Thermal Effects in High-Power Continuous-Wave Single ... - IEEE Xplore

IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 20, NO. 5, SEPTEMBER/OCTOBER 2014

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Thermal Effects in High-Power Continuous-Wave Single-Pass Second Harmonic Generation Saeed Ghavami Sabouri, Suddapalli Chaitanya Kumar, Alireza Khorsandi, and Majid Ebrahim-Zadeh, Member, IEEE

Abstract—We present a theoretical model which describes the effects of thermal load distribution on single-pass second harmonic generation (SP-SHG) of high-power continuous-wave (cw) radiation in MgO:sPPLT nonlinear crystal to provide green output at 532 nm. Numerical simulations are performed based on real practical values and actual operating conditions associated with a recent SP-SHG experiment, generating 10 W of cw green radiation using a Yb-fiber laser. The model is used for four oven configurations to simulate the implications of thermal effects on SH power. The observed asymmetric feature of the phase-matching curves, particularly at higher fundamental powers up to 50 W, are characterized and explained by considering the generation of heat due to crystal absorption. The concept of optical path difference (OPD) is introduced to study the formation of thermal lens and its effects on the displacement of focal point inside the thermally loaded crystal. We further study the dependence of the SH power on the different oven schemes by increasing the input fundamental power up to 50 W. It is found that a top-sinked oven design is the optimum configuration for achieving maximum SHG efficiency without saturation. Comparison of the simulation results with experimental data confirms the validity of the theoretical model. Index Terms—Nonlinear optical devices, nonlinear materials, second harmonic generation, thermal effects.

I. INTRODUCTION IGH-POWER continuous-wave (cw) laser sources in the green are of great interest for a wide range of applications including human surgery [1], display technology [2], [3], for pumping of Ti:sapphire lasers [4], as well as optical parametric oscillators (OPOs) [5]. Such applications require sufficiently

H

Manuscript received September 30, 2013; revised November 5, 2013 and November 18, 2013; accepted December 9, 2013. Date of publication December 18, 2013; date of current version January 15, 2014. This work was supported by the Ministry of Science and Innovation, Spain, under Project OPTEX (TEC2012-37853) and the Consolider program SAUUL (CSD200700013). This work was also supported by the European Office of Aerospace Research and Development (EOARD) under Grant FA8655-12-1-2128 and the Catalan Agència de Gestió d’Ajuts Universitaris i de Recerca (AGAUR) under Grant SGR 2009-2013. S. Ghavami Sabouri and A. Khorsandi are with the Department of Physics, University of Isfahan, Isfahan 81746-73441, Iran (e-mail: [email protected]; [email protected]). S. Chaitanya Kumar is with ICFO-Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860 Castelldefels, Barcelona, Spain (e-mail: [email protected]). M. Ebrahim-Zadeh is with ICFO-Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860 Castelldefels, Barcelona, Spain and also with the Institucio Catalana de Recerca i Estudis Avancats (ICREA), Barcelona 08010, Spain (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.2013.2295359

high output power, excellent beam quality, and long-term stability. In the absence of suitable laser gain materials, nonlinear optical techniques, in particular second harmonic generation (SHG) [6], [7] of 1-μm band solid-state lasers, have been established as the most viable approach to provide green radiation. A practical approach, which offers great simplicity and reduced cost, is external single-pass SHG (SP-SHG) in quasiphase-matched (QPM) ferroelectric crystals [8]. Crucial to the attainment of practical powers and efficiencies in this scheme is the ability to exploit long interaction lengths under noncritical phase-matching and access to the highest nonlinear tensor coefficients. Among the QPM crystals, periodically-poled LiNbO3 (PPLN) has been the most widely established material, due to a mature fabrication technology, large effective nonlinearity (deff ∼16 pm/V), and widespread availability in interaction lengths up to 80 mm, but its potential for SHG into the visible is constrained by photorefractive damage. Doping with MgO increases this damage threshold, but stable generation of visible radiation is still increasingly problematic at higher powers [9]. Periodically-poled KTiOPO4 (PPKTP) is another QPM material candidate for green generation, offering moderate nonlinear coefficient (deff ∼10 pm/V) and wide temperature acceptance bandwidth, but its low thermal conductivity makes it more sensitive to thermal effects, limiting its use at higher powers [10]. An alternative QPM material with excellent properties for SPSHG into the green is MgO-doped stoichiometric periodicallypoled LiTaO3 (MgO:sPPLT). It also possesses a moderate effective nonlinear coefficient (deff ∼9 pm/V), but offers superior thermal conductivity and increased resistance to photorefractive damage than both MgO:PPLN and PPKTP [11]. Using SP-SHG of a high-power cw Yb-fiber laser in sPPLT, 19 W of diffraction-limited radiation at 532 nm was generated at 15% conversion efficiency [12]. Our recent studies have also shown MgO:sPPLT to be a promising candidate for cw SP-SHG into the green [13]–[15]. Using a 30-mm-long MgO:sPPLT crystal and a cw Yb-fiber laser, we achieved a cw SP-SHG conversion efficiency of 32.7%, providing 9.6 W of green power at 532 nm for 29.5 W of fundamental power [16]. Subsequently, by deploying cascaded multicrystal scheme based on 30mm-long MgO:sPPLT crystals, we achieved efficient cw SPSHG of the same Yb-fiber laser, providing more than 13 W of cw output power at 532 nm at 55% conversion efficiency [17]. With the rapidly continuing advances in cw fiber lasers offering unprecedented optical powers near 1 μm, and further potential for power-scaling, SP-SHG schemes offer increasing potential for the realization of compact, high-power cw green sources in simplified, cost-effective, and practical designs.

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At the same time, the attainment of maximum efficiency and green output power in SP-SHG with high stability and optimum beam quality, at increasing fundamental powers, is ultimately limited by thermal loading in nonlinear crystals. Essentially, due to the absorption at the fundamental and SHG wavelengths in the nonlinear crystal, strong thermal de-phasing of the interacting waves leads to saturation of SH power at higher fundamental powers. This phenomenon has been observed and studied in several earlier reports [18]–[20]. Owing to the comparatively low thermal conductivity of nonlinear materials, it is difficult to maintain perfect phase-matching condition along the full interaction length of the crystal. Particularly, in the presence of thermal loading due to finite absorption, it is even more challenging to maintain the entire nonlinear crystal at a constant temperature, using a simple thermoelectric oven. Moreover, tight focusing of the fundamental radiation inside the medium, and the variation of its beam waist along the direction of propagation through the nonlinear crystal, results in a non-uniform 3-dimensional thermal distribution. In addition, as the major part of SHG radiation is generated in the radially central section of the crystal and amplified towards the output end of the crystal, the absorption of SHG will further exacerbate the non-uniformity of thermal distribution inside the medium. As a result, temperature variation along the fundamental beam propagation causes the perfect phase-matching condition to be confined to a particular zone, while in other parts of the crystal it cannot be fulfilled due to the thermal de-phasing. In a recent work, such an inhomogeneous temperature distribution of 0.2 K between the center and entrance face of a 3-cm-long PPLN crystal was measured, while generating 1.3 W of SH power at 488 nm using 9.4 W of fundamental power in single-pass configuration [21]. In order to minimize thermal non-uniformity and control the thermal saturation effect, the authors used three resistance heaters connected directly to individual Pt100 detectors. Such a special and intricate scheme was then capable of increasing the SHG saturation level. Accordingly, many theoretical approaches and simulation models have been developed to investigate and characterize the thermal effects in high-power cw SHG observed in the experiments [22]–[24]. By solving the coupled-wave equations for SHG in combination with heat equations, it has been shown that in the case of PPLN and PPLT nonlinear crystals, the maximum attainable SH power is limited by thermal loading at high input powers [25]. An approximated theoretical model has also been proposed [26], to verify observed experimental results in PPKTP by considering the effect of two-photon nonlinear absorption on de-phasing in the crystal, and subsequently on the saturation of SHG output power. However, using an analytical model to simulate the thermal distribution, the authors made a relatively large number of approximations, resulting in significant deviation from the experimental results. In this study, for the first time to our knowledge, the coupledwave equations for SHG and the heat equations are solved exactly and simultaneously, in the presence of linear and nonlinear absorption of the fundamental and second harmonic (SH) waves, without approximations. Thermal saturation effect and the observed asymmetry in temperature phase-matching curve, which

have been reported in many experiments, are deduced from the presented model. By using the concept of optical path difference (OPD), thermal lensing in the nonlinear crystal is fully characterized and introduced as one of the major effects contributing to the SH power saturation. We show that by proper choice of oven configuration for the nonlinear crystal, the level of saturation can be passively controlled at moderate input fundamental powers. II. THERMAL MODEL FOR SHG In order to model the detrimental thermal effects in a typical SHG process, we start by rewriting the coupled-wave equations for SHG in the presence of linear and nonlinear absorption of fundamental and SH beams, while including a differential equation describing the heat transfer inside the medium. Since the linear absorption coefficient of the fundamental is nearly an order of magnitude lower than that of the second harmonic, the two-photon absorption at the fundamental wavelength is negligible, and hence can be ignored in the model. Unlike earlier presented models [22]–[26], we include the diffraction term in the coupled-wave equations in order to consider the effects of beam waist variation, and hence to specify the thermal loading area. Therefore, the coupled-wave equations governing the SHG process become: i 2deff ωf dEf = − Es Ef∗ ∇2 Ef + i dz 2kf T nf c × exp −i

z =z

z =0

1 Δk(z , T )dz − αf Ef 2

i dEs 2deff ωs 2 = − Ef ∇2T Es + i dz 2ks ns c z =z 1 × exp i Δk(z , T )dz − (αs + β |Es |2 )Es 2 z =0 (1) where Ef ,s , nf ,s and αf ,s are the field amplitude, refractive index and linear absorption of the nonlinear crystal for the fundamental and SH wave, respectively, while deff and β are the effective nonlinear coefficient and nonlinear absorption of the SH wave in the crystal, respectively. Here, ∇2T denotes the transverse Laplacian acting on x- and y-axis, normal to zaxis as propagation direction. This term is responsible for the beam diffraction and Δk is the temperature- and wavelengthdependent phase-mismatch parameter, given by 2nf ns 1 − − Δk = 2π λf λs Λ where Λ is the grating period of the QPM nonlinear material. The heat equation associated with the absorption of fundamental and SH beams can be expressed as [25]: ∇2 T(r) = Q (r) /K Q (r) = −

2C εo 2 2 4 (α f n f |E f (r)| + α s n s |E s (r)| + βn s |E s (r)| ) K

(2)

SABOURI et al.: Thermal Effects in High-Power CW Single-Pass Second Harmonic Generation

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TABLE I CHARACTERISTICS OF FUNDAMENTAL AND SHG BEAMS BASED ON MGO:SPPLT CRYSTAL ASSOCIATED WITH THE REPORTED EXPERIMENT [13]

Fig. 1. Schematic representation of the four different heating configurations used in the simulation: (a) oven-surrounded scheme, (b) open-top, (c) close-top, and (d) top-sinked scheme. The propagation direction of the fundamental beam is perpendicular to the page.

where the constant parameters, C and K, are specific heat and thermal conductivity of the nonlinear crystal, respectively, and Q(r) is the thermal load per unit volume inside the medium, which can be considered as heat source due to absorption. By simultaneously solving Eqs. (1) and (2), heat distribution in the crystal volume, as well as fundamental and SHG intensities across the medium, can be obtained. In addition, phasemismatch due to inhomogeneous heating of the crystal can be evaluated. However, thermal distribution, and consequently the phase-matching condition, will be altered by the generated SH and fundamental waves. Therefore, voluminous and repetitive calculations must be performed until thermal distribution and field intensities reach steady-state values. For faster calculations, and to reach the required accuracy, parallel computing toolbox in MATLAB environment is implemented to transform x and y coordinates to the Fourier space and restoring the data in the real space. In solving the above equations by split-step Fourier method, initial values and boundary conditions are required. We assume that the SH wave is initially at the quantum noise level at the crystal entrance and fundamental wave has Gaussian intensity distribution with a finite beam waist. Boundary condition for the heat equation is strongly dependent on the oven configuration, which in turn determines the mechanism of heat transfer between the crystal and its surrounding environment. Assuming the crystal has a rectangular cross-section, several heating configurations may be employed, depending on the particular oven design, as illustrated in Fig. 1: 1) Oven-surrounded scheme: Crystal sides are enclosed by the oven while the entrance and exit facets of the crystal are open, enabling direct heat convection between the crystal and its surrounding environment, Fig. 1(a). 2) Open-top scheme: The bottom of the crystal is in contact with the oven and all other sides are in convection with surrounding air, Fig. 1(b). 3) Close-top scheme: Open-top scheme with a heat isolator on the top to prevent the heat escaping from the crystal, Fig. 1(c). 4) Top-sinked scheme: Open-top scheme with a heat-sink on the top to direct the heat flow toward the environment, Fig. 1(d).

Heat convection between the crystal and its surrounding environment can be described by the following expression: ∂T (x, y, z) = h[T (x, y, z = 0) − T0 ] (3) −K ∂z z =0 where h is known as the convection coefficient, which depends on the utilized oven configuration, T0 is the constant temperature of the environment, and K is the thermal conductivity of the nonlinear crystal. The temperature non-uniformity inside the thermally loaded area is attributed in part to the oven configuration. In addition, beam waist variation of focused fundamental and generated SH radiation along the propagation direction (z-axis) also play an important role in the non-uniform heat distribution in the crystal. Thus, the mesh size of the crystal must be reduced as much as possible to include very small changes in the calculations, which in turn drastically increases the computation time. In order to save time, we used very small mesh size, as compared to the two beam waists, at the center of the crystal, and relatively larger mesh outside the region near the end faces of the crystal. III. SIMULATIONS AND RESULTS To compare the experimental results obtained earlier with our simulation, we performed the calculations for SP-SHG in MgO:sPPLT as the nonlinear crystal and used the actual practical values associated with the experimental setup in Ref. [13], which are listed in Table I. In order to simultaneously solve the coupledwave and heat equations, Eqs. (1) and (2), we used the relevant Sellmeier equations for MgO:sPPLT reported in Ref. [27], to calculate the required phase-mismatch parameter, Δk. Fig. 2 shows the schematic of the nonlinear crystal and the respective coordinate axes used in this investigation. The input power and focusing parameter, ξ, play an important role in obtaining the optimum conversion efficiency. The focusing parameter is defined as ξ = 2Lc /kw02 , where, Lc and w0 are the crystal length and beam waist, respectively. According to the Boyd and Kleinman criteria [29], the maximum efficiency for a typical SHG interaction in the absence of double-refraction occurs at ξopt = 2.84. However, in our earlier experiments, the fundamental beam waist at the center of crystal is chosen such that the Rayleigh range of the fundamental is equal to

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Fig. 2. Schematic of the nonlinear crystal with the corresponding coordinate axes used to investigate the effect of thermal loading due to the tight focusing of interacting beams. Fundamental beam is focused at center of the crystal and the variation of the beam waist is included in the model. Thermal distribution in the crystal is monitored along the x-cut and y-cut planes.

Fig. 3. Simulated temperature tuning curves at different fundamental power levels for the open-top configuration as used in the experimental work reported in Ref. [13]. Each data point is a simulation and the solid curves are the fit to the simulation data.

the crystal length. To account for this, and achieve consistency with experimental conditions, we introduce a more practical parameter, ξRay . By using the values from Table I, we arrive at ξRay = 1.23. At low power levels, where the absorption of fundamental and SH beams can be neglected, the SHG phasematching condition can be readily achieved by adjusting the crystal temperature in a given oven configuration. However, as the input fundamental power is increased, the phase-matching temperature deviates from the set condition and, hence, to maintain maximum SH power, the temperature of the oven has to be precisely re-adjusted. This optimization procedure is commonly necessary in all high-power SHG experiments. Using our theoretical model, we can calculate the behavior of the SHG phase-matching profile based on real experimental parameters. In Fig. 3, we show simulation results for the temperature dependence of normalized SH power at three different levels of input fundamental power, assuming ξRay = 1.23. This simulation is performed for the open-top configuration, depicted in Fig. 1(b). During this simulation, the temperature of the nonlinear crystal is always optimized for maximum SH power at each step, similar to the experimental optimization procedure. The data points in Fig. 3 correspond to the results from the simulation using temperature steps of 0.1 ◦ C, similar to that used in our experiments [13], [16]. The temperature tuning curves provide crucial information about the thermal loading of the nonlinear crystal, originating from various sources such as absorption at the fundamental and SH wavelengths. The shift in the peak of the temperature tuning curve towards the lower temperature, with increased fundamental power level, is the primary indication of thermal loading

in the nonlinear crystal. Indeed, as can be seen in Fig. 3, the absorption of the fundamental and SHG beams, particularly at higher fundamental powers, cause the peak of the phasematching curve to be displaced toward lower temperatures, underlining excellent consistency between our theoretical model and the experimental results reported in Ref. [13]. For a fixed fundamental power of 1 W, the maximum SH power is achieved at an oven temperature of 57.1 ◦ C, while this temperature reduced down to 56.3 ◦ C for an increased fundamental power of 50 W, indicating that the nonlinear crystal experiences a temperature rise of 0.8 ◦ C due to the increased fundamental and SH power levels. Another important parameter for efficient SHG is the temperature acceptance bandwidth. Ideally, a large temperature acceptance bandwidth is desirable to ensure that the generated SH power is less susceptible to any temperature fluctuations. As evident from Fig. 3, the temperature acceptance bandwidth for a fixed fundamental power of 1 W is estimated to be 0.95 ◦ C, which decreases to 0.56 ◦ C for a fundamental power of 50 W. Fig. 3 also shows that the temperature tuning curve tends to become increasingly asymmetric at higher fundamental powers, a behavior that has also been reported in earlier experimental works [13], [26]. This asymmetry in the tuning curve is another indication of thermal loading in the nonlinear crystal and leads to a reduction in the temperature bandwidth, thus making the SH power highly vulnerable to any temperature fluctuations while operating at high power levels. Hence, it is worthwhile to study the various issues associated with the origin of the thermal loading in the nonlinear crystal. A physical explanation for the asymmetry in the temperature tuning curve can be understood by a careful observation of Fig. 3. Let us consider the temperature tuning curve for a fixed fundamental power of 50 W and scan the temperature from low to high values. As we increase the oven temperature from T , by an amount of ΔT , approaching the phase-matching temperature of the nonlinear crystal, we begin to generate more SH power. As the fundamental power is kept constant, the heat generated due to the absorption of the fundamental raises the crystal temperature to a steady-state value. However, the absorption at the generated SH wavelength results in additional rise in the temperature, which pushes the temperature of the crystal even closer to the phase-matching temperature, leading to the generation of more SH power. This process continues until the system reaches steady-state. The net change in the temperature of the crystal is ΔT , which is greater than ΔT . Hence, an increase of ΔT in the oven temperature from T results in a new crystal temperature of T + ΔT . As a result, the SH power reaches the maximum value very quickly. Further increasing the temperature of the oven beyond the phase-matching temperature results in the decrease in the SH power, which in turn reduces the generated heat, and hence curbs the rise in the crystal temperature. It is to be noted that the value of ΔT is initially close to ΔT , but begins to significantly deviate from ΔT as we approach the phase-matching temperature. As we move past the phase-matching temperature, the value of ΔT begins to approach ΔT again. The variation in the value of ΔT on either side of the phase-matching temperature is asymmetric due to the thermal load in the nonlinear crystal. A similar argu-


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Fig. 4. Saturation of SHG efficiency at higher pumping levels for both focusing parameters, ξR ay and ξo p t . The crystal oven is operating in open-top scheme and its temperature is adjusted to obtain phase-matching temperature of 56.9◦ C. Calculations are performed for two different conditions with and without absorption of fundamental and SH beams inside the MgO:sPPLT crystal.

ment also holds when we scan the temperature of the oven from high to low values. The generation of high SH power depletes the fundamental, reducing the thermal load due to the fundamental power. However, given the ∼9 times higher absorption coefficient at the SH wavelength, the amount of heat generated, and hence the rise in the temperature of the nonlinear crystal, is dominated by the SH power. It is quite reasonable to expect that if the optimum phase-matching temperature can be maintained when increasing the fundamental power up to about 50 W, higher SH powers can be achieved and conversion efficiency can be enhanced. However, this expectation is not consistent with the observed experimental results, where at a certain input fundamental power, the SH power begins to saturate [26]. This is partly associated with the absorption of fundamental and SHG beams leading to non-uniform thermal load inside the nonlinear crystal. It can be explained by following the basic dependence of the refractive indices and the resultant phase-mismatch, Δk, on the adjusted temperature, which leads to the dependence of the SHG efficiency on the generated heat inside the medium at higher fundamental powers. In Fig. 4 the effect of different pumping levels on the SH power saturation is simulated for theoretical and practical confocal parameters, ξopt and ξRay , respectively, with and without considering the absorption at the fundamental and SHG wavelengths. As illustrated in Fig. 4, in case of the practical confocal parameter, ξRay , the SHG efficiency deviates by ∼8.6% at a fundamental power of 20 W, beyond which a significant saturation and roll-off is clearly observed. This is due to the growth of thermal effects inside the crystal, which increase the absorption of fundamental and SHG beams at higher input powers. In this case, the maximum attainable SH power is achieved for smaller practical ξRay compared with ξopt . This is because, for a larger fundamental beam waist, the generated heat can be distributed in a larger volume around the center of the crystal, and hence, the SH power is less significantly affected by the thermal dephasing phenomenon. Therefore, with the characterization of thermal distribution inside the crystal, one can investigate the effect of the generated heat on the SHG saturation. It can be further observed that in case of ξopt the SHG intensity saturates faster than in the case of ξRay . This results in lower efficiency when using the theoretically optimum confocal parameter, ξopt , which can be associated with the stronger thermal effects due to the tighter focusing of fundamental beam.

Fig. 5. Simulation results for y-cut thermal distribution inside the ovensurrounded MgO:sPPLT nonlinear crystal pumped by 5 W and 50 W for the absorption at (a) fundamental, (b) SHG, (c) both fundamental and SHG wavelengths, and (d) for no absorption inside the crystal. Oven temperature is set at 56.9 ◦ C. The practical confocal parameter, ξR ay , is used in the simulation.

Fig. 6. Simulation results for y-cut thermal distribution inside the open-top MgO:sPPLT nonlinear crystal pumped by 5 W and 50 W for the absorption at (a) fundamental, (b) SHG, (c) both fundamental and SHG wavelengths, and for (d) no absorption inside the crystal. Oven temperature is set at 56.9 ◦ C. The practical confocal parameter, ξR ay , is used in the simulation.

Further, the heat distribution is strongly dependent on the input fundamental power and the oven configuration. To clarify the contribution of fundamental and SHG absorption in the crystal, we have simulated the heat distribution at two different pumping levels for the four oven configurations under four conditions: (a) when there is only absorption of fundamental; (b) when there is only absorption of SHG; (c) when there is no absorption; and, finally (d) when there is absorption for both fundamental and SHG. The results are shown in Figs. 5–8. It is to be noted that

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Fig. 9. Temperature variations along z-axis versus the MgO:sPPLT crystal length for different pumping levels. Oven configuration is open-top and focusing parameter is ξR ay .

Fig. 7. Simulation results for y-cut thermal distribution inside the close-top MgO:sPPLT nonlinear crystal pumped by 5 W and 50 W for the absorption at (a) fundamental, (b) SHG, (c) both fundamental and SHG wavelengths, and (d) for no absorption inside the crystal. Oven temperature is set at 56.9 ◦ C. The practical confocal parameter, ξR ay , is used in the simulation.

Fig. 8. Simulation results for y-cut thermal distribution inside the top-sinked MgO:sPPLT nonlinear crystal pumped by 5 W and 50 W for the absorption at (a) fundamental, (b) SHG, (c) both fundamental and SHG wavelengths, and (d) for no absorption inside the crystal. Oven temperature is set at 56.9 ◦ C. The practical confocal parameter, ξR ay , is used in the simulation.

these color-scale plots should be viewed together with the temperature scale to obtain a clear understanding of the thermal distribution in the crystal. As can be seen from following Figs. 5(a) to 8(a), in the case of fundamental beam absorption only, for all oven configurations, the generated heat is almost concentrated in the center of MgO:sPPLT crystal at both low and high fundamental powers. On the other hand, in the case of SHG beam absorption only, as shown in Figs. 5(b) to 8(b), by increasing the fundamental power

up to 50 W, the thermal load area is dramatically deformed and begins to divide into two different zones, with the stronger thermal zone tending toward the output end of crystal where the nonlinear crystal is subjected to increasing SHG intensity. This is to be expected, as the crystal has stronger absorption at the SHG wavelength. In contrast, at the lower input power of 5 W, heat distribution is determined only by fundamental beam, as the SHG intensity is too low to affect the thermal load in the crystal. Moreover, in oven configurations shown in Figs. 5(c), 6(c) and 8(c), even at 50 W of input fundamental power, the generated heat in the second zone, near the output end of crystal, can be more easily exchanged with surrounding environment. As a result, the SHG efficiency can be increased almost without influencing the phase-matching condition. In fact, comparison of Figs. 5(b)–8(b) and Figs. 5(c)–8(c) shows that thermal effects are dominated by absorption at the second harmonic wavelength and the maximum temperature is always recorded towards the output end of the crystal, except in case of the top-sinked oven configuration where thermal lensing and de-phasing are minimized due to efficient extraction of the excess heat enabled by the proper design of the oven. Fig. 9 shows the temperature variation along the length of the MgO:sPPLT crystal (z-axis), when it is configured in opentop oven scheme, Fig. 6(c). As can be seen, for a fundamental power of 1 W, where the generated heat is negligible, the thermal distribution inside the medium is uniform. By increasing the input power toward 50 W, the generated heat either on axis or near the output end of the crystal increases, and the thermal distribution pattern gradually begins to change. Moreover, the illustrated results show that at higher fundamental powers the temperature maximum corresponding to the focal point is displaced toward the entrance face of the crystal. This can be considered as the most important finding of this work. It leads to the introduction of a novel interpretation, namely, the formation of a thermal lens inside the medium due to the refractive index gradient induced by the thermal distribution. Therefore, it can be concluded that at higher fundamental powers, the greater shift in the temperature maximum is due to the decrease in the focal length associated with the formed thermal lens. This results in the movement of the beam waist toward the entrance face of the crystal, as well as a drop in the SHG efficiency. This is also consistent with Boyd and Kleinman theory [29], where


the maximum SHG efficiency is obtained when the fundamental beam waist is focused at the center of the nonlinear crystal. In summary, therefore, the saturation and drop in the SHG output power at higher input powers can be understood in terms of two inter-dependent explanations: 1) Non-uniformity of thermal distribution in the nonlinear crystal, resulting in the formation of non-zero phasemismatch parameter, Δk. At the same time, perfect phasematching condition cannot be established because at each point in the crystal the temperature varies during the pumping process. 2) Thermal lens formation and movement of fundamental beam waist from the center of nonlinear crystal along the propagation direction. In the following section, we investigate the effects of thermal lensing and OPD values on the SHG conversion efficiency.

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Fig. 10. Calculated OPD values for different level of pumping power in the xcut plane of thermally loaded MgO:sPPLT nonlinear crystal. Oven configuration is the open-top scheme and the practical focusing parameter is ξR ay = 1.23.

IV. OPD AND THERMAL LENSING As discussed above, thermal loading causes a movement of the focal point inside the crystal. To investigate the fundamental basis of this effect, we start by introducing the concept of OPD and thermal lensing effect inside a thermally loaded nonlinear crystal. Thermal lensing in a nonlinear crystal can be better understood by considering the heated crystal as a thin lens with a certain focal length. Thermally induced OPD is a complex quantity, because it is associated with the non-uniform formation of refractive index inside the medium. It causes refractive index variation, bulging out of the end-faces, and stress-induced birefringence inside the crystal. Of the three effects, the latter is less important and may be neglected, so that the OPD distribution in a small slice lying between z to z + dz can be written as [30] dn(r, T ) (T (r) − T (0))dz (4) dOP D = n(r, T )α(T ) + dT where n(r, T ) is the refractive index, r is the radial distance from the axis, T is the local temperature, and α(T ) = 2.2 × 10−6 (T − 298) − 5.9 × 10−9 (T − 298)2 is the thermal expansion coefficient of the MgO:sPPLT nonlinear crystal [31]. By integrating Eq. (4) over the entire length of the crystal, OPD can be computed using the obtained information concerning the temperature distribution inside the medium. It strongly depends on the temperature distribution and can be affected by local expansion and refractive index variation. Fig. 10 shows the OPD calculated for different fundamental powers in the x-cut plane for the open-top oven scheme. In a medium possessing greater OPD values, thermal effects are quite significant and thermal lensing becomes a very important issue, particularly in the attainment of efficient SHG when high-power lasers are utilized. The OPD of an ideal lens is parabolic and it can be associated with an equivalent thin lens with a focal length of [32] OP D (r) − OP D (0) = −

r2 2f

(5)

Fig. 11. Fitting an appropriate parabola in the region of parabolic trend of OPD curve indicated in Fig. 10 for 30 W input power. The fitting is performed for r = w 0 , 2w 0 , 3w 0 . The fundamental beam waist at the focal point is assumed w 0 = 50.17 μm, corresponding to ξR ay = 1.23.

where f is the focal length of the equivalent lens. For example, the shift in the fundamental beam waist shown in Fig. 6 can be associated with a thermal lens inside the crystal with a measurable focal length. In order to achieve the equivalent focal length of the formed thermal lens inside the medium, the regions of OPD values that coincide with parabolic trends of the plotted curves in Fig. 10 are preferred. Fig. 11 illustrates the results of fitting an appropriate parabolic curve to the calculated OPD values inferred from Fig. 10 for a fundamental power of 30 W and for various radial distances from the z-axis. As can be seen from the plot, due to the development of a non-uniform OPD in the thermally loaded crystal, in the region with a diameter equal to the fundamental beam waist, the trend of OPD is parabolic. It is found that the size of parabolic region is independent of pumping level and the oven configuration used. It is due to the fundamental beam profile, which keeps the thermal distribution unchanged in the region of beam waist. Therefore, by equating r = w0 in Eq. (5), one can obtain the focal length of the formed thermal lens for different input powers and oven schemes. The equivalent focal length of the thermal lens formed inside the thermally loaded MgO:sPPLT is calculated, and shown in Fig. 12. As clearly evident from the plot, the focal length of equivalent thin lens exponentially decreases with increasing fundamental power. This can be connected to the results in Fig. 5, where the focal point is seen to move toward the input face of the crystal. For the confocal parameter, ξRay , as the fundamental power increases from 5 to 50 W, the thermal lens focal length decreases from 4.66 to 0.37 mm, while for ξopt , it decreases from 2.78 to 0.14 mm. We further observed that in both cases, tighter focusing of the fundamental beam in the nonlinear crystal

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Fig. 12. Calculated equivalent focal length of the formed thermal lens with increasing fundamental power up to 50 W. The calculation is performed for two focusing parameters, ξR ay and ξo p t , and assuming an open-top oven scheme.

leads to stronger thermal lensing. As a result, the variation of the thermal lens focal length with fundamental power is the factor responsible for the displacement of the focal point observed in Fig. 9. Therefore, the OPD approach provides a reliable method in the physical treatment and explanation of the reduction of SH power in a thermally loaded crystal. V. OPTIMUM OVEN CONFIGURATION To achieve high-power SHG radiation at high efficiency, it is thus necessary to find a solution to compensate and control thermal effects. In this section, through the design of different oven configurations as a passive solution, we attempt to minimize thermal non-uniformities in high-power cw SP-SHG. To this end, we modeled the four different oven configurations to obtain maximum SHG efficiency inside a thermally loaded MgO:sPPLT crystal by applying the OPD approach and investigated the thermal lensing effects. It is worthwhile to note that temperature distribution in the x-cut plane is symmetric, while in the y-cut plane it is asymmetric. Figs. 13(a) and (b) illustrate the calculated OPD values in x- and y-cut planes of the crystal for the four different oven configurations. As evident in Fig. 13, the OPD values are as low as ∼0.06 μm in both planes when using top-sinked oven configuration. This means that the generated heat can be further released from the crystal and, therefore, the SHG output power is minimally affected by thermal loading in this scheme. In contrast, the OPD value is ∼1.3 μm for closetop scheme, which in comparison shows a growth of ∼20 times. However, due to the asymmetric thermal distribution in the ycut plane of the crystal, except for the oven-surrounded scheme, OPD trends are also almost asymmetric. Since we are dealing with a region of radius equal to the focused beam waist, as confirmed by the results in Fig. 8, without loss of generality, it is quite reasonable to assume the same OPD values for the two planes. Based on the above assumption, Fig. 14 depicts the variation of thermal lens focal length on a logarithmic scale, as a function of input fundamental power, for the four oven configurations included in the simulation. As can be seen from the plot, when the crystal is supported by the top-sinked scheme with free facets, its temperature can be better controlled, and thus an equivalent focal length of less than ∼100 mm and ∼6 mm is formed for 5 and 50 W of fundamental power, respectively. However, in the close-top and open-top

Fig. 13. Calculated OPD values in (a) x-cut, and (b) y-cut planes for different oven configurations. The input fundamental power is set for 30 W and practical focusing parameter is ξR ay = 1.23.

Fig. 14. Variation of the focal length of thermal lens with different input powers for four oven configurations. Confocal parameter is ξR ay = 1.23.

configurations, the thermal load causes the formation of a very strong lens with varying from 4.67 to 0.35 mm when the fundamental power is increased from 5 to 50 W. This, in turn, leads to the generation of SHG radiation with reduced efficiency. This can be explained by the displacement of the fundamental beam waist from the center of crystal. Therefore, the maximum SH power can be achieved by use of an optimum oven configuration to minimize the thermal effects. Finally, the effect of using different oven configurations on the SHG efficiency is shown in Fig. 15. As can be seen in Fig. 15, in the top-sinked scheme, the SHG efficiency has a similar behavior to a non-absorbing crystal. It deviates from the non-absorbing case only by 3% at 20 W of fundamental power, and subsequently no saturation effect is observed throughout the full range of pumping levels up to 50 W. On the other hand, for the oven-surrounded and open-top schemes, the SH power begins to saturate at fundamental powers beyond 20 W and the deviation from the non-absorbing case at 20 W of fundamental power is 6.76% and 8.82%, respectively.


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generating significantly high SH power without saturation, through more effective and rapid exchange of the generated heat with the surrounding environment. Simulation results have confirmed the validity of the presented model and remarkable consistency with previously obtained experimental results.

REFERENCES Fig. 15. Variation of SHG efficiency with input fundamental power for the different oven configurations. The practical confocal parameter is ξR ay = 1.23.

In contrast, for the close-top oven scheme, saturation occurs even at lower input powers with a deviation as large as 48.23% at 20 W of fundamental power. In the close-top oven configuration, the drop in SHG efficiency beyond the saturation level is also more rapid than the other configurations, which is consistent with the simulated results shown in Figs. 5–8. The results, therefore, show that using a top-sinked oven scheme provides the optimum configuration for exchanging the generated heat in the crystal with the surrounding environment, thus maximizing the SH power and efficiency when pumping at high input powers, without significantly altering the phase-match condition. VI. CONCLUSIONS In conclusion, we have developed a theoretical model to investigate the effects of thermal loading and heat distribution in cw SP-SHG based on MgO:sPPLT crystal, for the generation of high-power green radiation at 532 nm. The study has been performed by deriving exact numerical solutions of the coupled-wave equations for SHG combined with a standard heat equation, using real experimental parameters reported in Ref. [13]. Real boundary conditions, as well as different heating oven configurations, were used to characterize the asymmetric behavior of the phase-matching curve, and the subsequent reduction in the SHG output power. It was theoretically shown that by increasing the input fundamental power up to 50 W, the phase-matching temperature shifts toward lower values, in excellent agreement with recent experimental results [13], [26]. The absorption of fundamental and SHG beam inside the crystal at high fundamental powers was concluded to be the main cause in altering the phase-matching condition as well as disrupting the formation of a uniform thermal distribution. The most important effect of the non-uniform heat distribution was shown to be the displacement of the focal point toward the entrance face of the nonlinear crystal. Using the theoretical model, for the first time to our knowledge, we further studied the effects of thermal flow on OPD values for different proposed oven configurations. This resulted in the calculation of an equivalent focal length associated with a thermal lens formed inside the nonlinear crystal. The effect of thermal lensing on the focus displacement was fully modeled and schematic plots were generated from the results. It was shown that the effect of thermal lensing is quite significant and responsible for the above displacement. It was also found that the top-sinked oven scheme is capable of

[1] R. J. Rockwell, “Design and functions of laser systems for biomedical applications,” Ann. New York Acad. Sci., vol. 168, pp. 459–471, Feb. 1969. [2] X. Chang-Qing, “Periodically poled optical nonlinear crystals for laser display applications,” in Proc. Opt. Fiber Commun. Optoelectr. Expo. Conf., Oct. 2008, pp. 1–3. [3] G. J. Dixon, “Compact blue-green lasers get down to business,” IEEE Circuits Devices Mag., vol. 9, no. 6, pp. 18–22, Nov. 1993. [4] S. C. Kumar, G. K. Samanta, K. Devi, S. Sanguinetti, and M. EbrahimZadeh, “Single-frequency, high-power, continuous-wave fiber-laserpumped Ti:sapphire laser,” Appl. Opt., vol. 51, no. 1, pp. 15–20, Jan. 2012. [5] G. K. Samanta, S. C. Kumar, R. Das, and M. Ebrahim-Zadeh, “Continuous-wave optical parametric oscillator pumped by a fiber laser green source at 532 nm,” Opt. Lett., vol. 34, no. 15, pp. 2255–2257, Aug. 2009. [6] A. Ashkin, G. D. Boyd, and J. M. Dziedzic, “Resonant optical second harmonic generation and mixing,” IEEE J. Quantum Electron., vol. 2, no. 6, pp. 109–124, Jun. 1966. [7] C. Gmachl, A. Belyanin, D. L. Sivco, M. L. Peabody, N. Owschimikow, A. M. Sergent, F. Capasso, and A. Y. Cho, “Optimized second-harmonic generation in quantum cascade lasers,” IEEE J. Quantum Electron., vol. 39, no. 11, pp. 1345–1355, Nov. 2003. [8] M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phasematched second harmonic generation: Tuning and tolerances,” IEEE J. Quantum Electron., vol. 28, pp. 2631–2654, Nov. 1992. [9] T. Volk, N. Rubinina, and M. Wöhlecke, “Optical-damage-resistant impurities in lithium niobate,” J. Opt. Soc. Amer. B, vol. 11, no. 9, pp. 1681– 1687, Sep. 1994. [10] A. Zukauskas, V. Pasiskevicius, and C. Canalias, “Second-harmonic generation in periodically poled bulk Rb-doped KTiOPO4 below 400 nm at high peak-intensities,” Opt. Exp., vol. 21, no. 2, pp. 1395–1403, Jan. 2013. [11] N. E. Yu, S. Kurimura, Y. Nomura, and K. Kitamura, “Stable high-power green light generation with thermally conductive periodically poled stoichiometric lithium tantalate,” Jpn. J. Appl. Phys., vol. 43, pp. L1265– L1267, Sep. 2004. [12] S. Sinha, D. S. Hum, K. E. Urbanek, L. Yin-wen, M. J. F. Digonnet, M. M. Fejer, and R. L. Byer, “Room-temperature stable generation of 19 watts of single-frequency 532-nm radiation in a periodically poled lithium tantalate crystal,” J. Lightw. Technol., vol. 26, no. 24, pp. 3866–3871, Dec. 2008. [13] S. Chaitanya Kumar, G. K. Samanta, and M. Ebrahim-Zadeh, “Highpower, single-frequency, continuous-wave second-harmonic-generation of ytterbium fiber laser in PPKTP and MgO:sPPLT,” Opt. Exp., vol. 17, no. 16, pp. 13711–13726, Jul. 2009. [14] G. K. Samanta, S. Chaitanya Kumar, K. Devi, and M. Ebrahim-Zadeh, “Multicrystal, continuous-wave, single-pass second-harmonic generation with 56% efficiency,” Opt. Lett., vol. 35, no. 20, pp. 3513–3515, Oct. 2010. [15] G. K. Samanta, S. Chaitanya Kumar, M. Mathew, C. Canalias, V. Pasiskevicius, F. Laurell, and M. Ebrahim-Zadeh, “High-power, continuous-wave, second-harmonic generation at 532 nm in periodically poled KTiOPO4 ,” Opt. Lett., vol. 33, no. 24, pp. 2955–2957, Dec. 2008. [16] G. K. Samanta, S. Chaitanya Kumar, and M. Ebrahim-Zadeh, “Stable, 9.6 W, continuous-wave, single-frequency, fiber-based green source at 532 nm,” Opt. Lett., vol. 34, no. 10, pp. 1561–1563, May 2009. [17] S. Chaitanya Kumar, G. K. Samanta, K. Devi, and M. Ebrahim-Zadeh, “High-efficiency, multicrystal, single-pass, continuous-wave second harmonic generation,” Opt. Exp., vol. 19, no. 12, pp. 11152–11169, May 2011. [18] J. H. Lundeman, O. B. Jensen, P. E. Andersen, and P. M. Petersen, “Threshold for strong thermal dephasing in periodically poled KTP in external cavity frequency doubling,” Appl. Phys. B, vol. 96, no. 4, pp. 827– 831, Sep. 2009.

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[19] R. Bhandari and T. Taira, “6 MW peak power at 532 nm from passively Qswitched Nd:YAG/Cr4+:YAG microchip laser,” Opt. Exp., vol. 19, no. 20, pp. 19135–19141, Sep. 2011. [20] K. Regelskis, J. Eludeviius, N. Gavrilin, and G. Raiukaitis, “Efficient second-harmonic generation of a broadband radiation by control of the temperature distribution along a nonlinear crystal,” Opt. Exp., vol. 20, no. 27, pp. 28544–28556, Dec. 2012. [21] A. Sahm, M. Uebernickel, K. Paschke, G. Erbert, and G. Tränkle, “Thermal optimization of second harmonic generation at high pump powers,” Opt. Exp., vol. 19, no. 23, pp. 23029–23035, Oct. 2011. [22] J. Zheng, S. Zhao, Q. Wang, X. Zhang, and L. Chen, “Influence of thermal effect on KTP type-II phase-matching second-harmonic generation,” Opt. Commun., vol. 199, no. 1, pp. 207–214, Nov. 2001. [23] M. Sabaeian, L. Mousave, and H. Nadgaran, “Investigation of thermallyinduced phase mismatching in continuous-wave second harmonic generation: A theoretical model,” Opt. Exp., vol. 18, no. 18, pp. 18732–18743, Aug. 2010. [24] N. E. Yu, C. Jung, D. Ko, J. Lee, O. A. Louchev, S. Kurimura, and K. Kitamura, “Thermal dephasing of quasi-phase-matched secondharmonic generation in periodically poled stoichiometric LiTaO3 at high input power,” J. Korean Phys., vol. 49, no. 2, pp. 528–532, Aug. 2006. [25] O. A. Louchev, N. E. Yu, S. Kurimura, and K. Kitamura, “Thermal inhibition of high-power second-harmonic generation in periodically poled LiNbO3 and LiTaO3 crystals,” Appl. Phys. Lett., vol. 87, no. 13, pp. 131101–131103, Sep. 2005. [26] Z. M. Liao, S. A. Payne, J. Dawson, A. Drobshoff, C. Ebbers, D. Pennington, and L. Taylor, “Thermally induced dephasing in periodically poled KTP frequency-doubling crystals,” J. Opt. Soc. Amer. B, vol. 21, no. 12, pp. 2191–2196, Dec. 2004. [27] A. Bruner, D. Eger, M. B. Oron, P. Blau, M. Katz, and S. Ruschin, “Temperature-dependent Sellmeier equation for the refractive index of stoichiometric lithium tantalate,” Opt. Lett., vol. 28, no. 3, pp. 194–196, Feb. 2003. [28] A. Bejan and A. D. Kraus, Heat Transfer Handbook. Hoboken, NJ, USA: Wiley. [29] G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys., vol. 39, no. 8, pp. 3597–3639, Jul. 1968. [30] S. Ghavami Sabouri, A. Khorsandi, and M. Ebrahim-Zadeh, “Thermal mode-switching of a passively Q-switched microchip Nd:YAG laser,” Appl. Phys. B, vol. 108, no. 2, pp. 261–268, Aug. 2012. [31] D. Nikogosian, Nonlinear Optical Crystals: A Complete Survey. Berlin, Germany: Springer, 2005. [32] S. Fan, X. Zhang, Q. Wang, S. Li, S. Ding, and F. Su, “More precise determination of thermal lens focal length for end-pumped solid-state lasers,” Opt. Commun., vol. 266, no. 2, pp. 620–626, Oct. 2006.

Saeed Ghavami Sabouri was born in 1983 in Mashhad, Iran. He received the Ph.D. degree from the University of Isfahan, Iran, in 2013. His subjects of investigation included nonlinear optics especially optical parametric oscillator, laser spectroscopy, and fiber lasers. He is currently an Assistant Professor of atomic and molecular physics in University of Isfahan, Isfahan, Iran.

Suddapalli Chaitanya Kumar received the B.Sc. degree in mathematics, physics, and electronics from Acharya Nagarjuna University, Andhra Pradesh, India, in 2003, and the M.Sc. degree in physics from Indian Institute of Technology Guwahati, Assam, India, in 2006. He received the Ph.D. degree in photonics (Excellent Cum Laude with Hons.) from ICFOThe Institute of Photonic Sciences, Barcelona, Spain, in 2012, for his thesis on high-power, fiber-laserpumped optical parametric oscillators from the visible to mid-infrared. He is currently a postdoctoral researcher at ICFO. His research interests include fiber-based optical frequency conversion sources and continuous-wave and ultrafast optical parametric oscillators (OPOs) from the ultraviolet to mid-infrared. He has coauthored more than 30 peer-reviewed and invited papers in leading international journals in Photonics, with over 280 citations, and has presented more than 50 contributed, post deadline and invited talks at major international conferences such as CLEO-USA and CLEO-Europe. Throughout his career, he has worked on several funded projects with active international collaborations, and some of his research works have been highlighted in Laser Focus World and Nature Photonics. He is a professional member of Optical Society of America (OSA), The International Society for Optics and Photonics (SPIE), and European Physical Society (EPS). Currently, he is also serving as an International Outreach Project Manager at the Knowledge and Technology Transfer office of ICFO.

Alireza Khorsandi was born in 1966 in Isfahan, Iran. He received the Ph.D. degree from the Technical University of Clausthal, Clausthal-Zellerfeld, Germany, in 2001. His subjects of investigation included nonlinear optics especially frequency mixing in nonlinear crystals, laser spectroscopy, and fiber sensors. He is an Associate Professor of atomic and molecular physics in University of Isfahan, Isfahan, Iran.

Majid Ebrahim-Zadeh received the Ph.D. degree from St. Andrews University, St. Andrews, U.K., in 1989. He is an Institucio Catalana de Recerca i Estudis Avancats (ICREA) Professor at ICFO-The Institute of Photonic Sciences, Barcelona, Spain. He was a Royal Society of London Research Fellow at St. Andrews from 1993 to 2001, and a Reader from 1997 to 2003. He has been active in the advancement of nonlinear optics and parametric sources for over 20 years. His research has led the realization of new generations of innovative light sources from the UV to mid-IR and from the cw to femtosecond time-scales. He has published more than 460 journal and peer-reviewed papers, including 75 invited and 12 post-deadline papers at major international conferences, has edited 2 books and authored 12 major invited book chapters and reviews in volumes such as Science, OSA Handbook of Optics, Springer, Handbook of Laser Technology and Applications, and Laser and Photonics Reviews. He has been a regular instructor of the short course on OPOs at CLEO/USA since 1997 and at CLEO/Europe since 2007, has served over 40 times on the technical, organizing, and steering committees of major conferences worldwide, and has chaired three international conferences. He is an Associate Editor of the IEEE PHOTONICS JOURNAL and has served as advisory and topical editor of Optics Letters, and Guest Editor of the Journal of Optical Society of America B. He is the Cofounder, President, and Chief Scientist of Radiantis, a company created from his research laboratory in 2005. He received several honors and awards, including Royal Society of London university fellowship and merit awards (U.K.: 1993–2001), Innova Prize for technology innovation and enterprise (Spain: 2004), and Berthold Leibinger Innovation Prize (Germany: 2010). He is a Fellow of OSA and SPIE.