cavity models and with varying degrees of phase coherence. Operating modes with high degrees of phasing will achieve higher levels of feedback compared to modes that have low degrees of phasing, thus achieving higher round-trip gain than poorly phased modes. REFERENCES [1] IPG Photonics. (2008). [Online]. Available: www.ipgphotonics.com [2] M. Wickham, J. Anderegg, S. Brosnan, D. Hammons, H. Komine, and M. Weber, “Coherently coupled high power fibre arrays,” in Proc. SPIE, Fiber Lasers III: Technol., Syst., Appl, vol. 6102, 2006, pp. 3–5.

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Christopher J. Corcoran received the Ph.D. degree in physics from Massachusetts Institute of Technology (MIT), Cambridge. He is currently with Corcoran Engineering, Inc., Waltham, MA. His current research interests include fiber optic sensors, electromagnetic shielding, permanent magnet design, automotive actuators, microelectromechanical (MEM) systems, and railroad applications.

Frédéric Durville (M’03) was born in France in 1960. He received the Ph.D. degree in materials sciences from the Université Claude Bernard, Lyon, France, in 1984. He was the Research Manager at Laser Application, France, where he was responsible for the development of high average power industrial Nd:YAG lasers (over 500 W) and their applications. He was the Director of Advanced Research and Development at Cynosure, Inc., where he was responsible for the development of new medical laser systems and applications. In 2000, he founded Optical Fiber Systems, Inc., Chelmsford, MA, where he is currently the President, and developed, manufactured, and marketed optical fiber assemblies and laser systems. He has authored or coauthored more than 25 publications, and holds four patents on lasers and laser applications. Dr. Durville is a member of the International Society for Optical Engineers (SPIE), the Optical Society of America (OSA), and the Directed Energy Professional Society (DEPS), and was selected by the Air Force Research Laboratory (AFRL) to be a member of a panel of experts during the 2000 Solid-State and Diode Laser Technology Review (SSDLTR) Conference, Albuquerque, NM.