1. INTRODUCTION

Light detection and ranging (LIDARs) are extensively used in a wide array of applications such as navigation [1], metrology [2], vegetation-assessment [3], and gas detection [4]. These have predominantly been implemented in land-based settings [1,2,4] but have also progressively been adapted for air- [3] and space-borne [5] applications. Underwater and air-to-water implementations, however, remain challenging. Such systems require high peak-power and high-brightness pulsed sources operating in the water transmission window between 440 and 490 nm [6]. It should be mentioned that over the last decade, there has been enormous progress in the development of light emitting diodes, based on organic and inorganic luminescent materials [7,8], as well as widespread commercialization of continuous-wave and gain switched picosecond lasers in the blue spectral region. These sources find applications in microscopy modalities such as spectroscopy and fluorescence lifetime imaging. Yet, scaling the low pulse energies from these sources to levels appropriate for LIDAR applications is not straightforward.

Systems based on Yb-doped fiber lasers operating at 977 nm followed by a second harmonic generation (SHG) stage have shown promise in potentially reaching the target specifications [9]. However, such sources are typically broadband, with linewidths of about 3 nm, which limit the achievable conversion efficiencies. Tm-doped fiber lasers operating at 1940 nm followed by two SHG stages have also been proposed [10]. However, the need for two free space SHG stages could prove challenging to stabilize sufficiently to achieve optimal performance in field conditions. Another approach relies on employing the weaker laser transitions in ${{\rm Nd}^{3 +}}$ around 940 nm to realize blue light through SHG [11]. Yet, none of these technologies are as widely commercially available or as well-developed as high pulse energy systems based on ${{\rm Yb}^{3 +}}$- and ${{\rm Nd}^{3 +}}$-dopants operating around 1030 nm and 1064 nm, respectively.

The output from such systems can be converted to the 970 nm region by exploiting four wave mixing (FWM) in multimode fibers [12], thus enabling further SHG towards the water window. Fiber optical parametric gain using FWM is achieved by exciting a higher order mode (HOM, in this case ${{\rm LP}_{0,7}}$) that exhibits a suitable dispersion profile for phase-matched generation of the anti-Stokes wavelengths around 970 nm from the 1064 nm input pump. Considering energy and momentum conservation for the FWM in the low intensity limit, where the power dependent phase-matching contribution can be neglected, leads to the following relation between the effective modal indices and the ratio of the pump and the anti-Stokes vacuum wavelengths (${\lambda _p},\;{\lambda _a}$, respectively):

In the intramodal FWM case, the anti-Stokes light occupies the same HOM as the pump, which is projected onto the Laguerre-Gaussian (LG) free space eigenbasis after outcoupling prior to collimation and focusing into the crystal. The fiber supported ${{\rm LP}_{0,p + 1}}$ modes can be mapped to ${{\rm LG}_{p,0}}$ modes with a more than 80% overlap when ${p}\; \lt \;{10}$ [14]. A pure LG mode with the radial index, ${p} = {0},{1},{2},{\ldots}$, and the azimuthal index, ${l} = {0},{1},{2},{\ldots}$, is characterized by a beam quality factor given by [15]:

Approximating the Laguerre-Gauss superposition of the ${{\rm LP}_{0,7}}$ mode, shown in Fig. 1(a), by its dominant component, i.e., ${{\rm LG}_{6,0}}$, gives a value of ${{\rm M}^2} = {13}$. Although the mode field is spatially coherent, the brightness of the beam decreases inversely proportionally to the square of the ${{\rm M}^2}$, $B = \frac{P}{{{{({{{\rm M}^2}\lambda})}^2}}}$ [16], where $P$ is the power of the beam at wavelength $\lambda$. Obviously, LIDAR applications would benefit from employing high-brightness, ideally, the lowest order Laguerre-Gauss mode.

 

Fig. 1. (a) Intensity distribution in an ${{\rm LP}_{0,7}}$ mode. (b) Cascaded Kerr nonlinearity at FH and SH frequencies as a function of phase mismatch $\Delta {\rm k}$ for the process of frequency doubling of 971 nm in a 10-mm PPKTP structure with a QPM periodicity of 6.66 µm. $n_{2,\rm FH}^I$ for the FH wave (blue curve), $n_{2,\rm SH}^I$ for the SH wave (red curve).

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Superficially, the intensity distribution of an ${{\rm LP}_{0,7}}$ mode shown in Fig. 1(a), and hence also its dominating free space mode ${{\rm LG}_{6,0}}$, is reminiscent of a pseudo-nondiffracting Bessel-Gauss beam [17], for which successful beam transformation into an axial Gaussian-like beam through SHG has been demonstrated [18]. As Bessel-Gauss beams are localized distributions, resulting from the interference of the waves located on different parts of a cone in a converging annular beam, the beam transformation during SHG can be readily achieved by designing the nonlinear interaction for appropriate noncollinear phase-matching conditions. By contrast, the ${{\rm LG}_{p,0}}$ modes are eigenfunctions of the Fourier transform, therefore, apart from appropriate scaling, they retain their spatial distribution during propagation [19,20]. In this case, the problem of nonlinear mode transformation becomes a much more intricate task, compared to the simple case of noncollinear interaction with well-defined angles when using Bessel-Gauss beams. In order to achieve mode transformation for ${{\rm LG}_{p,0}}$ modes, the radial phase distribution needs to be appropriately modified during the SHG process. From symmetry considerations, achieving a perfect mode transformation during SHG would require a 3D quasi-phase matched (QPM) structure with cylindrical symmetry designed for a specific fundamental harmonic ${{\rm LG}_{p,0}}$ mode. At present, fabrication technology for such structures in nonlinear materials, useful for blue light generation, does not exist. Therefore, we consider the brightness enhancement which could be achieved in a realistic 1D QPM structure.

In this work we theoretically and experimentally investigate conditions for maximizing brightness by direct SHG of sub-nanosecond near-infrared pulses at 971 nm generated from intramodal FWM in the ${{\rm LP}_{0,7}}$ mode, without additional mode manipulation. In particular, we show that the highest conversion efficiency does not provide higher brightness beams. In fact, phase-mismatched SHG, although with lower conversion efficiency, results in brightness enhancement owing to a nonlinear spatial phase imparted by a cascaded $\chi^{(2)}:\chi^{(2)}$ process, leading to mode transformation during SHG. We employ periodically poled KTP (PPKTP), which is characterized by a high nonlinearity and a high transmission in the 485 nm region, as the material of choice in this study.

2. CONCEPT AND NUMERICAL MODEL

The field generated by the SHG of an ${{\rm LG}_{0,p}}$ will be a sum of two fields: one, generated in collinear interactions, that will have a similar intensity distribution to that of the fundamental harmonic (FH), and, a second, arising due to noncollinear interactions between adjacent radial antinodes, containing phase shifts of $\pi$ and producing an additional ring structure as a result. Therefore, the total number of rings in the second harmonic (SH) intensity distribution can be larger compared to that of the FH distribution. Accordingly, the ${{\rm M}^2}$ of the SH beam can increase. This contrasts with the SHG of the lowest-order ${{\rm LG}_{0,0}}$ mode, where ${{\rm M}^2}$ remains the same if back conversion is avoided. We therefore define a brightness enhancement factor, BE, as the ratio of the second harmonic and the fundamental beam brightness,

It is clear from Eqs. (2) and (3) that the best way to substantially increase the brightness during SHG of an ${{\rm LG}_{p,0}}$ mode is by transforming it into a SH mode or a mixture of modes with lower mode indices. Our goal is achieving maximum BE without invoking any additional spatial beam processing, which typically requires additional optical elements that could lead to misalignment-induced reliability constraints.

Qualitatively, the concept for brightness enhancement during the SHG process of an ${{\rm LG}_{p,0}}$ mode can be understood as follows. The maximum conversion efficiency could be expected around the phase matching point, $\Delta kL = {0}$, with a small offset to compensate for the Gouy phase ($({2}p + {1})\pi$) [21,22]. Here, $L$ is the length of the nonlinear crystal and $\Delta k = {k_{{\rm SH}}} - 2{k_{{\rm FH}}} - {k_g}$ is the wavevector mismatch, with ${k_{{\rm SH}}},{k_{{\rm FH}}}$, ${k_g} = 2\pi /\Lambda$ denoting wavevectors of the SH, the FH and the QPM grating with a period of Λ, respectively. Here, the collinear interactions will be phase matched and the axial part of the beam as well as rings will generate SH without intermixing. The resulting ${{\rm M}^2}$ of the SH beam will be similar to that of the FH. Therefore little, if any, brightness enhancement, as defined by Eq. (3), would be expected. Increasing $|\Delta kL|$ will phase match noncollinear interactions among different ring structures. The phase of the SH, ${\varphi _{{\rm SH}}} = {\varphi _{F1}} + {\varphi _{F2}} + \pi /{2}$, [23], generated by the interaction of FH fields from adjacent rings will contain an additional phase shift of $\pi$ (the phase shift between the FH rings, ${\varphi _{F1}} = {\varphi _{F2}} + \pi$) compared to the collinearly generated SH field (${\varphi _{F1}} = {\varphi _{F2}}$). Thus, by detuning $|\Delta kL|$ to promote noncollinear interactions, one would expect an increase of the number of antinodes in the SH field and an associated increase in the beam quality factor ${{\rm M}^2}$. It might seem that brightness enhancement is an elusive goal. However, there is a useful complication concerning cascaded nonlinear interactions in the SHG process. Under phase-mismatched conditions, two $\chi^{(2)}:\chi^{(2)}$ cascade processes will take place, the FH cascade, $\omega \to 2\omega \to \omega$, [24], and the SH cascade, $2\omega \to \omega \to 2\omega$. These processes will introduce phase modulation on the FH and the SH fields proportional to the intensity of the FH beam, $I_{\rm FH}$. Formally, the effects are akin to those of self-phase modulation and cross-phase modulation, where the refractive index changes at the FH and SH frequencies can be expressed as $\delta {n_{{\rm eff},\rm FH}} = n_{2,\rm FH}^I{I_{{\rm FH}}}$ and $\delta {n_{{\rm eff},\rm SH}} = n_{2,\rm SH}^I{I_{{\rm FH}}}$, for the FH cascade and SH cascade processes, respectively. The effective Kerr coefficients, $n_{2,\rm FH}^I,\,n_{2,\rm SH}^I$ can be derived from the second order coupled wave equations, which, by using plane waves and negligible pump depletion approximations, give the following expressions:

Given the large parameter space of the nonlinear interactions, a numerical model was set up to probe possible interactions under different operating conditions. As we are interested in high-energy narrowband pulses on the 100’s of picosecond timescale and crystal lengths on the centimeter scale, the impact of group velocity dispersion and group velocity mismatch could be neglected. This allowed us to use a time-independent model to study the impact of different focusing conditions at varying power levels and phase mismatches [25]:

This model was solved by evaluating the transverse derivatives, describing diffraction, in the Fourier plane and the nonlinear coupling equations in the spatial coordinate plane employing a fourth order Runge-Kutta method. These terms were thus effectively considered separately and propagated independently over small distances with an alternation at every other step, similar to the conventional split-step method used to study nonlinear interactions in optical fibers [26].

3. SIMULATION RESULTS

The numerical model was initiated by an ${{\rm LP}_{0,7}}$ mode that was collimated and propagated through free space, in order to mimic steering optics, before being focused through a second lens prior to entering the crystal. This was carried out with a diffraction integral ABCD-matrix approach for the radial profile, as described in [27], which allows for convenient scaling of the spatial axes between the input and output planes—and thus enables a constant number of grid points for various beam sizes.

 

Fig. 2. Simulated conversion efficiencies (top panel), number of rings in the near- (second panel) and far-field (third panel) and ${{\rm M}^2}$-values (bottom panel) when using a 150 mm positive lens and having the crystal’s (a) entrance facet 25 mm before the focal plane, (b) center at the focal plane and (c) entrance facet 25 mm after the focal plane.

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The model has three variable parameters: the position of the crystal with respect to the focal plane, the instantaneous power of the FH beam and the phase mismatch, $\Delta {k}$. The brightness of the SH beam is related to the fraction of power contained in the axial peak in the far field. Number and power contents in concentric rings in the SH intensity distribution indicate the presence of higher order ${{\rm LG}_{p,0}}$, modes which directly relate to the beam quality factor, as discussed above.

It should be noted that due to orbital angular momentum conservation, only the modes of zero azimuthal index will be present in the SH field. Therefore, the number of concentric rings in the SH intensity distribution is a suitable proxy measure for the resulting SH beam quality and brightness. Radial and angular second moments of the SH field distribution, $\omega _r^2,\omega _k^2$, respectively, have been used to calculate the resulting ${{\rm M}^2}$ of the beam: ${{\rm M}^2} = {\omega _r}{\omega _k}/2$ [23].

The second moments are defined in the following manner [25] for radially symmetric beams:

The simulated results for the situation relevant to the experiment are shown in Fig. 2. Here, a lens with ${f} = {150}\;{\rm mm}$ was used to focus the FH mode into a 10-mm PPKTP crystal. Figures 2(a)–2(c) correspond to the crystal positions with the entrance face being 25 mm before the focal plane, the crystal center coinciding with the focal plane and the crystal entrance being 25 mm after the focal plane, respectively. The heat-maps display conversion efficiency (top row), number of concentric rings in the SH near-field (second row), number of concentric rings in the SH far-field (third row) and the ${{\rm M}^2}$ value (bottom row) as a function of instantaneous power and phase mismatch. The number of concentric rings account for the rings with local intensities at least 1% of the largest intensity in the beam. The FH wavelength is 971 nm and, $\Delta k = {0}\;{{\rm m}^{- 1}}$, corresponds to exact phase matching for plane wave collinear SHG. The simulated power range corresponds to the peak powers expected from the HOM fiber source.

The SHG efficiency maps on the top row of the Fig. 2 show that the maximum efficiency is reached close to $\Delta k = {0}\;{{\rm m}^{- 1}}$, for all positions of the crystal with respect to the focal plane. In fact, the model shows that high efficiencies, in the range of 40% should be attainable. For the focal plane position in the middle of the crystal [Fig. 2(b)], the area of high SHG efficiency is shifted slightly towards negative values of $\Delta k$ to compensate for the Gouy phase shift. As discussed above, operation close to the phase-matching point corresponds to mostly collinear interactions which produce SH beams with a ring structure similar to that of the FH and with comparable ${{\rm M}^2}$ value. This picture is affirmed by the ${{\rm M}^2}$ maps on the bottom row of the Fig. 2(b). Here, at $|\Delta k|\; \approx \;{0}\;{{\rm m}^{- 1}}$, the ${{\rm M}^2}$ of the SH beam remains similar to that of the fundamental. For larger detuning, notably at $\Delta k\; \approx \;{\pm}60{0}\;{{\rm m}^{- 1}}$, regions with higher ${{\rm M}^2}$ appear. These are related to noncollinear SHG interactions with intermixing of the SH from the axial part and from different rings in the FH. Specifically, we consider two noncollinear processes, schematically shown in Fig. 3. The longitudinal mismatch components (parallel to the QPM grating wavevector) for the noncollinear processes can be expressed as

 

Fig. 3. Noncollinear SHG geometries corresponding to (a) $\Delta {k}\; \gt \;{0}$ and (b) $\Delta {k}\; \lt \;{0}$. (c) Numerical simulation of SH beam evolution in the nonlinear crystal corresponding to the maximum FH power in Fig. 2(a) at a phase mismatch of $\Delta {k} = - {466}\;{{\rm m}^{- 1}}$. Qualitatively similar evolution at the highest power is obtained for a negative detuning $\Delta {k} = {403}\;{{\rm m}^{- 1}}$. The predicted noncollinear interaction angle, by Eqs. (9) and (10), of 7 mrad with respect to ${r} = {0}\;{\rm mm}$ is indicated by the black lines.

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Fig. 4. Evolution of (a) the FH and (b) the SH normalized intensity in the nonlinear crystal for a phase detuning of $\Delta { k} = - {675}\;{{\rm m}^{- 1}}$ and the FH power of 5.6 kW. The crystal entrance face is located 25 mm after the focal place of the ${ f} = {150}\;{\rm mm}$ lens.

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Fig. 5. (a) Pulse integrated conversion efficiencies $\eta$, (b) BE as defined in Eq. (3) for ${\rm M}_{{\rm FH}}^2 = 13$, and (c) near- and far-field radial beam profiles and beam profiles, at a phase mismatch of $\Delta {k} = - {675}\;{{\rm m}^{- 1}}$ shown in blue and red, respectively. ${{z}_0}$ in the legends denotes the position of the crystal entrance face with respect to the focal plane.

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The normalized SH intensity evolution in the nonlinear crystal at the highest FH power at a phase mismatch of $\Delta k = - {466}\;{{\rm m}^{- 1}}$ is shown in Fig. 3(c). Noncollinear generation of an additional SH ring structure is clearly visible. The noncollinear process shown in Fig. 3(b), corresponding to the negative $\Delta k$, is asymmetric and contains an uncompensated transversal phase mismatch of ${k_{{\rm SH}}}{\sin}\alpha$. It can be expected that the SHG efficiency for this process would depend on the wavefront curvature, and therefore on the position of the crystal with respect to the focal plane. Indeed, as can be seen in Fig. 2(c) the efficiency at negative values of $\Delta k$ increases when the crystal is positioned after the focal plane. At the same time, this also corresponds to a marked decrease in ${{\rm M}^2}$ of the SH at a detuning of $\Delta k\;\sim\; - {700}\;{{\rm m}^{- 1}}$, indicating that the radial $\pi$ phase jumps are compensated by the action of the cascaded $\chi^{(2)}:\chi^{(2)}$ process. The normalized SH and FH intensity evolutions in the nonlinear crystal for a detuning of $\Delta k = - {675}\;{{\rm m}^{- 1}}$, shown in Fig. 4, reveal self-focusing action of the cascaded nonlinearity on the fundamental beam, and predominantly noncollinear SHG process which generates a SH intensity distribution with much less pronounced ring structure and, therefore, a low ${{\rm M}^2}$ value. We can expect the largest brightness enhancement to be achieved under these operating conditions.

It is clear from the heatmaps in Fig. 2 that the SHG efficiency, output ring structure and ${{\rm M}^2}$ also depend on the FH power. Therefore, for pulsed FH sources, as in our experiment, the output beam distributions should be weighted by the instantaneous conversion efficiency and integrated over the temporal pulse profile of the FH pulse, in order to obtain the pulse-averaged values. The integrated values of the conversion efficiency and the brightness enhancement (assuming ${\rm M}_{{\rm FH}}^2 = 13$), as defined by Eq. (3), are shown in Fig. 5(a) and 5(b), respectively, for different positions of the nonlinear crystal with respect to the focal plane. Although the conversion efficiency, in the region of 30% can be achieved for small phase mismatches, $\Delta k\; \approx \;0\;{{\rm m}^{- 1}}$, corresponding to predominantly collinear SHG, the largest brightness enhancement for all crystal positions is obtained for the phase-mismatched SHG, where noncollinear interactions and cascaded $\chi^{(2)}:\chi^{(2)}$ processes play a dominant role. Figure 5(c) shows normalized near- and far-field radial intensity distributions and the corresponding beam profiles, at a phase mismatch of $\Delta k = - {675}\;{{\rm m}^{- 1}}$, in agreement with the cleanest on-axis SHG that was achieved experimentally. Simulations using tighter focusing, by employing an ${ f} = {75}\;{\rm mm}$ lens, reveal that, although the SHG efficiency could be enhanced, the brightness enhancement is reduced. In general, for tighter focusing the SH intensity distribution is substantially more structured and has higher ${{\rm M}^2}$ values.

 

Fig. 6. Schematic illustration of the experimental setup, where SLM, SPF, ${ \lambda}/4$, ${\lambda}/2$, and PPKTP, respectively, denote spatial light modulator, short-pass filter, quarter-wave plate, half-wave plate and periodically poled KTP.

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4. EXPERIMENTAL RESULTS

To experimentally confirm our numerical results, we built a setup based on a ${Q}$-switched Nd:YAG laser that was amplified in a polarization-maintaining Yb-doped double-clad fiber amplifier, which delivered 650 ps pulses at a repetition rate of 19.5 kHz. The output was then spatially reshaped from a Gaussian beam to an ${{\rm LP}_{0,7}}$-mode with a spatial light modulator, and pulses with peak powers of 45 kW were then coupled into a 1.6 m long step-index multimode fiber, which had a core diameter of ${47}{.5\,\,\unicode{x00B5}{\rm m}}$. A tunable Ti:Sapphire laser operating at 971 nm, delivering 70 mW, was used to seed the FWM and was consequently also converted to an ${{\rm LP}_{0,7}}$ mode before launching into the multimode fiber. More details about this part of the system can be found in [12].

The output from the multimode fiber was then collimated and a flip mirror was placed in the beam path to enable spatial, spectral and temporal characterization of the FWM process. The anti-Stokes part of the output was isolated with a short pass filter that had a cutoff wavelength of 1000 nm. A quarter-wave plate and a half-wave plate were then used to optimize the polarization for type-0 phase matching before focusing the short-pass filtered light into the crystal. The crystal was a 10 mm long PPKTP sample with a structure period of ${6}{\rm .66\,\,\unicode{x00B5}{\rm m}}$, which was placed in a copper holder on top of a thermoelectric element for temperature control to set the phase matching conditions. The QPM period was designed to achieve phase matching at a temperature of 48°C for plane-wave SHG of 971 nm radiation. A second short pass filter, with a cutoff wavelength of 900 nm, was then placed after the crystal to isolate the frequency-doubled light before temporal, spectral and spatial characterization. A schematic depiction of the setup is given in Fig. 6.

 

Fig. 7. (a) Temporal profile of a 971 nm pulse with a spatial ${{\rm LP}_{07}}$-mode shown in the inset. (b) FWM spectra showing a 43 dB contrast of the 971 nm signal to the spontaneous FWM.

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The anti-Stokes light generated from the seeded FWM gave rise to 329 ps pulses with an energy of 1.81 µJ and peak powers of 5.6 kW at 971 nm, as shown in Fig. 7(a). These pulses had bandwidths of 0.1 nm and exceeded the spontaneous FWM by 43 dB, as seen in Fig. 7(b), and exhibited an ${{\rm LP}_{07}}$-mode profile as seen in the inset to Fig. 7(a).

To probe the SHG parameter space experimentally, lenses with focal lengths varying from 25 to 400 mm were used for focusing the anti-Stokes pulses into the PPKTP sample, while varying its position relative to the focal plane as well as its temperature.

It was found that the highest conversion efficiencies were obtained under conditions with predominantly collinear phase matching, with the crystal placed close to the focal plane, which resulted in a SH beam with extensive ring structure. The highest obtained conversion efficiency was 48% and was achieved with a lens with a focal length of 75 mm and the crystal placed with its center at the focal plane with the crystal kept at a temperature of about 49°C. Yet, with this focusing configuration it was impossible to suppress the ring structure in the SH beam as expected from the numerical simulations.

The cleanest on-axis conversion, i.e., the highest degree of outer ring suppression, was obtained with a lens with a focal length of 150 mm and the crystal placed about 20 mm after the focal plane at a temperature of about 46°C. With this focusing arrangement, a maximum SH energy of 443 nJ was generated with an energy conversion efficiency of 24.5%. The measured SH FWHM pulse width was 260 ps, substantially shorter than the fundamental, and had a peak power of about 0.6 kW as seen in Fig. 8(a). The SHG efficiency’s temperature dependence at this crystal position is shown in Fig. 8(b). CCD images of the SH beam profiles associated with the numbered phase matching points in Fig. 8(b) are shown in Fig. 8(c). As expected from the simulations, the SH beam profile was rather sensitive to the crystal temperature. A temperature decrease by 4°C from the point of maximum SHG efficiency corresponds to the negative phase detuning of $\Delta k = - {675}\;{{\rm m}^{- 1}}$. In accordance with the numerical simulations, there is a clear asymmetry in how the SH beam structure develops with phase detuning. Namely, by tuning to the positive $\Delta k$ side, the SH beam structure acquires pronounced ring structure, while that structure gradually disappears as the temperature is tuned towards $\Delta k = - {675}\;{{\rm m}^{- 1}}$ leaving predominantly axial SHG. In terms of peak powers, the SHG efficiency at the detuned point 2 reached about 11% of the FH’s peak power. Comparing with the simulation results we can estimate a brightness enhancement of about 8. The quantitative discrepancies in terms of SHG efficiency and exact crystal position with respect to the focal plane are related to the experimental uncertainties in exact propagation distances, assumption of ideal collimation of the output beam from the multimode fiber as well as possible aberrations from the lenses. Nevertheless, the numerical model captures well this rather involved physical picture.

 

Fig. 8. (a) Temporal pulse profile of the second harmonic with clean on-axis conversion. (b) SH energy and conversion efficiency as a function of temperature with the crystal placed about 20 mm after the focal plane of the ${f} = {150}\;{\rm mm}$ lens. (c) SH beam profiles corresponding to the numbered points in (b).

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5. CONCLUSION

In this work we have shown that brightness enhancement by direct SHG of a near-infrared ${{\rm LP}_{0,7}}$ mode projected onto a superposition of LG free space modes is possible in 1D PPKTP structures. The physical mechanism relies on phase-mismatched SHG where noncollinear interactions become dominant and cascaded $\chi^{(2)}:\chi^{(2)}$ processes act to emulate a self-focusing Kerr nonlinearity, which results in acquired radial phase shifts. The SHG with ${{\rm LG}_{p,0}}$ modes is radically different from that employing pseudo-nondiffracting Bessel-Gauss beams although the near-field FH intensity distributions of such beams might appear similar. We focused on investigating SHG of a free space coupled ${{\rm LP}_{0,7}}$ mode which was the one generated in the experimental setup. We experimentally achieved an SH brightness enhancement of 8. The physical mechanism, however, should be applicable more generally to the class of ${{\rm LP}_{0,p}}$ modes generated by FWM in HOM interactions in optical fibers. The experimentally achieved brightness enhancement and the strategy for achieving it would be dependent on the mode order mostly due to the increase in ${{\rm M}^2}$ of the FH beam as the mode number increases. The radial phase compensation by cascaded $\chi^{(2)}:\chi^{(2)}$ processes mostly affects the axial part and the first two rings in the FH distribution which are the areas of highest intensity for all ${{\rm LG}_{p,0}}$ modes. The remaining parts of the FH beam are lost. However, this is not a strong limitation, considering that about 95% of the power in ${{\rm LG}_{6,0}}$ mode is contained within the axial part and three closest rings. In a sense, this method of lossy mode conversion is conceptually similar to the one using an axicon lens for direct FH mode transformation from ring structured to Gaussian like beams [28,29]. It should be mentioned that a brightness enhancement of speckled FH beams, and the resulting SH beams, has been observed through self-focused spatial beam cleanup at high intensities in unpoled KTP for phase mismatch values comparable to the ones studied here [30]. These results occurred at FH intensity levels of ${0.9}\;{{\rm GW/cm}^2}$, whereas our peak intensities did not exceed ${0.03}\;{{\rm GW/cm}^2}$. As such, we naturally operated below the intensity levels necessary to excite spatial solitons and did therefore not observe any spectral broadening as reported in [31]. Lastly, the beam cleanup results at high intensities in [30] are less sensitive to the exact phase mismatch than our results. Therefore, we conclude that polychromatic spatial soliton formation did not contribute in any essential way to our observations.

The simulations and experiments have shown that the far-field distribution, specifically, the power contained in the axial portion of the SH beam is quite sensitive to the phase mismatch. This fact can be used for modulation of the SH axial beam, e.g., by applying an external electric field on the PPKTP crystal and exploiting the relatively large electrooptic response in this material. For instance, changing the phase detuning from $\Delta k = - {675}\;{{\rm m}^{- 1}}$ to $\Delta k = - {775}\;{{\rm m}^{- 1}}$, i.e., $\delta (\Delta k) = - {100}\;{{\rm m}^{- 1}}$ would transform the axial maximum in the SH far field distribution to the axial minimum. Using the expression $\delta (\Delta k) = 2\pi {r_{33}}E({n_{{\rm SH}}^3 - n_{{\rm FH}}^3})/{\lambda _{{\rm FH}}}$ and the value for KTP’s electrooptic coefficient ${{r}_{33}} = {36.3}\;{\rm pm/V}$ we estimate that an electric field of $E = {556}\;{\rm V/mm}$ would be sufficient for achieving this phase detuning. This field is about 4-times lower than what is typically used for electric-field poling of this ferroelectric, therefore it will not detrimentally affect the nonlinear properties of the QPM structure.