1. Introduction

As excellent electro-optic (EO) and nonlinear optical materials, KH$_2$PO$_4$ (KDP), K(H$_{1-x}$D$_x$)$_2$PO$_4$ (DKDP), $\beta$-BaB$_2$O$_4$ (BBO), and LiNbO$_3$ (LN) are widely used in various optoelectronic devices such as optical switcher and frequency converter [1–3]. In particular, KDP-family crystals have key applications in high power laser facilities and ultrafast laser systems because of their favorable properties such as high transmittance from infrared to ultraviolet regions, adequate nonlinearity, high laser-induced damage threshold, good optical homogeneity, and availability in large size [4–7]. As for the EO materials, the EO coefficient is one of the basic physical parameters of KDP-family crystals. However, the current studies and applications of the EO effect of KDP-family crystals are mainly focused on KDP and DKDP with high deuterium content [8–12]. For the partially deuterated KDP crystals, their EO coefficients are sporadically reported, although they have crucial applications in many fields. For example, for the harmonic conversion of high-peak-power lasers at fusion level where large-aperture crystals are required, transverse stimulated Raman scattering (TSRS) in partially deuterated KDP crystal can be effectively reduced [13]. For DKDP crystal with a deuteration level of 12 $\pm$ 2%, broadband second harmonic generation (SHG) at 1053 nm can be achieved [14]. Moreover, non-critical phase-matched (NCPM) fourth harmonic generation (FHG) can be realized for the lasers with wavelength ranging from 1010 to 1070 nm in partially deuterated KDP by adjusting the deuterium content [15,16]. For example, NCPM FHG of 1053 nm laser can be achieved near room temperature in DKDP crystal with 70% deuteration [16]. Hence, the EO coefficients of different deuterated KDP crystals are necessary to be determined accurately.

For the measurement of EO coefficients, the methods commonly used today are mainly based on the linear optical effects, such as half-wave voltage, interferometry, ellipsometry, and the Senamon polarization methods [17–20]. Even though the half-wave voltage method has the advantages of simple operation, low cost, and wide application, its measurement sensitivity and accuracy are limited. The interference method can achieve high precision, but the system is sensitive to mechanical or environmental instabilities. For the elliptic photometry method, it has high precision, but the cost of this method is very high. While the Senamon polarization method is simple in structure and low in cost, it requires a complex operation to ensure the accuracy of the measurement and takes a long time [21]. All the above methods require the laser to strictly keep the stability of polarization in the whole measurement system. Small disturbances result from the thermal effect, optical, or air fluctuations in transfer processes will have a significant impact on the accuracy of the above methods [22]. In fact, in the process of nonlinear optical frequency conversion, phase matching is very sensitive to subtle refractive index changes [23,24], tiny phase mismatch would result in a significant drop in energy conversion efficiency. Moreover, the energy conversion efficiency only depends on the nonlinear process in nonlinear materials, and other transfer processes will not affect it. These characteristics have great advantages in the accurate measurement of the EO coefficients.

Here, we present a novel method with the advantages of high precision and sensitivity based on the nonlinear optical technology to measure EO coefficients. The core principle is that an external electric field is applied in the K(H$_{1-x}$D$_x$)$_2$PO$_4$ crystals to compensate for the phase mismatch caused by the variation of angle. According to the applied voltage and corresponding angle change in K(H$_{1-x}$D$_x$)$_2$PO$_4$ crystals, we can obtain the linear EO coefficient precisely. In theory, a new formula to exactly calculates the EO coefficients of KDP-family crystals is developed. In the experiment, the FHG characteristics of different deuterated KDP crystals at different wavelengths were demonstrated systematically, and the EO coefficients of a series of different deuterated KDP crystals were measured comprehensively. Finally, we comprehensively obtained the EO coefficients of KDP, 23%-, 45%-, 68%-, 82%-, and 90%- DKDP, and given the relationship between deuterium content ($x$) and EO coefficient ($\gamma _{63}$) of K(H$_{1-x}$D$_{x}$)$_2$PO$_4$ crystals, which provides a good reference for future EO applications with KDP-family crystals.

2. Theory

Taking $\bar {4}$2m point group crystals as an example, when they are used as the EO crystals, transverse EO modulation is usually adopted. If an external electric field is applied along the optical axis ($z$ axis), they transform into biaxial crystals from uniaxial crystals, and the relationships between the new and original principal refractive indices are satisfied the followed equations [24]:

When the crystals are used as the nonlinear materials for frequency conversion, the conversion efficiency is closely related to the phase mismatch ($\Delta k$) among the interacting waves. $\Delta k$ is determined by the refractive indices. For DKDP crystal, the refractive indices are the functions of external parameters such as wavelength ($\lambda$), crystal temperature ($T$), intensity of external electric field ($E$), and rotation angle ($\Delta \theta$), and internal parameter, deuterium content ($D$).

For the optical materials with $\chi ^{(2)}$ nonlinear effects, such as SHG, FHG, and optical parametric amplification, we can choose the appropriate nonlinear process and corresponding phase-matching scheme according to their characteristics. For simplicity, we take FHG with type-I NCMP as an example for theoretical analysis. In the case of type-I NCPM, i.e., 90$^\circ$ phase matching, the refractive indices of $\lambda _2$ and $\lambda _4$ for the DKDP crystal obey: $\displaystyle n_e \left ( \lambda _4 , D \right ) =n_o \left ( \lambda _2 , D\right )$, where $\lambda _2$ and $\lambda _4$ are the second and fourth harmonic wavelengths of the fundamental wave, respectively. When the angle changes by $\Delta \theta$, a phase mismatch results from the variation of extraordinary refractive index, and the conversion efficiency is hence significantly reduced [16].

If $E$ $\parallel$ $z$, the refractive indices of $\lambda _2$ and $\lambda _4$ are related to the electric-field intensity, while that of $\lambda _4$ is the function of angle, as well. The refractive indices of $\lambda _2$ and $\lambda _4$ and the phase mismatch can be described explicitly as follows:

In Eq. (5), the influence of the electric field on the refractive index is ignored, because it is much smaller than the angle-induced change. Take the case of $E=500\ V$/mm and $\Delta \theta = 1^{\circ }$ as an example, for the fourth harmonic, the impact of the external electric field on the extraordinary refractive index ($6.73 \times 10^{-9}$) is much smaller than the refractive index change caused by angle variation ($1.37 \times 10^{-5}$). Thus, it is rational to suppose that the extraordinary refractive index is independent of the electric field and adopts an approximate expression. In the case of voltage-tuning phase matching (VTPM) [23], an appropriate voltage can be chosen so that the refractive indices of the interacting waves satisfy: $\displaystyle n_4\left ( \lambda _4, \Delta \theta , D \right )=n_2\left ( \lambda _2, E, D \right )$. Therefore, the phase mismatch induced by angle variation can be precisely compensated. The EO coefficient dependence of the voltage and angle is determined by:

As can be seen from Eq. (7), the variables involved in this method are voltage and angle. At present, the high-precision voltage generator can achieve an accuracy of less than 0.1%, and the angle control accuracy can achieve 8 $\mu$rad or even lower. Considering these factors, theoretically, this method can achieve a measurement accuracy of less than 0.2%. This means that by applying the linear EO effect to the frequency conversion process, the EO coefficient can be accurately measured according to the voltage magnitude and angle variation.

3. Experiment

In the experiment, we take K(H$_{1-x}$D$_x$)$_2$PO$_4$ crystals as an example to verify the proposed method. For the EO coefficients measurement of K(H$_{1-x}$D$_x$)$_2$PO$_4$ crystals, comprehensively considering the parameters of the samples and the frequency conversion characteristics of KDP-family crystals, the FHG scheme was chosen. The layout of the experiment is shown in Fig. 1. The lasers with wavelengths of 1053 nm (Nd:YLF) and 1064 nm (Nd:YAG) were employed as the fundamental wave. For the two lasers, the repetition rates are 1 Hz and 10 Hz, respectively, and the pulse widths are 8.5 ns and 7.5 ns (full width at half maximum, FWHM), respectively. Their transverse profiles are circle spots with diameters of 6 mm and 7 mm, respectively. A half-wave plate (HWP) and a polarization beam splitter (PBS) were used to adjust the fundamental wave energy to ensure that the second harmonic energy incident into the FHG crystals is consistent (10 mJ). The SHG was obtained by a 25-mm-thick LBO crystal ($\theta _{\mathrm {SHG}} = 90^\circ$, $\varphi _{\mathrm {SHG}} = 11.5^\circ$, type-I phase matching). To perform the FHG experiment, a series of different deuterium content, 25 mm-thick DKDP crystals with an aperture of $15 \times 15$ mm$^2$ were cut at the type-I NCPM direction ($\theta _{\mathrm {FHG}} = 90^\circ$, $\varphi _{\mathrm {FHG}} = 45^\circ$). Both the surfaces perpendicular to the optic axis were mounted with electrodes, and the transmittance faces of the crystals were finely polished and coated with anti-reflection films. Raman spectrums of these DKDP crystals are shown in Fig. 2. With the deuterium content increasing, the spectral mean of the PO$_4$ vibration displays an approximately linear redshift and 1 cm$^{-1}$ shift represents about 2.684% deuterated level variation, which indicated the deuterium contents of the samples are 0%, 23%, 45%, 68%, 82%, and 90%, respectively [25]. The DKDP sample was placed in a rotation stage with an angle control accuracy of 8 $\mu$rad. A high-DC-voltage generator that varied the voltage from 0 to 10 kV with a control error of 1% and a temperature-controlling device with a practical operating range of 30-180$^\circ \mathrm {C}$ and a precision of $\pm 0.1^\circ \mathrm {C}$ was employed to adjust the electric field and temperature of the crystals.

 

Fig. 1. Experiment setup for measuring the linear EO coefficient of K(H$_{1-x}$D$_x$)$_2$PO$_4$ crystals.

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Fig. 2. Raman spectrums of different deuterated KDP crystals.

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First, we determine the NCPM temperature ($T_{\mathrm {NCPM}}$) of the different deuterated crystals. The temperature tuning curves of different deuterium content DKDP crystals were plotted in Fig. 3. As shown in Fig. 3(a), for KDP, 23%-, and 45%-DKDP, $T_{\mathrm {NCPM}}$ of 1053nm laser are $133^\circ \mathrm {C}$, $96.6^\circ \mathrm {C}$, and $63.6^\circ \mathrm {C}$, respectively. In Fig. 3(b), for 68%-, 82%-, and 90%-DKDP, $T_{\mathrm {NCPM}}$ of 1064nm laser are $107.4^\circ \mathrm {C}$, $77.6^\circ \mathrm {C}$, and $59.5^\circ \mathrm {C}$, respectively. Considering the temperature control range ($30-180^\circ \mathrm {C}$) of our equipment and the possible crystal damage caused by high temperature (>$150^\circ \mathrm {C}$), FHG of 1053nm laser was adopted in KDP, 23%-, and 45%-DKDP crystals, and FHG of 1064nm laser was adopted in 68%-, 82%-, and 90%-DKDP crystals. Once the temperature was fixed at its $T_{\mathrm {NCPM}}$, the crystal orientation was rotated, which would introduce phase mismatch and thus the efficiency would decrease inversely. Then the voltage was applied to the DKDP crystal, phase matching would be achieved again by adjusting the voltage magnitude. The energies of the fourth harmonic output from different deuterated KDP crystals were measured by energy meters and were averaged over 30 measurements at each angle to eliminate the influence of energy fluctuations.

 

Fig. 3. Temperature tuning curves of different deuterium content DKDP crystals at the fundamental wavelength of (a) 1053nm and (b) 1064nm. The dots are experimental results and the solid lines are fitted results.

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4. Results and discussion

The experimental and simulated results are given in Fig. 4 and the FHG efficiencies and angular bandwidth of six crystals are listed in Table 1. We defined the FHG efficiencies as $\eta = E_4 / E_2$, where $E_2$ and $E_4$ are the input energy of the second harmonic and the output energy of the fourth harmonic, respectively. The measured maximum efficiencies ($\eta _{\mathrm {max}}$) of KDP, 23%-, and 45%-DKDP crystals were about 23%, and for 68%-, 82%-, and 90%-DKDP crystals, the $\eta _{\mathrm {max}}$ were about 18%. As we can see, in the presence of an external electric field, the maximum angular acceptance bandwidth ($\Delta \theta _{\mathrm {FWHM}}$) of FHG can be increased by 2.2 times. More importantly, the FHG process can maintain the perfect phase matching in a wide range of angles by introducing the EO effect. This has important practical significance for improving the stability of the deep ultraviolet laser. Here, we define the angular range of efficiency stability ($\Delta \theta _{\mathrm {ES}}$) as the full width at which the efficiency falls to 97% of the maximum efficiency. Based on the NCPM, the FHG can only achieve stable efficiency in the angle range of $0.54^\circ$, while the maximum value of $\Delta \theta _{\mathrm {ES}}$ for VTPM FHG can be up to $2.35^\circ$, which is 4.4 times larger than that of NCPM FHG.

 

Fig. 4. Experimental and simulated FHG efficiencies obtained using (a) KDP, (b) 23%-, (c) 45%-, (d) 68%-, (e) 82%-, and (f) 90%-DKDP, with the corresponding voltages plotted versus the angle. The solid line and circular dots denote the simulated and experimental NCPM FHG efficiencies, respectively; the dashed line and square dots denote the simulated and experimental VTPM FHG efficiencies, respectively; the dots line and diamond dots denote the simulated and experimental voltages, respectively. All the efficiencies are normalized to their maximum values at the corresponding initial phase-matching condition.

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Table 1. Experimental results of FHG angular characteristics in different deuterated KDP crystals based on NCPM and VTPM.

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Figure 5 gives a complete comparison of the phase-matching characteristics for different deuterated KDP crystals. It can be seen intuitively that $\Delta \theta _{\mathrm {FWHM}}$ for VTPM FHG is increased with the increasing of deuterium content within the same voltage regulation range because the EO coefficient gradually increases as the deuterium content increases. The above results prove that a wider angular acceptance bandwidth can be achieved by using the EO properties of crystals, which means that the stability of output FHG energy can be greatly improved.

 

Fig. 5. FHG characteristics of different deuterium content DKDP crystals based on the VTPM. The solid lines are simulated results, and the dots are experimental results.

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As shown in Fig. 4, when the voltage was adjusted along with the rotation angle of the crystal to keep efficiency from falling, the EO coefficients can be obtained by bringing the corresponding voltage and angle into Eq. (7). Considering the errors that may be caused by the high-voltage generator and the rotation stage, we take the average value of the measurement results at ~20 different angles for each crystal. The linear EO coefficients of KDP, 23%-, 45%-, 68%-, 82%-, and 90%-DKDP are −10.21 pm/V, −12.72 pm/V, −17.65 pm/V, −21.22 pm/V, −23.00 pm/V, and −24.85 pm/V, respectively. It can be seen from the measurement results that $\gamma _{63}$ of 90%-DKDP crystal is more than twice that of KDP crystal. This is mainly because the $\gamma _{63}$ of KDP-family crystals is closely related to the static dielectric tensor ($\epsilon _{33}^T$) of the crystals, and they have the relationship of $\gamma _{63} \propto \epsilon _{33}^T - 1$ [11,26]. For the values of $\epsilon _{33}^T$, it increases significantly with the increase of deuterium content [27,28]. Specifically, for KDP and 99.8%-DKDP crystals, the values of $\epsilon _{33}^T$ are 20.8 and 50, respectively [28]. Accordingly, the $\gamma _{63}$ value of 99.8%-KDP crystal is about 2.5 times that of KDP crystal.

Based on the obtained EO coefficients and the nonlinear coupled-wave equations [24], we simulated the corresponding FHG processes, the results are shown by the lines in Fig. 4. It can be seen that the simulated results are in good agreement with the experimental results. According to our results, the linear EO coefficients of DKDP crystals with different deuterium content were plotted in Fig. 6. Since the relationship between dielectric constant and deuterium content is approximately linear [27], and we find that our experimental results also show approximate linear variation. Therefore, linear fitting was used for the measured results, and the relationship between deuterium content and EO coefficient can be described by the followed equation:

 

Fig. 6. Linear EO coefficients of K(H$_{1-x}$D$_x$)$_2$PO$_4$ crystals. The dots are experimental results, and the solid black line is a fitted result.

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Table 2 gives a detailed comparison between our results and the reported results. It can be seen that the EO coefficients of KDP and high-deuterated KDP crystals agree well with the data reported by Onuki, Volkova, Christmas, and Sliker [11,12,29,30], while the results of the DKDP crystal with deuterium content of 18%−86% are higher than Volkova’s [31]. Meanwhile, there are obvious differences among the results of different reports, such as Onuki and Myers’s report on the EO coefficient of KDP [30,31], and Christmas and Volkova’s report on the EO coefficient of 90%-DKDP [12,31]. Besides, there may be some errors in Volkova’s report, for example, the EO coefficient of 18%-DKDP crystal is almost the same as that of KDP, and the EO coefficient of 50%-DKDP crystal is lower than that of 40%-DKDP [31]. The reasons for these differences may due to the inhomogeneity of deuterium content, crystal growth methods as well as the actual conditions of the measurement schemes. Nevertheless, our results provide a reliable reference for the application of partial deuterated DKDP in the future.

Table 2. Comparison of experimental results and previously reported linear EO coefficients $\gamma _{63}$ of K(H$_{1-x}$D$_x$)$_2$PO$_4$ crystals.

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In our experiments, the measurement error mainly comes from the angle control of the crystal and voltage accuracy. For our devices, the angle of the samples can be controlled with an accuracy of 8 $\mu$rad. According to Eqs. (5) and (7), the measured error caused by the angle can be calculated as less than 0.1%. The applied voltage has a control precision of 1%, which may introduce an error of about 1%. Meanwhile, considering the error of 0.033% caused by the crystal size (15 mm $\pm$ 5 $\mu$m), the total error of our measurement results is about 1.13%. If a higher precision voltage source is employed, higher accuracy measurement results can be obtained.

Compared to traditional methods, this method has significant advantages, including high precision, low opto-mechanical noise, and no need to consider the polarization change in the transmission process. Meanwhile, this method can be well applied to the crystals of different crystal systems and multiple point groups, such as $\bar {4}$2m and 4mm point groups (tetragonal), 222 and 2mm point groups (orthorhombic), 32 and 3m point groups (trigonal), and 6mm and $\bar {6}$2m point groups (hexagonal). For the crystal with multiple EO components (such as 3m point group) [32], this scheme will involve multiple EO coefficients. Multiple phase-matching equations can be established by applying electric fields in different directions and adjusting the intensity of the electric field, and then EO coefficients can be solved. Although this will increase the complexity of the measurement and calculation, multiple coefficients can be obtained simultaneously. For the crystals with low symmetry, such as 1 point group (triclinic) and 3 point group (trigonal) [32], the EO effect is too complicated due to a large number of independent non-zero tensor elements. Therefore, it is difficult to measure the EO coefficients of the crystals with these point groups by the proposed method. Nevertheless, this method still has great application potential in the EO coefficient measurement of many crystals.

5. Conclusion

In conclusion, we proposed and demonstrated a novel measurement method of the EO coefficient based on the $\boldsymbol{\chi }^{(2)}$ nonlinear optical technology. The prominent advantage of this nonlinear optical measurement method is that phase matching is very sensitive to the change of refractive index, which is fully utilized, and the change of refractive index can be given by an exact analytical formula. More importantly, for traditional methods, the stability of the laser polarization is required to strictly keep in the whole measurement system, while the results of our method only depend on the $\boldsymbol{\chi }^{(2)}$ nonlinear process that occurred in the nonlinear materials. Therefore, the EO coefficient can be obtained with high precision. In addition, for the EO coefficients measurement of numerous crystals, this method can be well applied. Based on this method, the linear EO coefficients of KDP, 23%-, 45%-, 68%-, 82%-, and 90%-DKDP were measured precisely, and a conclusion formula for determining the linear EO coefficients of DKDP crystals with different deuterium content was given. In this work, not only the FHG characeristics of different deuterated KDP crystals at different wavelengths were comprehensively demonstrated, but also the EO coefficients of K(H$_{1-x}$D$_x$)$_2$PO$_4$ crystals were systematically obtained, which provides important references for improving the stability of deep ultraviolet laser and expanding the application of KDP-family crystals in the field of laser technology and nonlinear optics.