1. Introduction

There are increasing demands for photonic microwave sources [1–3] over the past decades due to their excellent performances over their electrical counterparts in low phase noise, ultrawide bandwidth, easy tunability, low propagation loss and immunity to electromagnetic interferences. Moreover, applications such as atomic clocks, radar, communications, and precision measurements in mobile, airborne, and space have even more requirements for their microwave sources, which should be compact, lightweight, ruggedized, power-efficient and low-noise all at once. Several photonic technologies, i.e., dual-frequency or multi-frequency lasers [4,5], modulator-based OEOs [6], soliton microcombs [7], monolithic femtosecond lasers [8] etc., have been developed to meet these requirements. Among these technologies, a dual-frequency laser may offer a simple, efficient and reliable scheme because its two frequencies are not necessarily phase-locked so that it is more immune to vibrations, shocks and temperature variations than other technologies such as microcombs [7] and femtosecond lasers [8].

In early studies, dual-frequency, dual-polarization solid-state lasers with Fabry–Pérot cavities have been applied for microwave or THz generation [9–11], where the signal noise levels were not evaluated at that time. Dual-frequency, dual-polarization DFB [12] lasers and VECSELs [13] were used to generate microwave signals with phase noise as low as -75 dBc/Hz (10 GHz carrier) and -100 dBc/Hz (< 3 GHz carrier) at 10 kHz offset, respectively. In a more recent work, a dual-frequency NPRO laser with LG modes could generate an 8.2 GHz signal with phase noise of -118 dBc/Hz at 10 kHz offset [14]. This noise reduction may benefit from in one hand the monolithic ring-cavity design [15], which intrinsically lowers the laser phase noise, and in other hand the good mode overlapping between the two frequencies, which enhances the common-mode-rejection effect [16]. Even though LG modes or other type of high-order modes have their indispensable applications [17], there are still limitations when considering cases that need the light propagates along a long fiber or a long distance in space. The relatively large mode area and divergence angle over a fundamental Gaussian mode deteriorate the optical power coupling coefficient to the fiber and increase the propagation loss to the remote target area.

Here, in this work, we show that stable DFFM laser could be generated directly in NPRO. The microwave signal generated using the DFFM laser shows lowered phase noise than using a dual-frequency LG laser. The article is organized as follow. We first show the method to stimulate dual-frequency or multi-frequency oscillations in NPRO and the subsequent microwave generation. We then build a theoretical model to analyze the phase noise of the microwave signal that considers the phase-intensity coupling between the dual-frequency and the thermal fluctuations caused by the pump laser. The experimental results validate the model and show reduced phase noise at low-Fourier frequencies of less than 1 kHz and high-Fourier frequencies of more than 1 MHz when compared with microwave signals generated from LG lasers. We finally show the frequency tunability of the microwave signal by strain (piezo) and temperature (thermoelectric cooler, TEC), respectively. Considering NPRO lasers have been successfully applied in industry [18], scientific applications [19] and space missions [20–24], we believe our investigations could facilitate NPRO-based microwave sources used in these areas.

2. Single-, dual-, and multi-frequency lasers produced by NPRO

The schematic of the experimental setup is illustrated in Fig. 1(a). The monolithic Nd:YAG (0.8 at.%) NPRO used in our experiments has a dimension of 3 mm × 8 mm × 12 mm and exhibits an incident angle of ∼30° [25], and an out-of-plane angle of 90°. The perimeter of the non-planar ring is approximately 28.5 mm, which corresponds to a free spectral range (FSR) of 5.7 GHz. The front surface is coated for anti-reflection at 808 nm pump wavelength and high-reflection (99.8% for S polarization and 94% for P polarization) at 1064 nm laser wavelength. The pump source is a free-space multimode laser-diode (LD) with emitter size of $100\,\mathrm{\mu}\textrm{m}\; \times \; 1\,\mathrm{\mu}\textrm{m}$ and continuous-wave output power of up to 1 W. The pump beam from the LD is first beam-shaped by a fast-axis-collimator, and then injected to the NPRO by a single focus-lens. The divergence-angle (0.1°< x < 2.0°, 2.2°< y < 13.8°) and beam-waist (48 µm < x < 350 µm, 34 µm < y < 130 µm) of the pump can be adjusted by selecting different focus-lens and controlling its distance to the LD. A pair of Neodymium magnets are used to supply enough magnetic field for unidirectional emission. Both the LD and the NPRO are temperature-controlled by TEC modules. Typical threshold power of 80∼150 mW and slope efficiency of 30∼50% are measured in our system. The corresponding output power is 75∼125 mW at around 1064.4 nm with fixed pump power of 400 mW in all this work if without specification. A non-polarizing beam splitter divides one part of the output beam to a CMOS camera (Dataray, WinCamD-UHR-1/2) to monitor the beam profile, and the other to a single-mode fiber collimator. A fiber beam-splitter is used to transfer the light to a scanning Fabry-Pérot interferometer (Thorlabs, SA210-8B, FSR 10 GHz, resolution 67 MHz) and to a fast photodetector (Keyang Photonics, KY-PRM-40 G). The spectra and phase noise of the microwave signal (beatnote) are measured by a spectrum analyzer (R&S, FSW13) and a phase-noise analyzer (R&S, FSWP26), respectively.

 

Fig. 1. (a) The schematic of the experimental setup. NPBS, non-polarizing beam splitter; SFPI, scanning Fabry-Pérot interferometer; PD, fast photodetector; CMOS, digital camera. Insert: the fundamental-mode Gaussian beam profile. (b) Laser mode information measured by the SFPI under different pump conditions. Blue curve: single-frequency lasing. Green curve: DFFM lasing with spacing of 5.7 GHz. Red curve: MFFM lasing with spacing of 5.7, 11.4 GHz and 17.1 GHz, respectively. The free spectral range of the SFPI is 10 GHz.

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Figure 1(b) shows the laser mode information measured by the scanning Fabry-Pérot interferometer and displayed on an oscilloscope. The blue curve represents a single-frequency emission, while the green curve and the red curve represent a dual-frequency and a multi-frequency emission, respectively. The transition from single-frequency to dual-frequency or multi-frequency is realized by increasing the divergence-angle and beam-waist radius of the pump beam and adjusting the relative position between the pump beam and the NPRO. Note that during this process the laser wavelength stay at around 1064.4 nm. The CMOS camera works simultaneously to monitor the beam profiles of the respective modes. We consciously avoid stimulating high-order modes (i.e., LG mode or Hermite Gaussian mode) during this process. A spectrum analyzer is utilized to monitor the beatnotes from the DFFM and MFFM lasers in the meantime. We confirmed the beatnotes mainly include 5.7 GHz and 11.4 GHz information (17.1 GHz is beyond the range of our spectrum analyzer), meaning that the laser modes are adjacent longitudinal-modes. Because NPROs are invented and approved to have high-performance single-frequency emission [25], it is counterintuitive to generate DFFM or MFFM lasers. We attribute these phenomena to the excessive divergence-angle and nonuniform energy-density in cross-section of the pump-beam [26], which might introduce inhomogeneous gain along the beam path and across the beam cross-section. This in turn may destroy the free-competition among the adjacent longitudinal-modes, and leaves two or more frequencies coexist. Subsequently, we will use these phenomena to generate high-purity photonic microwave signals.

3. Pure microwave generation from the NPRO laser

Optical frequencies in either DFFM or MFFM share the same ring resonator, it is possible to generate a pure microwave signal that is the beatnote of the optical frequencies through common-mode-rejection. The more similar features the two frequencies share, the more noise can be rejected. To this purpose, we first checked the polarization of the laser output from NPRO. We did not find character of orthogonal polarization between the two frequencies in a DFFM or MFFM but believe they share the same polarization (elliptical polarization in most cases). We then tried to match the optical power of each frequency to have higher beating efficiency. The RF spectra of a 5.7 GHz signal as well as an 11.4 GHz signal corresponding to a DFFM and a MFFM are shown in Fig. 2(a) and Fig. 2(b), respectively. The signal-to-noise ratio (SNR) of the 5.7 GHz signal is around 90 dB, while the SNR of the 11.4 GHz signal is only 75 dB. To investigate this obvious difference, we further measured the relative-intensity-noise (RIN) of the lasers in single-frequency mode, DFFM or MFFM with power meter (1 Hz - 500 Hz) and spectrum analyzer (500 Hz - 1 MHz). The results are shown in Fig. 2(c). There are only slightly differences in the low frequency band between a single-frequency fundamental-mode laser and a DFFM laser. However, a MFFM laser obviously have higher RIN from 1 Hz to 10 kHz means that there remains severe mode-competition that leads power fluctuations. This power fluctuations will add phase noise through phase-intensity coupling (as will be discussed later) and thus deteriorate the SNR of the 11.4 GHz signal. Note the noise peaks at around 200 kHz are relaxation oscillations that are intrinsic in solid state lasers if without active noise cancellations [27]. In the subsequent study, we will focus on the characterization of the 5.7 GHz signal. We will build a theoretical model to examine the noise sources and compare the model with the experimental measurements.

 

Fig. 2. Beatnotes spectrum of center frequency 5.7 GHz (a) and 11.4 GHz (b), measured by the spectrum analyzer with resolution bandwidth (RBW) = 200 Hz, video bandwidth (VBW) = 100 Hz. The light horizontal trace is the measurement noise floor by blocking light on the fast photodetector. (c) The relative-intensity-noise (RIN) of the laser in single-frequency, dual-frequency or multi-frequency modes, respectively. Red curve: Single-frequency fundamental-mode laser. Green curve: DFFM laser. Blue curve: MFFM laser.

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3.1 Theoretical model of phase noise for a DFFM microwave signal

The spectral purity of the DFFM microwave signal depends on the intensity and phase fluctuation correlation between the two frequencies generated from the same resonator [28]. Its phase noise is mainly contributed by the following two aspects: A. the mutual coupling noise of the phase and intensity of the DFFM; B. the noise from the fluctuations of the crystal temperature caused by the fluctuations of the pump power. The thermal noise induced by external factors such as temperature vibration and mechanical noise introduced from the environment would also add phase noise to the signal in the low frequency range, which are not considered here.

3.1.1 Transmission of intensity to phase noise through phase-intensity coupling

A phase noise model suitable for the DFFM laser generated in NPRO is developed. First, to study the coupling noise between the two frequencies, we refer to a theoretical model in [28] and simplify our laser from four-level to two-level laser system. We start with the simplest rate equation (1-4), carry out the theoretical derivation, and finally obtain the phase noise model suitable for the DFFM laser generated in NPRO. The two frequencies are distinguished by α and β, respectively. Here ${F_\alpha }$ and ${F_\beta }$ are the number of photons for the two frequencies, ${N_\alpha }$ and ${N_\beta }$ are the corresponding population inversions, assuming that the two frequencies are partially independent and partially coincident with each other. The ${G_\alpha }$ and ${G_\beta }$ represent the unsaturated population inversions, and ${\tau _\alpha }$, ${\tau _\beta }$ are the corresponding photon lifetimes inside the ring resonator. $\tau$ is the population inversion lifetime, $\varepsilon$ is the excited emission-coefficient (proportional to the excited emission-cross-section), and the coefficients ${\xi _{\alpha \beta }}$ and ${\xi _{\beta \alpha }}$ are the ratios of the cross-saturation coefficients to the self-saturation coefficients, which will characterize the mutual coupling between the two frequencies [29].

The steady-state values corresponding to the two frequencies can be obtained from (1-4) as follows,

The Eq. (9) and (10) lead to the output laser RIN of the two frequencies, where $i = \alpha ,\textrm{ }\beta $,

Combining with Eqs. (5), (6), (9), (10), we obtain the phase noise of the RF beatnote due to the phase-intensity coupling,

3.1.2 Thermal fluctuation caused by pump fluctuation

For the thermal noise induced by the pump power fluctuations, the relationship between the laser pump jitter and the phase jitter of the beatnote can be obtained by calculating the change of the optical path in the ring resonator due to temperature fluctuations as follows,

Here, ${\omega _0}$ is the output laser angular frequency. ${R_T}$ and ${\tau _T}$ are the respective thermal resistance and response time of the gain medium. ${P_\alpha }$ and ${P_\beta }$ are the corresponding pump powers to the two frequencies, which are derived from the ratios of the output powers of each frequency to the total pump power. ${\Gamma _T}$ is the thermal refractive index, which is determined by:

The two parts of noise given in Eqs. (16) and (17) must be coherently added, since they come from the same noise source (pump noise). The phase noise power spectral density of the beatnote is finally obtained,

3.2 Theoretical and experimental phase noises

Since pump power fluctuations affect both the phase-intensity coupling noise and the thermal noise, we first measure the pump noise. Figure 3(a) shows the RIN of our pump LD at a typical output power of 400 mW at 808 nm, where we used similar method as used in Fig. 2(c). Because the pump LD is working at a constant current mode and there is no any feedback loop for power stabilization, its RIN from -60 dB/Hz at 1 Hz to -140 dB/Hz at 10 MHz is not at a high-performance level. The RIN noise at low-frequency band might be limited by the thermal noise and pump current noise and has potential improvements in the future through active noise-cancellation loops.

 

Fig. 3. (a) Experimentally measured RIN spectrum of the pump LD. (b) Phase noise from experimental measurement and theoretical calculations of a 5.7 GHz DFFM beatnote. Parameter values used for the theoretical calculations: $\tau = 230\,{\mathrm{\mu} \mathrm{s}}$, ${\tau _\alpha } = {\tau _\beta } = 2\,\textrm{ns}$, ${r_\alpha } = 2.69$, ${r_\beta } = 2.31$, $\eta = 0.9$, ${P_\alpha } = 215\,\textrm{mW}$, ${P_\beta } = 185\,\textrm{mW}$, $\epsilon = 2.8 \times {10^{ - 8}}\,\textrm{cm}$, $C = 0.28$, ${R_T} = 7.14\,\textrm{K}.{\textrm{W}^{ - 1}}$, ${\tau _T} = 9.8\,\textrm{ms}$, $dn/dT = 7.8 \times {10^{ - 6}}\; {\textrm{K}^{ - 1}}$.

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In theoretical calculations, we substitute the measured pump noise above into the theoretically model. The calculated results and experimentally measured phase noise are shown in Fig. 3(b). The black solid curve is the experimentally measured phase noise for a 5.7 GHz carrier (as shown in Fig. 2(a)). The blue dotted curve is the noise from the phase-intensity coupling between the two frequencies. The green dotted curve is the phase noise coming only from thermal noise induced by pump noise. The total theoretical noise is shown as the red solid curve. The noise measurement limit of the instrument is also shown as the gray zone. From Fig. 3(b), it can be found that the thermal noise is dominant at low Fourier frequencies (1 Hz to 70 Hz), whereas the effect of the phase-intensity coupling between the two frequencies is dominant at Fourier frequencies higher than 70 Hz. The overall trend of theoretical prediction shows a good match with the experimental results and proves that the theoretical model is valid. The fast deviation between the theoretical noise and the experimental measurements at low Fourier frequencies (< 10 Hz) might come from the thermal noise and vibrational noise induced from environment that cannot be cancelled by common-mode-rejection. The departure between the theoretical prediction and experimental measurement at high Fourier frequencies (> 1 kHz) might be attributed to the RIN of the output laser which further contributes to phase noise through intensity modulation that are not considered in the model. Note the theoretical laser RIN (Eq. (15)) of the two frequencies can also be calculated which shows a good match (not shown here) with the experimental result as shown in Fig. 2(c) (the green solid curve).

4. Comparison with dual-frequency LG and other photonic microwave signals

A dual-frequency LG laser can also be generated in NPRO [31]. The formation of dual-frequency LG laser comes from the generation of different mode components at different frequencies. Different from the DFFM laser, the dual-frequency LG laser is formed by interference of two transverse-modes with the same order but different phase rotation directions (such as $LG_{01}^ + $ and $LG_{01}^ - $). Therefore, the beatnote of the DFFM laser is expected to have a higher level of common-mode-rejection and thus lower noise due to high-degree of mode overlapping. By adjusting the pump beam to the middle of the two optimal positions where it is allowed to simultaneously generate two LG modes with different phase rotation directions of the same order [14], we are able to generate two different dual-frequency LG laser modes, i.e., $L{G_{01}}$ and $L{G_{02}}$. The beam profiles are observed and distinguished [32] by the CMOS camera as shown as the inset of Fig. 4(a). The beam profile of the fundamental mode $TE{M_{00}}$ is also shown for comparison. The phase noise curves of beatnotes from the dual-frequency $TE{M_{00}}$ (DFFM), $L{G_{01}}$ and $L{G_{02}}$ lasers are shown in Fig. 4(a), whose corresponding carrier frequencies are 5.7 GHz, 4.3 GHz and 3.0 GHz respectively. One can see obvious noise reduction of the 5.7 GHz signal at low Fourier frequencies up to a few kilohertz. We explain this by the difference of the optical-to-optical conversion efficiency, i.e., the optical-to-optical conversion efficiencies of LG lasers are much lower (typically < 30%) than that of the DFFM laser. The residual pump noise in LG lasers contributes more thermal noise at low Fourier frequencies that cannot be common-mode-rejected. The noise floor differences between $L{G_{01}}$ and $L{G_{02}}$ lasers in the high frequency band (> 1 MHz) might due to their different noise levels in the RIN of the lasers themself. We further checked their laser RINs and indeed find that the $L{G_{02}}$ laser has higher noise floor of RIN at Fourier frequencies higher than 1 MHz.

 

Fig. 4. (a) Phase noise comparisons of dual-frequency laser microwave signals in different transverse-modes from an NPRO. Insets: the corresponding beam profiles of $TE{M_{00}}$, $L{G_{01}}$, and $L{G_{02}}$. (b) Phase noise comparisons of different compact photonic microwave generators, in which we scale all the carrier frequencies to 10 GHz. Blue dashed line with triangle: 1 GHz Dual-polarization DFB fiber laser [12], red dot-dashed line with dots: 3 GHz two orthogonally polarized modes using VECSEL [13], yellow dashed line with squares: 19.6 GHz microcomb in $\textrm{S}{\textrm{i}_3}{\textrm{N}_4}$ [7], green dashed line with inverted-triangle: 10 GHz microcomb in $\textrm{Mg}{\textrm{F}_2}$ [33], and black solid line: 5.7 GHz DFFM laser (this work).

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Figure 4(b) compares phase noise of our 5.7 GHz signal with photonic microwave signals generated from other compact devices, including VECSEL [13], DFB laser [12], soliton microcombs [7], and Kerr microcombs [33]. Here, we scale all the carrier frequencies to 10 GHz for convenient comparisons. One can see our device shows lower noise performance compared with VECSEL and DFB laser technologies, and has comparable noise level with microcomb technologies while using a quite simple setup. Therefore, it explicates an alternative method for generating low-noise microwave signals.

5. Tunability of the photonic microwave signal

The frequency tunability of a microwave signal is a prerequisite when considering metrology, precision measurement applications. In our experiments a thin slice of piezo is glued above the ring resonator for fast frequency tuning and a TEC element is glued below the ring resonator for temperature control and slow frequency tuning. The schematic is show as in Fig. 5(a). A sinusoidal signal of 10 Hz was applied to the piezo with peak-peak voltage (V_pp) of 10 and 20, respectively. By reducing the sampling rate of the spectrum analyzer, it is clear to see the beatnote envelope spreads from a spike to a saddle, as shown in Fig. 5(b). The 10 V_pp and 20 V_pp are corresponding to saddle widths of 151 Hz and 306 Hz respectively. The piezo tuning coefficient is thus estimated to be greater than 15 Hz/V. The 3dB-bandwidth of piezo tuning is also tested to be greater than 1 kHz in the experiment.

 

Fig. 5. (a) Schematic of the devices used for piezo tuning and temperature control. (b) Microwave signals (saddle envelope) with 10 Hz sinusoidal voltage modulation on the piezo. Black curve: 151 Hz tuning range with 10 Vpp input. Blue curve: 306 Hz tuning range with 20 Vpp input. The piezo tuning coefficient is estimated to be greater than 15 Hz/V. (c) Microwave signal center frequency shifting with temperature. Red dots: microwave signals with SNR greater than 75 dB. Blue inverted triangles: microwave signals with SNR less than 75 dB (RBW = 200 Hz, VBW = 100 Hz). Black dash line: linear fitting. The temperature tuning coefficient is given to be -60.5 ± 0.1 kHz/K.

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The temperature-tuning of the microwave frequency can be explained by the response of the FSR of the ring resonator to temperature. First, the FSR is a function of refraction of n of the gain medium and ring resonator length of L.

Then the variation of FSR due to temperature variation can be given as

Here, $dn/dT$ and $dL/dT$ are the thermal refractive index ($7.8 \times {10^{ - 6}}\textrm{ }/\textrm{K}$) and thermal expansion coefficient ($1.82 \times {10^{ - 7}}\textrm{ m}/\textrm{K}$) of the gain medium respectively. By introducing the relative parameters, a theoretical temperature-tuning coefficient is obtained as -60.1 kHz/K. Figure 5(c) shows the experimental measurements of the center frequency shift of the 5.7 GHz signal with temperature. Because there is mode hopping during the temperature change, the signal intensities have a variation, where the red dots represent SNR of above 75 dB, the blue triangles represent SNR of below 75 dB. A linear fitting of the data points gives a slope of -60.5 ± 0.1 kHz/K, which is consistent with the theoretical prediction. We had separately measured several NPRO samples and found the width of temperature window for high SNR (above 75 dB) microwave signals not only depends on the pump condition (that is mentioned in the former parts) but also depends on the Nd:YAG material itself used for producing the NPRO. Some NPROs have narrower temperature windows for DFFM lasing, which are typically more than 0.2 K and less than 0.5 K in each window. Fortunately, this is broad enough for maintaining the DFFM lasing (i.e., the microwave signal) infinitely due to the millikelvin temperature stability in the NPRO by using active temperature controller.

6. Conclusions

In conclusion, we have demonstrated stable DFFM lasing from NPRO laser. The DFFM share similar noise characters between the two frequencies due to their productions from the same ring resonator that allows to generate high-purity microwave signals through common-mode-rejection. A 5.7 GHz microwave signal that is the beatnote of the two frequencies from a DFFM has phase-noises as low as -112 dBc/Hz at 10 kHz offset and -150 dBc/Hz at 10 MHz offset. This performance surpasses their counterparts from dual-frequency LG modes, VECSEL and DFB lasers and is comparable with the best microcomb microwave sources. The phase noise is limited by thermal noise at low Fourier frequencies (less than 70 Hz) and by phase-intensity coupling noise at high Fourier frequencies (more than 70 Hz) from theoretical analysis. To further improve the stability of the signal, a pump source with lower RIN is expected, and a DFFM with very similar amplitude is preferred between the two frequencies. Because the microwave signal is frequency tunable by piezo and temperature, its noise can thus be further suppressed through active noise cancellation loops. Considering NPRO lasers have been successfully used in many areas including research, metrology, industry and space mission, etc., our findings are expected to facilitate these areas with NPRO-based low-noise microwave sources in the future.