## Abstract

We experimentally investigate ferroelectric domains close to the phase transition in a nonlinear photonic crystal exhibiting disordered domain distribution implementing Čerenkov second-harmonic generation. The interplay of domain structures and the second-harmonic signal is studied exemplarily in strontium barium niobate measuring the far-field distribution as well as imaging domain structures microscopically.

Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

## 1. Introduction

In recent years, much attention has been devoted to investigate frequency conversion in ferroelectrics with a random distribution of ferroelectric domains, i.e. with a random distribution of the *χ*^{(2)} nonlinearity [1–4]. This class of nonlinear photonic crystals offers a possibility to realize nonlinear optical processes over a wide spectral range [5, 6]. One of the most interesting phase-matching schemes to achieve second-harmonic emission in such media is the so-called Čerenkov second harmonic generation (ČSHG). Its phase-matching condition is noncollinearily fulfilled when the longitudinal phase mismatch is zero and the transverse phase mismatch is partially compensated by the nonlinear structure. It is automatically satisfied at *χ*^{(2)}-boundaries as for example at ferroelectric domain walls [6–10]. Due to its sensitivity to spatial disparities of the second-order nonlinearity, ČSHG has been recently developed as a versatile tool for ferroelectric domain diagnostics.

In two dimensional nonlinear photonic structures, a ČSH signal is emitted on a cone at an angle Θ = arccos(*n _{ω}/n*

_{2}

*) determined by the refractive indices of the fundamental and second-harmonic waves, respectively (see Fig. 1). The conical emission reduces itself to two bright second harmonic spots when a single domain wall is illuminated [11] (Fig. 1). Several studies have approved the direct relation of the spatial properties of ČSHG and ferroelectric domain structures. For instance, the intensity distribution of the second-harmonic cone can be attributed to the individual domain shape and the crystallographic symmetry, e.g. the four-fold symmetry in strontium barium niobate (SBN) [12, 13], or the hexagonal symmetry in LiNbO*

_{ω}_{3}and LiTaO

_{3}, respectively [14–17]. The nonlinear second-order tensor properties also have an impact on the azimuthal intensity distribution of the second harmonic ring [18]. Moreover, ČSHG has been employed to characterize the ferroelectric domain size distribution in random nonlinear photonic structures [19]. A further progress has been made by embedding Čerenkov SHG in a microscopic system to profile the domain structures in all three dimensions [20]. The developed technique recently allowed us profiling the ferroelectric domain kinetics during electrical switching in SBN [21].

As-grown SBN crystals typically have needle-like ferroelectric domains with a cross-section of a square shape with rounded corners [13, 22] (see Figs. 2(a)–2(c)). The domains are randomly distributed in size and space. The average size ranges between a few nanometers and a few micrometers [7, 23–25]. Due to this random domain distribution SBN is best suited for broadband quasi-phase matching processes [6, 7, 26, 27]. The distribution of domains can be influenced by electric-field poling and by heating. SBN shows a relaxor-type phase transition from the ferroelectric to the paraelectric phase, i.e., the Curie temperature *T _{c}* is not sharply defined [28, 29]. The Curie temperature of SBN crystals with the congruently melting composition Sr

_{0.61}Ba

_{0.39}Nb

_{2}O

_{6}is relatively low, about 70 °C–80 °C.

The evolution of ferroelectric domains with increasing temperature up to the phase transition is still an open question and under active debate. Specially, how the individual domains locally transform in shape near the relaxor phase transition.

In this article, we analyze Čerenkov SHG in random SBN in dependence of the temperature from the ferroelectric to the paraelectric phase. SBN is the ideal platform here because of its low Curie temperature. This makes SBN more suitable technically to investigate microscopically. In addition the domains are naturally distributed and spatially separated. The advantage here is that the domains can be traced individually during changing the temperature.

In the first part, we measure the overall second-harmonic intensity of Čerenkov SHG in the far-field while heating up a strontium barium niobate crystal over the Curie temperature *T _{c}*. This allows tracing the order parameter, i.e., the spontaneous polarization

*P*, which approaches zero at

_{s}*T*. In the second part, we use Čerenkov-type second-harmonic microscopy to image the cross section of ferroelectric domains in dependence of temperature. Finally, by comparing far-field and microscopic Čerenkov SHG we gain new insights in the domain dynamics near the phase transition.

_{c}## 2. ČSHG near the phase transition

#### 2.1. Far-field investigations

Our Sr_{0.61}Ba_{0.39}Nb_{2}O_{6} crystal has the dimensions of 6.6 × 6.6 × 1.6 mm^{3} and is doped with Cr_{2}O_{3}. The large surfaces perpendicular to the c-axis are polished to optical quality. We use femtosecond laser pulses with a wavelength of *λ* = 1200 nm, a pulse duration of about 80 fs, pulse energies up to 100 µJ, and a repetition rate of 1 kHz. The laser pulses are collimated, having a beam diameter of about 1.5 mm. The beam is propagating along the c-axis of the crystal illuminating a large number of domains. The second harmonic signal is emitted in this configuration on a Čerenkov cone that is projected on a color CCD camera. A simplified schematic is depicted in Fig. 2(2).

In order to ensure the same electrical and thermal history, the samples were initially heated up to 200 °C for 2 hours to erase any spurious polarization. This process is necessary since the phase transition in SBN strongly depends on the poling history [30]. In this as-grown case, the domains are smaller than 1 µm, and the intensity of the second harmonic on the ring is homogeneous (see Fig. 2(a) inset). Before recording the dynamics of the second harmonic on the ring, the crystal is partially repoled several times at room temperature leading to micro-scaled squircle-shaped domains (see Fig. 2(b)). The intensity of the second harmonic on the ring shows a four-fold modulation attributed to the domain shape (inset in Fig. 2(b)). To measure Čerenkov SHG on a cone in dependence of temperature, the SBN sample is sandwiched between two controlled Peltier elements. A thermistor at the sample face controls the temperature (Fig. 2(d)). The sample faces along the optical axis are kept free for the propagation of the fundamental beam. Čerenkov SHG is recorded *in situ* while increasing the temperature of the sample above Curie temperature and cooling it down. The temperature is increased in steps of 1 °C/s starting from room temperature regularly up to 100 °C. The temperature is then decreased to room temperature with the same velocity. The exposure time of the CCD camera is kept fixed during the experiment. The results are depicted in Fig. 3.

Figure 3 shows the evolution of the modulated Čerenkov SH cone while increasing and decreasing the sample temperature. At room temperature (Fig. 3(a)), the modulation of the azimuthal second harmonic intensity can clearly be seen. The total intensity becomes weaker with increasing sample temperature. At about 75 °C, the second harmonic signal vanishes completely (Figs. 3(e)–3(f)).

Subsequently, the sample is cooled down. The second harmonic signal appears again at about 70 °C and increases with decreasing sample temperature (see Figs. 3(g)–3(h)). The intensity of Čerenkov SHG on the ring is clearly weaker than in the repoled case at the beginning, and appears to be homogeneous, i.e., the four-fold modulation is no longer visible.

#### 2.2. Microscopic view

To combine the far-field analysis with the corresponding microscopic measurements of the domain structures in dependence of temperature we use a Čerenkov SHG microscope that is described in details elsewhere [21]. The fundamental beam of a Ti-Sapphire laser (800 nm, 80 MHz repetition rate, about 60 fs pulse duration, and up to 3.5 nJ pulse energy) is coupled into a commercial laser scanning confocal microscope (Nikon eclipse, Ti-U) and tightly focused by the microscope objective with a numerical aperture of 0.8 to a near diffraction limited spot in the sample to generate the second-harmonic signal at the domain walls individually. The position of the focus is raster-scanned in the *xy*-plane by a piezoelectric table (P-545, PI nano). The intensity of the second-harmonic signal is collected by a condenser lens (NA 0.9) and measured by a photomultiplier as a function of the focus position.

To control the temperature, a copper plate is mounted on the top of the sample. This plate is equipped with two thermal resistors and a temperature sensor (PT100), and has a hole allowing to pass the second-harmonic signal to the photomultiplier. For this experiment an SBN sample with the dimensions 6.0 × 6.0 × 1.1 mm^{3} is employed. The large surfaces perpendicular to the c-axis are polished to optical quality. The crystal is prepared to have micro-scaled domains by several poling cycles in the same manner as before (cfg. Fig. 2(b) and Fig. 4(a)).

An area of 40 µm × 40 µm is raster-scanned, while heating up the sample to reach the paraelectric phase. To avoid loosing the observed structures in the scanned area due to the temperature-dependent drift while heating, the crystal is readjusted between the scans in the *xy*-plane to compensate for this shift.

The results are shown in Fig. 4. Starting from room temperature (Fig. 4(a)) the crystal has random square micro-scaled domains. With increasing temperature the domains become smaller. The mean diameter shrinks from <*d*> ≈ 4.9 µm down to about 3 µm at 55 °C (Fig. 4(e)) and further down to about 1.8 µm at 75 °C (Fig. 4(f)). At 80 °C, the intensity of the ČSHG signal has decreased to such a degree that the background noise becomes apparent (Fig. 4(h)). The ČSHG completely vanishes as the Curie temperature is exceeded at ≈ 85 °C (Fig. 4(i)).

## 3. Analysis of the dynamics

The intensity distribution of far-field ČSHG is directly related to the averaged domain size and shape of the ferroelectric domains in SBN. This SHG intensity in the undepleted pump approximation can be described in the following way [12]:

*I*

_{FW}is the intensity of the fundamental wave propagating along the

*z*-axis,

*d*

_{eff}is the effective nonlinear coefficient,

*L*is the average domain length, and

_{z}*S*(Δ

*k*) is the Fourier spectrum of the domain structure with the transverse phase mismatch vectors Δ

_{xy}*k*in the

_{xy}*x*- and

*y*-direction, respectively. At the Čerenkov angle Θ, the longitudinal phase mismatch Δ

*k*is 0. The four-fold modulation on the SHG ring (cfg. Fig. 3(a)) originates from the shape of the domains (cfg. Fig. 4(a)) which is included in the Fourier spectrum

_{z}*S*(Δ

*k*)(

_{xy}*T*). At the fundamental wavelength 1200 nm, the external Čerenkov angle Θ amounts to about 37.5° and the corresponding wave vectors of the fundamental and second-harmonic are 24.3 µm

^{−1}and 11.73 µm

^{−1}, respectively. In general, at room temperature, a strong ČSHG signal is expected for strong Fourier coefficients, i.e. when S(Δ

*k*) is maximum. The transverse mismatch Δ

_{xy}*k*=

_{xy}*k*

_{(2}

_{ω}_{)}sin

*θ*

_{internal}= 6.4 µm

^{−1}would be best compensated for by an inverted domain distribution with an average domain diameter of about 0.5 µm and a duty cycle of 50%. However, the length

*L*of the nanoscaled domains in the unpoled sample is considerably smaller and leads to a weaker SHG signal than for the microscaled domains in the repoled case.

_{z}When the temperature is increased, the domain size decreases. Simultaneously, the order parameter, i.e. the spontaneous polarization *P _{s}*, and therefore the nonlinear coefficient

*d*

_{eff}decreases. Above the Curie temperature, the crystal is in the paraelectric phase (point group 4/mmm), and the nonlinear susceptibility tensor, i.e.,

*d*

_{eff}is zero, hence no second harmonic is emitted anymore. When the temperature is subsequently decreased, the domain distribution in the ferroelectric phase is different compared to the original repoled distribution. After this heating treatment, the domains are much smaller, and on the order of a few hundred nanometers. The SHG ring is not modulated anymore due to the changed domain shape, i.e., the Fourier spectrum

*S*(Δ

*k*)(

_{xy}*T*) cannot be completely resolved for such small domains. Also, the total ČSHG intensity is smaller for nanodomains than for microdomains as can be seen in Fig. 5 where the results of the far-field and the microscopic measurements are compared. The ČSHG intensity of the complete ring is plotted for the heating and cooling process as well as the mean domain diameters determined from the microscope images.

The phase transition can be described by the temperature dependence of the spontaneous polarization *P*(*T*) = *P*_{0}[1 − (*T* + 273)/(*T _{c}* + 273)]

*with the parameter*

^{β}*β*, which describes the type of the phase transition. The second-harmonic intensity is in turn proportional to the square of the nonlinearity:

*I*

_{(2}

_{ω}_{)}(

*T*) =

*I*

_{(2}

_{ω}_{)},

_{0}[1 − (

*T*+ 273)/(

*T*+ 273)]

_{c}^{2}

*. Applying this relation to our measured data from Fig. 4, we find the best fit for*

^{β}*β*= 0.16±0.03 and

*T*= 75 °C, which is very close to its value in [31]. The minimal deviation may be attributed to using different SBN crystals with different pre-history. Nonetheless, this supports the hypothesis that the relaxor-type phase transition of SBN can be described by a 3D Ising model. Above the Curie temperature the ferroelectric looses its polarization information with further heating up. This makes the cooling down process of a different prehistory from that of the repoling state at the beginning. Thus, the comparison of the heating and cooling processes does not necessarily obey the same fitting parameters.

_{c}Figure 5 also shows that the raising of the temperature is associated with decreasing not only the nonlinearity represented by the signal to noise ratio (Fig. 4(i)), but also with decreasing the local spontaneous polarization represented by shrinking of the individual domains, and therefore its spatial distribution in the far-field too. This non-separable process explains the switch-off of the four-fold modulation on the ring in the far-field and its total intensity.

## 4. Conclusion

In conclusion, we have analyzed optically the ferroelectric domains near the phase-transition region in a relaxor strontium barium niobate using the combination of two methods based on Čerenkov SHG. From the SHG intensity in the far-field, we could determine the Curie temperature and the critical exponent *β*. The phase transition in SBN can be characterized by a 3D random-field Ising model in accordance with literature. We also found that the decreasing of the nonlinearity strength at the domain walls is accompanied with the shrinkage of the domains locally at the microscopic level when increasing the temperature towards phase transition, i.e. with changing the Fourier spectrum of the domain distribution.

## Funding

Open Access Publication Fund of University of Muenster.

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