1. Introduction

The laser wavelengths obtained using the nonlinear optical frequency conversion technology cover most of the optical waveband from infrared to visible and ultraviolet [1–3]; therefore, the applications of the laser have been greatly broadened in many fields [4–6].With the unceasing progress of laser technology and expansion of application fields, laser power and repetition rate are continuously increased [7–10]. As a result, the thermal effect of the laser is becoming increasing serious owing to the optical absorption of materials [11,12]. In the frequency conversion system, part of the laser energy absorbed by the nonlinear crystal is converted into heat energy, hence resulting in increased crystal temperature. With the coaction of the laser and cooling system or surrounding environment, a temperature gradient will be formed in the crystal. In the high average power regime, the change of crystal internal temperature will be more drastic. Since the refractive index of nonlinear crystals is closely related to the temperature [13], the thermal effect will lead to uneven distribution of refractive index and seriously impact the phase-matching condition required for highly efficient frequency conversion, resulting in reduced conversion efficiency reduction and deteriorated beam quality [12,14,15]. Thus, the laser thermal effect has become an important factor restricting the efficiency and stability of frequency conversion, which is a problem that must be solved in the high-power and high-repetition-rate laser frequency conversion system.

In order to mitigate the laser thermal effect, several cooling methods and heat management measures have been put forward to control the crystal temperature [15–17]. Although the crystal temperature can be reduced by actively removing the generated heat, these methods require additional devices that complicate the system and are not easy to apply, and may exacerbate the uneven distribution of crystal temperature, especially for large-aperture crystals. Recently, researchers began to look for new ways to reduce the influence of the thermal effect on the nonlinear optical interaction process using nonlinear crystal characteristics. Some temperature-insensitive frequency conversion designs with a simple structure were reported [18–22]. These designs are based on the cascaded crystal structure to compensate for the phase mismatch using different methods. In [18,19], phase mismatch compensation was achieved by cascading two different types of crystals, which have opposite signs of the first temperature derivation of the phase mismatch ($\partial \Delta k/\partial T$). For the method reported in [20,21], it is based on the electro-optic effect to compensate for the phase mismatch. In [22], a phase mismatch compensation method based on the combination of three crystals was reported. The main principle of this method is based on the analysis of $\partial \Delta k/\partial T$ in different crystals with different propagation directions by which crystals with opposite signs of $\partial \Delta k/\partial T$ are selected and cascaded, thereby achieving phase mismatch compensation.

For the designs reported in [18–22], two different types of crystals are used. For example, the second harmonic generation at 1064 nm with the combination of KTiOPO₄ and LiB₃O₅ (LBO) crystals, and 532 nm with the combination of BaB₂O₄ (BBO) and LBO crystals were theoretically analyzed in [22]. Since there are great differences in many properties among different crystal types, including the optical transparency range, phase-matching type and waveband, and the maximum growth size of the crystal [13], these differences result in more constraint conditions and greatly limit the application range of the designs. Moreover, these designs assume that each crystal has the same temperature variation and temperature gradient. In fact, for different crystal types, their optical absorption coefficient, specific heat capacity and thermal conductivity coefficient generally have great differences [13]. In practical application, the temperature variation and temperature gradient of each crystal induced by the thermal effect are not the same. Thus, the effectiveness of these designs cascaded different types of crystals will be reduced.

For cascaded schemes using different crystal types, the problems mentioned above are inevitable. Here, we present a new scheme that is capable of self-compensating for thermally induced phase mismatch using the same type of crystals, thereby achieving temperature-insensitive frequency conversion. The basic structure of the scheme is shown in Fig. 1. Three crystals with the same type are cascaded. Crystals A and C located at both ends are used for frequency conversion, and the middle crystal B is used for phase mismatch compensation. By configuring the polarization states of the optical waves in crystal B, the signs of $\partial \Delta k/\partial T$ in adjacent crystals are opposite, resulting in opposite phase mismatch. In this way, the phase mismatch caused by the laser thermal effect in crystal A can be self-compensated in crystal B, so the interacting waves can still achieve efficient frequency conversion in crystal C. Since the types of the crystals used are the same, the aforementioned problems can be fundamentally overcome.

 

Fig. 1 Configuration of polarization states for temperature-insensitive frequency conversion. (a) Type-I second harmonic generation (SHG). (b) Type-II third harmonic generation (THG).

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2. Basic principle and crystal parameters analysis

In order to present the principle of the scheme clearly, we take KH₂PO₄ (KDP) crystal as an example to analyze type-I SHG and type-II THG. In crystals A and C, type-I SHG allows an ordinary light (o) of frequency ${\omega }_1$ incidents to the crystal and generates an extraordinary light (e) of frequency${\omega }_2=2{\omega }_1$, i.e., $o\left({\omega }_1\right)+o\left({\omega }_1\right)\to e\left({\omega }_2\right)$. The phase mismatch ($\Delta k$) and the first temperature derivative of phase mismatch ($\Delta k^{\prime }$) can be expressed as [23]:

For crystal B, by selecting the appropriate cutting direction, the polarization states of the fundamental wave (FW) and the second harmonic (SH) are e polarization and o polarization, respectively, as shown in Fig. 1(a). The corresponding $\Delta k$ and $\Delta k^{\prime }$ are given by:

For type-II THG in crystals A and C, an e light of frequency ${\omega }_3={\omega }_2+{\omega }_1$ is generated from an e light of frequency ${\omega }_1$ and an o light of frequency ${\omega }_2$, i.e., $e\left({\omega }_1\right)+o\left({\omega }_2\right)\to e\left({\omega }_3\right)$. In crystal B, the FW, SH and third harmonic (TH) are o light, e light, and o light, respectively, as shown in Fig. 1(b). Thus, for Eqs. (1)–(4), similar equations can be written as:

In this scheme, the parameters of crystal B are a key to accurately compensate for the thermally induced phase mismatch and achieve temperature-insensitive frequency conversion. At the phase-matching temperature ($T_0$), since $\Delta k_{\mathrm{A}}\left(T_0\right)=\Delta k_C\left(T_0\right)=0$, the phase mismatch value [$\Delta K_B\left(T_0\right)$] generated in crystal B should be an integral multiple of 2π. Meanwhile, the lengths of crystals A and B should satisfy special proportion, as indicated by the following equations:

According to Eqs. (2)–(4) and Eqs. (6)–(8), it can be found that $\Delta {k^{\prime }}_{\mathrm{A}}$, $\Delta {k^{\prime }}_B$, and $\Delta k_B$ are related to the cutting angles of the crystals. However, for a given frequency conversion, the required cutting angle (i.e., phase-matching angle) and lengths of crystals A and C are determined. Therefore, $\Delta {k^{\prime }}_{\mathrm{A}}$ and $L_{\mathrm{A}}$ are known initial conditions when calculating the parameters of the crystal B. Since crystal B is used only to compensate for the phase mismatch, the cutting angle can be flexibly chosen and has the optimal value. For simplicity, we take the KDP crystals and a laser at wavelength of 1053 nm as an example to analyze quantitatively and assume that crystal A is 10 mm in length and the initial phase-matching temperature is 20°C.

According to the temperature dependent dispersion equations of KDP crystal [24], it is convenient to plot the curves of $\Delta {k^{\prime }}_{\mathrm{A}}$ and $\Delta {k^{\prime }}_B$ varying with the crystal angle, as shown in Fig. 2(a) and 2(b). For type-I SHG and type-II THG at wavelength of 1053 nm, the phase-matching angles of KDP crystal are 41.0° and 59.0°, respectively, and the corresponding $\Delta {k^{\prime }}_A^{SHG}$ and $\Delta {k^{\prime }}_A^{THG}$ are 53.6/(m⋅°C) and 93.3/(m⋅°C). Using Eq. (9) the lengths of crystal B at different crystal angles can be obtained, as shown in Fig. 2(c) and 2(d).

 

Fig. 2 Values of $\Delta k^{\prime }$, $L_B$, and $\Delta K_B\left(T_0\right)$ as a function of angles. (a), (c), and (e) Type-I SHG. (b), (d), and (f) Type-II THG.

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It can be seen that $\left|\Delta {k^{\prime }}_B\right|$has a maximum value, and the required length of crystal B is the shortest at the cutting angle of 90°. We also note that ${\partial \Delta {k^{\prime }}_B/\partial \theta |}_{90°}=0$, ${\partial L_B/\partial \theta |}_{90°}=0$, and $\Delta {k^{\prime }}_B$ and $L_B$ are insensitive to the change of the angle in the limit of ${\theta }_B\to 90°$. Consequently, the optimal choice for the cutting angle of crystal B should be near 90°. In this case, the crystal not only has a lower material cost but also is easier to obtain and apply. Considering the symmetry of the refractive index ellipsoid of the crystals [23], this conclusion is universal.

We then consider Eq. (10); the change of the phase mismatch value generated in crystal B with the angle can be calculated according to the dispersion equation [24]. Because our focus is on the variation of $\Delta K_B\left(T_0\right)$ with a period of 2π, in order to intuitively analyze, $\Delta K_B\left(T_0\right)$ was translated in units of 2π. The obtained results are shown in Fig. 2(e) and 2(f), and the corresponding angles and lengths at the integral multiple of 2π are the required crystal parameters.

In fact, the condition described by Eq. (10) can be easily satisfied by slightly rotating or tilting crystal B. If the cutting angle of crystal B is chosen near 90°, the required length of crystal B varies very little with the angle, and Eq. (9) can thus be regarded as always satisfied during the process of adjusting crystal B. Therefore, in practical applications, it is easy to implement this scheme just by choosing the crystal cutting angle near 90° and determining an appropriate length. Additionally, in this case, crystal B has a low requirement for processing precision.

3. Experimental equipment

In order to prove the validity and universality of the proposed scheme, temperature-insensitive SHG and THG were experimentally demonstrated based on the above analysis. In the experiment, a Nd:YLF laser (wavelength of 1053 nm) with a repetition rate of 1 Hz was used as the fundamental laser source. The transverse profile of the output FW is a circular spot with a diameter of 6.5 mm. The time waveform is an 8.5-ns (full width at half maximum, FWHM) Gauss pulse. Two KDP crystals with lengths of 18 mm and two KDP crystals with lengths of 14 mm were cut for type-I and type-II phase matching to generate SH and TH at 1053 nm, respectively. Their cutting angles are ${\theta }_I=41.3°$, ${\phi }_I=45°$ and ${\theta }_{II}=59.2°$, ${\phi }_{II}=0°$, respectively. An 8.2-mm-long and a 6.1-mm-long KDP crystals with the same cutting angle of 87° were used to compensate for the phase mismatches of SH and TH, respectively. The optical surfaces of the abovementioned crystals were polished and coated antireflection films at wavelengths of 1053, 526.5, and 351 nm. The schematic diagrams of the experimental equipment used to verify the proposed scheme are shown in Fig. 3.

 

Fig. 3 Schematic diagrams of temperature-insensitive (a) SHG and (b) THG experiments.

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In the temperature-insensitive SHG experiment, three KDP crystals were placed in the temperature-control devices, which have a temperature regulating range of 40°C–180°C and a control accuracy of ± 0.1°C. A prism was used to separate the SH from the remaining FW, which was emitted from the last KDP crystal, and then the energy of SH was measured by an energy meter, as shown in Fig. 3(a). For the THG experiment, a 10-mm-thick BBO crystal was used to generate SH. Compared with the KDP crystal, the BBO crystal has a larger nonlinear coefficient [13] and can thus achieve higher conversion efficiency. In order to specifically analyze the influence of temperature on the THG efficiency and the effect of the proposed scheme, the KDP crystals were placed in the temperature control devices while the BBO crystal was always at room temperature. The beams emitted from the last KDP crystal were incident into a prism to separate the TH from the remaining FW and SH, as shown in Fig. 3(b).

In this paper, the efficiencies of SHG and THG are defined as ${\eta }_2=E\left({\omega }_2\right)/E\left({\omega }_1\right)$ and ${\eta }_3=E\left({\omega }_3\right)/E\left({\omega }_1\right)$, respectively, where $E\left({\omega }_1\right)$, $E\left({\omega }_2\right)$, and $E\left({\omega }_3\right)$ represent the initial energy of the FW and the energies of the generated SH and TH, respectively. Considering the regulating range of temperature control devices, the initial phase-matching temperatures of the SHG and THG were both set as 60°C. In the experiments of SHG and THG, the measured FW pulse energies output from the Nd:YLF laser were 33.76 and 32.63 mJ, respectively. At each temperature, the energies of the SH and TH were measured 50 times and averaged to eliminate the influence of laser energy fluctuation on the experimental results.

4. Experimental results

According to the theoretical analysis in section 2, we know that the dispersion equations and the thermo-optic coefficients of the crystal directly determine the first temperature derivative of phase mismatch and the length of the phase mismatch compensating crystal. However, by comparing the different references [13,24,25], it can be found that there exist differences in the dispersion equations and the thermo-optic coefficients of the KDP crystal, especially the thermal-optic coefficients. The main reason is that the optical properties of the KDP crystal are closely related to its growth method and growth process, including growth rate, temperature control, concentration of the growth solution, and so on [26,27]. Therefore, the optical properties of the KDP crystals grown in different ways are different, and there may even be differences between the crystals obtained from different growth batches. In addition, the reported optical parameters of the crystal are also affected by the measurement method, accuracy, and experimental conditions. Especially, the measurement of the thermo-optic coefficients has high requirements for the experimental conditions, and the measured results are very susceptible to the external environment.

In our experiments, $\Delta {k^{\prime }}^{SHG}=28.77/\left(\text{m}\cdot °\text{C}\right)$ and $\Delta {k^{\prime }}^{THG}=94.63/\left(\text{m}\cdot °\text{C}\right)$ were used to calculate the lengths of the phase mismatch compensating crystals based on the data reported in [13,21,24,25]. In order to compare the consistency between the adopted data and the actual thermo-optic characteristics of the employed KDP crystals, we experimentally analyzed the temperature-dependent properties of a single KDP crystal first.

4.1 Thermo-optic characteristics of KDP crystals used for SHG and THG

In this experiment, an 18-mm-long and a 14-mm-long KDP crystals were used as SHG and THG, respectively. The measured conversion efficiencies vary with temperature as shown in Fig. 4. In order to directly compare the experimental results with the simulated results, all conversion efficiencies were normalized to the corresponding maximum efficiencies obtained at the initial phase-matching temperature (60°C).

 

Fig. 4 Measured and simulated efficiencies of (a) SHG and (b) THG vary with temperature using a single KDP crystal.

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Figure 4(a) shows the variation in the efficiencies of type-I SHG at 1053 nm with temperature. At the phase-matching temperature, the measured conversion efficiency is 1.46%. The green circles plot that the measured temperature acceptance bandwidth ($\Delta T_{Experiment}^{SHG}$) was 10.47°C, as defined by the FWHM. According to $\Delta T_{Experiment}^{SHG}$, the first temperature derivative of the phase mismatch ($\Delta {k^{\prime }}_{Experiment}^{SHG}$) can be calculated to be $29.54/\left(\text{m}\cdot °\text{C}\right)$. For the numerically simulated results corresponding to the experimental conditions, they were indicated by a dashed line in Fig. 4(a). The corresponding $\Delta T_{Simulation}^{SHG}$ and $\Delta {k^{\prime }}_{Simulation}^{SHG}$ were 10.74°C and $28.80/\left(\text{m}\cdot °\text{C}\right)$. By comparing $\Delta {k^{\prime }}^{SHG}$, $\Delta {k^{\prime }}_{Experiment}^{SHG}$, and $\Delta {k^{\prime }}_{Simulation}^{SHG}$, it can be seen that the experimental result are in good agreement with the simulated results and the adopted data.

For type-II THG at 1053 nm, the measured efficiencies varying with temperature are indicated by violet circles in Fig. 4(b). The maximum conversion efficiency is 1.95%. The corresponding $\Delta T_{Experiment}^{THG}$ and $\Delta {k^{\prime }}_{Experiment}^{THG}$ were 2.94°C and $135.25/\left(\text{m}\cdot °\text{C}\right)$, respectively. We can find that there is an obvious difference between the experimental result ($\Delta {k^{\prime }}_{Experiment}^{THG}$) and the adopted data ($\Delta {k^{\prime }}^{THG}$). Based on $\Delta {k^{\prime }}^{THG}$, the simulated results are also obviously different from the experimental results, as shown by the black dashed line and violet circles in Fig. 4(b). If the experimental data are used for simulation, the numerically calculated results are in good agreement with the experimental results, as shown by the red dotted line in Fig. 4(b), the temperature acceptance bandwidth is 2.99°C.

Based on the above experimental analysis and numerical simulation, it can be seen that the thermo-optic coefficients of the KDP crystal used for THG in the experiment are different from the data reported in [24]. These differences will affect the determination of the parameters for the phase mismatch compensating crystal. In practical applications, accurately obtaining the thermo-optic coefficients of the employed crystals is a key factor to successfully implement the proposed scheme.

4.2 Temperature-insensitive SHG and THG

In the temperature-insensitive SHG and THG experiments, the temperatures of the KDP crystals were changed in the ranges of 52°C–68°C with increments of 0.5°C and 57°C–63°C with increments of 0.2°C, respectively. The experimental results are shown in Fig. 5.

 

Fig. 5 Measured and simulated efficiencies vary with temperature based on the cascaded KDP crystals. (a) SHG and (c) THG without phase mismatch compensation. (b) SHG and (d) THG with phase mismatch compensation. All the efficiencies were normalized to their maximum values at the initial phase-matching temperature.

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When the KDP crystals were at the phase-matching temperature, the measured maximum efficiency of SHG was 4.14%. If the temperatures of KDP crystals were changed, and the generated phase mismatch was not compensated, two cascaded KDP crystals used for SHG can be considered as a single crystal. In this case, the efficiencies of SHG decreased rapidly with the change of the temperature, as indicated by red circles in Fig. 5(a). The measured temperature acceptance bandwidth was 5.24°C. If the thermally induced phase mismatch is compensated using the proposed frequency conversion scheme in this paper, the temperature acceptance bandwidth can reach 10.52°C, which is two times larger than that of the conventional scheme using a single crystal, as revealed by green squares in Fig. 5(b). By comparing Fig. 5(a) and 5(b), it can be observed that the temperature acceptance bandwidth is significantly increased, which proves the validity and feasibility of the proposed self-compensation scheme of thermally induced phase mismatch.

In the THG experiment, a BBO crystal was used to generate SH. The measured and numerically calculated SHG efficiencies were 14.93% and 15.28%, respectively. Figure 5(c) and 5(d) plot the THG efficiencies change with temperature. At 60°C, the maximum conversion efficiency (${\eta }_{\max }$) was measured to be 5.11%. For two cascaded KDP crystals without phase mismatch compensation, the temperature acceptance bandwidth is only 1.49°C, as shown by blue circles in Fig. 5(c). Based on our proposed scheme, the obtained temperature acceptance bandwidth is 1.7 times larger than that of Fig. 5(c), increasing to 2.6°C, as shown by violet squares in Fig. 5(d).

These experiments were numerically simulated, and the results were plotted together with the experimental results, as revealed by the black dashed lines in Fig. 5. The calculated maximum efficiencies of SHG and THG with phase mismatch compensation are 4.23% and 5.22%, respectively. By compensating for the phase mismatch, the simulated temperature acceptance bandwidths of the SHG and THG are increased from 5.36°C to 10.85°C and from 1.48°C to 3.08°C, respectively. The related data are shown in Table 1 to allow a complete comparison between the experimental and simulated results of the traditional frequency conversion scheme and the scheme proposed in this paper.

Table 1. Experimental and simulated results of the conventional scheme and the proposed scheme

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According to Fig. 5 and Table 1, it can be seen that the simulated results of SHG are in good agreement with the experimental results, whereas the temperature acceptance bandwidth of THG in Fig. 5(d) is obviously smaller than that of the simulation. The main reason is that there exist obvious deviations between the reported thermo-optic coefficients, which are used to calculate the length of the phase mismatch compensating crystal, and the actual thermo-optic characteristics of the KDP crystals used for THG, as demonstrated by experimental results in section 4.1. Consequently, the calculated phase mismatch compensating crystal length is inaccurate, so the thermally induced phase mismatch cannot be compensated very well. Nevertheless, both the experimental and simulated results indicate that the proposed scheme can achieve self-compensation of thermally induced phase mismatch, thus increasing the temperature acceptance bandwidth of frequency conversion.

In addition, it can be seen from Table 1 that the experimentally measured maximum conversion efficiencies of SHG and THG increase slightly when the phase mismatch compensation is employed, while the simulated results are slightly reduced. The main reason is that the phase mismatch caused by air dispersion cannot be eliminated accurately in the experiment when only two nonlinear crystals are cascaded without crystal B. If crystal B is used, the phase mismatch caused by air dispersion can be compensated accurately. Therefore, in the experiment, the conversion efficiency is slightly improved. For simulated results, in the absence and presence of crystal B, the phase mismatches caused by air dispersion are both assumed to be completely eliminated. If crystal B is used, the reflection loss of laser energy increases. Therefore, the maximum conversion efficiency of simulation reduces slightly with crystal B.

5. Discussion

5.1 Multi-crystal cascade for temperature-insensitive frequency conversion

For all the above experiments, the initial phase-matching temperatures of KDP crystals were set as 60°C. Since the refractive indexes of the crystals vary approximately linearly with the temperature [13], similar experimental results can be obtained for the temperature variations in other ranges. For the temperature-insensitive SHG and THG in section 4.2, only two KDP crystals were used for frequency conversion in each experiment. Based on our proposed scheme, if multiple frequency conversion crystals are cascaded (the total length is constant), and a phase mismatch compensating crystal was used between each pair of frequency conversion crystals, the temperature acceptance bandwidth can be further increased.

In order to give quantitative results, we take SHG as an example to simulate the cases of multi-crystal cascade. For simplicity, the adopted laser parameters and the initial conditions of the KDP crystals are consistent with those aforementioned. It is important to note that the energy loss caused by crystal surface reflection should be considered for multi-crystal cascade. If the crystals were coated with antireflection films, the transmittance of a single surface can reach 99.5% [28]. Therefore, the transmittance of each surface was assumed to be 99.5% in the simulation, and the obtained results are shown in Table 2 and Fig. 6(a).

Table 2. Simulated results of crystal cascade with different number

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Fig. 6 (a) Simulated results of crystal cascade and (b) schematic diagrams of crystals sequence with different number. (c) The change of SHG efficiency with $\Delta K_{Air}$ induced by air dispersion and the change of $\Delta K_B$ induced by crystal B with the rotation angle. (d) Variation of the crystal length in the direction of beams propagation. In (a) and (c), all the efficiencies were normalized to the maximum values. In (b), dark blue rectangles indicate the crystals used for SHG, and light blue rectangles indicate the phase mismatch compensation crystals.

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According to Fig. 6(a) and Table 2, it can be observed that the temperature acceptance bandwidth increases significantly with the increase in the number of cascaded crystals. If four KDP crystals are used to generate SH, the temperature acceptance bandwidth is 21.76°C, which is 4.1 times larger than that of using a single crystal (5.36°C), as shown by the green dotted line in Fig. 6(a).

5.2 Compensation of phase mismatch caused by air dispersion

For frequency conversion scheme employing the crystal cascade structure, the conversion efficiency is affected by the phase difference caused by the air dispersion between the crystals. Therefore, in practical applications, the distance between crystals is also an important factor that cannot be ignored. For the frequency conversion scheme reported in [18,19], the spacing of the crystals must be adjusted to a proper distance and be strictly controlled; otherwise, the conversion efficiency will reduce. For our proposed scheme, by fine-tuning the phase mismatch compensating crystal (i.e., crystal B in Fig. 1), it can simultaneously compensate for the phase mismatches induced by the thermal effects and air dispersion.

Herein, we still use the above experimental parameters and take SHG as an example to analyze. In order to be more pertinent, we assume that both crystals A and C (as indicated in Fig. 1) are at the phase-matching temperature, and the condition described by Eq. (10) is also satisfied. The change of SHG efficiencies with phase mismatch value ($\Delta K_{Air}$) induced by air dispersion is shown by the red solid line in Fig. 6(b). It can be seen that the air dispersion can significantly affect the conversion efficiency. However, in our proposed scheme, $\Delta K_{Air}$ can be well compensated by adjusting the crystal B. The corresponding mathematical expression can be written as:

For a small rotation angle, the change of the beam propagation length in the crystal is very small. It is assumed that the rotation angle varies in the range of −1.2°–1.2°. The corresponding variation of the beams propagation length is less than 0.022% (i.e., $\Delta L<0.00022L_B$) while the change of $\Delta K_B$ is remarkable, as shown by the black dashed line in Fig. 6(b). For convenient analysis, the values of $\Delta K_B$ have been translated in units of 2π. From Fig. 6(b), it can be found that the phase mismatch caused by air dispersion can be well compensated in crystal B. Therefore, this scheme cannot only compensate for the thermally induced phase mismatch but also eliminate the influence of air dispersion.

5.3 Spatial walk-off in crystal B

In general, the walk-off effect occurs when the extraordinary light wave transmits in nonlinear crystals [23]. Therefore, it is necessary to analyze the spatial walk-off in crystal B. In this paper, for the SHG, the FW is extraordinary light in crystal B; for the THG, the SH is extraordinary light in crystal B, as shown in Fig. 1. According to the dispersive equation of KDP crystal [13], the walk-off angles of the FW and SH varying with the cutting angle of the crystal B can be calculated [23], as shown in Fig. 7(a).

 

Fig. 7 (a) Walk-off angles of the FW and SH vary with the cutting angle of crystal B. (b) Spatial walk-off distances of the FW and SH.

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In our experiment, the cutting angle of crystal B is 87°. The corresponding walk-off angles of the FW and SH are $\rho \left({\omega }_1\right)=2.34$ mrad and $\rho \left({\omega }_2\right)=2.83$ mrad, respectively. In the SHG and THG experiments, the lengths of crystal B are $L_B^{SHG}=8.2$ mm and $L_B^{THG}=6.1$ mm, respectively. The corresponding spatial walk-off distances of the FW and SH can be calculated to be $d\left({\omega }_1\right)=19.2$ μm and $d\left({\omega }_2\right)=17.3$ μm, respectively, which are less than 0.3% of the beam spot (diameter of 6.5 mm). Therefore, the influence of the spatial walk-off is very little. If the cutting angle of crystal B is 90°, the walk-off angle is zero, which means that no spatial walk-off exists in crystal B.

5.4 Possible frequency conversion in crystal B

In the application of this scheme, we should consider and avoid the possible frequency conversion occurred in crystal B. In order to intuitively explain how to avoid the possible frequency conversion in crystal B, we take KDP crystal as an example to analyze.

There are two basic preconditions for effective frequency conversion [23,29]. The first is that the phase matching condition must be satisfied between the interaction waves, and the second is that the effective nonlinear coefficient cannot be zero.

According to the conclusion in section 2, the optimum choice of the cutting angle of crystal B should be near 90°. For most of the wavelengths, the phase-matching angles are not near 90° [13]. Therefore, for crystal B with cutting angle near 90°, most of the wavelengths cannot meet the phase matching condition, and the frequency conversion does not occur. Take the SHG process at wavelength of 1053 nm as an example, the required phase-matching angle of type-II sum-frequency generation in crystal B is 59.2°. However, in our experiment, the cutting angle of crystal B is 87°. Obviously, this process does not satisfy the phase matching condition. Therefore, sum-frequency generation doesn't occur in crystal B.

For a few particular wavelengths with particular phase-matching type, their phase-matching angles are near 90° [13]. For these particular cases, we can choose an appropriate azimuth of the crystal B so that the effective nonlinear coefficient is zero.

If the type-I phase-matching angle near 90°, the effective nonlinear coefficient of KDP crystal can be expressed as [13]:

If the type-II phase-matching angle near 90°, the effective nonlinear coefficient of KDP crystal can be expressed as [13]:

We choose $\phi =45°$, $d_{oee}=d_{eoe}=0$. Therefore, the frequency conversion with type-II phase matching doesn't occur in crystal B.

In brief, we could suppress the possible frequency conversion occurred in crystal B. For crystal B with cutting angle near 90°, most of the wavelengths cannot meet the phase matching condition. Therefore, the frequency conversion does not occur in crystal B in general. For a few particular wavelengths with particular phase-matching type, the phase-matching angles are near 90°. For these particular cases, we can choose an appropriate azimuth make effective nonlinear coefficient to zero. Thus, frequency conversion is avoided in crystal B.

6. Summary and conclusions

In summary, we proposed and experimentally investigated a new scheme aimed for the frequency conversion of the high-power and high-repetition-rate laser. It can self-compensate for the thermally induced phase mismatch and achieve temperature-insensitive frequency conversion. The core design of this scheme is based on the cascade structure using the same type of crystals. By inversing the polarizations of the interaction waves in adjacent crystals, the signs of the first derivative of the phase mismatch with respect to temperature are opposite, resulting in opposite phase mismatch. Thus, the self-compensation of thermally induced phase mismatch is achieved.

In order to verify this scheme, taking SHG and THG at 1053 nm as examples, the principle of this scheme was analyzed in detail, and the temperature-insensitive SHG and THG using KDP crystals were experimentally demonstrated. The experimental results show that the temperature acceptance bandwidths of SHG and THG are increased to 2.0 and 1.7 times larger, respectively. The sensitivity of frequency conversion to the temperature variation is effectively reduced. Furthermore, we also analyzed and discussed the multi-crystal cascade, reflection loss of laser energy, the compensation of the air dispersion induced phase mismatch, and spatial walk-off and suppression of frequency conversion in crystal B. The results indicate that this scheme can eliminate the influence of air dispersion, and the temperature acceptance bandwidth can be further improved by cascading more crystals. Considering the reflection loss of the crystal surface, if four KDP crystals are used to generate SH, the conversion efficiency is decreased by only 8.43%, and the temperature acceptance bandwidth can be increased from 5.36°C to 21.76°C.

Since the frequency conversion crystals and phase mismatch compensating crystal are of the same type, this scheme is completely unlimited by the optical transparency range, crystal type, phase-matching type, and laser wavelength. It can be widely used in various frequency conversions at different wavebands. Moreover, in the schemes currently reported [18–22], both temperature variation and temperature gradient are inconsistent between the cascaded crystals. The proposed scheme can fundamentally overcome these problems, which induced by numerous factors, including the differences of the optical absorption coefficient, specific heat capacity and thermal conductivity coefficient between different crystal types. Therefore, the proposed scheme has excellent versatility and a more significant practical effect, which may provide a new and feasible approach for the design of high-power laser frequency conversion system.