Thermal characteristics of second harmonic generation by phase matched calorimetry

Hwan Hong Lim; Sunao Kurimura; Keisuke Noguchi; Ichiro Shoji

doi:10.1364/OE.22.018268

1. Introduction

High power green lasers have numerous applications such as in laser displays, materials processing, laser pumping, and medical surgery. Second harmonic generation (SHG) by the quasi-phase-matching (QPM) technique, allowing the use of highly nonlinear materials, has been an attractive method to realize an efficient and compact continuous wave (CW) green laser. Compared with an intra-cavity SHG, a single-pass SHG is more stable and has simpler architecture. Several groups have achieved efficient CW single-pass SHG with QPM crystals [1–6]. Among QPM materials, a periodically poled Mg-doped stoichiometric lithium tantalate (PPMgSLT) crystal has been demonstrated as most promising for 10W-level CW green SHG, because of a high thermal conductivity and high nonlinearity [7]. However, thermal effects depending on SHG (green) and fundamental (F) power obstruct higher power operation in CW green QPM SHGs.

Thermally induced index change is a limiting factor when accessing the high power region in SHG. Thermal dephasing decreases conversion efficiency, indicated by the saturation of output power. Thermal lensing also deteriorates beam quality and increases instability. The thermal effects in SHG materials can be reduced by: 1) the material choice for low absorption and high thermal conduction, 2) improved boundary conditions allowing for higher heat spread, and 3) the choice of focusing conditions. For item 1), PPMgSLT can be a solution because of its low absorption coefficient (< 0.01 cm⁻¹) and high thermal conductivity (~8.4 $W / m \cdot K$ ). A large effective nonlinear coefficient (~10 pm/V) is also helpful for reducing the F power through an efficient conversion. As for the item 2), the boundary condition surrounding a SHG crystal is crucial to heat spread, as demonstrated in the previous paper [5]. A carefully constructed four-sided heat spreader was proposed for a solution to item 2) with a reduced heat flow distance to the metal as short as 0.15 mm [5].

To generate a high SH power, a reasonable normalized conversion efficiency η_norm ( = P_SH/P_F² [W⁻¹]) is primarily required. After fabrication, η_norm of the QPM devices is determined by their length and the focusing parameter ξ ( = L/b) used; here, L is the device length and b is the confocal parameter. To reduce thermal dephasing for high-power operation, it seems reasonable to use a long device length and a loose focusing. The long length gives the required η_norm and the loose focusing leads to a low beam intensity that minimizes the temperature rise. However, it might fail to decrease the thermal dephasing because of the narrowed temperature acceptance bandwidth as extending the length, aside from the difficulty of fabricating long and wide (large-aperture) devices. For such optimization problems, full understanding of the temperature rise in SHG materials for various conditions is required.

In addition, evaluation of the thermal performance of SHG devices is also important for developing their thermal performance. Because the overall heat removal performance is related to the above items, a method that characterizes the overall performance of final heat-removal modules is necessarily required. In this paper, we examine the item 3) above to improve the thermal performance after fabricating the module. We suggest a solution of the heat equation for SHG with a focused Gaussian beam, to quantitatively understand the temperature rise in the materials. The heat removal performance of SHG devices is also quantitatively characterized with an effective heat capacity by phase matched calorimetry (PMC) [5]. Finally, a strategy to manage the thermal dephasing for high power operation in SHG is discussed with the simulation results.

2. Temperature in SHG material with focused beam

The complex amplitudes of F and SH waves must obey the paraxial wave equation for SHG [8]

2 i k_{S H} \frac{\partial E_{S H}}{\partial z} + \nabla_{T}^{2} E_{S H} = \frac{ω_{S H}^{2}}{c^{2}} χ^{(2)} E_{F}^{2} e^{i Δ k z},

where Δk = 2k_F-k_SH and χ⁽²⁾ is the second order nonlinear susceptibility. For focused Gaussian beams, the complex spatial amplitudes of F and SH waves in the constant-fundamental approximation can be written as follows [8]:

E_{F} (r, z) = \frac{A_{F}}{1 + i τ} \exp (\frac{- r^{2}}{ω_{0}^{2} (1 + i τ)}),

E_{S H} (r, z) = \frac{A_{S H} (z)}{1 + i τ} \exp (\frac{- 2 r^{2}}{ω_{0}^{2} (1 + i τ)}),

τ = \frac{2 (z - z_{0})}{b} = \frac{2 (z - z_{0})}{k ω_{0}^{2}},

where ω₀ is the beam waist, z₀ is the focusing position from the input surface, b is the confocal parameter and A_SH (z) is a function of z. After substituting Eq. (2) and (3) to the paraxial wave Eq. (1), the equation can be integrated directly to obtain [8]

A_{S H} (z) = \frac{i 2 π}{λ_{F} n_{S H}} χ^{(2)} {| A_{F} |}^{2} \int_{z 0}^{z} \frac{\exp (i Δ k z^{'})}{1 + 2 i z^{'} / b} d z^{'},

P_{S H} (z) = \frac{1}{2} π n_{S H} c ω_{0}^{2} ε_{0} {| A_{S H} (z) |}^{2},

where λ_F is the wavelength of fundamental, n_SH is the refractive index of SH wavelength, c is the speed of light, and

ε_{0}

is the permittivity of free space.

For a single pass CW SHG scheme, the temperature distribution T (r, z) in SHG materials satisfies the heat equation under steady state conditions and with cylindrical symmetry, given as follows;

\frac{1}{r} \frac{\partial}{\partial r} (r \frac{\partial T}{\partial r}) + \frac{\partial^{2} T}{\partial z^{2}} = - \frac{h}{κ},

where h is the thermal power (or thermal load) per unit volume in W/m³ by absorbed F and SH lights and κ is the thermal conductivity of material in

W / m \cdot K

. Here, an isotropic thermal conductivity has been assumed. In our SHG scheme, the SHG crystal for CW operation has relatively much greater length in z than r. From the viewpoint of cooling the SHG crystal, the radial periphery of the SHG crystal can be contacted to a metal heat sink with a high thermal conductivity, resulting in a higher heat transfer rate by conduction than would be available via convection, as shown in Fig. 1.Compared with the heat transfer along the radial direction, heat transfer from the input and output surfaces is insignificant. If the heat flow in z is ignored, one can obtain an analytical solution by neglecting the longitudinal derivative in Eq. (7) with a defined h (r, z), as previously reported by several researchers [9–11]. In SHG materials, the thermal power h (r, z) can be expressed as

\begin{array}{l} h (r, z) = α_{F} \frac{2 P_{F}}{π ω_{F}^{2} (z)} \exp (\frac{2 r^{2}}{ω_{F}^{2} (z)}) \exp (- α_{F} z) \\ + α_{S H} \frac{2 P_{S H} (z)}{π ω_{S H}^{2} (z)} \exp (\frac{2 r^{2}}{ω_{S H}^{2} (z)}) \exp (- α_{S H} z) . \end{array}

Then, the temperature distribution in SHG materials is expressed as follows:

T (r, z) = T_{b} + Δ T (r, z),

\begin{array}{l} Δ T (r, z) ​ = Δ T_{F} (r, z) + Δ T_{S H} (r, z) \\ = \frac{α_{F} P_{F} \exp (- α_{F} z)}{4 π κ} \times [\ln (\frac{r_{b}^{2}}{r^{2}}) + E_{1} (\frac{2 r_{b}^{2}}{ω_{F}^{2} (z)}) - E_{1} (\frac{2 r_{}^{2}}{ω_{F}^{2} (z)})] \\ ​ ​ ​ + \frac{α_{S H} P_{S H} (z) \exp (- α_{S H} z)}{4 π κ} \times [\ln (\frac{r_{b}^{2}}{r^{2}}) + E_{1} (\frac{2 r_{b}^{2}}{ω_{S H}^{2} (z)}) - E_{1} (\frac{2 r_{}^{2}}{ω_{S H}^{2} (z)})], \end{array}

where α is the absorption coefficient of crystal, r_b is the radius of crystal, ω is the Gaussian radius of beam, E₁ is the exponential integral function, P_F is the input power of fundamental, and P_SH (z) is the SH power change in crystal.

Fig. 1 Schematic of the structure of the SHG material used in the simulation with a focused Gaussian beam. TEC is a thermoelectric cooler.

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3. Phase matched calorimetry (PMC)

To compare and improve the thermal performance of SHG devices, a method to evaluate their thermal performance is required. Therefore, it is important to determine a quantitative value which represents the heat-removal performance of devices, and the temperature rise in them. As discussed in section 2, the temperature in SHG crystals has a distribution along the z direction due to the beam focusing and the z-dependent SH power. However, in real SHG experiments, the optimum temperature to generate the maximum output power can be determined by measuring temperature tuning curves due to temperature distributions smaller than QPM bandwidth.

Figure 2(a) shows a typical measurement of the TEC temperature versus SH power. For an optimum output power, the initial QPM temperature of the TEC should be decreased so as to compensate for the temperature rise in the SHG material. Therefore, the decreased TEC temperature indicates an effective temperature rise for a given SH power. We proposed a fitting equation to determine a quantitative value, referred to as the effective heat capacity C_α, derived from the decreased TEC temperature data [5]. We have assumed identical beam sizes of F and SH, and the use of a collimated beam. The fitting equation is as follows [5]:

T_{T E C} = T_{T E C}^{0} - Δ T_{T E C},

Δ T_{T E C} = \frac{1}{C_{α}^{}} [R \sqrt{\frac{P_{S H}^{O u t}}{η_{n o r m}}} + P_{S H}^{O u t}],

C_{α}^{} = C \frac{π ​ ω_{0}^{2}}{α_{S H}},

where

R = α_{F} / α_{S H}

,

η_{n o r m} = P_{S H}^{O u t} / {(P_{F}^{I n})}^{2}

. The heat capacity C_α indicates the containable power per unit temperature increase (W/°C). A larger C_α value guarantees a higher heat removal performance. However, in practice, C_α is limited in its use as an absolute quantity to indicate the heat removal performance of devices, because C_α is a function of beam waist size. To compare C_α values of devices with the fixed length, one should use the same focusing parameter ξ in each SHG experiment. However, it is possible to simply eliminate the dependence of C_α on ξ from Eq. (13). Because the C_α and ξ has the inverse relation as follows:

C_{α}^{} = C \frac{π ​ ω_{0}^{2}}{α_{S H}} = C \frac{π ​ L}{α_{S H} k ξ},

the product value

ξ \cdot C_{α}

becomes an absolute quantity for determining the heat removal performance of devices for the fixed device length.

Fig. 2 Measured TEC temperature versus SH power (a) and SH power versus F power (b).

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We experimentally examined the dependence of C_α on the focusing parameter ξ. For single-pass SHG experiments, we used a 1 mol% Mg-doped PPSLT crystal with a QPM period of 8.4 μm and with dimensions 0.3 (width) $\times$ 0.5 (height) $\times$ 20 (length) mm³, embedded in a metal module for efficient heat disposal, as shown in Fig. 3.Details of the fabrication processes are reported elsewhere [2]. We used a CW Yb-doped fiber laser (1083 nm) for a Gaussian F beam, and changed the focusing parameter using five lenses with different focal lengths.

Fig. 3 Photograph of PPMgSLT modules that allow four-side heat spread.

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Figure 2 shows the data set and fitting results at a focusing parameter ξ = 0.41. In Fig. 2(a), the decreased TEC temperature data were fitted with Eq. (11), giving the values C_α = 16.1 W/°C and $T_{Q P M}^{0}$ = 47.9°C, where a constant value η_norm of 0.19%/W was experimentally obtained [Fig. 2(b)]. Figure 4 shows the focusing parameter dependence of C_α (squares) and η_norm (triangles). C_α becomes larger as the focusing becomes looser, as shown by the red line, while η_norm decreases with looser focusing, this having an inverse relation to the focusing parameter ξ as compared with C_α. With this result, we propose the $ξ \cdot C_{α}$ value as a quantity to characterize the thermal behavior of SHG devices. The clarified inverse relation and the trade-off relation between C_α and η_norm could also help in selecting an appropriate focusing parameter for any required power level and conversion efficiency.

Fig. 4 Dependent of C_α (square) and η_norm (triangle) on the focusing parameter, ξ. The error bar is the standard error in each fitting.

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It will be noted that C_α includes an unknown parameter, C [Eq. (13)]. If we can replace the optimum TEC temperature change ΔT_TEC in Eq. (11) with the peak temperature change in the focal plane, the quantity C can be defined approximately as follows (see Appendix):

C ≅ \frac{2 κ}{r_{b}^{2}},

According to Eq. (13) and (15), C_α is related to: 1) the physical quantities (κ, α_SH); 2) the aperture size (r_b), and; 3) the beam waist size (ω₀). The aperture size dependence of C_α has already been reported in [5], where the size dependence was practically examined by reducing only the crystal widths, from 0.5 mm to 0.3 mm, and keeping the crystal height constant at 0.5 mm. A linear increase of C_α was found when reducing the crystal width.

Finally in this section, we wish to point out the limit of our PMC method. Our model assumes that the linear absorption of F and SH light should be dominant and the change of beam waist size should be negligible. For a high SH power region, there are possibilities of nonlinear absorption processes such as green-induced infrared absorption (GRIIRA), and a change in the beam waist size caused by the thermal lensing effect. For comparing the heat removal performance of SHG devices, we recommend that our PMC method is applied in a low SH power region, satisfying the above preconditions.

4. Simulation results and discussion

In this section, we firstly investigate the temperature distribution along the device length with the aim of reducing thermal dephasing. To examine the temperature rise in the device on SH output power and focusing condition, we numerically calculated the temperature distribution for weak and strong beam divergence using Eq. (10) for different SH output powers of 1, 5, and 10 W. The focusing position is fixed at the center of a 10 mm-long SHG material. The parameters used in simulations are listed in Table 1.Figure 5 shows the calculated ΔT (r, z) for ξ = 2.84 [η_norm = 1.1%/W] (a) and ξ = 0.5 [η_norm = 0.49%/W] (b). With increased SH power, the gap between the temperature at the beam center (solid line) and at the beam periphery (dotted line) is widened in both cases. However, the temperature distributions are different in the two cases. Compared with the loose focusing case (b-1), the temperature rise for tight focusing (a-1) starts from the input side at a lower temperature because of the lower F power due to the higher η_norm. In addition, the temperature rise around the output side is saturated for the tight focusing case due to stronger beam-divergence, leading to the intensity reduction.

Table 1. Relevant physical parameters

View Table

Fig. 5 Calculated ΔT (r, z) for a strong (ξ = 2.84) (a) and a weak (ξ = 0.5) (b) focusing. (a-1), (b-1): ΔT (z) at r = 0 (solid) and r = ω (z) (dotted) for the same output SH powers, 1, 5, and 10 W. (a-2), (b-2): ΔT (r, z) - ΔT (r, 0).

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From the viewpoint of dephasing, the difference between the maximum temperature and the minimum temperature in the whole laser beam should be small. In practice, the mean value of the temperature difference can be offset by decreasing the TEC temperature. To compare the temperature difference along z-direction in both cases, we calculated the difference between the temperature rise ΔT (r, z) and the temperature rise at the input position ΔT (r, 0) for each SH output power, as shown in Fig. 5(a-2) and 5(b-2). Compared with the loose focusing (b-2), the difference for the tight focusing (a-2) increases more rapidly along the z-direction and then becomes saturated for each power level. Thermal dephasing could appear early in the tight focusing case, interrupting further efficient conversion along the z-direction. These simulations with various parameters - such as focusing parameter, focusing position, crystal length and size, and so on - could help us to understand the thermal behavior of SHG crystals.

Finally, we discuss the strategy for reducing thermal dephasing during high power operation in SHG. Figure 6 shows the temperature distribution along the device length for various calculation parameters. Here we have assumed an SH power of 30 W with a F power of 80 W during single pass CW SHG. The minimum normalized conversion efficiency is determined by the output power, leading to the minimum focusing parameter ξ. As discussed in the section 3, the use of the smallest ξ can increase C_α and reduce thermal dephasing. The longer length allows looser focusing for the same SH output power. We consider the range of crystal length from 10 to 40 mm. Figure 6 shows the calculated ΔT (0, z) [solid curves] and ΔT (ω (z), z) [dotted curves] for the different crystal lengths of 10, 20 and 40 mm. The smallest possible values of the focusing parameters are 0.48 (ω₀ = 40.8 μm), 0.23 (ω₀ = 83.8 μm), and 0.11 (ω₀ = 168.6 μm) for 10, 20, and 40 mm for 30 W output power, respectively. The temperature increase becomes smaller for the longer length crystals because the larger beam size (smaller ξ) is applicable to the fixed output power. However, one should consider the QPM temperature bandwidth (ΔT_QPM) because we need to compare ΔT_QPM with the local temperature increase. The temperature rise is normalized by the ΔT_QPM, as shown in Fig. 6(b). Although the absolute temperature rise for the longer device is smaller, the relative rise to the ΔT_QPM is larger. Therefore, the use of longer devices with loose focusing is not be the best solution to suppress thermal dephasing; this arises because of the inverse proportionality of ΔT_QPM to the length. In other words, the combination of a short length and a moderated focusing parameter could be better for suppression of thermal dephasing in QPM SHG.

Fig. 6 (a) Calculated ΔT (r, z) needed to generate SH power of 30 W with F power of 80 W for different crystal lengths of 10, 20 and 40 mm at the smallest possible ξ. Solid: ΔT (0, z), dot: ΔT (ω (z), z). Horizontal straight lines indicate the calculated QPM temperature acceptance bandwidths at FWHM with the dispersion of MgSLT [12]. (b) Normalized ΔT (r, z) in each QPM temperature acceptance bandwidth versus normalized z for each length.

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The second strategy for reducing thermal dephasing is to choose an appropriate focusing position. Since the beam size change along the length is small for a loose focusing, the focusing position change doesn’t affect the thermal performance much. However, in a tight focusing case, the beam size distribution along the length strongly depends on the focusing position. The output SH power is also determined by the position. The center focusing has the maximum conversion efficiency, and the front and the rear focusing exhibit the lower efficiency as far from the center position. We calculated ΔT (0, z) [solid curves] and ΔT (ω (z), z) [dotted curves] for a SH output power of 30 W at ξ = 1 (ω₀ = 28.2 μm) for three focusing positions referred as front (−25%), center (0%), and rear ( + 25%) in a 10 mm long crystal, as shown in Fig. 7.Although the front focusing requires the higher F power than the center focusing for generating the same SH power, the front focusing contributes to the suppression of thermal dephasing. In particular: 1) the maximum value in the temperature rise becomes smaller; 2) the gap between the maximum and minimum temperature becomes narrower and; 3) the temperature rise at the front focusing position is lower than that for the other focusing positions. The contribution of the focal area to SH power is greater for tighter focusing. By shifting the focusing position from the output surface, it is possible to separate the location of the maximum intensity of the F beam from that of SH beam. Therefore, the front focusing can partly suppress thermal dephasing in SHG when using tight focusing.

Fig. 7 Calculated ΔT (0, z) for output SH power of 30 W at ξ = 1 (ω₀ = 28.2 μm) for three focusing positions, referred as front (−25%), center (0%), and rear ( + 25%) in a 10-mm long crystal. The arrows indicate the focusing positions.

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5. Conclusion

We have proposed the solution of the heat equation for SHG with a focused Gaussian beam and analyzed the temperature rise behavior in the SHG crystal depending on SH output power, focusing parameter, and focusing position. We also proposed the quantity $ξ \cdot C_{α}$ to characterize the heat removal performance of SHG devices. We finally discussed a strategy for suppressing thermal dephasing in SHG that depends on the combination of a short crystal length, a moderated focusing parameter and the use of front focusing We believe that the proposed solution, the quantity, and the discussed strategies will be helpful in the production of green SHG lasers and in the application of SHG devices.

6. Appendix

For the SHG with low conversion efficiency, the effective TEC temperature to obtain the optimum SH power can be related on the temperature rise in the focal plane, because the temperature in the focal plane has the maximum value. Even if the temperature rise on the output side is higher than the temperature rise at the focal plane by the high SH power at the output side due to high conversion efficiency, the temperature rise at the focal plane is still primarily important because the contribution of the intense focal plane to SH power conversion is biggest. In this manner, if the effective temperature can be represented by the peak temperature in the focal plane, the fitting Eq. (11) can be explained by deriving from Eq. (10). The temperature increase ΔT in Eq. (10) can be simplified as

\begin{array}{l} Δ T (r, z) ≅ \frac{α_{F} P_{F} \exp (- α_{F} z)}{2 π κ ω_{F}^{2} (z)} [r_{b}^{2} - r^{2}] \\ + \frac{α_{S H} P_{S H} (z) \exp (- α_{S H} z)}{2 π κ ω_{S H}^{2} (z)} [r_{b}^{2} - r^{2}], \end{array}

where the series expansion formula of the exponential integral was used.

E_{1} (z) = - γ - l n z - \sum_{n = 1}^{\infty} \frac{{(- 1)}^{n} z^{n}}{n n!} .

Then, the peak temperature in the focal plane expresses

Δ T (0, z_{0}) = \frac{α_{S H} r_{b}^{2}}{2 κ π ω_{0}^{2}} [R P_{F} (1 - α_{F} z_{0}) + 2 P_{S H} (z_{0}) (1 - α_{S H} z_{0})],

where z₀ is the length from input surface to the focal plane, and we assumed the next relation

ω_{F} = \sqrt{2} ω_{S H}

. For QPM SHG crystals with α < 0.01 cm⁻¹ and L < 4 cm, the power reduction terms

\exp (- α_{F (S H)} z_{0})

have the ratio to be less than 2%. When one neglects the power reduction terms by absorption, the final fitting equation can be obtained as follows:

T_{T E C} = T_{T E C}^{0} - Δ T (0, z_{0}),

Δ T (0, z_{0}) = \frac{1}{C_{e f f} (α_{S H}, κ, r_{b}, ω_{0})} [R \sqrt{\frac{P_{S H}^{O u t}}{η_{n o r m}}} + \frac{1}{2} P_{S H}^{O u t}],

C_{e f f} = \frac{2 κ π ω_{0}^{2}}{α_{S H} r_{b}^{2}},

where we used the relation

P_{S H} (z_{0}) = 0.25 \times P_{S H}^{O u t}

for center focusing. Compared with Eq. (12), the contribution of SH power to the temperature rise in Eq. (20) is reduced by the factor of 1/2. This difference is attributed to different assumptions concerning the F and SH beam size and the considered SH power at the different positions –namely, at the center, and the output surface. Nevertheless, Eq. (21) can help us to understand the physical aspect of C_α.

Acknowledgments

This work was partially supported by a Grant-in-Aid 20244062 from the Ministry of Education, and from JST CREST.

References and links

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Symbol	Quantity	Unit	Value
α_F₍_SH₎	absorption coefficient	cm⁻¹	0.002 (0.008)
κ	thermal conductivity	$W/m \cdot K$	8.4
n_F₍_SH₎	refractive index	1	2.1338 (2.2002)
r_b	radius of crystal	μm	250
L	length of crystal	mm	10-40
T_b	temperature at crystal boundary	°C	30
d_eff	effective nonlinear optical coefficient	pm/V	10

Thermal characteristics of second harmonic generation by phase matched calorimetry

Abstract

1. Introduction

2. Temperature in SHG material with focused beam

3. Phase matched calorimetry (PMC)

4. Simulation results and discussion

5. Conclusion

6. Appendix

Acknowledgments

References and links

Cited By

Figures (7)

Tables (1)

Equations (21)

Optics Express