Investigation about the influence of longitudinal-mode structure of the laser on the relative intensity noise properties

The influence of the longitudinal-mode structure (LMS) of the laser on the relative intensity noise (RIN) properties was investigated in this paper after an all-solid-state continuouswave (CW) single-frequency 1064 nm laser with output power of 50.3 W was achieved. The LMS of the laser was manipulated by controlling the temperature of the nonlinear lithium triborate (LBO) crystal deliberately introduced to the resonator. When the laser worked with single-longitudinal-mode (SLM) operation, the stable RIN spectrum was observed and measured. Moreover, with the decrease of the nonlinear conversion efficiency (NCE), the frequency and amplitude of the resonant-relaxation oscillation (RRO) peak regularly shifted to the higher frequencies and increased, respectively. However, with further decrease of the NCE, the laser began to work with the multi-longitudinal-mode (MLM) or mode-hopping operation and the unstable RIN spectra of the laser were both observed not only at low frequencies but also at high frequencies. Once the NCE was moved away, the MLM or mode-hopping can only enhance the fluctuation of the laser RIN spectrum at the low frequencies (lower than the frequency of RRO). The experimental results directly revealed the relationship between the LMS of the laser and the RIN spectra, and confirmed definitely that the key to achieve a stable high power laser with low intensity noise was to realize single-frequency operation of the laser with free MLM and mode-hopping. © 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

transmission at 1064 nm and HT at 532 nm (T 532nm > 95%). The gain medium was an α-cut composite YVO 4 /Nd:YVO 4 rod (3 mm× 3 mm × (3+20) mm), including an un-doped end cap of 3 mm and Nd-doped part of 20 mm with concentration of 0.8% and a wedge angle of 1.5 degree at the second end face to keep stable polarization [42]. The front end-face of the Nd:YVO 4 crystal was coated with anti-reflection (AR) films at 1064 and 888 nm (R 1064nm;888nm < 0.25%), and the second end-face was coated with AR films at 1064 nm (R 1064nm < 0.25%). An optical diode comprised of an 8 mm long terbium gallium garnet crystal and a half-wave plate (HWP 1 ) was applied to eliminate the spatial hole burning effect and realize the unidirectional operation of the laser [43,44]. A type-I non-critically phase-matched lithium triborate (LBO) crystal [45] (S 1 , S 2 : AR 1064nm;532nm ) with a dimensions of 3 mm× 3 mm× 18 mm was deliberately inserted into the resonator to introduce the intra-cavity nonlinear loss [46,47]. The LBO crystal was placed at the beam waist between M 3 and M 4 , and the temperature of which was controlled by a homemade temperature-controller with the precision of 0.01 • C. The output second-harmonic-wave (SHW) was separated from the fundamental-wave (FW) by a dichroic mirror and monitored by a power meter (PM 1 ) (LabMax-Top, Coherent). The main part of the FW laser leaked from S 1 was detected by another power meter (PM 2 ) (LabMax-Top, Coherent). A small part of the FW laser reflected by S 1 and S 2 was injected into a scanned confocal Fabry-Perot (F-P) cavity to monitor the LMS of the laser. The free spectrum range and finesse of the F-P cavity were 375 MHz and 200, respectively. The transmitted signal from F-P cavity was detected by a photodetector and the transformed signal was displayed on a digitizing oscilloscope (OSC 1 : MSO 5204B, Tektronix). The leaked part from S 2 was equally divided into two parts by a 50/50 beam splitter composed of HWP 2 and polarization beam splitter and reflected by S 3 and S 4 and focused by f 4 and f 5 into a homemade balanced homodyne detector (BHD) to measure the RIN spectrum of the laser. The employed BHD was composed of two InGaAs photodiodes (ETX500, JDSU Corporation) in a single electronic board. The achieved common mode rejection ratio of the BHD was 60 dB and high enough to accurately calibrate the QNL [48]. The DC (direct current) output of the BHD was connected with the digitizing oscilloscope OSC 2 (TBS 1022, Tektronix) to monitor whether the light is totally received by each photodiode, the AC (alternating current) output was connected with a spectrum analyzer (N9320A, Agilent) to read the intensity noise spectrum of the laser, respectively.

Experimental results and analyses
In the experiment, based on the accurate measurement method of the intra-cavity linear loss of lasers by means of a nonlinear loss deliberately introduced to the laser resonator [49], the transmission of the output coupler was optimized and output coupler M 4 employed in the experiment was coated with the transmission of T 1064nm =25%. When the LBO crystal was   working at the optimal phase-matching temperature of 149.0 • C, a stable single-frequency CW 1064 nm laser with the output power of 50.3 W was achieved under the pump power of 113 W. To the best of our knowledge, it was the highest output power single-frequency all-solid-state CW 1064 nm laser in a single resonator without any amplifiers. The output powers as a function of incident pump power were recorded in Fig. 2. It was depicted that a single-frequency 532 nm laser beam with the maximal output power of 1.91 W was simultaneously generated because of the existence of the LBO crystal. The corresponding optical-optical conversion efficiency was 46.20%. The beam quality M 2 was measured by a M 2 meter (M2SET-VIS, Thorlabs) and the measured values of M 2 x and M 2 y are 1.08 and 1.10, respectively. The measured degree of polarization of 1064 nm laser at the maximum output power was more than 110:1. The long-term power stability of 1064 nm laser was measured (Fig. 2 inset) and the peak-to-peak power fluctuations was less than ±0.50% during 5 h. The linewidth of the single-frequency 1064 nm laser was also measured by means of a delayed self-heterodyne interferometer [50]. The path length imbalance of 25 km between the long and short branches of the employed interferometer was long enough to measure the linewidth of the 1064 nm laser. A small part of 1064 nm laser was firstly divided into two equal parts by a polarization maintaining coupler and separately injected into two branches of the interferometer. Then the transmitted 1064 nm laser from two branches were combined in another polarization-maintaining coupler and the beat signal was directly detected by a fast detector (S/N29637, Thorlabs). Lastly, the detected signals was recorded and analyzed by another spectrum analyzer. In the process of linewidth measurement, to avoid the 1/f noise of detection circuits, a polarization maintaining fiber-coupled acousto-optic modulator (T-M150-0.4C2G-3-F2P, Gooch & Housego) was utilized in the short branch to shift the beatings by 150 MHz from the baseband. The measured self-heterodyne linewidth data (shown as line (a) in Fig. 3) was fitted by the Gaussian lineshape function, and the fitted result was shown in curve (b) in Fig. 3. It was clear that the measured spectral linewidth of 1064 nm laser was 206 kHz. Fig. 4. The LMS of the laser with different operation conditions. (A), the NCE η as a function of the temperature of the LBO crystal, (a 1 )∼(a 9 ) recorded the NCE η with respect to the temperature of the LBO crystal at 149.0 • C∼152.5 • C, respectively. (B), the LMS of the laser with respect to different NCE η. Curve (b 1 ), the LMS of the single-frequency laser with NCE η ≥ 0.28%; curve (b 2 ), the initial LMS of the laser with NCE η = 0.21%, which would gradually evolve into the curve (b 3 ); when the NCE η of the laser was 0.16%, the LMS of the laser rapidly alternated between the curve (b 3 ) and curve (b 4 ); curve (b 4 ), the LMS of the laser with NCE η = 0.028%; when the NCE η of the laser was zero, the LMS of the laser randomly switched between curves of (b 1 ), (b 2 ), and (b 3 ).
The LMS of the laser was manipulated by changing the phase-matching temperature of the nonlinear LBO crystal which directly decided the value of the intra-cavity nonlinear loss. The nonlinear loss could be simply marked by the nonlinear conversion efficiency (NCE) η which was defined by the ratio between the power of 532 nm laser and that of the intra-cavity 1064 nm laser. The values of η as a function of the temperature of the LBO crystal and the corresponding LMS were shown in Fig. 4(A) and Fig. 4(B), respectively. When the NCE η ≥ 0.28%, the laser can work with single-longitudinal mode (SLM) operation as shown in (b 1 ) of Fig. 4(B), where the nonlinear loss was high enough to suppress the oscillation of non-lasing mode. However, once the temperature of the LBO crystal was detuned more than 2 • C (η < 0.28%), the laser started to work in multi-longitudinal-mode (MLM) region. At this moment, the nonlinear loss supplied by the nonlinear conversion process was too low to suppress the oscillation of the non-lasing mode. The curve (b 2 ) of Fig. 4(B) showed the initial LMS of the laser when the NCE η = 0.21%, where another two side longitudinal-modes with low intensity appeared and oscillated together with the dominant mode. Through the long-term observation, however, the LMS of the laser as shown in curve (b 2 ) of Fig. 4(B) gradually evolved into the curve (b 3 ) of Fig. 4(B). With decreasing NCE η to 0.16%, the intensity of the side modes rapidly increased and mode-hopping phenomenon occurred, and the LMS of the laser rapidly alternated between the curve (b 3 ) and curve (b 4 ) of Fig. 4(B). When the NCE was further decreased to η = 0.028%, the LMS of the laser was always like curve (b 4 ) of Fig. 4(B), where it was difficult to distinguish the dominant and side modes because of the strong intensity fluctuation of all the modes. When the NCE η was decreased to zero, the LMS of the laser randomly switched between that of (b 1 ), (b 2 ), and (b 3 ) shown in Fig. 4(B) in a long operating time. We summarized the relationship between the LMS of the laser and the intra-cavity NCE η. When the intra-cavity NCE η was high enough to suppress the oscillation of the non-lasing mode, introducing a nonlinear loss to the laser resonator was a great way to achieve the SLM operation of the laser. However, if the intra-cavity NCE η was too low to suppress the oscillation of the non-lasing mode, the laser run with the MLM operation and intra-cavity nonlinear conversion process could directly induce the strong coupling between different longitudinal-modes. As a result, the frequent and strong mode-hopping appeared.
While the LMS of the laser was monitored, the RIN spectrum of the laser was simultaneously measured corresponding to the different intra-cavity NCE η in the experiment. In the process of the RIN spectrum measurements, the resolution bandwidth and the video bandwidth of the spectrum analyzer were set to be 30 kHz and 30 Hz, respectively. Firstly, to calibrate the QNL, two photodiodes of the BHD were simultaneously illuminated with 1 mW 1064 nm laser, respectively. The obtained spectrum V 1 of the subtracted signal was directly characterized as QNL of the 1064 nm laser under the detection power of 2 mW. Then one of the photodiodes was blocked and the other was illuminated with the power of 1 mW. In this case, spectrum V 2 can be read out by the spectrum analyzer. Lastly, the RIN spectrum of the 1064 nm laser was extracted according V obs =V 2 -(V 1 -3dB) [48]. In addition, the overall measured RIN spectra had been normalized to that with the largest values of NCE η according to the normalized equation [11,12], where V n was the normalized RIN spectrum, k represented the normalizing factor which was obtained with the intra-cavity FW power corresponding to η of (a 1 )∼(a 9 ) divided by that with η of (a 1 ). Figure 5 firstly displayed the evolution of RIN spectra of the laser in the case that the laser always worked with SLM operation where the NCE η was decreased from 1.01% to 0.28%. When the NCE η was 1.01% and 0.93%, the RRO peak of the laser was completely suppressed according to the RIN spectrum of (a) and (b) in Fig. 5. When the NCE η was decreased to 0.76% (curve (c) of the Fig. 5), the RRO peak began to pop out and the measured frequency and amplitude of the RRO peak were 788.20 kHz and 25.75 dB, respectively. When the NCE η was further decreased to 0.51% (curve (d) of the Fig. 5), the RRO peak of the laser was so dramatic that it was easy to be observed and measured. At that time, the measured frequency and amplitude of the RRO peak were 840.36 kHz and 28.04 dB, respectively. When the NCE η was decreased to 0.28% (curve (e) of the Fig. 5) corresponding to the edge of the SLM region of the laser, the frequency and amplitude of RRO peak were 937.86 kHz and 32.21 dB, respectively. It was clear that the frequency and amplitude of the RRO peak shifted to the high frequencies and increased with the decrease of the NCE η. The RIN properties of the single-frequency laser with different NCE η can also be well theoretically predicted according to intensity noise spectrum (V f ) of the FW laser [16] where V vac1 and V vac2 were the vacuum noise caused by the second-harmonic generation and the output coupler, respectively; V p , V spont , V dipole and V losses were corresponding to the noises coming from the pump source, spontaneous-emission, dipole fluctuation and intra-cavity losses, respectively. The coefficients of k 1 (ω f , γ f ), k 2 (ω f , γ f ), k 3 (ω f , γ f ), k 4 (ω f , γ f ), k 5 (ω f , γ f ) and k 6 (ω f , γ f ) in V f were the function of ω f and γ f . The frequency of the RRO ω f and the damping rate of the oscillation γ f were given by and where 2κ m was the photon decay rate induced by the output losses; G was the stimulated-emission rate per photon; Γ was the pump rate; γ t was the atomic spontaneous emission rate from upper level to lower level; 2κ t was the total photon decay rate, which was expressed by 2κ t =2κ m +2κ l +2κ n , where 2κ l was the photon decay rate induced by the intra-cavity losses, 2κ n was the photon decay rate induced by the nonlinear loss, which was expressed by 2κ n =2µα 2 , µ and α 2 were the nonlinear conversion rate and the intra-cavity photon number, respectively. According to the equations (2)-(4), the evolution trends of the amplitude and the frequency of the RRO were obtained with the variation of the NCE η, which was completely consistent with the law presented in our previous work [16]. The measured results once again verified that the nonlinear conversion process can modify the laser dynamics and manipulate the RIN of the laser.
However, the stable evolution law of the RIN spectra of the laser did not continue when the NCE η was decreased to 0.21%, where the nonlinear loss is too low to suppress the oscillation of the non-lasing mode. As a result, the side mode began to oscillate together with the dominant oscillating mode and the laser began to work with MLM operation. Figure.   beginning of the appearance of the side modes (LMS of the laser was like curve (b 2 ) in (B) of Fig. 4), the intensity of the side modes was small and could be ignored, so the measured RIN spectrum of the FW laser was relatively stable as shown in curve (a) of Fig. 6. With the increase of the intensity and number of the side mode (LMS of the laser was like curve (b 3 ) in (B) of Fig.  4), the measured RIN spectrum of the FW laser became instable, which was depicted in curve (b) of Fig. 6. The measured results shown in Fig. 6 illustrated that the RIN spectrum of the laser was directly decided by the LMS of the laser. In order to verify the relationship between the RIN spectrum and the LMS of the laser, the RIN spectra of the laser were also measured when the NCE η were separately 0.16%, 0.028% and 0, and the measured results were shown in Fig. 7. Not only NCE η was equal to 0.16% but also 0.028%, the LMS of the laser was both unstable as shown in curve (b 4 ) in (B) of Fig. 4. In that case, the strong intensity fluctuation of the overall longitudinal-modes and frequent mode-hopping occurred and the measured RIN spectra of the laser were displayed in curve (a) and (b) of Fig. 7. It was clear that the RIN spectrum of the laser strongly fluctuated at low frequencies as well as the high frequencies. However, when the NCE η was decreased to zero, the stable RIN spectrum of the laser was observed once again, which was depicted in curve (c) of Fig. 7. However, the stable RIN spectrum was temporary because the LMS of the laser randomly switched between SLM, MLM and mode-hopping when the NCE η was zero. In consequence, the unstable RIN spectra of the FW laser were observed once more at different instants of time, which were depicted in Fig. 8. The curve (a), (b) and (c) in Fig. 8 were corresponding to the SLM, MLM and mode-hopping, respectively. It was clear that the RIN increase of the laser occurred only at the low frequencies when the NCE η was equal to zero. Compared Fig. 7 to Fig. 8, it was revealed that when the nonlinear loss existed but too low to suppress the oscillation of the non-lasing modes, the different longitudinal modes could strongly couple between each other through the nonlinear process (which was named "green problem" [51,52]) so that the RIN spectrum of the FW laser strongly fluctuated not only at low frequencies but also high frequencies. In stark contrast, the RIN spectrum fluctuation of the FW laser was observed only at low frequencies when the NCE η was moved away. The measured results also revealed that the RIN spectrum of a high power single-frequency laser with free MLM and mode-hopping was as stable as that of a low power single frequency laser which was often used as the seed source for implementing the injection-locking amplification. It was further verified that introducing large enough nonlinear loss to a resonator was a good way to effectively suppress the MLM and mode-hopping as well as achieve SLM operation of a high power laser.

Conclusion
In summary, we firstly constructed a highly stable, low intensity noise all-solid-state singlefrequency 1064 nm laser with output power of 50.3 W and perfect beam quality as well as high degree of polarization. Then the influence of LMS of the laser on the laser RIN property was experimentally investigated. The LMS of the laser was manipulated by controlling the temperature of the nonlinear crystal LBO deliberately introduced to the resonator. When the laser worked with SLM operation, the stable RIN spectrum was observed and measured. Moreover, with the decrease of the NCE η, the frequency and amplitude of the RRO peak regularly shifted to the higher frequencies and increased, respectively. The contribution was ascribed to the enough large NCE η which can effectively suppress the oscillating of the non-lasing modes. However, with further decrease of the NCE η, the laser began to work with the MLM or mode-hopping operation and the unstable RIN spectrum of the laser was observed. Once the MLM or mode-hopping phenomenon occurred in the laser, the strong intensity fluctuations would appear not only at low frequencies but also at high frequencies. The reason was ascribed to the coupling of different longitudinal-modes through the nonlinear process. Once the NCE η was moved away, the MLM or mode-hopping can only enhance the fluctuation of the laser RIN spectrum at the low frequencies (lower than the frequency of RRO). Our work essentially provided an important reference for the influence of LMS on the RIN properties of a high power laser, which revealed that the key to achieve a stable high power laser with low intensity noise was to realize single-frequency operation of the laser with free MLM and mode-hopping.

Funding
Key Project of the Ministry of Science and Technology of China (2017YFB0301403).