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Abstract
Engineered domain structures play an essential role in nonlinear optics for quasi-phase-matched parametric processes. Pyroelectric field-assisted domain inversion with focused femtosecond laser pulses is a promising approach to create arbitrary two-dimensional nonlinear photonic structures in a large volume without externally applied electrical fields. We fabricate lattices of ferroelectric domains by patterning lithium niobate crystals with femtosecond laser pulses and then heating them to elevated temperatures. After cooling to room temperature, domains form below and above the laser-induced seeds. We investigate the effect of temperature and seed spacing on the number and size of inverted domains. In a temperature range of 220 °C-300 °C all domains are inverted in a two-dimensional lattice with periods of 15 µm × 6.3 µm. Smaller lattice periods result in a smaller fraction of inverted domains. Measurements with conducting, nonconducting, and short-circuited crystal surfaces reveal the influence of surface charges during the domain formation process. From the obtained domain widths and spacings, we calculate the effective nonlinear coefficient of quasi-phase-matched second-harmonic generation in two-dimensional nonlinear photonic structures.
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1. Introduction
Domain engineered structures are widely used in nonlinear optics for quasi-phase-matched parametric processes such as second-harmonic generation (SHG), sum- and difference frequency generation, and spontaneous parametric down-conversion. A domain is a space in a ferroelectric crystal where the spontaneous polarization is homogeneous. Adjacent domains are separated by each other by domain walls. The spontaneous polarization between neighboring domains can be orientated in different directions depending on the type of crystal. Common orientations include $180^{\circ }$, $90^{\circ }$, head-to-head, and tail-to-tail domains. In particular, 180$^{\circ }$ domains can be structured to build nonlinear photonic crystals because the sign of the $\chi ^{(2)}$ nonlinearity is reversed when the domain is inverted. A common way to switch ferroelectric domains is to apply an electric field that exceeds the coercive field [1]. This makes it possible to create a variety of one- and two-dimensional nonlinear photonic structures [2–5]. Among all nonlinear optical crystals, periodically-poled lithium niobate (LiNbO$_3$) and lithium tantalate (LiTaO$_3$) are widely used because of their wide transparent spectral range and high nonlinearity. There is a great interest in domain inversion techniques based on all-optical processes which do not require external electric fields [6]. Traditional poling techniques for instance require static masks to pattern the electrodes on the crystal surface, limiting flexibility. Lately, promising results have been obtained in structuring the nonlinearity in LiNbO$_3$ with focused femtosecond laser pulses. These include all-optical poling [7–9] and $\chi ^{(2)}$ depletion [10–14]. Direct femtosecond-laser writing is commonly used to fabricate waveguides [15,16] and gratings [17] in LiNbO$_3$ which offers the possibility to easily combine it with the engineering of ferroelectric domains.
Recently, we have discovered a pyroelectric field-assisted domain engineering approach that can be used to invert ferroelectric domains in 2D in magnesium doped lithium niobate (LiNbO$_3$:Mg) without using an external applied electric field [18]. The inversion process works in the following way. First, permanent defects are induced by focused femtosecond laser pulses along the polar ${z}$-axis (Fig. 1(a)). Then, the crystal is heated up above 200 $^{\circ }$C and subsequently cooled down to room temperature (Fig. 1(b)). After this heating-cooling cycle, ferroelectric domains are inverted below and above the defects that act as seeds (Fig. 1(c)). Domain inversion occurs if a certain pulse energy and defect length are exceeded. We generated two-dimensional domain structures with periods of $20\,\mu \text {m} \times 20\,\mu \text {m}$ with all domains inverted. However, we have observed that the width of the domains is larger at the boundaries of a 2D lattice compared to the inner part. Therefore, the question naturally arises how small the lattice period can be. The answer to this question is crucial because for nonlinear 2D photonic crystals a certain length of a reciprocal lattice vector is required for a specific quasi-phase-matching process. In addition, a detailed study of the influence of heating temperature on the process of domain inversion is lacking.
Fig. 1. Scheme of the experimental procedure. (a) Permanent defects in the form of elongated filaments (grey) are induced with focused femtosecond laser pulses along the $z$-axis. (b) The sample is heated to elevated temperatures and cooled down to room temperature. (c) After cooling down, ferroelectric domains (green) are inverted below the filaments.
Here, we examine the pyroelectric field-assisted domain inversion in LiNbO$_3$:Mg, dependent on the temperature for 2D rectangular lattices of different periods. We measure the fraction of domains that are inverted and the domain diameters. Measurements have been performed for single and multiple heating cycles. Furthermore, we have examined the influence of the electrical termination of the crystal surfaces such as conducting, nonconducting, and short-circuited on the inversion process. Finally, we calculate the effective nonlinear coefficient of quasi-phase-matched SHG in two-dimensional nonlinear photonic structures with the given lattice periods from the obtained domain diameters.
2. Quasi-phase-matched SHG in a rectangular nonlinear photonic lattice
We consider a weakly focused fundamental wave with wave vector $\mathbf {k}_\text {Fun}$ and a polarization parallel to the $z$-axis which propagates either along the $x$- or $y$-axis below the laser-induced filaments through the rectangular nonlinear photonic lattice, which is built by the inverted ferroelectric domains (see Fig. 1(c)). A second-harmonic with wave vector $\mathbf {k}_\text {SHG}$ will be efficiently generated if the phase mismatch $\Delta \mathbf {k}=\mathbf {k}_\text {SHG}-2 \mathbf {k}_\text {Fun}$ is compensated for by a reciprocal lattice vector $\Delta \mathbf {k}=\mathbf {K}_{mn}$. We restrict ourselves to the case of collinear phase matching of the first order $(m, n)=(1,0)$ with $K_{10}=2\pi /\Lambda _x$ or $K_{10}=2\pi /\Lambda _y$, because it is the most efficient one. Here, $\Lambda _x$ and $\Lambda _y$ denote the distance between the domains in $x$- and $y$-direction, respectively. In Fig. 2 the lattice period $\Lambda _{x,y}$ is plotted in dependence of the the wavelength of the fundamental wave for quasi-phase-matched SHG in LiNbO$_3$:Mg [19]. Quasi-phase-matched SHG can be realized for fundamental waves between 1000 nm and 1600 nm with grating periods ranging from 6 $\mu$m to $20\,\mu$m. If the nonlinear photonic lattice consists of a rectangular domain grating with two different periods along the $x$- and $y$-axis, two fundamental waves of different wavelength can be frequency doubled simultaneously. If one fundamental wave propagates for instance along the $y$-axis in a periodically poled lattice with $\Lambda _y=6.3\,\mu$m, a second harmonic at 515 nm is quasi-phase-matched. If a second fundamental wave propagates along the $x$-axis with $\Lambda _x=10\,\mu$m, a second harmonic at 600 nm is generated simultaneously. For the case of perfect quasi-phase matching of a plane fundamental wave and considering an undepleted pump beam, the intensity of the second harmonic is:
Fig. 2. Domain period $\Lambda _{x,y}$ in dependence of the wavelength of the fundamental wave for quasi-phase-matched SHG of the order $(m,n)=(1,0)$ in LiNbO$_3$:Mg at room temperature. The marked points are the periods 6.3 $\mu$m (SHG at 515 nm), 10 $\mu$m (SHG at 600 nm), and 15$\mu$m (SHG at 695 nm), respectively.
3. Materials and methods
We use 500 $\mu$m thick $z$-cut lithium niobate crystals doped with 5 mol. % magnesium oxide with the dimensions of 1 cm $\times$1 cm. The z-surfaces are polished to optical quality. First, permanent microstructures at the desired positions of the ferroelectric domain patterns are induced in the crystals by direct femtosecond laser writing. The laser writing setup mainly consists of a 10 W femtosecond fiber laser (Amplitude Satsuma HP) with a central wavelength of 1030 nm, a microscope objective with a numerical aperture of 0.8, and a motorized 3D translation stage with an accuracy of 300 nm (Aerotech FA130). Femtosecond laser pulses with a pulse width of 250 fs and a repetition rate of 1 kHz are focused along the polar axis in a depth of 220 $\mu$m. The laser focus is moved to the top $-z$-surface and stopped in a distance of about 40 $\mu$m before reaching the surface. Thus, permanent defects are induced in the form of elongated filaments (Fig. 1(a)). Nine different rectangular lattices with the dimensions of $150\,\mu$m $\times \,150\,\mu$m are induced in five different samples. Every lattice has a different period along the $x$-axis of $\Lambda _x=$ 6 $\mu$m, $\Lambda _x=$ 10 $\mu$m, and $\Lambda _x=$ 15 $\mu$m, respectively, while the period along the $y$-axis was fixed to $\Lambda _y=$ $6.3\,\mu$m. We have chosen this fixed period because it allows for quasi-phase-matched SHG of 1030 nm (cfg. Figure 2). Every lattice was induced with a different writing energy of 50 nJ, 250 nJ, and 400 nJ, respectively. After inducing the microstructures, a sample is heated to an elevated temperature $T_\text {set}$ on an aluminum hot plate (HS60 Torrey Pines, $\Delta T=\pm 1\,^{\circ }$C) in air (Fig. 1(b)), thus creating a pyroelectric field. During cooling, the domain inversion is driven by a space-charge field that locally exceeds the threshold field of domain nucleation (Fig. 1(c)). Čerenkov second-harmonic generation microscopy allows visualizing the generated ferroelectric domains and laser-induced seeds in 3D [21,22]. For this purpose, femtosecond laser pulses with a central wavelength of 800 nm, a repetition rate of 80 MHz, a pulse length of 100 fs, and pulse energies of less than 3 nJ are focused in the laser scanning SHG microscope by a microscope objective (NA=0.79) into the sample along the $z$-axis. The diameter (FWHM) of the focal spot is approximately 800 nm. The sample is scanned in the $xy$-plane in different depths by a piezo-driven $xyz-$stage. Whenever the laser focus passes a domain wall, Čerenkov SHG is generated. A condenser lens images the second harmonic, and its intensity is measured with a photomultiplier. Structural modifications in the crystal volume written by focused femtosecond laser pulses can also be detected in the laser scanning SHG microscope. The difference is that a laser-induced filament always gives a signal, while in the case of a ferroelectric domain, only the domain boundaries lead to Čerenkov SHG with the laser parameters used [12,23].
Each single crystal is modulated with the same nine gratings to study the temperature dependence of local domain inversion step by step. The $+z$-surface of sample #1 is in direct contact with the aluminum hot plate and heated once (single heating) to $T_\text {set}=$220 $^{\circ }$C with a rate of $4.4\,^{\circ }\text {C}/\text {min}$ and then cooled down. After heating up, the hot plate is switched off and the sample remains on the hot plate. The time constant for cooling down from 200 $^{\circ }$C to room temperature is about 19 minutes (cfg. Figure 1(b)). Sample #2 is put also in direct contact with the aluminum hot plate but heated up multiple times (multiple heating). We start at a low temperature of $T_\text {set}=50\,^{\circ }$C and after cooling down the inverted domains at the bottom $+z$-surface are detected by Čerenkov SHG microscopy. Afterwards we start the next heating-cooling cycle with the same sample and increase the set temperature by 10 $^{\circ }$C. After cooling down we take again images of the domains. This process is repeated until $T_\text {set}=400\,^{\circ }$C is reached. The other three crystals have different electrical terminations of the surfaces. Each of these crystals is heated once to $T_\text {set}=220\,^{\circ }$C and cooled down. Sample #3 is heated to 220 $^{\circ }$C while it is electrically isolated from the aluminum hot plate by a thin glass slide. Sample #4 is coated on the upper $-z$-surface with conducting silver while the boundary $+z$-surface is in contact with the aluminum hot plate. Sample #5 is completely short-circuited by conducting silver.
4. Results
4.1 Examples of domain inversion for different lattice periods
Two examples of femtosecond laser-induced microstructures and inverted ferroelectric domains are shown in Fig. 3. The seeds are written in sample #1 with a pulse energy of 250 nJ and the lattice periods are $\Lambda _x=15\,\mu$m (Fig. 3(a)) and $\Lambda _x=6\,\mu$m (Fig. 3(b)), respectively. The sample was directly heated to 220 $^{\circ }$C. In Fig. 3(a) the seeds range from 40 $\mu$m to 220 $\mu$m from the top $-z$-surface as indicated by the red lines. The lower contrast visible in the $xy$-plane at a particular $z$-position inside the seeds is due to the stacking of two measurements. The domains are inverted below and above the seeds and are extended to the surfaces. In this example 100 % of the domains are inverted (cfg. Figure 4(a)). In Fig. 3(b) the $+z$-surface is seen. The domains are mainly inverted at the border. In the central part, where domains have not been inverted, one can see the ends of the seeds. In this lattice 62 % of the domains are inverted (cfg. Figure 4(c)). Both examples show that the domains have straight boundaries and are approaching the $+z$-surface once they are inverted. We want to point out that the process of domain inversion shows similar behavior when the seeds are induced at the $+z$-surface and is independent of the writing direction along the $z$-axis [18]. Fig. 4 shows SHG microscope images of the $+z$-surface of three lattices with different periods $\Lambda _x$ induced with a pulse energy of 250 nJ in sample #1 (cfg. Figure 3). The width of every detected domain at the bottom of a crystal is determined by digital image analysis. We assume a round domain shape for the analysis since the typical hexagonal shape is only evident for larger domains located at the edges and corners of a lattice. Because of the different statistics of edge and center domains, we refer to the lattice’s two outermost rows and columns as edge domains and all other domains are center domains.
Fig. 3. Čerenkov SHG microscope images of domains in the volume of lattices with different periods $\Lambda _x$ in sample #1. The pulse energy is 250 nJ and the period along the $y$-axis $\Lambda _y=6.3\,\mu$m. (a) $\Lambda _x=15\,\mu$m. Laser-induced seeds are seen from the top ($-z$-surface). Ferroelectric domains completely form below the seeds. (b) $\Lambda _x=6\,\mu$m. Viewed from the bottom ($+z$-surface). Domains mainly form at the edges of the lattice while some domains are not inverted in the center part.
Fig. 4. Čerenkov SHG microscope images of domains at the bottom $+z$-surface of sample #1 for different periods $\Lambda _x$, constant $\Lambda _y=6.3\,\mu$m, and a constant pulse energy of 250 nJ. (a) $\Lambda _x=15\,\mu$m. (b) $\Lambda _x=10\,\mu$m. (c) $\Lambda _x=6\,\mu$m. The width of the scale bar is $20\,\mu$m.
4.2 Evaluation of number and diameter of inverted domains
In the following, we evaluate the domain properties at the $+z$-surface, where the longest domains with a length of approximately 250 $\mu$m below the seeds end, for various experimental parameters such as temperature, pulse energy, lattice period, and electrical contacts of the crystal surfaces. The fraction of all inverted domains and the average diameter of center and edge domains are summarized in Tables 1, 2, and 3, respectively.
Table 1. Fraction of all domains, that have been inverted after a single heating to $\text {T}_{\text {set}}=220\,^{\circ }\text {C}$ (samples #1, #3, #4, #5) and after multiple heating cycles to $\text {T}_{\text {set}}=220\,^{\circ }\text {C}$ (sample #2), respectively. The bold lattice parameters are used for the calculation of the normalized effective nonlinear coefficient in Section 4.3.
Table 2. Average diameter of edge domains, that have been inverted after a single heating to $\text {T}_{\text {set}}=220\,^{\circ }\text {C}$ (samples #1, #3, #4, #5) and after multiple heating cycles to $\text {T}_{\text {set}}=220\,^{\circ }\text {C}$ (sample #2), respectively.
Table 3. Average diameter of center domains, that have been inverted after a single heating to $\text {T}_{\text {set}}=220\,^{\circ }\text {C}$ (samples #1, #3, #4, #5) and after multiple heating cycles to $\text {T}_{\text {set}}=220\,^{\circ }\text {C}$ (sample #2), respectively. The bold lattice parameters are used for the calculation of the normalized effective nonlinear coefficient in Section 4.3.
We have measured the domain inversion in dependence of the temperature in sample #2 for three writing energies and three lattice periods. The lattices that have been induced with 50 nJ do not result in a regular domain inversion. Already at 120 $^{\circ }$C some domains are formed but in unpredictable places. Therefore, in the following we focus on evaluating the lattices written with energies of 250 nJ and 400 nJ. These lattices did not show any domain formation at 120 $^{\circ }$C. At 170 $^{\circ }$C a small fraction of domains is inverted, as can be seen in Figs. 5 and 6. The percentage of all inverted domains, including edge and center domains, increases with increasing temperature. At a threshold of approximately 190 $^{\circ }$C, the rate of domain inversion is greatest. Above 220 $^{\circ }$C no additional domains are inverted anymore. 100% of domains are inverted if the lattice period is larger than $\Lambda _x=15\,\mu$m. The fraction of inverted domains mainly depends on the lattice periods and not on the pulse energy used, as shown by comparing Figs. 5 and 6. As the distance between the domains becomes smaller, fewer domains are inverted. Especially less center domains are inverted for smaller lattice periods, because edge domains are larger and more stable than center domains. Above 300 $^{\circ }$C, some already inverted domains start to disappear, especially within lattices with short periods. We have compared the domain statistics of the multiple heating cycle of sample #2 at 220 $^{\circ }$C with sample #1 heated directly to a temperature of 220 $^{\circ }$C. The results are summarized in Table 1.
Fig. 5. Fraction of inverted domains in sample #2 in dependence of the set temperature for a pulse energy of 250 nJ and for three periods $\Lambda _x$. The period in $y$-direction is always $\Lambda _y=6.3\,\mu$m. The lines are a guide for the eyes.
Fig. 6. Fraction of inverted domains in sample #2 in dependence of the set temperature for a pulse energy of 400 nJ and for three periods $\Lambda _x$. The period in $y$-direction is always $\Lambda _y=6.3\,\mu$m. The lines are a guide for the eyes.
Fig. 7. The average domain diameter of edge domains in dependence of the maximum heating temperature. (a) Writing energy 250 nJ. (b) Writing energy 400 nJ. The lines are a guide for the eyes.
The number of heating cycles shows only a minor influence on the fraction of inverted domains. The most distinct discrepancy of 11% is observed between single and multiple heating to 220 $^{\circ }$C. The influence of the electrical termination of the crystal surfaces on the fraction of inverted domains is also low, except for a completely short-circuited crystal. There is no significant difference in domain formation if the crystal is electrical isolated during heating or if both surfaces are conducting. Since the images of the domains at the $+z-$surface in sample #3 and #4 look basically the same as those in sample #1, they are not shown explicitly here. Notably, a completely short-circuited crystal (sample #5) does not show any domain formation. The average domain diameter at the edge of a lattice is on the order of 3.4 $\mu$m for single heating and 9% smaller (3.1 $\mu$m) for multiple heating (see Table 2). The diameter is almost independent of temperatures between 200 $^{\circ }$C and 300 $^{\circ }$C (Fig. 7). At higher temperatures than 300 $^{\circ }$C the average edge domain diameter decreases slightly. This decreasing behavior is less pronounced for a lattice written with a higher energy of 400 nJ. A larger spacing of laser-induced seeds results in slightly larger domains, while the pulse energy does not influence the diameter.
An evaluation of the diameter of center domains shows a very similar behavior compared with edges domains (Fig. 8 and Table 3). The main difference is that center domains are smaller than edge domains. For single heating, the average center domain diameter is 2.38 $\mu$m and for multiple heating 1.76 $\mu$m.
Fig. 8. The average domain diameter in the center of the lattices in dependence of the maximum heating temperature. (a) Writing energy 250 nJ. (b) Writing energy 400 nJ. The lines are a guide for the eyes.
4.3 Effective nonlinear coefficient
In the following we calculate the normalized effective nonlinear coefficient for a first order QPM process. The largest Fourier coefficients are expected for lattices with motif sizes not much smaller than the area of a unit cell. Therefore, we select two lattices with almost 100% of inverted domains and a large domain radius. These are lattices in sample #3 that have been heated up once to 220 $^{\circ }$C while the bottom of the crystal was electrically isolated from the hot plate (see Tables 1 and 3). The first grating has the parameters $\Lambda _x=15\,\mu$m, $\Lambda _y=6.3\,\mu$m, $R=1.24\,\mu$m, and was written with 250 nJ pulse energy. The center domain diameter is used here, because the small number of larger edge domains has no big influence on the conversion efficiency. The corresponding normalized effective nonlinear coefficient is:
The second lattice was induced with 400 nJ and has a slightly smaller domain radius of $R=1.10\,\mu$m and also a smaller unit cell with $\Lambda _x=10\,\mu$m and $\Lambda _y=6.3\,\mu$m. The normalized effective nonlinear coefficient is: This coefficient is slightly higher than the coefficient of the lattice with the larger spacing. The lattices with smaller domains and less inverted domains will result in a smaller $|G_{10}|^2$.5. Discussion
5.1 Domain formation process
The heat-assisted domain formation process can be understood within the framework of a model considering a local charge compensation of the pyroelectric field [18]. The spontaneous polarization $P_s$ in lithium niobate in the form of bound charges leads to a depolarizing field $E_\text {dep}$ which can be screened by $E_\text {scr}$ due to bulk or surface charges. The final pyroelectric field is:
At room temperature, the net pyroelectric field is zero because $E_\text {dep}$ is completely screened. Heating leads to a decrease in polarization because of the displacement of bound charges. Hence, the temperature change $\Delta T$ results in the pyroelectric field: Here, the dielectric constant of LiNbO$_3$ is $\epsilon =28$ and the pyroelectric coefficient $\partial P_S/\partial T=-0.83 \times 10^{-4}\,\mathrm {C/(m^2K)}$ [24]. The pyroelectric field causes negative charges to accumulate at the $+z$-surface and positive charges at the $-z$-surface. The charges accumulate during the heating-cooling procedure at the surfaces at the location of the induced defects. These surface charges are essential for domain inversion because the short-circuited sample did not show any inverted domains. However, at this heating stage, the pyroelectric field cannot cause a local inversion of the polarization because it is oriented in the same direction as the polarization. Therefore, we assume that bulk mobile charges at elevated temperatures partly compensate for the pyroelectric field. The conductivity of these thermally activated charges depends on temperature $T$ and activation energy $E_\text {A}$: with the Boltzmann constant $k_B$.During cooling, the depolarizing field returns to its original value while the space-charge field remains. At a certain temperature, the space-charge field locally exceeds the field for domain nucleation resulting in domain inversion [25]. This temperature of domain inversion is supposed to be at about 190 $^{\circ }$C as at this temperature, the rate of domain inversion was highest in the multiple-heating experiment. We have also used slower heating and cooling rates. However, the domain formation process seems to be not strongly depended on the rate of temperature change. Furthermore, the coercive field of LiNbO$_3$:Mg decreases with increasing temperature and is about 1.4 kV/mm at 190 $^{\circ }$C [26]. At lower temperatures than 220 $^{\circ }$C, the conductivity of the charges is lower and the final space-charge field is smaller, which results in fewer inverted domains.
If we assume that the concentration and/or the mobility of bulk charges are higher in the vicinity of the laser-induced seeds, the space-charge field would be compensated preferably in these regions. Thus, the laser-induced seeds spatially modulate the space-charge field. It is known, that femtosecond-laser-pulses lead to a distortion of the crystal lattice and an induction of stress [27]. These modifications are also called Type II modifications in comparison to Type I modifications. Type I modifications are reversible when heating the crystal while Type II modifications remain stable. These permanent defects are commonly used to write for instance waveguides [15,16] and gratings [17] in LiNbO$_3$ with femtosecond-laser pulses. It has been observed that femtosecond-laser written structures show an increased absorption in the visible spectral range [28]. This change could partly be reversed by a thermal treatment. Such induced absorption could be attributed to the formation of polarons. Especially bi-polarons are known to increase the conductivity. However, the pronounced temperature dependence might also lead to the assumption that ions such as protons or lithium ions with an activation energy of about 1.2 eV become mobile and compensate for the pyroelectric field resulting in a space-charge field. For instance, a temperature of about 200 $^{\circ }$C is also used to thermally fix volume phase holograms in photorefractive LiNbO$_3$ [29]. In this case, a drift of protons compensates for the space-charge field.
Since a pulse energy higher than 250 nJ has no significant influence on the domain size in our experiment, a minimum pulse energy should be sufficient to control the domain inversion process. This threshold energy is assumed to be at about 70 nJ when self-focusing takes place leading to permanent defects of Type II. For lower energies such as 50 nJ no defects are visible in the bright field microscope. The observed backswitching of already inverted domains at temperatures above 300 $^{\circ }$C might be due to a similar process responsible for the inversion process but involving a different kind of charges. Such backswitching of domains at elevated temperatures was observed in 2D periodically-poled LiNbO$_3$ and LiTaO$_3$ [25,30].
The domain nucleation starts at the seeds and then domains grow longitudinal to the crystal surface [31]. However, compared to traditional electric field poling the final transverse domain wall movement is strongly hindered in our experiment because of the limited space-charge field. As a result, the electric potential at the location of the seeds is higher at the corners of the lattice compared to the potential at the edges, which in turn is higher than the potential in the center of the lattice [25]. This directly corresponds to the different domain diameters at these locations.
Pyroelectric field-assisted domain engineering is not limited to LiNbO$_3$:Mg. This technique can also be applied to other ferroelectric crystals with a high pyroelectric coefficient and a low coercive field if focused femtosecond-laser pulses lead to a locally increased conductivity in these crystals. Therefore, the effect is expected also to occur in stoichiometric lithium tantalate with comparable experimental parameters, since the material properties are similar to LiNbO$_3$:Mg. However, the use of a crystal with a larger coercive field like congruent LiNbO$_3$ would require an adaptation of the parameters.
5.2 Quasi-phase-matched SHG in rectangular pyroelectric field-assisted poled lattices
The fabricated 2D nonlinear photonic lattices allow for simultaneous quasi-phase matched SHG of 1030 nm ($\Lambda _x=6.3\,\mu$m) and 1200 nm or higher ($\Lambda _y>10\,\mu$m). The maximum normalized effective nonlinear coefficient in our 2D nonlinear lattice is limited to approximately 1% because the domain size decreases with decreasing domain distance in the heat-assisted inversion approach described here. For comparison, a 2D lattice which is quasi 1D in one direction $R/\Lambda _y=0.5$ results in a normalized effective nonlinear coefficient of 34% for the optimum ratio $R/\Lambda _x=0.29$. Moreover, a 1D rectangular grating with a duty cycle of 50% has the highest normalized effective nonlinear coefficient of $(2/\pi )^2=41\%$. This becomes clearer, recognizing that the lattices have a small duty cycle with $R \ll \Lambda _x$ and $R \ll \Lambda _y$. Then, the first order Fourier coefficient (Eq. (3)) can be approximated as
This is two times the motif area divided by the unit cell area. Please note that this Fourier coefficient for small duty cycles is of the same order of magnitude for other motif shapes such as hexagons. In our experiment, the duty cycle is limited by the maximum space-charge field that builds up during the cooling. A way to get larger domains could be to bias the crystal with an external electric field. This bias field could be smaller than the coercive field because it adds up to the pyroelectric field. We want to point out that the presented results are valid for large 2D lattices. However, in a quasi 1D lattice the domain period can be as small as 2 $\mu$m using pyroelectric field-assisted domain inversion [18]. In most cases, periodically poled LiNbO$_3$ is operated at a fixed temperature to achieve quasi-phase-matching for the desired wavelength. To avoid any photorefractive changes, this temperature is typically chosen between 150 $^{\circ }$C and 250 $^{\circ }$C. The nonlinear photonic structures studied here can be used in this temperature regime under the condition that the domain inversion processes have been completed above 220 $^{\circ }$C. Then, all domains remain stable.6. Conclusions
The domain inversion assisted by pyroelectric fields was studied in 2D laser written lattices as a function of temperature and lattice periods during post-heat treatment. We have identified a threshold temperature and determined an optimal temperature regime between 220 $^{\circ }$C and 300 $^{\circ }$C. In this temperature range, all domains can be inverted in a 2D rectangular lattice with periods of 15 $\mu$m $\times$ 6.3 $\mu$m. Such a nonlinear photonic lattice allows for simultaneous quasi-phase-matched SHG of 1030 nm and 1200 nm, respectively. Smaller lattice periods and lower temperatures result in fewer inverted domains. Above 300 $^{\circ }$C the already inverted domains switch back in the center of the lattices. The average diameter of the central domains is approximately 2.38 $\mu$m. This size is almost independent of temperature and lattice periods. The electrical termination of the crystal surfaces has no significant influence on the domain formation process unless the crystal is completely short-circuited. In this case, no domains are inverted, indicating that surface charges are essential for the inversion process. The calculated normalized effective nonlinear coefficient $|G_{10}|^2$ of quasi-phase-matched second-harmonic generation is on the order of 1% in the lattices studied. This coefficient is limited because the ratio of domain area to the area of the unit cell is also limited in the pyroelectric field-assisted domain inversion process. Larger duty cycles and therefore higher conversion efficiencies can be expected by using a bias electric field in addition to the pyroelectric field during poling.
Acknowledgments
We thank Lauriane Gugnot for supporting the 3D SHG laser scanning microscopy measurements.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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