Since the first demonstration in 1961, the second harmonic generation (SHG) [1] process has been widely used in an exceedingly wide array of different applications. The usefulness of the process stems primarily from the ability to generate new wavelengths where lasers are unavailable [2]. Moreover, close to degeneracy, the cascaded χ⁽²⁾:χ⁽²⁾, where SHG and difference frequency mixing are combined in the same medium, provides a method for controlling temporal and spatial phases of laser radiation. These effects have been used in phase noise suppression [3], optical communications [4,5], laser mode-locking [6,7], spatial mode conversion [8], and other areas. Continuous wave SHG efficiencies approaching 90% in bulk nonlinear crystals have been demonstrated by employing resonant enhancement cavities [9,10], while 99% pump depletion has been reported using single-pass SHG in a periodically poled lithium niobate waveguide [11]. Quasi-phase-matching (QPM) in structured χ⁽²⁾ allows further design of the spatial and spectral properties of the SHG process beyond the capabilities of homogeneous media [12–14].

Backward SHG (BSHG) is the process where the frequency-doubled light is generated in a counter-propagating direction to the input fundamental frequency field. BSHG is one of the processes that cannot be realized in known homogeneous χ⁽²⁾ materials due to a large momentum mismatch. Birefringence cannot address the momentum mismatch, as the phase mismatch is the sum of all three interacting waves. Recently, stacked metasurfaces have been proposed to achieve phase-matching, but the conversion efficiency still needs to be improved for these materials [15]. QPM remains the most viable route to achieve highly efficient BSHG. Phase matching of SHG processes using QPM is schematically shown in Fig. 1, where Fig. 1(a) displays QPM SHG in the forward co-propagating geometry (FSHG), while Fig. 1(b) corresponds to BSHG. The QPM condition for efficient SHG requires minimizing the wave vector mismatch, with the “+” and “−” signs corresponding to the BSHG and the FSHG case, respectively. Here, the usual definition of the wave vector k_i= 2πn_i/λ_i, with refractive index, n_i, is used, and the subscripts, p, SH, correspond to the waves at the fundamental and second harmonic frequencies. The design parameter in the QPM is the structure grating wave vector K_G= 2π/Λ. Spatial modulation of the χ⁽²⁾ sign with appropriate period Λ, allows phase matching any process, including BSHG, provided the structure could be practically realized [16,17]. As can be seen from Fig. 1(b), the structure grating periodicity, $\Lambda = {{{\lambda _p}} / {2({n_p} + {n_{SH}})}}$, needs to be substantially smaller than the fundamental wavelength for the most efficient, first-order QPM BSHG. Due to substantial fabrication difficulties, only low-efficiency higher-order QPM BSHG has been demonstrated before this work [18,19].

 

Fig. 1. Phase-matching diagrams for (a) FSHG and (b) BSHG.

Download Full Size | PDF

Due to stringent phase-matching conditions, the BSHG is expected to have a much narrower spectral acceptance bandwidth than the co-propagating SHG geometry [20]. The efficient BSHG process would enable the exploration and application of novel nonlinear dynamic regimes, including self-pulsing, limiting, and bistability [21–23], novel ways for phase conjugation [24], and ultrashort pulse shaping [25].

We report on the first-order QPM BSHG realized in periodically poled Rb-doped KTP (PPRKTP) in the QPM structure with a periodicity of Λ=317 nm and a structure length of 12.6 mm. To the best of our knowledge, this is the first demonstration of the first-order QPM BSHG process. The BSHG efficiency of 18.7% was reached for narrowband ns pulses. The substantial efficiency of the first-order QPM BSHG allowed us to investigate the process's spectral and temperature tuning properties, which was not possible in earlier higher-order demonstrations. The BSHG results are compared with the QPM FSHG process realized in a PPRKTP sample with Λ=40.67 µm, which phase-matched co-propagating frequency doubling in the same spectral range. Due to the obvious challenges of fabricating first-order BSHG structures, we discuss the fabrication process in some detail.

In the experiments, we employ QPM structures fabricated in RKTP. The PPRKTP structures with the periodicity of Λ=317 nm (QPM length of 12.6 mm) and Λ=40.67 µm (QPM length of 17 mm) were fabricated which allowed first-order phase-matched BSHG and FSHG, respectively, at the fundamental wavelength of 2309 nm.

We fabricated high-quality sub-µm QPM structures using a coercive field engineering technique [26]. As our expertise and understanding of the technique [27] grow, we can successfully produce smaller grating pitches. The RKTP samples were photolithographically patterned on the c⁻ polar surfaces. With the extreme difference periods, the crystal, phase-matching FSHG, was patterned by a maskless lithography system (Heidelberg Instruments, MLA 150). The BSHG crystal was patterned with an in-house UV-interference system [19]. The patterned and unpatterned surfaces of the crystals were exposed to oxygen plasma etching, creating ion exchange stop layers where there was no resist. The crystals were then ion-exchanged by 4-hours immersion in molten (330°C) nitrate salt containing 20-mol% KNO₃, 73-mol% RbNO₃, and 7-mol% Ba(NO₃)₂, which locally increased the coercive field. Lastly, the RKTP crystals were periodically poled using planar, liquid KCL electrodes, and a single triangular-shaped electrical-pulse of 1.25- to 5-ms duration with a peak magnitude between 5.5 and 9 kV/mm. The resulting ferroelectric domain grating of the BSHG structure is shown in the piezoresponse force microscopy (PFM) image in Fig. 2. Ferroelectric domains merged approximately 500 µm below the BSHG sample c⁻ polar surface. Introducing Rb ions deeper into the RKTP crystal can improve the coercive field grating, extend the poling through the crystal, and reduce duty-cycle errors due to lateral growth [27]. Nevertheless, the resulting QPM structure had an optical aperture of 0.5 mm×4 mm and a length of 12.6 mm, sufficient for obtaining an efficient BSHG process. Two such samples were fabricated, one of them being used for generating a broadband chirped wave around 2309 nm in the backward wave optical parametric oscillation (BWOPO) process [28], which was, in turn, used as a fundamental frequency wave to pump the second BSHG sample.

 

Fig. 2. PFM image of the patterned polar surface. The bright regions correspond to the inverted ferroelectric domains.

Download Full Size | PDF

The broadband fundamental frequency excitation allowed us to verify the phase matching point of the BSHG process and to characterize the temperature tuning behavior. As a pumping source, we used the forward-traveling idler of a BWOPO that had the following characteristics: a Ti:Sapphire regenerative amplifier operating at 782 nm and delivering stretched 655-ps (FWHM) long pulses with a positive linear frequency chirp of 19.4 mrad/ps² and spectral bandwidth of 5 nm pumped a periodically poled BWOPO PPRKTP crystal (Λ=317 nm). The BWOPO generated a narrow backward wave at 1180 nm and a broadband forward wave (∼30 nm) centered at 2320 nm with a pulse length of 410 ps (FWHM) and a frequency chirp similar to that of the pump [29,30].

Figure 3(a) compares the measured SHG spectra generated by the same fundamental wave in the BSHG and FSHG processes. Spectra were measured using an optical spectrum analyzer (ANDO AQ-6315A) with 0.05-nm resolution. As expected, the BSHG generates a much narrower spectrum (0.09-nm FWHM) than the FSHG process. The FSHG spectrum reflects within the acceptance bandwidth the spectral distribution of the chirped fundamental wave generated by the BWOPO. The narrow BSHG spectra can be understood by realizing that the fundamental acceptance bandwidth depends inversely on the sum of the group velocities of the interacting waves. We estimate [17] the fundamental acceptance bandwidth (δλ_p= 2δλ_SH) of 53 pm for the BSHG and 13.2 nm for the FSHG case. The measured FSHG spectral extent is in accordance with theoretical estimates. The BSHG spectrum is broader than expected for monochromatic plane waves. This is because the fundamental wave is chirped, and the effective interaction length will be shorter than the physical length of the QPM structure. Moreover, the duty cycle errors and grating defects seen in Fig. 2 would contribute to broadening. In this wavelength range, the theoretically expected FSHG/BSHG acceptance bandwidth ratio is equal to 336 for the same crystal length.

 

Fig. 3. (a) Spectra of BSHG (red) and FSHG (black) pumped by the broadband (30 nm) source. (b) BSHG temperature tuning curve.

Download Full Size | PDF

Figure 3(b) shows the measured temperature tuning of the BSHG. From a linear fit, the BSHG tunes at 17.1 pm/K between 20 and 60°C. The measurement fit compares well to the theoretical value of 16.7 pm/K, using thermo-optic and thermal expansion coefficients [31–33]. Thermal expansion plays a significant role in BSHG tuning compared with the usual FSHG case. For instance, in the BSHG structure used in this work, 48% of the tuning rate is attributed to the change in the QPM periodicity due to thermal expansion. While in the FSHG case, the contribution of thermal expansion is 11%.

To characterize the BSHG efficiency, we built a narrowband source at 2309 nm. The source was based on a plane–plane cavity, singly resonant optical parametric oscillator (OPO) employing type-II QPM in PPRKTP with a grating periodicity of 75.18 µm [34]. Large differences in the group velocities of the signal and idler generated in perpendicular polarizations narrow the OPO bandwidth dramatically, even close to degeneracy. The pump was an injection-seeded Q switched Nd:YAG laser operating at 100 Hz (Innolas, Spitlight). The idler output pulses at 2309 nm had a pulse length of 6.8 ns (FWHM) with a bandwidth of 0.2 nm (Yokogawa AQ6376 OSA, resolution 0.1 nm). The OPO generated a maximum of ∼0.2 mJ of idler. An f = 50 mm CaF₂ lens focused the idler to a beam waist radius of approximately 48 µm. The idler wave was polarized parallel to the BSHG crystal's z axis to exploit the d₃₃ nonlinear coefficient. The maximum fundamental fluence was ~2 J/cm², below the nanosecond laser damage threshold of 12 J/cm² at 2 µm [35].

Figure 4(a) shows the measured BSHG average output power as a function of the fundamental power. The output power follows the typical square law between input and output in the un-depleted pump regime for SHG. The conversion efficiency reached 18.7%; at this point, the efficiency was limited by the output of the OPO. Note that the maximum fundamental power was lower than the OPO-generated power due to Fresnel losses in the focusing lens and the uncoated surface of the BSHG crystal. From the quadratic fit, we estimate the effective nonlinearity, d_eff, to be 2.4 pm/V. This value is lower than the theoretical limit of a PPKTP type-0 QPM device (10.76 pm/V [36]), likely due to non-optimum average duty cycle and poling defects in the BSHG structure.

 

Fig. 4. (a) BSHG average power with the quadratic fit (b) BSHG (red) and FSHG (black) normalized power dependence on temperature.

Download Full Size | PDF

Figure 4(b) displays the SHG power dependence on the crystal temperature in the BSHG and the FSHG structures, excited by the narrowband ns OPO. The measured temperature bandwidth in the FSHG structure was $\mathrm{\delta T} = {\; }35^\circ \textrm{C}$, which is close to the expected theoretical value of $\mathrm{\delta T} = {\; }33.7^\circ \textrm{C}$. The Peltier element's cooling range was insufficient to cover the FSHG bandwidth fully. Meanwhile, for the BSHG structure, we determined a FWHM bandwidth of $\mathrm{\delta T} = 8.1^\circ \textrm{C}$. The bandwidth expected from theory for the monochromatic fundamental is $\mathrm{\delta T} = 1.3^\circ \textrm{C}.$ The difference between the measured and theoretical value can be attributed to the OPO idler bandwidth (0.2 nm), which is still larger than the acceptance bandwidth of the BSHG structure (0.053 nm). Considering the measured SHG temperature tuning rate, we estimate the expected temperature bandwidth to be $\mathrm{\delta T} = 6^\circ \textrm{C}$, close to the experimentally observed value.

The measured ns OPO idler spectrum and the generated BSHG spectrum at a fixed crystal temperature are shown in Figs. 5(a) and 5(b), respectively. The BSHG spectral measurement is limited to the OSA resolution of 0.05 nm (∼11 GHz at 1154 nm). By considering the fundamental acceptance bandwidth of 53 pm, the bandwidth expected in a perfect, 12.6-mm-long BSHG structure would be approximately two times smaller than the resolution of the measurement. The QPM structure used in the experiments is far from perfect. Nevertheless, the spectral measurements demonstrate the potential for high spectral brightness SHG using the BSHG process.

 

Fig. 5. Measured spectra of (a) ns OPO fundamental and (b) BSHG.

Download Full Size | PDF

In conclusion, we have demonstrated that current ferroelectric structuring technology in RKTP allows realization of first-order BSHG processes, which can generate powers suitable for practical applications. The BSHG has a great potential for narrowband, high spectral brightness second harmonic generation. In the BSHG structure used here, we attained 18.7% conversion efficiency in a 12.6-mm-long PPRKTP structure using the narrowband nanosecond source with 200-pm bandwidth. The BSHG spectral bandwidth in such a structure should be approximately 27 pm. For comparison, the FSHG reached 50% saturated conversion efficiency for the same fundamental power, but the FSHG bandwidth was approximately 100 pm. That shows that BSHG, even at the current pioneering stage, can reach a similar or higher spectral brightness than FSHG. This property would benefit applications in precision interferometric measurements, generation of squeezed states, and others. Counterpropagating upconversion might be valuable for single-photon detectors, which increase the measurement range of efficient low-cost avalanche Si-photodiodes [37]. Moreover, the spectral noise scales inversely with the number of domains in QPM devices [38], where the main fabrication error is duty cycle variations. The backward processes using QPM increase the number of domains over the same length, therefore, have the potential to reduce the noise pedestal outside the QPM bandwidth. For instance, the ratio of the domain numbers in the BSHG and the FSHG structures used in this work is approximately 100, giving better spatial averaging for the second-order parasitic processes. BSHG is a novel nonlinear optical geometry enriching the optics and photonics toolbox. The natural next steps for studying the BSHG process are the temporal and beam dynamics and pushing BSHG processes into the visible spectral range.