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Abstract
Thermal-induced phase-mismatch distortion, which will dramatically deteriorate the efficient energy transfer, has become a critical obstacle to power scaling of optical parametric amplifiers. To ease this efficiency deterioration, the noncollinear optical parametric amplification (OPA) configuration widely employed to achieve broadband phase-matching (PM) may also serve as a promising approach to optimize the temperature acceptance. In this paper, starting from the noncollinear wave-vector equations, a required thermo- and angle-relationship analogous to that of noncollinear broadband PM is firstly inferred. Based on the presented mathematical criterion, we have explored the potential spectral boundaries of this ingenious temperature insensitive OPA scheme. Full-dimensional simulations of two types of typical OPA processes were quantitatively discussed. Compared with traditional collinear PM designs, the presented noncollinear PM configurations show significant common characteristics on improving the temperature acceptance and subsequently the overall amplification efficiency. For a typical high power parametric process of the 532 nm pumped near-IR OPA at 800 nm especially, incredible temperature bandwidth exceeding 8000 °C was obtained while a YCOB (xz plane) crystal is adopted. What is more, it can also be applied to ameliorate the gain-spectrum thermo-instability of OPA.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Optical parametric amplification (OPA) is a typical quadratic nonlinear interaction to amplify a lower-frequency signal wave from visible to far-infrared and even the Thz spectral region [1,2]. In contrast to the population-inversion-based solid-state laser amplifiers, OPA offers an enormous gain bandwidth and low levels of amplified spontaneous emission, which makes it advantageous for generating few-cycle laser pulses. Furthermore, benefiting from its instantaneous nature, thermal effects are normally negligible in moderate power OPAs [3], which makes it also attractive candidates for power scalable laser amplifiers [4–6]. However, thermal gradients arise ultimately when the pump laser is sufficiently powerful, leading to a nonuniform distribution of refractive-index [7,8]. Perfect phase-matching (PM) thus could not always be satisfied in the overall parametric process. It dramatically deteriorates the efficient energy transfer in OPA and places a ceiling on the maximum conversion efficiency, especially for the cases where the interacting waves may be intensively absorbed [9,10]. As reported, a purposeful use of suitable nonlinear crystals to absorb the by-produced idler wave is conducive to inhibiting the detrimental back-conversion effect, and thus obtaining higher conversion efficiency and better beam quality [11,12]. Whereas, the significantly enhanced thermal loading gets more serious in such type of OPAs. In summary, thermal issues have become the critical obstacle to the further development of high average power OPA systems. To alleviate that thermal nonuniform distribution, a variety of optimization techniques have been proposed, for instance, suppressing the thermal gradients by beam shaping [13] or maximizing the heat dissipation by gas-cooled multiple-plate configurations [14]. On the other hand, developing temperature insensitive PM schemes is another promising approach which can fundamentally ease the thermal-induced phase-mismatch. For other typical quadratic nonlinear processes of harmonic generations, compared with a conventional PM scheme, the temperature acceptance bandwidth has been effectually broadened via the multi-crystal design [15–17] or the compensation method based on electro-optic effect [18]. Recently, Tang et al. presented that the noncollinear configuration widely employed to achieve broadband PM was also applicable to realizing a temperature insensitive OPA [19]. By means of the specified noncollinear configuration, 6 times larger temperature bandwidth was successfully obtained. Whereas, the detailed analyses on this noncollinear design have not been addressed similar to its conventional broadband PM counterpart.
In this paper, we perform comprehensive analyses on this ingenious temperature insensitive OPA configuration. Starting from the wave-vector equations of a noncollinear PM geometry, a mathematical thermo- and angle-relationship analogous to that of broadband noncollinear PM is firstly inferred. Based on common nonlinear crystals of lithium borate (LBO) and yttrium calcium oxyborate (YCOB), potential spectral boundaries of that temperature insensitive PM scheme are explored, which shows that this approach can be applied to constructing high average power OPAs across an ultrabroad spectrum range from visible to mid-infrared. On this basis, two types of typical OPA processes are numerically studied. Specially, nonlinear media with strong absorption to the respective idler waves are chosen. Compared with traditional collinear designs, the presented noncollinear PM configurations can significantly mitigate the conversion efficiency deterioration caused by uneven temperature distribution and deliver more powerful outputs. What is more, it can also be applied to ameliorate the gain-spectrum thermo-instability of OPA.
2. Underlying principles
PM condition is an intrinsic factor for all quadratic nonlinear interactions. To fulfill this PM requirement, the wave vectors of pump (kp), signal (ks), and idler (ki) should satisfy that
For those high-average power OPA systems, however, optical absorption of the interacting waves leads to detrimental thermal gradients within the associated nonlinear medium. The ideal perfect PM thus could not be always satisfied at a fixed PM angle. The resulting thermal-induced phase-mismatch of Δk(T), which gets more serious with the increasing pump power, plays a crucial role on the efficiency degeneration and places an inherent limitation on power scaling of the OPAs.As shown in Figs. 1(a) and 1(b), for an initially satisfied collinear PM geometry, as the crystal temperature shifts from the PM value, the generated idler wave still has to propagate in collinear with the signal and pump waves to minimize the thermal-induced Δk(T). However, the wave-vector derivatives to temperature commonly may not be just neutralized by each other in this condition, resulting in a limited thermal acceptance. It is well-known that the noncollinear PM geometry has an excellent ability to improve the PM spectral bandwidth by virtue of introducing an additional degree of freedom. Intuitively, extremely broadened temperature acceptance bandwidth would be also conceivable if the noncollinear angles are properly reset. As that shown in Fig. 1(c), α is the internal noncollinear angle between pump and signal waves, and independent to the operation temperature. Then, the idler wave emits at a resulting internal angle of β with respect to the pump wave. In this case, the momentum conservation condition can be thus decomposed into the components parallel and perpendicular to kp :
Where kp(T), ks(T), and ki(T) respectively represent the temperature-dependent wave vectors of pump, signal, and idler. Here, Eqs. (2) and (3) would be primarily discussed in a non-steady temperature situation. It should be noted that the deviated angle β is not constant, but auto-adjusts with the operation temperature so that the overall Δk(T) is minimum. Supposing perfect PM is satisfied at an initially set phase-matching temperature T0 (i.e., Δk∥(T0) = Δk⊥(T0) = 0), when the temperature deviation is ΔT, the wave vector mismatches along these two orthogonal directions can be approximated, to the first order, as Where T0 represents the perfect PM temperature. ∂kn(T)/∂T|T = T0 is the first wave-vector derivative to temperature at T0, n = i, p, s refers to the idler, pump, and signal waves respectively. Noted that, in addition to temperature T, the value of ki may also vary with its transmission direction (i.e., the deviated angle β), for example, when the idler wave is extraordinary. The derivation presented above is based on the premise that ki is independent on β, and only considering the change in ki caused by the temperature shifting. This setting is applicable to the cases that the idler wave is ordinary, or in a QPM situation.
Fig. 1 Schematic of the collinear wave-vector geometries under (a) the PM temperature and (b) the shifted one; (c) Illustration of the presented temperature insensitive OPA scheme based on a noncollinear PM geometry. Note that, β is temperature-dependent and auto-adjusts to minimize the thermal-induced Δk (T).
To achieve a broad temperature bandwidth, Δk∥ and Δk⊥ must simultaneously vanish. Similar to the derivation procedure presented in [20], multiplying Eq. (4) by sinβ and Eq. (5) by cosβ and subsequently adding these results, we get a thermo- and angle-relationship as
Obviously, it indicates that the primary PM condition may be impervious to temperature shifting provided that a specific noncollinear configuration is satisfied, where the projections of the first wave-vector derivatives should be matched along the initial propagation direction of idler. Then, the first temperature derivatives of all the involved wave-vectors can just be eliminated by each other and temperature insensitive phase-matching can be achieved. Under these circumstances, the accessible temperature acceptance would be only limited by the second- and higher-order terms of the temperature derivatives. From a practical point of view, Eq. (6) allows to handily determine the noncollinear angles required for broadband temperature PM or choose appropriate media for various OPA interactions.Although Eq. (6) presents a clear mathematical criterion for that temperature insensitive PM geometry, there is not necessarily a right solution for any OPA and medium combination. Based on this equation, the potential spectral boundaries of the noncollinear temperature insensitive PM scheme are explored. As shown in Fig. 2, the required noncollinear angle α is plotted as a function of signal wavelength for different high average power pump sources [21-22]. Here, nonlinear crystals of LBO and YCOB commonly used are considered. For each wavelength-pair of the pump and signal waves, there is only one specified noncollinear geometry in which the temperature insensitive OPA can be achieved. And the available PM spectrum varies with the pump wavelengths or the employed crystals. For a Type-I LBO crystal in the xz plane as an example, the temperature insensitive PM can be realized for broadband signal wavelengths from ~600 nm to ~3.1 μm, when the pump wavelength is fixed at 532 nm. As presented, relying on only two conventional nonlinear crystals, this approach is applicable across an ultrabroad spectrum range from visible to mid-infrared. Noted that, the shadowed area indicates the signal wavelengths whose idler-counterparts just lie in the absorption spectrum of the associated nonlinear crystal. As reported, purposely absorbing the by-produced idler wave and thus blocking the interactions between the idler and the signal waves are conducive to inhibiting the detrimental back-conversion effect, but definitely gives rise to more severe thermal loading. Combined with that versatile temperature insensitive PM scheme, it provides a potential opportunity on constructing such an idler-abortion OPA system be free of thermal effects.
Fig. 2 Calculated noncollinear angle α for various pump wavelengths under the Type-I PM condition. (a) LBO in the xz plane; (b) YCOB in the xz plane. The shadowed area indicates the signal wavelengths whose idler-counterparts just lie in the absorption spectrum of the associated crystal. Here, the idler-absorption spectrum of that Sm3+: YCOB crystal used in [11] is also presented compared with the common YCOB crystal.
3. Proof of principle
In the following, to demonstrate the optimized performance of the proposed PM configurations, two types of typical OPA processes, i.e., the 800 nm pumped near-IR OPA at 1030 nm and the 532 nm pumped near-IR OPA at 800 nm, were numerically studied.
According to the results given in Figs. 2(a) and 2(b), nonlinear mediums of LBO [23] and Sm3+-doped YCOB (Sm3+:YCOB) were respectively chosen to absorb the redundant idler wave. It should be noted that YCOB is generally transparent to an idler wave at ~1.6 μm [24], but its absorption edge can be adjusted to cover this wavelength by doping the rare-earth ion Sm3+ [11]. Nevertheless, the temperature-relevant Sellmeier equation of Sm3+:YCOB has not been available so far. As an alternative, one of the common YCOB crystal is adopted in our study [25-26]. To construct temperature insensitive OPA configurations, based on a Type-I PM condition, special noncollinear angles of α = 2.4° and α = 6.0° must be chosen. Correspondingly, LBO and YCOB crystals cut along the directions of (θ = 90°, φ = 45.8°) and (θ = 90°, φ = 52.6°) in the xz plane were respectively adopted.
Under the slowly varying envelope approximation and quasi-CW assumption, all the temporal effects (e.g., group-velocity mismatch, etc.) are neglected and the noncollinear coupled-wave equations that govern the evolution of three parametrically interacting waves can be written as [27]
where Ej(x, y, z) represents the field envelope normalized to the input pump field E0, when j = s, i, p refers to the signal, idler, and pump waves, respectively. deff denotes the effective nonlinear coefficient and αidler is the idler absorption coefficient. Δk(T) = kp(T) - ks(T)·cosα - ki(T)·cosβ is the parallel thermal-induced wave vector mismatch, while the perpendicular part is always set to zero under an assumption that ks(T)·sinα = ki(T)·sinβ is always satisfied. In this manuscript, the thermal-induced wave vector mismatch Δk(T) was calculated based on the published Sellmeier equations, wherein, besides the first-order temperature derivatives, the other high order terms have all been included. Incident pump and signal waves are both set to be quasi-CW Gaussian lasers, that I(x, y) = I0·exp[-2(x ∕σ)2]·exp[-2(y ∕σ)2] with a uniform temporal distribution. I0 = ncε0E02/2 represents the peak intensity and σ is the beam radius in half-width at 1/e2 maximum of the radial intensity distribution. Specifically, diffraction and birefringent walk-off terms are both neglected based on an assumption that the beam radius is relatively large for the length of crystal. In addition, given that discussed noncollinear OPA configuration, noncollinear angles are included.Ignoring the thermal gradients in high average power OPAs, temperature acceptance bandwidth was first addressed in a small signal situation for both collinear and noncollinear PM cases. In the simulations, nonlinear coupled-wave equations of Eq. (7) were numerically solved by a standard split-step method. Figures 3(a) and 3(b) show the normalized temperature-dependent conversion efficiency. The perfect PM temperature was set at a room temperature of 20 °C and the crystal length of L was selected to be 15 mm. As the crystal temperature deviates from that PM value, the resulting thermal-induced |Δk(T)| appears and degenerates the energy transfer efficiency. In contrast to the conventional collinear PM cases, both presented noncollinear PM designs show remarkably improved thermal tolerance. For the 1030 nm near-IR OPA, the temperature acceptance bandwidth gets significantly extended from ~12 °C to ~46 °C by almost 4 times. In comparison, for that YCOB based 800 nm OPA, the promotion would be more pronounced. An incredible temperature bandwidth more than 8000 °C can be achieved when that of the collinear case is only ~42 °C, which indicates that the thermal-induced phase mismatch will no longer be a limiting factor for those high average power OPAs. The considerable discrepancy on the optimized performance for various parametric processes is mainly determined by the thermo-optic property of nonlinear mediums and will be discussed in detail in subsequent sections.OPCPA being features of high signal-to-noise ratio and broadband gain spectrum is a robust technique for producing ultra-intense lasers [28–31]. When it comes to those narrowband parametric processes, Δk(T) will only lead to a descent in amplification efficiency. As for high average power OPCPAs, however, thermal effects would also remodel the gain spectrum and eventually destroy the spectral information of the amplified signal wave. Thus then, thermal stability of the gain spectrum is also an important parameter for the high power OPCPA systems.
Fig. 3 Normalized conversion efficiency versus temperature of both collinear and the presented noncollinear PM schemes, for (a) a 800 nm pumped near-IR OPA at 1030 nm and (b) a 532 nm pumped near-IR OPA at 800 nm. 15-mm long LBO and YCOB crystals were respectively adopted and the PM temperature was set at a room temperature of 20 °C. ΔT = T - T0 denotes the temperature deviation.
For broadband OPOCA, noncollinear PM are usually adopted, and a suitable non-collinear angle (α1) needs to be selected to obtain the maximum gain bandwidth. For the YCOB crystal in a given 532 nm pumped OPA, α1 = ~3.6° at a signal wavelength of 800 nm。As for the presented LBO case, however, noncollinear broadband PM cannot be realized in the xz plane because of νi<νs, where νi and νs present the group-velocity of the idler and signal waves respectively. As a result, the gain spectrum of a collinear PM configuration (α1 = 0°) would be the broadest one it can achieve. Obviously, in general, the noncollinear angles of α1 are not equal to those of α2 needed to get the maximum temperature bandwidth.
Figures 4(a)-4(f) illustrate respective temperature-dependent gain spectra of the mentioned parametric processes, and make a comparison of the traditional noncollinear broadband PM scheme and the noncollinear temperature insensitive PM scheme. As it shows, in general, as the operation temperature deviates from the initially set PM value, different levels of overall-shifted appeared in the resulting gain spectra, depending on the thermal stability of gain spectrum of the OPA configuration. Though this shifting has little impact on the envelope of gain spectrum, dislocating from the incident signal-spectrum would still give rise to the conversion efficiency degeneration and more importantly, the thermal-induced spectrum narrowing. For the noncollinear broadband OPA configuration of YCOB (Fig. 4(b)), the presented gain spectrum shows even more terrible thermal stability, showing a significant spectrum distortion instead of a simple spectrum shifting as the collinear and the other non-collinear OPA configurations. This unusual characteristic can be roughly explained by the arithmetic expression of Δk. The temperature-depended Δk at an off-center wavelength can be approximately expressed as Δk(ΔT, Δλ) = ∂Δk/∂λ·Δλ + ∂2Δk/∂λ2·Δλ2 + ∂Δk/∂T·ΔT + ∂2Δk/∂T2·ΔT2. Since it is a broadband PM structure (∂Δk/∂λ = 0) and the ∂2Δk/∂T2 of YCOB is minimal, which we will discuss in the next section, it can be simplified that Δk(ΔT, Δλ) = ∂2Δk/∂λ2·Δλ2 + ∂Δk/∂T·ΔT. Clearly, the center-symmetric wavelengths ( ± Δλ) thus will present a similar gain variation with the temperature, and the Δk(ΔT, Δλ) = 0 may be simultaneously satisfied on both sides of the central wavelength with a certain ΔT. However, in a more general non-broadband PM structure, the existence of ∂Δk/∂λ·Δλ will destroy this symmetry.
Fig. 4 Temperature-dependent gain spectra of the mentioned [(a), (c), (e)] 1030 nm and [(b), (d), (f)] 800 nm OPA processes with distinct noncollinear angles of α. Same parameters were adopted as those given in Fig. 3. (a) The collinear PM configuration; (b) The broadband PM configuration; (e), (f) The presented temperature insensitive PM configurations; (c), (d) The other cases differ from the broadband or the temperature insensitive PM designs.
By contrast, the presented noncolinear geometry shows an extraordinary ability in improving the thermal stability of gain spectrum so that almost stabilized gain spectra were obtained across a considerable temperature range larger than 20 °C. In general, due to the inconsistency of α1 and α2, based on a noncollinear PM scheme, the temperature insensitive and broadband PM generally cannot be simultaneously satisfied. To realize the temperature insensitive and broadband OPCPA, an appropriate angular dispersion should be introduced [19], or optionally, we can select another appropriate noncollinear angle between α1 and α2, and make a trade-off between the temperature acceptance and the PM bandwidth, as shown in Fig. 4(d).
4. Full-dimension simulations
From the perspective of temperature acceptance bandwidth, preliminary results have clearly demonstrated that thermal insensitive OPA can be realized in a specific noncollinear PM geometry. Compared with the cases based on small signal approximation, in high average power OPAs, however, crystal temperature does not vary globally. Steady thermal gradients would be established along both the radial and the propagation directions when environmental temperature is constant. To characterize the potential applications of the temperature insensitive PM scheme in high average power regime, full-dimension simulations were performed as well. To imitate those thermal gradients, heat-transfer equations should be simultaneously solved with the nonlinear coupled-wave equations of Eq. (7). In this study, a heat-transfer model based on rod-type crystals is adopted [32, 33]. Assuming a constant heat conductivity κ, the steady-state temperature profile in an axisymmetric rod can be described as [32]
Where rb represents the radius of crystal rod and Q(r, z) is the heat source density in nonlinear medium. Although it may differ from other crystal shapes such as the familiar rectangular ones, the results presented here can still be taken as a useful reference to the potential applications of the noncollinear temperature insensitive PM design. For a conventional edge cooling situation, the heat-flow in longitudinal direction can be approximated as adiabatic, due to the cooling-surface convection coefficient is orders of magnitude larger than the natural conversion coefficient on the crystal-ends [33]. Moreover, Q(r, z) mainly comes from absorbed energy of the idle wave, and the power of the idler wave is closely related to the conversion efficiency. Therefore, in our opinion, under the sustained nonlinear interaction of high repetition rate lasers, power evolution of the idler wave with interaction length is a primary factor to determine the longitudinal temperature gradience of the crystal.In our simulations, the incident pump and signal waves are both quasi-CW lasers with a Gaussian-profile in space. Q(r, z) can be also approximated to be Gaussian. Under this approximation, the radial temperature profile may be expressed as [34]
Where Ph = αider·Pidler represents the total heat-deposit power per unit volume and wp is the beam radius. T0 denotes the environment temperature and is set at 20 °C.To numerically demonstrate the high-average-power OPA process, on the basis of a standard split-step method, a simultaneous calculation of the temperature profile at varying crystal positions is included: Based on Eq. (9), temperature profile T(r, z0) and the corresponding Δk(r, z0) are first calculated by using the current idler power at z0, and then Δk(r, z0) will be adopted to calculate the field envelopes of all of the pump, signal and idler waves at the next step (z0 + Δz), until the end of crystal. Since the by-produced idler wave is a major heat source for those considered OPA process, the temperature profiles and the phase mismatch in turn will strongly depend on the conversion efficiency and vice versus. Thus then, an iterative algorithm was performed until the temperature gradient reached steady-state [35].The simulated quantum conversion efficiency along with the amplified signal power are individually plotted in Fig. 5, as a function of the incident pump power. The essential crystal parameters, calculated based on the temperature-dependent Sellmeier equations, are all listed in Table 1 [11, 23, 25, 26, 36, 37]. In those numerical simulations, we assume a fixed crystal radius of rb = 12.5 mm, and the beam radius wp is one third of the rb that rb = 3wp. Pump intensity of ~9.5 and ~4.5 GW/cm2 is respectively chosen for the LBO- and Sm3+:YCOB- based parametric processes while the initial Gaussian signal intensity is fixed at 1% of that of the pump intensity. In such given conditions, the associated OPAs are able to achieve a similar nearly saturated efficiency of ~50% when the thermal effects are neglected. As presented, for those conventional collinear PM schemes, resulting from a severer phase- mismatch distortion, the conversion efficiency declines dramatically with the increasing pump power. Whereas, when the OPA configurations of specific noncollinear geometries are applied, both of the conversion efficiency and the consequent signal output are significantly improved, although there is a slight decrease in thermal-absent maximum efficiency because of the spatial walk-off. For instance, in the case of the LBO-based near-IR OPA at 1030 nm, conversion efficiency of the temperature insensitive PM scheme could still maintain at ~35% when the incident pump power is ~120 W, which is nearly two times as high as that of the conventional collinear PM case. As for the other parametric process of Sm3+:YCOB based near-IR OPA at 800 nm, the more intriguing feature of the noncollinear configuration is clearly illustrated by the nearly untouched conversion efficiency. This outstanding optimized performance could be attributed to its incredible temperature acceptance bandwidth mentioned above. For comparison, a drastic efficiency declines from ~50% to only ~20% is predicted in the collinear situation.The steady-state temperature distributions and signal-beam profiles are also studied comparatively and presented in Figs. 6(a)-6(h) and Figs. 7(a)-7(h). As shown, more serious temperature gradients appear in noncollinear PM configurations, particularly for the higher incident pump powers. Obviously, it is a consequence of the higher conversion efficiency which indicates more powerful idler wave has been produced and subsequently leads to greater thermal-accumulation. The combined higher amplification efficiency and more severe temperature distortion have sufficiently demonstrated the prominent performance of those noncollinear PM configurations. But in general, the promotion to overall amplification efficiency is not so inspiring as that to temperature bandwidth as previously discussed. This contrast may be interpreted in terms of non-equilibrium temperature distributions. The considered parametric processes are both idler-absorptive and then the thermal-accumulation mainly comes from energy absorption of the idler wave. As the temperature distributions shown in Figs. 6(a)-6(h), there is little temperature discrepancy for different OPA configurations at the outset, until the existing idler wave gets sufficiently powerful near the end of the crystal. Only in that moment can this temperature insensitive PM design show its superiority. By comparison, temperature bandwidth is acquired under a precondition where the whole crystal works at a uniform deviated temperature. As a consequence, the overall optimization on amplification efficiency is definitely less prominent than the enhancement on temperature bandwidth. But all in all, the prominent improvement in presented noncollinear configurations is fundamentally attributed to its larger temperature acceptance bandwidth.
Fig. 5 Dependence of quantum conversion efficiency and amplified signal power on the incident pump power, for both collinear and the presented noncollinear PM schemes. The perfect PM temperature was set at a room temperature of 20 °C. (a), (c) 800 nm pumped near-IR OPA at 1030 nm; (b), (d) 532 nm pumped near-IR OPA at 800 nm.
Table 1. Nonlinear Crystal Parameters of Type-I LBO (λp = 800 nm, λs = 1030 nm, λi = 3.58 μm) and Sm3+:YCOB (λp = 532 nm, λs = 800 nm, λi = 1.59 μm) at a temperature of 20 °C)
Fig. 6 Steady temperature distributions of the LBO and Sm3+:YCOB crystals in their respective parametric processes, for both (a-d) collinear and (e-h) the presented noncollinear PM schemes. The incident beam radius wp was ~4 mm.
Fig. 7 Beam profiles of the amplified signal wave in presented noncollinear PM schemes. (a)-(d) 1030 nm; (e)-(h) 800 nm. Note: the incident beam radius wp was ~4 mm [(a-c), (e-g)] and ~2 mm [(d), (h)] respectively; (c) and (g) are fictitious cases with same operation parameters as (b) and (f), but the idler wave is non-absorbable (i.e., αidler = 0) and the thermal effects are absent.
Beam quality is another critical issue in high-average power OPA. To avert the spatial distortion caused by the unidirectional lateral walk-off, signal and pump waves with large beam aperture should be configured. Here, the beam aperture is set to 2-3 times larger than the walk-off slipping. As shown, no clear beam-quality degradation appears even at a high average power of 150 W. It can be interpreted by the achievement of temperature insensitive PM condition which then minimizes the thermal-induced phase-mismatch distortions. In addition, the considered parametric processes are both idler-absorptive and the back-conversion effects are thus fundamentally suppressed, which is also helpful to obtain better beam profiles. To obtain the lower limit of the beam radius, we performed another simulation case when the beam radius is 1/2 of the original, namely 2 mm. As presented, nearly undistorted beam profiles are obtained, although the conversion efficiency has severely decreased from ~40% to ~25% due to the beam splitting of the pump and the signal waves. It can be attributed to a combined result of the temperature insensitive PM condition and the back-conversion inhibited characteristic of the discussed parametric processes. To prove this inference, we also performed a fictitious case for comparison, with same operation parameters (wp = ~4 mm) but the idler wave is non-absorbable (i.e., αidler = 0) and the thermal effects are absent. As shown in Figs. 7(c) and 7(g), we can see that considerable beam distortion appears as the common noncollinear OPA process should present.
Finally, the reasons why the optimized performances of various parametric processes or crystals aroused by the temperature insensitive PM scheme are different are explored in detail. As the crystal temperature deviates from the initially set perfect PM value, the resulting Δk (T) can be expressed in a Taylor series as
Where ΔT = T - T0 denotes the temperature deviation. In addition to the first temperature derivative, the second- and the higher-order terms are also included. Figure 8 shows the individual first- and second-order temperature derivatives for both LBO and YCOB crystals in their respective collinear OPA configurations. As presented, for the LBO crystal, the first temperature derivative, which increases linearly with ΔT, plays a dominant role when ΔT is less prominent. However, the second term increases with the square of ΔT and exceeds the former one eventually, showing a growing gap between these two types of phase mismatches. Although the proposed noncollinear PM scheme can completely eliminate the detrimental effects of the first temperature derivative and achieve a temperature insensitive OPA, this approach is still subject to the other higher order temperature derivatives. The inspiring temperature-insensitivity of YCOB crystal in 800 nm near-IR OPA happens to be a superior result benefiting from its minimal second-order temperature derivative.Fig. 8 Dependence of thermal-induced phase-mismatch (black lines) on temperature deviation for both (a) LBO and (b) YCOB crystals in their respective collinear OPA configurations. In addition, the first- (red lines) and second-order (blue lines) contributions are individually exhibited.
5. Conclusion
In summary, the noncollinear PM configuration with a proper angle-design is a potential technology to ease the thermal-induced phase-mismatch distortions in high average power OPAs. Starting from the noncollinear wave-vector equations, a mathematical thermo- and angle-relationship, which is analogous to that of the broadband noncollinear PM scheme, is firstly demonstrated. As predicted, it can be applied across an ultrabroad spectrum range from visible to mid-infrared to construct temperature insensitive OPA systems. Two typical OPA processes were numerically studied. Although differentiated optimized performances of various parametric processes or crystals are clearly observed, the presented noncollinear configurations show both significant characteristics on improving the temperature acceptance and subsequently the overall amplification efficiency. In particular, for a typical OPA process of 532 nm pumped near-IR OPA at 800 nm, the temperature bandwidth can be increased to incredible ~8000 °C while a YCOB crystal (xz plane) is adopted. In this case, thermal-induced phase mismatch will no longer be a block to average power boosting of such OPA process. Furthermore, in addition to the amplification efficiency improvement, more importantly, it is able to ameliorate the thermo-instability of gain-spectrum as well. Conclusively, the noncollinear temperature insensitive PM configurations may serve as a promising approach for the further development of high average power OPAs.
Funding
Natural Science Foundation of China (NSFC) (61505113); Science and Technology Project of Shenzhen (JCYJ20160308091733202, ZDSYS201707271014468); Science and Technology Planning Project of Guangdong Province (2016B050501005); Educational Commission of Guangdong Province (2016KCXTD006).
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