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Abstract
Low-coherence tunable visible light sources have a wide range of applications in imaging, spectroscopy, medicine, and so on. Second harmonic generation (SHG) based on a superfluorescent fiber source (SFS) can produce high-brightness visible light while retaining most of the characteristics of superfluorescent sources, such as low coherence, low intensity noise and flexible tunability. However, due to the limitations in phase matching conditions, SHG based on SFS is difficult to reach an equilibrium between high efficiency and robustness of phase matching to temperature variation. In this paper, based on a spectral tunable SFS, we provide a comprehensive analysis, both experimental and theoretical, of the impact of wavelength, linewidth, and temperature on the output performance of SHG. Our findings indicate that broader linewidths adversely affect conversion efficiency, yet they enhance the capacity to withstand temperature variations and central wavelength detuning, which is an advantage that traditional SHG methods do not possess. This work may pave the way for utilizing low-coherence visible light in domains and extreme environments where robust output stability becomes imperative.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
High-brightness visible light sources are widely used in underwater communication, optical coherence tomography and many other fields [1,2]. The second harmonic generation (SHG) of near-infrared lasers is one of the most frequently used methods to obtain a visible light source [3–8]. However, due to self-mode-locking, sustained self-pulsing and speckle noise caused by high coherence [9–12], SHG based on conventional lasers is limited to some extent in nonlinear imaging, uniform lighting and other aspects [13,14]. In contrast to lasers with traditional resonator structures, the superfluorescent fiber source (SFS) based on amplified spontaneous emission (ASE) has the advantages of wide spectral coverage, low temporal coherence, no longitudinal mode and excellent temporal stability [15,16]. These distinctive characteristics make SFS promising for a wide range of applications, such as fiber optic gyroscope [17], ghost imaging [18,19], pumping Raman fiber laser and mid-infrared optical parametric oscillator [20–26], ultra-flat supercontinuum generation [27], spectral beam combining [28] and so on [29]. Notably, the exceptional time-frequency characteristics of SFS significantly augment the output performance of nonlinear frequency conversion. For example, in 2015, A. J. Torregrosa et al. [30] primarily presented an up-conversion imaging system in the eye-safe spectral region around 1550 nm, which utilized the sum-frequency mixing of an ASE fiber source and a laser in a PPLN crystal. The work proved that an ASE source can reduce potential speckles in upconverted images. Theoretically, the conversion efficiency of SHG from a narrowband ASE is twice that of a narrow-linewidth single-frequency laser [31]. What’s more, expanding the spectral width of incoherent light can reduce the sensitivity to phase-mismatching, as a result, the tolerance to the detune of temperature and central wavelength is greatly relaxed in optical second nonlinearity [32].
In recent decades, a few works have been dedicated to SFS-based SHG, which show two notable features. Firstly, the effects of wavelength, linewidth and temperature were investigated independently. Secondly, compared with SHG based on narrow-linewidth fiber lasers which can achieve several tens percent of efficiency [33,34], the efficiency and output power of SFS-based SHG are much lower due to the lower incident power or broader linewidth. For example, in 2022, Eunji Park et al. [35] used an ytterbium-doped ASE to pump PPSLT crystals and obtained a second harmonic (SH) power of 7.1 mW and a conversion efficiency of 1.3%. Some researchers attempted to relax the phase-mismatching condition by optimizing the frequency doubling crystals. For example, X. Feng et al. [36] presented a novel fabrication method of a PPLN thin film ridge waveguide with a trapezoidal substrate, and obtained a SH power of 2.55 µW and a bandwidth of 11 nm based on a C-band ASE source. Overall, most SHG-related works in recent years have focused on using a laser with high coherence as the pump source and obtained high power and high conversion efficiency. By contrast, there are few reports focus on SHG based on low-coherence sources such as ASE [31], tungsten halogen lamps [32], superluminescent diodes [37,38] and so on [39].
In this paper, the quasi-phase matching theory in SFS-based SHG was first investigated, and the relationship between the linewidth, central wavelength, power of fundamental wave (FW) and the output performance of SHG, as well as quasi-phase matching temperature were analyzed in detail. Then, we utilized a wavelength- and linewidth-tunable SFS to pump a PPLN crystal, and experimentally investigated the output performance of SHG dependence on the FW properties. The experimental results agree well with the theoretical predictions. We believe this work can not only provide a reference for SHG based on low-coherence light sources, but also pave the way for utilizing low-coherence visible light in domains and extreme environments where robust output stability becomes imperative.
2. Theoretical analysis
Under the conditions of a Gaussian beam, small signal approximation and monochromatic light, the relationship between SH power and incident power can be expressed as [40]:
According to Eq. (1), the relationship between conversion efficiency and incident power can be obtained as:
There exists a linear relationship between conversion efficiency and incident power. The Sellmeier equation for e light in PPLN crystal is [41–43]
The change in the inverted lattice vector G(T0+δT) is caused by temperature variations, resulting in expansion or contraction of the crystal, which further affects the poled period. Due to the smaller thermal expansion coefficient of the PPLN crystal, the impact on Δk can be neglected, therefore,
When ΔkL/2 = 1.39156, sinc2(ΔkL/2) = 0.5, thus,
The temperature 3-dB width is given by
Similarly, under the condition of quasi-phase matching, when the efficiency drops to half, the linewidth is
The above calculation does not consider the influence of the linewidth of the FW on the result. Now, it is assumed that the linewidth of the FW is Δω, and the power spectral density function of the SH wave can be derived as [13]:
where |H(Ω)|2 = L2sinc2(ΔkL/2) is the filter function of the crystal, as shown in Fig. 1(a). The linewidth of the filter function Δλ can be calculated using Eq. (8). S(ω) represents the spectrum of FW. For the convenience of calculation, the spectrum of FW is assumed to be rectangular-shaped, with a linewidth of Δω, height of P, center angular frequency of ω0, and a triangular auto-correlation at 2ω0, as depicted in Fig. 1(b).Under the rectangular spectral approximation, the power spectral density of the SH wave can be given by:
Consider setting the linewidths of FW as 0.5Δλ, Δλ, 2Δλ, 4Δλ, 8Δλ, and 16Δλ, calculating the power spectrum function of the SH using Eq. (10), as presented in Fig. 2, the blue and the yellow line respectively represents the SH spectrum before and after filtering. When the ratio between the linewidth of the FW and that of the filter function is smaller, the difference between the SH and FW spectrum is also smaller. Conversely, as the ratio increases, the SH spectrum becomes closer to the filter spectrum.
When the conversion efficiency is high enough, the energy of FW transfers more to the SH, which means the consumption of FW cannot be ignored. As a result, the small signal approximation condition is no longer applicable, and the three-wave coupling equation, which describes the frequency conversion, must be used for calculations. However, this equation involves a rigorous analytical solution that includes elliptic integrals, making the solution and calculation process more complex. Therefore, further discussion regarding this equation is worth further exploration in future work.
3. Experimental results
3.1 Experimental setup
The experimental setup is illustrated in Fig. 3. The seed is a self-built low-power SFS with a backward pumping method [44]. The pump source is a 976 nm laser diode, whose pigtail is connected to the pump arm of the (2 + 1) × 1 combiner. A double-clad ytterbium-doped fiber (YDF) with a core/inner cladding diameter of 10/130 µm and absorption coefficient of 4.1 dB/m (@975 nm) is used as the gain medium and is connected to a signal arm of the combiner. To ensure complete absorption of the pump energy, the length of YDF is 15 m, and an 8° angle is cut in the rear end to reduce the feedback, to improve the threshold of self-stimulation oscillation of the SFS. At the output end of the combiner, a circulator is connected to prevent the back-propagated light. To obtain a narrow-band SFS, a bandwidth- and wavelength-tunable filter is connected to port 2 of the circulator. The output narrow-band ASE is then amplified in a fiber amplifier up to 8.34 W before entering a polarization beam splitter (PBS), where it is split into two beams of orthogonal linear polarization. Then one of the resulting beams serves as the FW and enters the frequency doubling module for SHG. The pigtail fiber of the frequency doubling module is PM980 fiber, the frequency doubling crystal is a single-pass PPLN crystal using type 0 quasi-phase matching (e + e = e) with a length of 25 mm, a nonlinear coefficient of 11.8 pm/V, a focusing radius ${\omega _0}$=50 µm, and a poled period of about 6.92 µm. The temperature control module is matched with the frequency doubling module, offering a temperature control range of 15°C-70°C.
3.2 Effect of central wavelength
We adjusted the central wavelength of the FW to 1061.5 nm, 1062.0 nm, 1063.0 nm, 1064.0 nm, 1064.6 nm, 1065 nm, 1065.5 nm and 1066 nm while maintaining a constant FW linewidth of 0.6 nm. The FW and SH spectra are shown in Fig. 4(a) and Fig. 4(b), respectively. Consequently, the corresponding SH central wavelengths are 530.7 nm, 530.9 nm, 531.3 nm, 531.8 nm, 532.2 nm, 532.3 nm, 532.6 nm and 532.8 nm, respectively. The discrepancy between the SH wavelength and half of the FW wavelength is approximately 0.2 nm, while the linewidth of the SH remains constant at 0.08 nm.
In the experiment, it was observed that the SH power has a quadratic relationship with the incident power, as shown in Fig. 5(a). The incident power refers to the power measured at port 2 in advance and is used to calculate the conversion efficiency. Additionally, Fig. 5(b) indicates that the central wavelength of FW, which exhibits the highest phase matching extent, is 1064.0 nm. However, as the FW wavelength deviates further, the phase matching becomes worse, leading to a reduction in conversion efficiency.
Fig. 5. (a) SH power at different central wavelengths. (b) SHG efficiency at different incident power.
3.3 Impact of spectral linewidth
The output power of different linewidths at the quasi-phase matching temperature was experimentally measured when the central wavelength was fixed at 1064.6 nm. Figure 6(a) shows the performance of SHG in different linewidth and incident power. When the linewidth is 0.45 nm and the pump power is 3.5 W, the SH power can no longer properly hold the square relationship to the incident power. This occurs due to the sufficiently high conversion efficiency, leading to a substantial deviation between the theoretical and actual values when using the small signal approximation. Subsequently, the conversion efficiency was calculated, as shown in Fig. 6(b). In the condition of a broadband pump, the conversion efficiency is significantly low, but we noticed that the effect of a slight temperature detunes on the SH power is considerably less. To prevent damage to the device, measurements of power and efficiency were not taken at higher pump power. The highest conversion efficiency achieved is 8.81% with an SH power of 320 mW, for a linewidth of the FW of 0.45 nm and a pump power of 3.63 W. Equations (1) and (2) are used to calculate the SH power and conversion efficiency, in which the phase mismatch factor is set to 0. The linewidth of the filter function is calculated to be about 0.08 nm using Eq. (8). Notably when the linewidth is 0.45 nm, the data is closest to the computation. As the linewidth decreases under a fixed pump power, the conversion efficiency gradually approaches a fixed value instead of increasing infinitely. Narrowing the linewidth, increasing the pump power and setting the multi-pass configuration are effective ways to improve the conversion efficiency.
Fig. 6. Relationship between (a) incident power and SH power (b) linewidth and conversion efficiency.
The graphs of the relationship between the measured conversion efficiency and incident power showed approximately linearity as demonstrated in Eq. (2). We linearly fitted the graphs and calculated the slope of each straight line. Table 1 shows the approximately equal product of the slope and the linewidth. The reason for this phenomenon can be explained as follows: in the experiment, the minimum linewidth of the FW is 0.45 nm, which corresponds to about 6Δλ. Therefore, the linewidth of SH is close to that of the filter function, namely 0.08 nm. Under a fixed pump power, each time the linewidth is doubled, the power spectral density can be roughly halved. Consequently, the power and conversion efficiency of the SH is also halved, assuming the linewidth of the SH remains fixed at 0.08 nm.
Table 1. Product of linewidth and frequency doubling efficiency slope
In the case of the broadband pump method, the linewidth and conversion efficiency are approximately inversely proportional. Furthermore, the product of the conversion efficiency slope and the linewidth remains constant for different pump powers.
Figure 7(a) and Fig. 7(b) show the spectra of FW and SH at different linewidths. The linewidth of SH is 0.16 nm, which is obviously wider under a broadband pump compared to 0.08 nm output under a 0.45 nm pump mentioned above, this means a broadband pump method can expand the SH linewidth to a certain extent. The central wavelength of SH maintains 532 nm. Notably, as the linewidth of the FW increases, the wings of the SH spectrum widen.
3.4 Impact of temperature
The experiment determined the central wavelength of the FW to be 1064.4 nm, the 3-dB linewidth to be 0.54 nm, and the pump power to be 4.17 W. By adjusting the crystal temperature, the maximum SH power was measured to be 306.9 mW at 50.1°C, corresponding to a conversion efficiency of 7.36%.
As depicted in Fig. 8(a) and Fig. 8(b), the SH spectrum generally exhibits two peaks. With the temperature gradually rising from a low level, one of them shifts red while another one remains unchanged. Once the temperature aligns with the quasi-phase matching temperature, the two peaks coincide, resulting in the highest SH power output.
The reasons for this phenomenon are as follows: the deviation of temperature leads to changes in the refractive index and poled period of the PPLN crystal, which subsequently affects the central wavelength of the filtering function. Therefore, the central wavelength of the filtered SH wave also changes accordingly, moving towards a long-wavelength direction with the increase in temperature. Since the central wavelength of the FW is 1064.4 nm, the SHG is fixed at a wavelength of 532.2 nm. Figure 9 shows the central wavelength and root-mean-square linewidth of SH at different temperatures, the more detuning from the matched temperature, the wider the linewidth.
To extend the temperature tuning range while maintaining power stability, we further explored the output performance under temperature variation. We respectively changed the incident linewidth to 1 nm, 2 nm, 4 nm, and 8 nm with the fixed pump power of 3.04 W and measured the SH power. The spectra are depicted in Fig. 10. The graphs indicate that the broadband pump method extends the spectral coverage. With the widening of the incident linewidth, the pedestal of the spectrum becomes noticeably broader. What’s more, the increment in temperature redshifts the peaks. Therefore, tuning the temperature under a broadband pump method is a measure to realize spectral manipulation.
Fig. 10. Wavelength-tunable SHG spectra with different FW linewidths of (a) 1 nm (b) 2 nm (c) 4 nm (d) 8 nm.
Figure 11(a) displays the power decline at different incident linewidth. It shows that with the pump linewidth of 0.54 nm, the SH power decreases rapidly with the temperature deviation, the amplitude of decline reaches almost 30 dB at 20°C instead of about 5 dB under an 8 nm pump. Specifically, under the broadband pump condition, the SH power exhibits excellent temperature stability, the temperature 3-dB width is nearly 23°C when the incident linewidth is 4 nm, rather than 1.2°C calculated by Eq. (4) under the monochromatic pump condition. We cannot obtain this property under an 8 nm pump, due to the higher SH power when the temperature reaches the limit of regulation. Even so, we can ensure that the temperature 3-dB width is at least 42°C. Therefore, a broadband pump method is an efficient path to enhance the temperature stability of output power, although it will reduce the conversion efficiency.
Fig. 11. (a) Power decline at different pump linewidth. (b) Quasi-phase matching temperature corresponding to different central wavelengths.
When setting the phase mismatch factor Δk = 0, a linear relationship can approximately exist between the quasi-phase matching temperature and the central wavelength within a certain range. We experimentally demonstrated this, as shown in Fig. 11(b), and found that the quasi-phase matching temperature changes by approximately 11.8°C for every 1 nm shift in wavelength, this means substantially extending the temperature tuning range is an effective method to enormously improve the span of wavelength [45]. In addition, noticed that the measured curve is flatten than the theoretical one, there exists an important indication: on the one hand, identic central wavelength detune needs less temperature variation compensation under the broadband pump condition, on the other hand, identic temperature tuning range results in a broader central wavelength tuning range of SHG. Due to the limitations of the temperature control module, we were unable to measure data at higher or lower temperatures. Therefore, a conclusion can be further summarized: the broadband pump method can significantly enhance the capacity to withstand phase-mismatching, including temperature variations and central wavelength detuning, which is also an advantage of the SFS-based SHG method.
4. Conclusion
In summary, we thoroughly investigated the effect of spectral linewidth, central wavelength, and quasi-matched temperature on the SHG of SFS, both in experiment and theory. To achieve this, we utilized a PPLN crystal in a single-pass configuration and a spectrum-manipulable SFS, and successfully generated low-coherence visible green light with a high conversion efficiency of 8.8% at a low pump power of 3.63 W. Additionally, we achieved a wavelength tuning range of 2.1 nm. Our findings reveal that narrower linewidths result in higher conversion efficiency that tends to stabilize at a fixed value. Moreover, we observed that the ability to tolerate temperature and central wavelength detuning significantly improves as the linewidth increases. This suggests the potential application of low-coherence visible light in domains and extreme environments where strong output stability is crucial.
Funding
National Postdoctoral Program for Innovative Talents (BX20190063); National Natural Science Foundation of China (61635005, 61905284, 62305391).
Disclosures
The authors declare that there are no potential conflicts of financial interest or personal relationships related to this work.
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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