# Ultrafast Third-Order Nonlinear Optical Response Excited by fs Laser Pulses at 1550 nm in GaN Crystals

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{g}= 3.4 eV) plays a key role in light emitting diodes, laser diodes, and detectors in the blue spectral range of the optical domain [5,6,7]. GaN possesses a large transparency, covering the visible and the near and mid infrared spectral domains [8]. It is chemically stable, has a high optical damage threshold, a weak material dispersion, and a low thermo-optic coefficient [9]. Its wide bandgap makes it also very promising for applications at the telecommunication wavelength of 1550 nm, for which both the two- and the three-photon absorption cannot take place. Electromagnetic interference shielding in an ultra-broad range of frequencies, distributed Bragg reflectors and UV-light driven fluorescent microengines are among the emergent applications of this compound when engineered in three-dimensional nanoarchitectures [10,11,12,13,14,15].

_{ph}= 0.8 eV, which is more than four times lower than the GaN bandgap (E

_{ph}< E

_{g}/4). Thus, the optical nonlinearity excited in bulk GaN crystal by single-photon absorption at this wavelength is a non-resonant one. Moreover, neither the absorption of two photons nor that of three photons can excite resonant transitions in GaN. Consequently, the third-harmonic generation excited in GaN by fs laser pulses at this wavelength is extremely fast (response time < 10

^{−15}s [33]).

## 2. Direct Extraction of the Third-Order Nonlinear Optical Susceptibility from the Third-Harmonic Generation

^{(3)}is induced in a NL optical material by a fundamental harmonic (FH) laser beam with the frequency ω

_{FH}= ω, incident on it [32]:

_{0}is the dielectric permittivity of vacuum, χ

^{(3)}is the third-order NL optical susceptibility of the NL material, associated with the THG process, and E

_{ω}is the electric field of FH beam.

_{TH}= 3ω (or, alternatively at a wavelength λ

_{TH}= λ

_{FH}/3). In contrast to the second-order NL optical process involved in the second harmonic generation, the NL process of THG takes place for any symmetry of the NL optical material and its characteristic NL optical parameter is the third-order NL optical susceptibility, χ

^{(3)}. The THG process is illustrated in Figure 1.

_{3}

_{ω}(L) is the intensity of the TH beam generated inside the NL material, at the exit face of the sample, I

_{ω}(0) is the intensity of the FH beam, at the entrance face of the sample, 3ω is the optical frequency of the TH wave, n

_{ω}and n

_{3}

_{ω}are the refractive indices of the NL optical material at the FH and TH frequencies, respectively, L is the thickness of the sample, c is the speed of light in vacuum, and Δk is the phase mismatch, which is defined as [32]:

_{ω}and λ

_{3}

_{ω}being the wavelengths of the respective waves in vacuum.

_{3}

_{ω}(z), is proportional to the third power of the FH beam intensity, I

_{ω}(0), and to the square of the z coordinate along the propagation path through the NL medium. When the phase mismatch Δk = 0 (phase matching), the square of the sinc function (sinc(x) = sin(x)/x) from Equation (2) equals 1, sinc

^{2}(Δk⋅z/2) = 1, and the TH beam intensity is maximum. For Δk ≠ 0, the TH beam intensity oscillates along the propagation path as a consequence of the sinc

^{2}(Δk⋅z/2) function.

_{C}= π/Δk represents the thickness of the NL material for which the argument of the square sinc function from Equation (2) is equal to π/2. When the thickness of the NL sample exceeds L

_{C}, the intensity of the TH decreases significantly, vanishes at 2L

_{C}, and then oscillates as the thickness L increases.

_{R}[37]:

_{0}is the waist of the focused beam and λ is the wavelength of the beam inside the material (the vacuum wavelength divided by the refractive index n of the material). The waist w

_{0}is directly related to the corresponding full width at half-maximum (FWHM) of the beam, ${\omega}_{0}=FWHM/\sqrt{2\mathrm{ln}2}$ [37]. The Rayleigh length represents the distance between the focal plane of the focusing lens and the plane in which, in propagation, the radius of the beam becomes $\sqrt{2}$ times larger than the waist w

_{0}of the focused beam (Figure 2).

^{(3)}, can be directly computed from the Equation (2), as:

## 3. Experimental Details of the Third-Harmonic Generation in GaN Crystal

#### 3.1. Experimental Setup

_{FH}= 1550 nm, with a repetition rate f

_{rep}= 76 MHz and the maximum average power P

_{FH,av,max}= 228.3 mW as the source of the FH beam.

_{1}(5 cm focal length) and L

_{2}(3.5 cm focal length) mounted on micrometric translation stages for the fine tuning of their positions relative to the sample. The lens L

_{1}focuses the FH beam down to a spot of 26 μm (FWHM) on the sample placed in its focal plane.

_{2}, which collects the entire TH beam, the spot size of TH beam generated inside the sample is adjusted relative to the size of the camera photosensitive array, ensuring high grey levels on the camera pixels illuminated by TH beam, yet avoiding their saturation.

_{ND}) calibrated at the FH wavelength. The FH beam, which is exiting from the sample, is cut off with two IR blocking filters (F

_{IR}) with known transmission at the wavelength of the TH. An additional filtering of IR radiation is provided by the camera itself, as it is possible to see in the Figure 4, in which the quantum efficiency of the photosensitive array of this camera is shown, available from https://www.thorlabs.com (accessed on 14 May 2020).

_{FH,av}, of the incident FH beam have been measured with an optical power-meter (Coherent, FIELDMAXII-TOP with OP-2 IR sensor), before the focusing lens L

_{1}.

#### 3.2. Measurement of Very Low Optical Powers Using a Camera as Power-Meter

_{2}, maintaining the linear operation regime with no saturated pixels. By collecting the signal from a software generated window that contains the pixels illuminated by the measured beam, a large signal-to-noise ratio can be maintained even at very low incident powers. Related to this aspect, at a commercial power-meter the electrical signal is generated by the entire sensor area, no matter how small the size of the measured spot is.

_{2}in another plane than the TH beam spot is.

#### 3.3. The Investigated Sample

^{7}cm

^{−2}. Previous experiments revealed fine spatial modulation of the electrical conductivity in this crystal attributed to instability in the growth direction [38,39].

_{o}refractive index of the GaN sample (positive uniaxial crystal, n

_{e}> n

_{o}) is accessed by both FH and TH light beams. At normal incidence of the FH beam, with this particular geometry of the GaN sample, the n

_{e}refractive index of GaN cannot be accessed, independently on the rotation around the c-axis of the sample or, which is equivalent, of the FH beam polarization.

_{ω}and n

_{3}

_{ω}) of the NL material at the FH and TH frequencies, respectively, were calculated using the Sellmeier equation [40] for the ordinary refractive index, n

_{o}, given by the Equation (6):

_{ω}= 2.32604 at λ

_{ω}= 1550 nm and n

_{3}

_{ω}= 2.45316 at λ

_{3}

_{ω}= 517 nm. Using these values of the refractive indices, we determined the value of the phase mismatch and the value of the coherence length, L

_{C}, for the investigated sample: $\left|\Delta k\right|$ ≅ 1.5458 × 10

^{4}cm

^{−1}, L

_{C}≅ 2 μm. The Rayleigh length, given by Equation (4), for our investigated sample is z

_{R}≅ 980 μm. The corresponding confocal parameter is b ≅ 1960 μm, which is more than four times higher than the sample thickness. So, we can assume a satisfactory validity of the plane wave approximation along the propagation path of the FH beam through the sample. The assumption of negligible absorption in GaN, at both FH and TH wavelengths, is well fulfilled [25,41,42] as well as the assumption of the undepleted pump approximation due to the very low conversion efficiency (<10

^{−8}in our experiment) of the FH in TH (no phase-matching), for which the intensity of the TH beam (Equation (2)) was analytically obtained.

## 4. Results and Discussion

_{TH}= 517 nm.

_{av}of a train of ultrashort laser pulses with Gaussian transversal spatial profile and sech

^{2}temporal profile to the peak intensity I

_{peak}, based on [37], is:

_{0}is the waist of the laser beam, τ is the laser pulse duration, and f

_{rep}is the repetition rate of laser pulses. The peak intensities of the TH beam for several peak intensities of the FH beam have been calculated from the corresponding TH average powers, in order to determine the dependence I

_{3}

_{ω}(L) = f (I

_{ω}(0)), where I

_{3}

_{ω}(L) ≡ I

_{TH,peak}(L) and I

_{ω}(0) ≡ I

_{FH,peak}(0). From the fitting of the experimental dependence I

_{3}

_{ω}(L) = f (I

_{ω}(0)) with a third grade polynomial, I

_{3}

_{ω}= C

_{THG}· I

_{ω}

^{3}, we determined the coefficient C

_{THG}= I

_{3}

_{ω}/I

_{ω}

^{3}, then from the Equation (5) we calculated the third-order NL optical susceptibility corresponding to the THG process, χ

^{(3)}, in the c-plane GaN crystal. It characterizes the ultrafast nonlinear response of solely electronic origin in this material excited with ultra-short laser pulses at the wavelength of 1550 nm.

_{FH,av,max}(0) = 228.3 mW. The corresponding I

_{FH,peak}(0) of the FH laser pulses, considered with a Gaussian transversal spatial profile and sech

^{2}temporal profile is I

_{FH,peak,max}(0) = 2.29 GW/cm

^{2}. The range of the incident FH peak intensities, considered in our experiments, is I

_{FH,peak}(0) = (0.99 ÷ 2.29) GW/cm

^{2}.

_{TH,av}(L) have been measured with the calibrated CMOS camera used as a very sensitive power-meter [35]. Each of these powers has been determined in two ways: by considering the entire photosensitive array of the camera used (1280 px * 1024 px = 1,310,720 pixels) as well as considering a software generated window of 200 px * 200 px (40,000 pixels), surrounding the TH beam spot, which represents only 3% of the surface of the photosensitive array of the camera. When measuring very low optical powers by the method described above, the use of a software generated window that surrounds the measured beam spot is preferable in terms of accuracy, as it is discussed in detail in [35]. The necessary corrections for Fresnel reflections of the FH beam on lens L

_{1}and of the TH beam on lens L

_{2}have been made. The reflection losses of the FH beam at the air–GaN interface (15.9%) and of the TH beam at the GaN–air interface (17.7%) have been taken into account. The attenuation of TH beam by the two IR blocking filters, F

_{IR}, has been also taken into account.

_{FH,peak}(0) = 0.99 GW/cm

^{2}of the FH beam, is not shown in the Figure 7, as it is difficult to be seen by the naked eye due to the very low grey levels of its pixels.

_{TH,peak}(L) = f (I

_{FH,peak}(0)), were obtained.

^{(3)}= (1.30 ± 0.15) × 10

^{−20}m

^{2}/V

^{2}, which is obtained when the software generated window is taken into account. To the best of our knowledge, this is the first direct measurement, by THG, of the ultrafast optical nonlinearity of solely electronic origin in c-plane GaN crystal.

^{(3)}obtained by us is in qualitative agreement with several values of the third-order NL susceptibility obtained by other experimental techniques, as Z-scan and wave mixing, at similar wavelengths. Thus, in 2019, in [25] the nonlinear refractive index of a ~10 μm GaN layer grown epitaxially on a sapphire substrate was measured by Z-scan technique performed with 120 fs laser pulses in the wavelength domain (1550 nm ÷ 550 nm), and the value obtained at 1550 nm was n

_{2}≈ 90 × 10

^{−20}m

^{2}/W. The corresponding value of the nonlinear optical susceptibility responsible for this intensity dependent refractive nonlinearity is proportional to the nonlinear refractive index n

_{2}[33] and has the magnitude χ

^{(3)}≈ 1.71 × 10

^{−20}m

^{2}/V

^{2}. In 2018, in [23] the nonlinear refractive index n

_{2}was measured in a GaN ridge waveguide by FWM with ~30 ns laser pulses at the telecom wavelength of 1550 nm and the estimated value was n

_{2}≈ 3.4 × 10

^{−18}m

^{2}/W. The corresponding value of the nonlinear susceptibility is χ

^{(3)}≈ 6.45 × 10

^{−20}m

^{2}/V

^{2}. We have to mention that this comparison of the value of χ

^{(3)}obtained by us with the values from [23,25] is only a qualitative one, as different NL optical processes are involved in the nonlinear optical response measured in these techniques [23,25,31].

## 5. Conclusions

^{(3)}, responsible for the nonlinear optical process of frequency conversion was directly determined in c-plane GaN crystal from the third-harmonic generation experimental results. Its value, χ

^{(3)}= (1.30 ± 0.15) × 10

^{−20}m

^{2}/V

^{2}, is in qualitative agreement with several values of the third-order nonlinear optical susceptibility obtained at similar wavelengths by other experimental techniques (Z-scan, four-wave mixing), in which different nonlinear optical processes of the third-order are involved. The obtained results are important for the potential use of GaN in ultrafast photonic functionalities.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 5.**Schematic representation of crystallographic a, c, and m axes relative to the investigated GaN sample plane. The light propagation direction and its polarization are also shown.

**Figure 7.**Images of the TH beam spot generated in GaN crystal, framed in a 200 px ∗ 200 px window at several peak intensities of the incident FH. The exposure times corresponding to the recordings are mentioned.

**Figure 8.**The experimental curves I

_{TH,peak}(L) = f (I

_{FH,peak}(0)), considering the entire photosensitive array of the camera (

**a**), and a 200 px * 200 px software generated window surrounding the measured beam spot (

**b**), respectively.

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Petris, A.; Gheorghe, P.; Braniste, T.; Tiginyanu, I.
Ultrafast Third-Order Nonlinear Optical Response Excited by fs Laser Pulses at 1550 nm in GaN Crystals. *Materials* **2021**, *14*, 3194.
https://doi.org/10.3390/ma14123194

**AMA Style**

Petris A, Gheorghe P, Braniste T, Tiginyanu I.
Ultrafast Third-Order Nonlinear Optical Response Excited by fs Laser Pulses at 1550 nm in GaN Crystals. *Materials*. 2021; 14(12):3194.
https://doi.org/10.3390/ma14123194

**Chicago/Turabian Style**

Petris, Adrian, Petronela Gheorghe, Tudor Braniste, and Ion Tiginyanu.
2021. "Ultrafast Third-Order Nonlinear Optical Response Excited by fs Laser Pulses at 1550 nm in GaN Crystals" *Materials* 14, no. 12: 3194.
https://doi.org/10.3390/ma14123194