1. Introduction

Delay-line oscillators such as lasers and optoelectronic oscillators can potentially oscillate in a large number of cavity modes that are supported by their amplifier. However, in homogeneously broadened oscillators that generate a continuous wave (CW) signal, the oscillating mode is limited to one or few cavity modes that have the largest gain. When the oscillator is turned on, modes that experience the highest small-signal gain will grow faster than other modes. The strongest mode will eventually cause the saturation of the whole gain spectrum, thus preventing oscillation from other cavity modes [1, 2]. Often, effects such as spatial hole burning, time-dependent gain fluctuation, and cavity changes due to environmental conditions may cause mode-hopping.

To select a particular oscillating mode, a narrowband filter should be inserted into the oscillator cavity [3]. However, in long-cavity oscillators, the filter bandwidth is usually insufficient to select a specific cavity mode. Therefore, the oscillating mode is randomly selected when the oscillator is turned on and mode-hopping may later occur.

Wave mixing in optical amplifiers [4–6] and in optical semiconductor amplifiers [7–10] have been intensively studied. The interference of waves with different frequencies causes a change in the refractive index and in the gain of the amplifiers. This effect may help to achieve mode-hop-free single-mode operation [7, 9, 10]. However, since wave mixing causes asymmetry in the gain spectrum with respect to the oscillating mode, the bandwidth of the cavity filter should be on the order of the mode spacing to ensure a single-mode operation without mode-hopping. Moreover, if the filter bandwidth is broad, the oscillation mode may be different each time when the laser is turned on.

Injection of another wave into the oscillator can be used to select the oscillating mode in lasers [3] and in OEOs [11–13]. However, injection locking may degrade the phase noise of the oscillator due to noise added by the external signal. It also requires a complex external feedback to dynamically lock the oscillator to the injected signal.

In this manuscript, we demonstrate, theoretically and experimentally that the oscillation mode of a homogeneously broadened delay-line oscillator can be selected from a large number of cavity modes by a short-time injection of an external signal into the oscillator. After the injected signal is turned off, the stand-alone oscillator continues generating the selected mode for an unlimited duration even in modes that can not build up without the short-time injection of the external signal. The stability of the selected mode is obtained even in oscillators that generate non-stationary signals when it is turned on. The effect is obtained due to gain saturation in the oscillator amplifier that occurs on a time scale that is faster than the inverse of the effective bandwidth of the oscillator cavity. In our theoretical analysis, we study the stability of a single-frequency solution of a stand-alone oscillator to a small perturbation. The initial condition of the stand-alone oscillation is set by the short-time injection of the external signal. We show that a saturated amplifier in the oscillator couples between perturbations at symmetrical frequencies with respect to the oscillating mode. We give an explicit solution to the case of deeply saturated amplifiers. In this case, the oscillation amplitude is stable. However, the phase may become unstable, depending on the amplifier saturation and on the second order frequency derivative of the cavity filter rather than on the filter transmission as in semiconductor lasers [7, 9, 10].

We have demonstrated mode selection in an OEO. Such an oscillator generates high-frequency microwave signals with ultra-low phase noise due to its ultra-long cavity length that can be on the order of few kms [14]. However, in such long delay-line oscillators, the frequency difference between adjacent cavity modes is on the order of tens of kHz, which is several orders of magnitude lower than their carrier frequency of about 10 GHz. To enforce the OEO to oscillate in a single mode, a narrowband RF filter is inserted into its cavity. However, the bandwidth of such filter is much broader than the OEO mode spacing and when the OEO is turned on, the oscillation mode is randomly selected from few cavity modes where the filter transmission is close to its maximum value [15]. Therefore, RF filter in the OEO cavity can not precisely set the oscillation frequency. The OEO cavity contains homogeneously broadened RF amplifiers such that the saturation due to a signal at one frequency affects the gain for signals at any other frequency. The saturation response time of the amplifiers is faster than the inverse of the OEO filter bandwidth. Hence, nonlinear interaction in the OEO amplifiers can suppress non-oscillating modes that we refer to as spurious modes or spurs.

In the OEO that we used in our experiments, the mode spacing was only 27.3 kHz. We could set the frequency of the oscillation mode to any desired mode out of about 400 cavity modes inside a wide frequency region, which is broader than the 3-dB bandwidth of the OEO filter. After temporarily setting the OEO frequency by a short-time injection of an external signal into the cavity, the stand-alone OEO continued oscillating in the selected mode, without jumping to another mode, for unlimited period of time. The OEO could oscillate even at frequency of 10.007 GHz, where its filter transmission was about 6 dB below its maximum transmission at 10 GHz. Such mode can not build up when the stand-alone OEO is turned on. The output power and the phase noise of the OEO did not significantly depend on the oscillation frequency. However, the spur powers of the stand-alone OEO were dependent on its carrier frequency. The minimum power of the first two spurs was obtained when the oscillation frequency was about +6 MHz or −3 MHz with respect to the central frequency of the filter. At those frequencies, the spurs were about 12 dB weaker than their largest amplitude at +1 MHz frequency offset. Although the OEO filter bandwidth is significantly broader than the OEO mode spacing, the nonlinear interaction in the deeply saturated OEO amplifier causes the observed reduction in the gain of the spurs. In previous works, the instability of a stand-alone OEOs has been studied theoretically and experimentally [15–17]. In those works, the Mach-Zehnder modulator (MZM) caused the dominant saturation effect. Due to the periodic dependence of the MZM transmission on the signal power, amplitude instability at sufficiently high small-signal cavity gain caused instability of a CW solution, while the signal phase was neutrally stable. In the OEO described in this manuscript, the dominant saturation effect is caused by the RF amplifiers and hence, the phase of a CW signal may become unstable while its amplitude is always stable. We note that phase instability in Surface Acoustic Wave (SAW) oscillators that can oscillate in a few modes has been studied in [18, 19].

By adding a controllable phase shifter to the OEO cavity, we could adjust the frequencies of the cavity modes such that the frequency of one of the modes becomes almost equal to that of the external signal. Hence, besides a small thermal drift and a small locking error, the OEO could preserve the frequency of the external signal after this signal was turned off. This "memory" effect is useful for setting the OEO frequency with an accuracy that is far beyond the accuracy that can be obtained just by the RF filter in the OEO. It is also important for applications that require reliable synchronization, such as communication systems, global positioning systems (GPS), and radars [20] that should operate even when the signal used for synchronization is not available for a short duration.

2. Principle of operation

Figure 1 illustrates the principle of operation of the OEO mode selection. Dashed-dotted curve illustrates the transfer function of the OEO filter that determines the open-loop gain. The frequency mode spacing of the OEO is much narrower than the bandwidth of the transfer function. At the initial state (a), the free-running OEO oscillates at the j-th cavity mode with a frequency f_j. To select an oscillation mode with a frequency f_m, an external signal with a frequency f_sig ≅ f_m is injected for a short time into the OEO cavity. By adding a controllable phase shifter to the OEO cavity, the frequency f_m can be always adjusted to be approximately equal to an arbitrary signal frequency, f_sig. After the OEO oscillation was built up, the external signal is turned off and the stand-alone OEO continues oscillating in the selected OEO mode with a frequency f_m (b).

 

Fig. 1 Schematic illustration of mode selection in OEOs. (a) A free-running OEO oscillates at frequency f_j. An external signal with a frequency f_sig, which is approximately equal to the frequency of the m-th cavity mode, is injected into OEO for a short time. (b) After the external signal is switched-off, the oscillation frequency equals f_m. Dashed curve indicates the transfer function of the open-loop cavity gain.

Download Full Size | PDF

The minimum injected power ratio, which is needed to set the OEO frequency, can be very low, on the order of −46 dB in our experiments, in case that the OEO is turned off before the external signal is fed into its cavity. If the OEO is turned on in the absence of an injected signal, the oscillation is built up from noise. However, if the OEO is turned on in the presence of the injected signal, only a very weak injected power is needed to select the oscillation mode, since the injected signal is significantly stronger than the internal noise of the OEO. The injected power may be significantly lower than required to injection-lock the OEO to the external source [21]. In such case, when the OEO is turned on in the presence of the injected signal, it changes its oscillation frequency and becomes quasi-locked to the injected signal [22–25]. In this operating region, the OEO oscillates in the cavity mode, with the nearest frequency to that of the external signal. The phase difference between the OEO and the external signal has a quasi-periodic behavior with phase slips, where in relatively short time intervals, the phase increases (or decreases) by 2π [22–25]. The average oscillation frequency is slightly pulled from the frequency of the corresponding mode of the stand-alone OEO. After the external signal is turned off, the stand-alone OEO continues oscillating in the selected cavity mode.

3. Experiment

In this section, we describe the experimental demonstration of mode selection in OEOs. This device is a delay-line homogeneously broadened oscillator. The main saturation effect in the OEO, described in this manuscript, is caused by its RF amplifiers. These amplifiers are homogeneously broadened and have a saturation response time on the order of the carrier frequency. The saturation decreases the amplifier gain and it does not affect the phase of the amplified signal.

The experimental setup of the OEO is shown in Fig. 2. The external signal was used to select the oscillating mode. An electronic switch was used to turn the external signal injection on and off. In order to change the oscillation mode of the OEO, the external signal was fed for a short time of about 10 ms. We have used an I/Q mixer to accurately measure the time evolution of the frequency and phase difference between the OEO and the external signal, used for the injection. A CW laser with a power of 14 dBm was fed into a linearly biased MZM with a V_π = 4 V. The modulator output was connected to a L = 7.5 km length fiber that was split into a 6-km fiber that was connected through an optical isolator to a 1.5 km fiber. The isolator was used to prevent back reflections. The light was converted into electrical signal by a photo-detector (PD) with conversion efficiency of 0.75 A/W and amplified by an amplifier G₁ with small-signal gain of 37.7 dB and a saturation power P_1dB of 13.8 dBm. Part of the amplified signal was fed to a signal source analyzer (SSA) (Keysight E5052A + E5053B) to measure its phase noise and the other portion passed through a bandpass filter (BPF) with a full width at half maximum (FWHM) bandwidth of 10 MHz around a central frequency of 10 GHz. A voltage-controlled phase shifter (PS) was used to adjust the frequency of the OEO modes. An external signal was injected through a voltage-controlled RF switch. The combined signal was then amplified by amplifier G₂ with a gain of 29 dB and P_1dB point of 27 dBm. The output of G₂ was tapped for measurements and then fed into the MZM.

 

Fig. 2 Schematic description of the experiment setup. Laser is a CW laser source, MZM is a Mach-Zehnder modulator, L is a 7.5-km fiber, PD is a photo-detector, G₁ and G₂ are RF amplifiers, C₁, C₂, C₃ and C₄ are directional couplers with coupling ratios of −6 dB, −6 dB, −20 dB and −5 dB, respectively, BPF is a bandpass filter with a central frequency of 10 GHz and a bandwidth of 10 MHz, PS is an electronically controlled phase shifter, SSA is a signal source analyzer, and "Switch" is an RF switch that is controlled by the computer (PC). The outputs of a I/Q mixer are sampled and used to measure the relative phase and frequency between the OEO signal and the signal generator.

Download Full Size | PDF

In the stand-alone OEO, the typical measured output powers of amplifiers G₁ and G₂ were 15.7 dBm and 20 dBm respectively. Hence, the gain of amplifier G₁ was about 30 dB in comparison with its small-signal gain of 37.7 dB and therefore, this amplifier was deeply saturated. We note that the MZM transmission was also close to its saturation point. However, since amplifier G₁ was deeply saturated, the change in its output power was very small and hence this amplifier effectively determined the saturation of the OEO power in the cavity.

The transfer function of the OEO cavity is determined by its narrowband bandpass RF filter. Figure 3 shows the power and the phase transmission of this filter as a function of the frequency offset from its central frequency. We have used an RF Network Analyzer (NA) to measure the filter transmission with a frequency resolution of 12.5 kHz. We note that the transfer function of the bandpass filter is not symmetric around its central frequency of 10 GHz. The maximum group delay of the filter is about 100 ns, which is negligible with respect to the delay induced by the fiber (36 μs). Hence the OEO mode spacing, of about 27.3 kHz, was determined by the fiber.

 

Fig. 3 Power transmission (solid blue) and phase (solid back) of the intracavity OEO filter, measured by using network analyzer, around a central frequency of 10 GHz with a frequency resolution of 12.5 kHz.

Download Full Size | PDF

The external signal was injected to the OEO for a short duration of about 10 ms in order to select the oscillation mode. After the injected signal was turned off, the OEO continued oscillating in the selected mode for an unlimited duration. If the external signal is fed to the OEO that oscillates in an arbitrary cavity mode, the injection ratio should be about −10 dB. Such a signal enables to injection-lock the OEO to the external source if the detuning between the frequencies of the signal and the corresponding OEO mode is lower than about 700 Hz. The theoretical locking range for −10 dB injection ratio and a 7.5 km long fiber equals 1.4 kHz [21]. The locking range was measured by changing the external signal frequency after the OEO was injection-locked to this source. The measured locking range bandwidth was about 1.3 kHz and it did not depend on the oscillation frequency.

An injection ratio on the order of −35 dB was sufficient to injection-lock the OEO [13] at a frequency where the OEO filter transmission is close to its maximum value, with a maximum frequency detuning of about 80 Hz. However, modes with frequencies located close to the FWHM bandwidth of the filter have lower gain and hence, the minimum injection ratio should be −10 dB. However, in order to select the OEO mode, there is no need to injection-lock the OEO to the external source. The injection power can be very low if the OEO is switched off each time that a new oscillation mode is selected. In our experiments, we switched off the oscillation for about 2 ms by turning off the photo-detector bias. An injection ratio of −46 dB was sufficient to select any desired OEO mode even if the frequency detuning between the stand-alone OEO mode and the external signal was as high as 5 kHz. For such a high detuning, the minimum theoretical injection ratio that is needed to injection-lock the OEO to the external source equals −1 dB [21]. However, if the OEO is turned on in the presence of the injected signal, a very weak injected power is required to quasi-lock the OEO and thus to select its oscillating mode, as explained in section 2. The external signal was injected in our experiments to the OEO for about 10 ms. This duration was significantly longer than the maximum measured duration, required for the oscillation buildup. This duration increased as the injection ratio was decreased. For an injection ratio of −70 dB, which was required to quasi-lock the OEO at the central frequency of its filter, the measured buildup duration was approximately 5 ms.

To select a specific oscillation mode, the frequency of the external signal should be approximately equal to the mode frequency. In our experiments, described above, we were able to select any desired OEO mode with a low injection ratio of −46 dB and with a maximum frequency detuning of 5 kHz between the signal and the corresponding OEO mode. However, for an external signal with an arbitrary frequency, the electrically controlled phase shifter in the OEO cavity should be adjusted. The spurs in the OEO spectrum correspond to the OEO cavity modes. The frequency difference between the external signal and the nearest spur, δ f, can be directly measured from the corresponding beating signal at output of the mixer. Therefore, before the external signal was supplied to the OEO, we sampled one of the outputs of the I/Q mixer with a sampling rate of 1 MSamples/sec. A peak at the frequency difference δ f was obtained in the spectrum of the sampled signal. The frequency of this peak, which is lower than half of the mode spacings, was used to adjust the PS voltage until the peak frequency became lower than 10 Hz.

If the frequency of the OEO should be changed by more than about 1.5 MHz, to an unknown frequency, the beating signal at the mixer output may be too low to be detected. However, to quasi-lock the OEO, the frequency difference δ f should be lower than about 5 kHz, while the mode spacing equals 27.3 kHz. Therefore, the PS phase can be adjusted by a maximum number of three iterations. In each iteration, the PS phase is changed by 2π/6 and if the OEO becomes quasi-locked to the external source, a strong beating signal is obtained at the mixer output. Then, the exact oscillation frequency can be set, with an accuracy higher than 10 Hz in our experiments, by switching off the injected signal and adjusting the PS voltage in order to shift the beating signal to a low frequency.

After the injection is switched off, the stand-alone OEO slightly changes its frequency to that of the nearest oscillating mode of the stand-alone oscillator. To measure the change in the OEO frequency after the external signal was disconnected, we measured the beating signal between the OEO signal and the external signal, which was used for the injection, by sampling the mixer outputs. Figure 4a shows the measured beating frequency after the injection was turned off at t = 1 s that is marked by a dashed vertical red curve. The injection ratio in this experiment was −10 dB and the oscillation frequency was equal to 10.001 GHz. This high injection ratio was sufficient to injection-lock the OEO to the external source before the injection was switched off. After the injected signal was switched off, the frequency of the stand-alone OEO changed from the frequency of the external signal to the mode frequency of the stand-alone oscillator. The measured change in the OEO frequency was about 5 Hz. This frequency change and its sign varied from experiment to experiment and it is caused due to experimental errors in adjusting the PS and by a small thermal drift of the OEO modes. After an initial frequency transient, the OEO frequency continues to drift slowly with time due to changes in environmental conditions. The OEO in our experiments was not thermally stabilized and we measured an average drift on the order of 0.5-2 Hz/sec.

 

Fig. 4 (a) Beating frequency between the OEO signal and the signal generator after the external signal is switched off at t = 1 s, marked by a dashed vertical red curve. The beating frequency is calculated by performing a derivation to the measured outputs of the I/Q mixer that are sampled at a rate of 40 kSamples/sec. The frequency difference between the signal generator and the stand-alone oscillator is only about 5 Hz. The injection ratio was equal to −10 dB and it enabled to injection-lock the OEO before the injected signal was turned off. (b) Transient response of the OEO after the injected signal was turned off at t = 0 s, in case that the frequency detuning between the injected signal and the corresponding OEO cavity mode was intentionally increased to 130 Hz. The figure shows the voltage of one of the outputs of the I/Q mixer that was sampled at a rate of 250 kSamples/sec. A transient behavior to a stand-alone oscillation occurs over a period of 30 μs, as marked by the two dashed red vertical lines.

Download Full Size | PDF

In order to accurately measure the transient behavior after the injected signal was turned off, we intentionally increased the frequency detuning between the external signal and the corresponding OEO mode to about 130 Hz. Figure 4b shows the time dependence of one of the outputs of the I/Q mixer that was sampled with a rate of 250 kSamples/sec. The transition from an injection-locked state to a stand-alone oscillation, marked by the two red dashed vertical lines, occurs over a period of about 30 μs after switching off the injection at t = 0. This duration is approximately equal to the round-trip time of 36 μs.

Figure 5 shows frequency drift of the oscillating signal over a long time scale for three carrier frequencies of 9.997 GHz, 10 GHz and 10.005 GHz. The frequency was calculated every 30 seconds from the measured beating signal between the OEO and the synthesizer that was used for injection. The OEO in our experiments was not thermally stabilized and its frequency smoothly changed due to slow variations in the environmental conditions. The change in the oscillation frequency in 30 seconds was on the order of 100 Hz, which is much smaller than the mode spacing of 27.3 kHz. Hence, we can deduce that the oscillation remained stable and no mode-hopping was obtained during our experiment duration of 24 hours.

 

Fig. 5 Frequency drift of the stand-alone OEO versus time for a carrier frequency of 9.997 GHz (blue), 10 GHz (red) and 10.005 GHz (green) calculated every 30 seconds from the measured beating signal between the OEO and the synthesizer that was used for injection. The oscillation frequency changes smoothly due to variations in the environmental conditions. No mode-hopping is observed during the measurement duration of 24 hours.

Download Full Size | PDF

The spectral region where the OEO frequency could be controlled was measured by changing the external signal frequency and measuring the OEO RF signal spectrum and the beating frequency after the external signal was turned off. In each frequency, we measured the oscillator frequency for about half an hour and verified that the OEO did not change its oscillating mode. The frequency region where we could set the OEO frequency was between 9.996 to 10.007 GHz. We have also verified that we could set the OEO frequency with a resolution of its mode spacing that was equal to 27.3 kHz. Figure 6 shows the RF spectra of the stand-alone OEO after it was set at twelve different frequencies. Each spectrum was measured by using an RF spectrum analyzer with resolution bandwidth (RBW) of 10 kHz. Although the 3-dB bandwidth of the OEO filter is equal to about 400 times the OEO mode spacing, we could obtain a stable operation in about 400 modes. We could also obtain a stable oscillation at a frequency of 10.007 GHz where the OEO filter transmission is 6 dB smaller than its maximum value. We note that in semiconductor lasers, the saturation effect causes changes in the refractive index of an amplifier and not only in its gain. Hence, the filter bandwidth should be on the order of the mode spacing to ensure mode-hope-free operation [7].

 

Fig. 6 RF spectra of the stand-alone OEO after setting its frequency by a short-time injection of an external signal at twelve different frequencies in the region of 9.996–10.007 GHz. The difference between the oscillation frequencies was chosen to be 1 MHz. Different colors of the plots correspond to different oscillation frequencies. The spectra were measured by using an RF spectrum analyzer with a resolution bandwidth of 10 kHz.

Download Full Size | PDF

Since the OEO amplifier G₁ operated in deep saturation, the OEO output power that was measured at the output of coupler C₁, did not significantly depend on the OEO frequency. Along the frequency bandwidth of 9.996 to 10.007 GHz, the output power was about 8±0.5 dBm which is within our measurement errors.

Figure (7) shows the phase noise measured at four different frequencies in the region of 9.996 to 10.007 GHz. The phase noise was measured by using a signal source analyzer. By using a sufficient number of 1000 correlations for each measurement, we have verified that the measured phase noise is not affected by the noise floor of the SSA. The peaks at 27.3 kHz, 54.6 kHz and 81.9 kHz correspond to the first three spurious modes.

 

Fig. 7 Phase noise spectra of a stand-alone OEO that was set to oscillate at frequencies of 9.997, 10, 10.003 and 10.006 GHz. Peaks at frequency offsets of 27.3 kHz, 54.6 kHz and 81.9 kHz correspond to the spurious modes of the OEO. Besides spurs power, the phase noise does not significantly depend on the oscillation frequency. The frequency resolution of the SSA was automatically set around the first three spurs to about 500 Hz, 1 kHz and 1.5 kHz respectively.

Download Full Size | PDF

The frequency resolution was automatically set by the SSA and it changed versus the frequency offset from the carrier. The frequency resolutions were 500 Hz, 1 kHz and 1.5 kHz near the first three spurs, respectively. The results indicate that the OEO phase noise did not significantly depend on the selected OEO frequency. However, the spurs powers changed as a function of the oscillation frequency. We have estimated the bandwidths of the first two spurs by measuring the phase noise around each spur with a frequency resolution of 0.5 Hz. The FWHM bandwidth of those spurs was lower than about 1 Hz. Then, we have measured the phase noise spectral density around each spur with a resolution of 1.8 Hz that exceeds the spurs bandwidth. Figure 8 shows the phase noise spectral density of the first spur (black diamond markers) and the second spur (blue circular markers) as a function of the frequency offset of the carrier frequency with respect to the central frequency of the filter. Peak power densities of −42 dBc/Hz and −51 dBc/Hz were measured for the first and second spurs at +1 MHz frequency offset. At frequency offsets of +6 MHz and −3 MHz, the power of the first spur decreases to about −55 dBc/Hz and −52 dBc/Hz, respectively. At the boundaries of the operating bandwidth, a sharp increase in the spurs power is obtained. Outside this frequency region, we could not set the frequency of the stand-alone oscillator. The spurs power in Fig. 8 shows the effective gain in non-oscillating modes in the presence of a strong oscillating mode. In the theoretical model presented below, we show that amplifier saturation reduces the gain for those spurs. In an OEO with deeply saturated amplifiers, the second order frequency derivative of the filter transmission determines the spurs suppression rather than the negligible change in the filter transmission for the different spurs. In semiconductor lasers, the suppression of non-oscillating modes is determined by the oscillation frequency with respect to the resonance frequency of the amplifier and by the cavity filter transmission.

 

Fig. 8 Phase noise power spectral density of the first (black diamond) and the second (blue circle) spurious modes located at frequency offsets of about 27.3 kHz and 54.5 kHz with respect to the carrier frequency, respectively. The power densities are shown as a function of the frequency offset of the carrier frequency with respect to the central frequency of the OEO filter (10 GHz). The powers were measured around each spur with a frequency resolution of approximately 1.8 Hz. Minimum powers of the spurs are obtained at frequencies that are different than the central frequency of the OEO filter. A sharp increase in the spurs power is obtained at the boundaries of the stable operating region.

Download Full Size | PDF

4. Theoretical model

In this section, we study the stability of a stand-alone delay-line oscillator after its oscillation mode was set by a short-time injection of an external signal. The results can be also used to study the OEO, described in section 3. We find a necessary stability condition for a single-mode oscillation and calculate the noise for oscillation at frequencies that are different from the central frequency of the oscillator filter, in case that the oscillator amplifiers are deeply saturated. The amplifier saturation effect that we added to the model, as described below, provides the nonlinear interaction that stabilizes the oscillating mode. We would like to emphasize that we study the stability of modes in a stand-alone oscillator, which can not be invoked without the short-time injection of the external signal. The stand-alone oscillator may even generate a non-stationary signal when it is turned on [15–17].

We model an oscillator with a delay, a saturable amplifier, a phase shifter, and a filter. The delay is modeled by a time delay τ, and the filter is modeled by its impulses response function h(t) or by its transfer function H (ω) that is equal to the Fourier transform of h(t). Additional phase θ may be added by adding a phase shifter. We assume that the oscillator voltage v(t) at the input to the amplifier equals $v\left(t\right)=\mathrm{Re}\left\{x\left(t\right)e^{j{\omega }_{c}t}\right\}$, where x(t) = a(t)e^jφ(t), ω_c is high carrier frequency, a(t) and φ(t) denote the time-dependent amplitude and phase respectively. Due to the narrow bandwidth of the oscillator filter in comparison with its carrier frequency, we assume that |da(t)/dt| ≪ a₀ω_c/(2π), |dφ(t)/dt| ≪ ω_c, were a₀ is the average amplitude of the oscillation. We also assume that frequency components at high harmonics of ω_c can not propagate in the cavity due to the narrow bandwidth of the oscillator filter and the RF amplifiers.

The amplifier saturation in our model provides nonlinear interaction between the spurs and the oscillating signal. The response time of the amplifier saturation should be smaller than the response time of the filter h(t) such that the amplifier can be modeled by its instantaneous nonlinear response function a_out = f_G (a_in), where a_in and a_out are the amplitudes before and after the saturable amplifier, respectively. We assume a monotonous saturation such that $f_G^{\prime }\left(a_{\text{in}}\right)>0$, where $f_G^{\prime }\left(a_{\text{in}}\right)$ is the first order derivative of f_G with respect to the amplitude. Saturation in amplifiers such as RF amplifiers, also causes some phase changes in the output signal [26–28]. However, as also shown below, amplitude noise is highly attenuated in oscillators with deeply saturated amplifiers [2]. Therefore, we can neglect the amplitude to phase (AM-PM) conversion in the stability analysis and we keep only the gain saturation effect. Such an amplifier model that was used in [29] to model an OEO, gave an excellent quantitative agreement between the theoretical and the experimental results.

Single-mode oscillation with an amplitude a₀ and a frequency ω_c should fulfill the Barkhausen condition [2] such that the signal does not change after each round-trip in the oscillator cavity:

The phase offset φ₀ of the signal can have an arbitrary value and without loss of generality we can assume φ₀ = 0. The condition for the phase in Eq. (1) is obtained at frequencies ω_p such that θ − ω_pτ + ∡H(ω_p) = 2πp, where p is an integer number. Since we assume that the oscillator mode spacing is significantly narrower than its bandwidth, there is a large number of cavity modes denoted by m = 1, …, N that fulfill the Barkhausen condition with amplitudes a_0,m ≠ 0 and frequencies ω_m. By adjusting a phase shift θ, it is possible to set the frequency of one of the cavity modes, ω_m, to any given frequency between the modes. However, the Barkhausen condition is only a necessary condition. Oscillation at ω_m should also be stable. This occurs when other cavity modes at frequencies ω_m+k for k ≠ 0 have lower gain than that of the oscillating mode. We refer to such non-oscillating modes as spurious modes.

We assume that the oscillation frequency ω_c is selected from N modes by a short-time injection into the oscillator of an external signal, with a frequency that is approximately equal to that of the m-th cavity mode, ω_m. The injected power is sufficient either to injection-lock [21] or to quasi-lock [22–25] the oscillator to the injected signal. In case that frequency of the injected signal is not equal to ω_m, the oscillation frequency changes, in a short period of time after the injected signal is turned off, to the frequency of the m-th mode, as was obtained in our experiments, described in section 3. Therefore, we assume that the frequency of the stand-alone oscillator equals ω_c = ω_m, and we analyze the small-signal stability of this initial solution even in case that the small-signal gain for this mode is lower than that for other cavity modes.

To obtain a necessary stability condition for a cavity mode solution with frequency ω_m and amplitude a_0,m and to calculate the noise at other modes, we write the dynamics equations that describe the change in the oscillator signal in each round-trip in the presence of noise. The amplitude and the phase of the phasor $x_m\left(t\right)=a_m\left(t\right)e^{j{\phi }_m\left(t\right)}$ fluctuate in time, such that a_m(t) = a_0,m + δa_m(t) and φ_m(t) = φ_0,m + δφ_m(t), where δa_m(t) and δφ_m(t) are small perturbations of the amplitude and the phase, and we assume that |δa_m(t)| ≪ a_0,m, |δφ_m(t)| ≪ 1 and without loss of generality we set φ_0,m = 0. We also assume that the bandwidth of H(ω) is much larger than 2π/τ and hence the time support of h(t) is much smaller than τ. Therefore, the change in the signal can be calculated separately for each round-trip. Finally, we assume that the spur frequencies are symmetric with respect to the oscillation frequency such that ω_m −(ω_m+k − ω_m) − ≈ ω_m−k. In general, the filter may add different phases to different frequencies such that ω_m − (ω_m+k − ω_m) does not equal to ω_m−k. However, since the filter bandwidth is orders of magnitude broader than the mode spacing, this effect may become important only at frequencies that are significantly different than the oscillation frequency (|k| ≫ 1). The experimental results shown in Fig. 8 indicate that the oscillation instability is determined by spurs with frequencies that are close to the carrier frequency and hence, we will neglect this effect.

We denote by n_m(t) the phasor of the effective noise source that is added to the signal oscillating at frequency ω_m at the output of the amplifier. This noise is caused by thermal noise of the RF amplifiers [2], shot noise of the photo-detector [2], and noise due to double Rayleigh backscattering in the fiber [29].

The dynamics of the signal phasor of an OEO that oscillates in the m-th mode, x_m(t), can be described by:

Due to instantaneous response of the amplifier saturation to perturbations, we model the nonlinear amplifier response by using a first order approximation:

The AM-AM conversion efficiency is defined as ${\gamma }_{\text{am}}\left(a_{0,m}\right)=f_G^{\prime }\left(a_{0,m}\right)/\left[f_G\left(a_{0,m}\right)/a_{0,m}\right]$.

To derive the first order dynamic equations for the perturbed oscillation phasor x_m(t), we approximate the perturbation as δx_m(t) ≈ δa_m(t) + ja_0,mδφ_m(t) and obtain the first order response of the amplifier:

We then substitute $\delta a_m\left(t\right)=\left[\delta x_m\left(t\right)+\delta x_m^{*}\left(t\right)\right]/2$ and $\delta {\phi }_m\left(t\right)=\left[\delta x_m\left(t\right)+\delta x_m^{*}\left(t\right)\right]/\left(2j\right)$ into Eq. (4) and obtain

Using Eq. (2) and Eq. (5), we obtain dynamic equation of the phasor perturbation δx_m(t),

We decompose the perturbation phasor and the noise phasor near the spurious modes as

We note that index m corresponds to the oscillating mode with a frequency ω_m, and index k corresponds to the spurious mode at a frequency ω_m+k. Equation (9) indicates that spurious modes ±k are coupled by the amplifier saturation. If the amplifiers are not deeply saturated, γ_am(a_0,m) ≈ 1 and we obtain that |Δ⁺| ≪ |Δ⁻| such that spurs at different frequencies are not significantly coupled. However, in case that the amplifiers operate in deep saturation, γ_am(a_0,m) ≪ 1 and then Δ^± ≈ ± f_G(a_m,0)/(2a_m,0). Hence, the spurs at frequencies ω_m+k and ω_m−k are strongly coupled by the gain dynamics. This coupling stabilizes the oscillation as described below.

To calculate the dynamics of the amplitude and the phase of the oscillation, we decompose the perturbation phasor in Eq. (9) into amplitude variation δa_m(t) and phase variation δφ_m(t) of the cavity modes as defined in Eqs. (7)–(8) such that $\delta a_{m,k}\left(t\right)=\left[\delta x_{m,k}\left(t\right)+\delta x_{m,-k}^{*}\left(t\right)\right]/2$, and $\delta {\phi }_{m,k}\left(t\right)=\left[\delta x_{m,k}\left(t\right)+\delta x_{m,-k}^{*}\left(t\right)\right]/\left(2j{a}_{0,m}\right)$. Since δa_m(t) and δφ_m(t) are real-valued functions, their modes decomposition fulfills: $\delta a_{m,k}\left(t\right)=\delta a_{m,-k}^{*}\left(t\right)$ and $\delta {\phi }_{m,k}\left(t\right)=\delta {\phi }_{m,-k}^{*}\left(t\right)$. Therefore, Eq. (9) can be written as coupled equations for amplitude and phase perturbations,

To obtain simple results, we solve Eqs. (10)–(11) assuming the amplifier is deeply saturated. This assumption is in accordance with our experimental results, described in section 3, where amplifier G₁ was operated in a deep saturation region. Under deep saturation, γ_am(a_0,m) ≪ 1.

In case of deep saturation and a wide filter bandwidth, Eq. (11) can be approximated as

Equations (12) and (14) show that phase stability of the k-th spurs is determined by the filter transmission for this spur and the transmission for the (−k)-th spur. This result is obtained since both spurs are required to generate a phase perturbation and only such a perturbation is not suppressed by the amplifier saturation.

To obtain a condition for phase stability, we assume that the phase perturbation changes slowly on a time scale that is on the order of the cavity delay τ, and therefore we can approximate δφ_m,k (t−τ) by the first order time derivative, $\delta {\phi }_{m,k}\left(t-\tau \right)\approx \delta {\phi }_{m,k}\left(t\right)-\tau \delta {\phi }_{m,k}^{\prime }\left(t\right)$. Equation (14) can now be approximated as

The eigenfunctions of Eq. (15) are $e^{-{\lambda }_{m,k}t/\tau }$ with eigenvalues λ_m,k = g_m,k. Integrating Eq. (15) from t = 0 gives:

For the oscillating mode k = 0, we obtain g_m,k = 0 and Eq. (17) becomes:

Hence, the phase of the oscillating mode is neutrally stable, and its spectrum is inversely proportional to τ², as obtained in [14]. Stability of the modes for which k ≠ 0 is obtained when g_m,k > 0 such that

Equation (19) indicates that when the oscillator amplifiers are deeply saturated, the stability of oscillation at a frequency ω_m is determined by the ability of the filter to suppress perturbations at frequencies ω_m±k.

The eigenvalue for the amplitude perturbations equals [γ_am(a_0,m) β_k (ω_m)]⁻¹ 1 ≫ 0. Hence, the oscillator is stable to amplitude perturbation and $\delta a_{m,k}\left(t\right)\approx n_{m,k}^a\left(t\right)$ in Eq. (10).

Assuming that the mode spacing is orders of magnitude narrower than the filter bandwidth, as obtained in the OEO described in section 3, the filter transmission for spurs in the neighborhood of the oscillating mode m can be approximated as |H(ω_m+k)| ≈ |H(ω_m)| + |H(ω_m)|′ (ω_m+k − ω_m) + (1/2) |H(ω_m)|″ (ω_m+k − ω_m)² O[(ω_m+k − ω_m)³]. Using this approximation, transmission of the phase perturbations [Eq. (12)] can be written as

The power of the non-oscillating modes, for the case that amplifiers are deeply saturated, can be calculated from Eq. (14). By taking the Fourier transform of both sides of Eq. (14), we obtain the power spectral density around the spurious mode k (k ≠ 0),

The amplitudes of the adjacent spurs can be approximated using Eq. (20):

Since ω_m+k − ω_m ≈ 2πk/τ, Eq. (22) indicates that the power of adjacent spurs decreases as 1/k⁴ as k is increased. Equation (22) gives the effective gain suppression of non-oscillating modes due to combination of the saturation dynamics in the amplifier and the filter response.

In order to quantitatively compare our theoretical and experimental results for the OEO described in section 3, there is a need to measure the second order derivative of the filter transfer function |H(ω)|″. However, the accuracy of a network analyzer is insufficient to estimate |H(ω)|″. In our experiments, the FWHM of the filter bandwidth equals 10 MHz while the mode spacing equals 27.3 kHz. The change of the filter transmission over a frequency region on the order of 1 kHz is only on the order of 10⁻⁴. However, we could verify the theoretical results for the ratio between the spur powers given in Eq. (22). The equation indicates that the power ratio between the first and second spurious modes is equal to 12 dB and it depends only on the ratio between the fourth order of the spurs numbers k⁴ and not on the oscillation frequency or the filter transmission. This result is confirmed by the measurements shown in Fig. 8 where the measured power ratio between the first two spurs changes between 10 to 12 dB. Equation (22) also indicates that the spur amplitudes are inversely proportional to the curvature of the filter, ${\tilde{H}}_m$. At the boundaries of the stable operating region, the curvature is close to zero. Hence, the spur amplitudes at the boundaries of the stable operating region sharply increase, as can be observed in the experimental results, shown in Fig. 8.

Figure 8 indicates that the spurs power changes as a function of the carrier frequency. Equation (21) indicates that the spurs power depends on both the noise sources and on the second order derivative of the filter transmission. To show that this dependence is mainly determined by the filter, we estimated the noise sources as a function of the oscillation frequency. At frequency offsets above few kHz, the main noise sources are the thermal noise of the RF amplifiers, shot noise of the photo-detector [29], and noise induced by Rayleigh scattering in the fiber [29]. The power spectral density of the phase noise due to thermal noise of the RF amplifiers equals ρ_thr = NFk_BT₀/P₀ [2], where NF is the noise figure, k_B is the Boltzmann coefficient, T₀ is the amplifier temperature, and P₀ is the input power to the amplifier. Assuming a noise figure of NF = 6, room temperature of T₀ = 300 K, and input power of P₀ = −15 dBm at the input of amplifier G₁ and P₀ = −2 dBm at the input of amplifier G₁ and P₀ = −2 dBm at the input of amplifier G₂, we obtain that ρ_thr = −150 dBc/Hz. The contribution of the shot noise is equal to ρ_shot = 2eI_PFR/P₀ [2], where e is the electron charge, I_PD is the average photo-current, and R = 50 Ω is a resistance connected to the photo-detector. For measured I_PD = 2.6 mA, we obtain ρ_shot = 148 dBc/Hz. The phase noise spectral density that is needed to obtain a good fit between the theoretical results given in Eq. (21) and experiments equals $N_{m,k}^{\phi }\left(\mathrm{\Delta }\omega \right)=-135\text{dBc}/\text{Hz}$ at about 30 kHz frequency offset and it gradually decreased to −146 kHz at frequency offset above about 400 kHz. Therefore, at frequency offsets above 400 kHz the thermal and shot noise are the dominant noise sources, while at frequency offsets lower than 100 kHz another noise becomes the dominant noise source. We attribute the additional noise to double Rayleigh scattering of the laser wave in the fiber [30]. Since the frequency offset of the first two spurs is below 60 kHz, the dominant phase noise around these spurs in our experiments is not caused due to phase noise of amplifiers. Similar result was also obtained in [29], where the dominant noise source was the shot noise in the photo-detector. Hence, the change in the measured power of the spurs as a function of the oscillation frequency, which is shown in Fig. 8, is not caused due to changes in the OEO noise sources. It is caused due to changes in the second order derivative of the filter transmission function, as given in Eq. (22). We also attribute the asymmetric spur power as a function of the carrier frequency to the asymmetry of the filter transmission, shown in Fig. 3.

5. Conclusions

In this manuscript, we have demonstrated, theoretically and experimentally that a delay-line oscillator can oscillate at a frequency where its small-signal gain is significantly below its maximal value. The oscillation mode is selected by a short-time injection of an external signal into the oscillator. After the external signal is switched off, the oscillator does not change its oscillation mode. The stability of this mode is obtained due to an amplifier with a fast gain saturation. The oscillator can oscillate even in modes that can not build up when the stand-alone oscillator is turned on.

We have demonstrated mode selection in an optoelectronic oscillator. The OEO oscillation mode could be set to each one of hundreds of its cavity modes. After switching off the external signal, the OEO continued oscillating in the selected mode for an unlimited duration, without switching to another mode. Although the OEO amplifiers are homogeneously broadened, the OEO could oscillate even at frequencies where the OEO filter transmission is significantly lower than its maximal value. The phase noise of oscillation is not affected by the external signal that is used for selecting the frequency of oscillation. The spurs power changed as a function of the oscillation frequency due to the dependence of non-oscillating mode suppression on the carrier frequency.

Using a perturbation analysis and a spectral decomposition of the perturbation, we have studied the necessary conditions required to generate a continuous wave signal at an arbitrary mode of a homogeneously broadened delay-line oscillator with a fast saturable amplifier. When the oscillator amplifier is deeply saturated, the signal phase determines the oscillation stability. In the OEO that we have studied, the phase noise and the OEO output power did not strongly depend on the OEO frequency. However, the power of non-oscillating modes that gives their effective suppression, depended on the oscillation frequency. This dependence is caused by the amplifier saturation and it is inversely proportional to the concavity of the OEO filter, as predicted by the theory.

By adding a phase shifter into the OEO cavity, it is possible to set the OEO frequency to any frequency in a spectral region that is slightly broader than the FWHM of the OEO filter. Therefore, after the external signal is turned off, the OEO will generate an ultra-low phase noise signal with a frequency that is approximately equal to the frequency of the external signal, used for the injection. No mode-hopping was observed during our maximal measurement time of about 24 hours. However, for a long time operation of the OEO, a frequency drift due to changes in environmental conditions was observed. This drift may be minimized by isolating the OEO and by refreshing the OEO frequency, from time to time, as performed in electronic memories.

The frequency "memory" effect is useful for applications that require reliable synchronization, even when the signal used for synchronization is not available for a short period of time. The effect is also important for setting the OEO frequency with high accuracy that can not be achieved by the RF filter in the OEO cavity. Without setting the OEO frequency, as described in this manuscript, the oscillation frequency is randomly selected between few cavity modes with the highest cavity gain. Hence, the uncertainty in the OEO frequency is equal to some multiple of its mode spacing, which is usually on the order of hundreds of kHz. The method described in this manuscript enables to set the frequency of a long-cavity OEO with high accuracy. It also enables reducing the power of the spurs without affecting the OEO ultra-low phase noise. The performance of the stand-alone OEO does not depend on the quality of the external source used to select the OEO frequency. Hence, tunable sources with high phase noise can be used to select the OEO mode.

The bandwidth, in which the OEO frequency can be set, is determined by the OEO filter. By designing the transfer function of the filter, the operating bandwidth can be increased and controlled. The method described in this manuscript can be applied to other delay-line oscillators such as lasers.