Experimental examination of frequency locking effect in acousto-optic system

Abstract

The optoelectronic system containing collinear acousto-optic cell fabricated on the base of calcium molybdate crystal and positive electronic feedback circuit was examined. The feedback signal is formed due to the optical heterodyning effect that occurs on the cell output and takes place in the special regime of collinear acousto-optic diffraction. It was discovered that three operation modes that may exist in this system. The boundaries between the modes were determined. The positions of the boundaries depend on the main parameters of the system—the incident light intensity and the feedback gain value. The new for acousto-optics phenomenon of acousto-optic system self-oscillations frequency locking by the RF generator signal was discovered and examined experimentally. Such an effect has never been observed before in the acousto-optic systems. It was experimentally shown that frequency locking effect may be used to select one of the multimode semiconductor laser longitudinal modes to improve laser radiation spectral composition.

1 Introduction

To date, acousto-optic devices have become quite widespread in science and technology. They allow controlling all main parameters of optical radiation—amplitude, phase, frequency, polarization, propagation direction, and spectral composition [1,2,3].

There is a special class among a variety of diverse acousto-optic (AO) devices (filters, modulators, deflectors, delay lines, etc.) that uses the optoelectronic feedback circuit for operation. A part of the light radiation intensity from the optical output of the AO cell is fed to the photodetector in such systems. The electric signal from the photodetector goes to the input of the feedback circuit connecting the output of the photodetector with the piezoelectric transducer of the AO cell [4]. This class of acousto-optic instruments expands substantially the range of optical information processing problems that can be solved by AO methods [5].

It was previously shown that feedback allows to enhance the characteristics of the conventional AO devices [6]. Some brand new AO devices for laser physics and optical information processing were constructed applying the feedback [7,8,9,10,11,12,13,14]. For example, it was proposed to use AO system with feedback to create the laser intensity stabilization system [15]. The idea to apply the chaotic oscillations aroused in such systems for information secure transmission also exists [16, 17] both for bulk and planar AO devices.

It was mentioned above that the feedback in such systems is hybrid: an optical signal in one of the AO cell output diffraction maxima is transformed by the photodetector into the electrical signal controlling the magnitude [6, 7, 9, 15] or the frequency [11, 12] of an acoustic wave excited in this AO cell by piezoelectric transducer and thus it effects the intensity of diffracted light on the AO cell output.

The analysis of the AO systems with feedback operation is a very complicated problem. Diverse oscillations (including harmonic, self-modulation, and chaotic ones) may be excited in the same AO system. Bistable and multistable regimes of operation with stable states distinguished by amplitude, frequency, and propagation direction of the diffracted waves can also be observed [5, 18,19,20,21,22].

The AO systems with feedback have two important peculiarities. Firstly, these systems are distributed as the time of acoustic wave propagation in the cell is comparable with the period of the oscillations aroused in the circuit. Secondly, these systems are principally nonlinear and the character of nonlinearities affects fundamentally the system behavior. The nonlinearities cause the variety of possible operation regimes, but, at the same time, they complicate essentially theoretical analysis of the systems. The main parameters for such systems are RF generator signal magnitude, feedback gain, and incident light radiation intensity [5].

In this paper, we continue to examine the optoelectronic system that applies electronic feedback jointly with an AO collinear cell [23, 24] fabricated of calcium molybdate crystal (CaMoO₄) [25, 26]. The various operation modes of the system were detected and borders between them were determined. The effect of AO system self-oscillations frequency locking that is absolutely new for acousto-optics was revealed and experimentally examined. The detailed theoretical analysis of this effect will be presented elsewhere.

2 The examined system description

The scheme of the examined optoelectronic system and the photo of the experimental setup optical part are presented at Fig. 1. The core element of the system is the pair of polarizers with the collinear AO filter mounted between them.

Fig. 1

a The schematic diagram of the experimental optoelectronic setup; b the optic part of the setup

An optical beam from a laser passes through a polarizer and enters the collinear AO cell (two types of laser sources were used—the single mode He–Ne laser with 632.8 nm optical radiation wavelength and chip multimode semiconductor laser with 657 nm central wavelength). The laser beam intensity is controlled by neutral density filter Thorlabs NDL-10S-4. A longitudinal acoustic wave is excited in the AO cell by a piezoelectric transducer and propagates at first along the crystallographic axis Z. Then it is reflected from the AO cell input optical face with transformation into a shear mode propagating along the X-axis. The light beam passes through the cell along the X axis collinear to the acoustic beam. The regime of traveling acoustic waves is ensured by an acoustic absorber. The cell is placed between a polarizer and an analyzer that specify the polarization of light at angles α and β with regard to the crystallographic Y-axis, respectively. In our experiments, we used the AO cell fabricated from a calcium molybdate crystal with $l=4$ cm AO interaction length.

The diffracted light passes through the Thorlabs EBS1 beam splitter and is registered by the Thorlabs PDA-10A photodetector connected to the input of the feedback circuit. The circuit includes phase shifter Mini-Circuits JSPHS-51+ and a pair of amplifiers Mini-Circuits ZHL-5W and ZFL-1200G+ that allows to tune the feedback circuit gain κ in the limits between 0 and 60,000.

Oscilloscope or RF signal spectrum analyzer may be connected to the feedback circuit for the visualization and analysis of the feedback signal characteristics. We also included the Agilent optical radiation spectrum analyzer with 0.06 nm spectral resolution to examine the spectral composition of the diffracted light. The signal from the feedback output is added to the signal from the RF generator in the summer and feeds the piezoelectric transducer of the AO cell.

3 Basic relations

The theoretical approach we apply in this paper is based on the model presented in [26, 27]. The operation of collinear AO cell placed between two polarizers was examined in these papers. It was shown that in general case of collinear AO interaction the light beam intensity after the polarizer mounted on the optical output of the AO cell (we call it analyzer) may be presented as the sum of three terms Eq. (1). The magnitudes of these terms depend not only on the AO diffraction efficiency but also on the input polarizer polarization plane orientation relative to the AO cell crystal optical axis and on the mutual orientation of polarizer an analyzer polarization planes [27, 28].

$${I_{\text{d}}}={I_0}+{I_1}\cos \left( {\Omega t+{\phi _1}+\Phi } \right)+{I_2}\cos \left( {2\Omega t+{\phi _2}+2\Phi } \right),$$

(1)

where ${I_0}$ is the component that is usually used in collinear AO filters [29], it obtains maximal value when polarizer is oriented along or orthogonally to the AO crystal optical axis and analyzer is perpendicular to the polarizer; ${I_1}$ component has modulation by intensity with $\Omega =2\pi f$ frequency that is equal to the ultrasound wave frequency aroused in the AO cell; ${I_2}$ is the component that has the amplitude modulation with doubled ultrasound frequency; Φ is the initial acoustic phase at the AO cell input; ${\phi _1}$ and ${\phi _2}$ are the additional phase shifts appearing at collinear AO interaction. In Eq. (1) all component magnitudes are the functions of polarizer and analyzer polarization planes orientation angles α and β [27, 28].

It was shown that special geometries of polarizer and analyzer mutual orientation exist when the 100% modulation of diffracted optical radiation with the $\Omega$ frequency without light losses can be achieved [27, 28]. Thus, Eq. (1) is transformed into ${I_{\text{d}}}={I_1}\cos \left( {\Omega t+{\phi _1}+\Phi } \right)$. This modulated signal may be used in the feedback circuit.

It was shown in [27, 28] that choosing the polarizer and analyzer orientations at angles $\alpha =90^\circ$ and $\beta =45^\circ$ or $\alpha =45^\circ$ and $\beta =0^\circ$ with regard to the Y crystallographic axis, we will obtain 100% modulation of diffracted light intensity after the analyzer with ultrasound frequency aroused in the AO cell. In these cases, the light intensity after the analyzer will be described by the following equation [23, 24]:

$${I_{\text{d}}}\left( t \right)={I_0}+{I_1}\cos \left( {\Omega t+{\phi _1}+\Phi } \right)=\frac{{{I_{\text{i}}}}}{2}+{I_{\text{i}}}\frac{\Gamma }{{{\Gamma ^2}+{R^2}}}\sin \left( {\frac{{\sqrt {{\Gamma ^2}+{R^2}} }}{2}} \right) \times \sqrt {{\Gamma ^2}{{\cos }^2}\left( {\frac{{\sqrt {{\Gamma ^2}+{R^2}} }}{2}} \right)+{R^2}} \cos \left( {\Omega t+{\phi _1}+\Phi } \right),$$

(2)

where ${I_{\text{i}}}$is the incident light intensity; $\Gamma =\left( {2\pi /\lambda } \right)l{{{\Delta}}}n$ is the Raman–Nath parameter (AO coupling coefficient) proportional to the acoustic wave amplitude aroused in the AO cell; $R=\left( {2\pi l/V} \right)\left( {f - {f_{\text{c}}}} \right)$ is the dimensionless AO phase mismatch; $V=2.9 \times {10^5}$ cm/s is the velocity of the acoustic wave involved into the AO interaction; λ is the optical wavelength; Δn is the maximal change of the crystal refractive index under the action of the acoustic wave; $f$is the ultrasound frequency; ${f_{\text{s}}}$is the phase matching ultrasound frequency defined by the equation ${f_{\text{c}}}=\left| {{n_{\text{e}}} - {n_{\text{o}}}} \right|V{\text{/}}\lambda$; ${n_{\text{e}}}$ and ${n_{\text{o}}}$ are the ordinary and extraordinary refractive indices of the AO crystal.

When the AO phase matching condition is fulfilled $R=0$ and ${I_1}=\left( {{I_{\text{i}}}{\text{/}}2} \right)\sin \Gamma$, the amplitude ${I_1}$ achieves maximal value 0.5 at the point $\Gamma =\pi /2$. Thus, at this point, the output intensity changes harmonically in time with the frequency of ultrasound $\Omega$ from zero to full intensity of the incident light ${I_{\text{i}}}$. This is the only case in the acousto-optics when AO cell produces amplitude modulation of the optical beam intensity not after the diffraction on the standing but on the travelling acoustic wave.

The additional phase shift ${\phi _1}$ appearing at collinear AO interaction is defined by the following equation:

$$\tan {\phi _1}=\frac{{\sqrt {{\Gamma ^2}+{R^2}} }}{R}\cot \left( {\frac{{\sqrt {{\Gamma ^2}+{R^2}} }}{2}} \right).$$

(3)

We need to include the phase shifter in the feedback circuit to fulfill the phase balance condition defined by Eq. (3).

The detector signal equals ${U_{\text{d}}}\left( t \right)=\sigma {\text{~}}{I_{\text{d}}}\left( t \right)$, where $\sigma$ is the sensitivity of the photodetector. After passing the feedback circuit ${U_{\text{d}}}\left( t \right)$ is added to the RF generator signal in the summer and feeds the AO cell piezoelectric transducer. Thus, the RF signal on the transducer may be described by the following equation:

$$U\left( t \right)={U_{\text{g}}}\cos \left( {{{{\Omega}}}t+\Phi } \right)+\sigma \kappa {I_{\text{i}}}\frac{\Gamma }{{{\Gamma ^2}+{R^2}}}\sin \left( {\frac{{\sqrt {{\Gamma ^2}+{R^2}} }}{2}} \right) \cdot \sqrt {{\Gamma ^2}{{\cos }^2}\left( {\frac{{\sqrt {{\Gamma ^2}+{R^2}} }}{2}} \right)+{R^2}} \cdot \cos \left( {\Omega t+{\phi _1}+\Phi +\chi } \right),$$

(4)

where U_g is the amplitude of harmonic signal from the RF generator; $\sigma$ is the sensitivity of the photodetector; $\kappa$ is the gain factor of the amplifier, and χ is the phase shift produced by the phase shifter. The sum of Eq. (4) terms will be maximal if

$$\chi = - {\phi _1}.$$

(5)

This equation indicates that the phase shifter has to compensate the AO phase shift ${\phi _1}$. And then it is possible to rewrite Eq. (4) as

$$U\left( t \right)={U_0}{\text{cos}}\left( {{{{\Omega}}}t+\Phi } \right),$$

(6)

$${U_0}=~{U_{\text{g}}}+\sigma \kappa {I_{\text{i}}}\frac{\Gamma }{{{\Gamma ^2}+{R^2}}}\sin \left( {\frac{{\sqrt {{\Gamma ^2}+{R^2}} }}{2}} \right) \cdot \sqrt {{\Gamma ^2}{{\cos }^2}\left( {\frac{{\sqrt {{\Gamma ^2}+{R^2}} }}{2}} \right)+{R^2}} .$$

(7)

The Raman–Nath parameter Γ is proportional to the acoustic wave amplitude and, consequently, to the electrical voltage amplitude U ₀ applied to the transducer:

$$\Gamma =\gamma {U_0},$$

(8)

where γ is transformation the coefficient determined by characteristics of the transducer and the AO cell.

Finally, it is possible to write the relation that describes the behavior of the AO filter with feedback:

$$\Gamma ={\Gamma _{\text{g}}}+{\rm K}\frac{\Gamma }{{{\Gamma ^2}+{R^2}}}\sin \left( {\frac{{\sqrt {{\Gamma ^2}+{R^2}} }}{2}} \right) \cdot \sqrt {{\Gamma ^2}{{\cos }^2}\left( {\frac{{\sqrt {{\Gamma ^2}+{R^2}} }}{2}} \right)+{R^2}},$$

(9)

where the ${{{{\Gamma}}}_{\text{g}}}=~\gamma {U_{\text{g}}}$ and ${\rm K}=\sigma \kappa \gamma {I_{\text{i}}}$ is the generalized feedback coefficient. Equations (3) and (9) form the phase and magnitude balance conditions for the examined system.

The operation of the system shown in Fig. 1 was experimentally and theoretically examined at relatively small values of the feedback gain in [23, 24]. These values were lower than the self excitation threshold of the system. Such a regime is in many ways analogous to the well-known phenomenon of regeneration in radio physics.

However, in such system, like in all systems with feedback, several modes of operation may exist. Depending on the magnitude of the gain and incident light intensity, it is possible to distinguish three operation modes in this system. All of them were observed experimentally. The experimentally determined boundaries between the operation modes are presented in Fig. 2. The gain ranges corresponding to various operation modes are numbered and marked by color.

Fig. 2

Experimentally defined borders between operation modes of the examined system. 1—regeneration; 2—self excitation with frequency locking; 3—generation

Area 1 corresponds to the regeneration mode and was previously examined in [23, 24]. In this mode, the optoelectronic system behaves like a tunable AO filter with characteristics (passband, spectral contrast, and AO diffraction efficiency) that may be controlled by varying the gain of the feedback circuit and the magnitude of the generator signal.

It was shown in [23, 24] that in this case, it is possible to achieve a significant (more than 50 times) narrowing of the transmission bandwidth of the system in comparison with a conventional collinear AO filter of the same interaction length and enhance the spectral contrast of the system by tens of times. The system operation is described by Eqs. (3) and (9) in this case.

The system operation mode 2 is located above the excitation threshold and has not been previously examined. The feedback gain is high enough to excite the system in this case, but its magnitude is not enough to maintain constant amplitude of oscillations in the system in the absence of the RF generator signal. The presence of the signal from the RF generator even with arbitrary frequency and very low magnitude plays the role of the additional noise source and allows maintaining constant amplitude of oscillations in the feedback loop.

Taking into consideration the existence of the separate signal from RF generator with the frequency that differs from the AO generator frequency, it is possible to rewrite Eq. (4) in this area as

$$U(t)=\left( {{U_{\text{g}}}\cos \left( {\Omega t+\Phi } \right)+\sigma \kappa {I_i}\frac{\Gamma }{{{\Gamma ^2}+{R^2}}}\sin \left( {\frac{{\sqrt {{\Gamma ^2}+{R^2}} }}{2}} \right) \cdot \sqrt {{\Gamma ^2}{{\cos }^2}\left( {\frac{{\sqrt {{\Gamma ^2}+{R^2}} }}{2}} \right)+{R^2}} } \right) \times \cos \left( {\Omega t+\Phi +{\phi _1}+\chi } \right)+\left( {\sigma \kappa {I_{\text{i}}}\frac{{{\Gamma _{\text{c}}}}}{{\Gamma _{{\text{c}}}^{{\text{2}}}+R_{{\text{c}}}^{2}}}\sin \left( {\frac{{\sqrt {\Gamma _{{\text{c}}}^{{\text{2}}}+R_{{\text{c}}}^{{\text{2}}}} }}{2}} \right) \cdot \sqrt {\Gamma _{{\text{c}}}^{{\text{2}}}{{\cos }^2}\left( {\frac{{\sqrt {\Gamma _{{\text{c}}}^{{\text{2}}}+R_{{\text{c}}}^{{\text{2}}}} }}{2}} \right)+R_{{\text{c}}}^{{\text{2}}}} } \right) \times \cos \left( {{\Omega _{\text{c}}}t+{\Phi _{\text{c}}}} \right),$$

(10)

where ${\Gamma _c}$, ${R_{\text{c}}}$, ${\Omega _{\text{c}}}$, ${\Phi _{\text{c}}}$ are the Raman–Nath parameter, phase mismatch, acoustic frequency, and total phase shift for the acoustic wave that appears in the case of system self excitation. The mismatch ${R_{\text{c}}} \to 0$ for every spectral component of the incident optical radiation.

The mismatch $R=\left( {2\pi l/V} \right)\left( {{f_{\text{g}}} - {f_{\text{c}}}} \right)$ value in the first term of equation is defined by the RF generator frequency.

Area 3 corresponds to the generation mode. The AO system behaves like the AO generator in this case [13, 14]. The feedback gain is high enough to maintain the constant magnitude of the oscillations in the feedback circuit without signal from RF generator. The first term in Eq. (9) equals zero in this case. The equations describing the system operation are the following:

$$\frac{{\text{K}}}{{\Gamma _{{\text{c}}}^{{\text{2}}}+R_{{\text{c}}}^{{\text{2}}}}}\sin \left( {\frac{{\sqrt {\Gamma _{{\text{c}}}^{{\text{2}}}+R_{{\text{c}}}^{{\text{2}}}} }}{2}} \right) \cdot \sqrt {\Gamma _{{\text{c}}}^{{\text{2}}}{{\cos }^2}\left( {\frac{{\sqrt {\Gamma _{{\text{c}}}^{{\text{2}}}+R_{{\text{c}}}^{{\text{2}}}} }}{2}} \right)+R_{{\text{c}}}^{{\text{2}}}=1,}$$

(11)

$$\tan {\Phi _{\text{c}}}=\tan \left( {{\Phi _{\text{c}}}+{\phi _{\text{c}}}+{\chi _{\text{c}}}} \right) - {R_{\text{c}}}.$$

(12)

The mismatch ${R_{\text{c}}} \to 0$ in this mode too. It is possible to obtain the gain values corresponding to the system excitation threshold from Eq. (11). We should consider the mismatch ${R_{\text{c}}}=0$ to define the excitation threshold. Then threshold generalized feedback coefficient values will be described by the following equation:

$${\rm K}=\frac{{2{\Gamma _{\text{c}}}}}{{\sin {\Gamma _{\text{c}}}}}.$$

(13)

And the corresponding feedback amplification (amplifier gain factor) will be

$$\kappa =\frac{{2{\Gamma _{\text{c}}}}}{{{{{\upsigma}{\upgamma}}}{I_{\text{i}}}\sin {\Gamma _{\text{c}}}}}.$$

(14)

Equation (14) describes the border between operation modes 2 and 3 in Fig. 2.

4 AO system frequency locking effect

It is known that if the additional external periodic force (driving force) acts on the nonlinear self-oscillatory system with the frequency close to the system eigenfrequency, the system oscillations synchronize with external force in some frequency band [30]. This effect is called synchronization of self-oscillations by external force, or self-oscillations frequency locking. Such effect exists only in the distinct band of the frequency difference between self- and external oscillations; the frequency band depends on the external force magnitude and feedback gain.

This effect exists in radio and electronic devices, lasers, mechanical systems, oscillating chemical reactions, and biological objects. It is applied also in the optoelectronics for radiation sources characteristics enhancement, frequency stabilization, and spectral resolution improvement [8, 31,32,33].

The AO cell plays the role of nonlinear element in our system, as the diffracted light intensity on the system output is defined by Eq. (2).

The frequency locking effect in the case when mismatch between external force (RF generator signal) frequency f_g and self-oscillations of AO generator with frequency f_с may be explained in the following way. The acoustic field inside the AO cell beyond the frequency locking band consists of two components—the wave aroused by RF generator signal with f_g frequency and the wave aroused by the feedback circuit signal with f_с frequency, in the case of excited AO system. The AO diffraction occurs at the both waves if the mismatch is small. Then the total signal on the AO cell transducer will be described by Eq. (10). The two terms with diverse magnitudes and frequencies form this equation. Thus, the beatings with period Т defined by the ${f_{\text{c}}} - {f_{\text{g}}}$ value appear. This results in the diffracted light intensity ${I_{\text{d}}}\left( {{{{\Gamma}}},R,t} \right)$ dependence on the Raman-Nath parameter averaged steepness periodical variation with ${f_{\text{c}}} - {f_{\text{g}}}$ frequency.

The self-oscillatory system has a finite inertia being defined in the examined case by the time delay $\tau$. In this system, it equals the time of acoustic wave propagation from the transducer to the light beam [13, 14]. If $T<\tau$ the system inertia causes the ${I_{\text{d}}}\left( {{{{\Gamma}}},R,t} \right)$ dependence steepness averaging over time. Some stationary value of the steepness is being set as a result. This value is the less, the more is the magnitude of oscillations produced by the driving force. The value of this stationary averaged steepness defines the magnitude of AO system self-oscillations. Thus, with the mismatch ${f_{\text{c}}} - {f_{\text{g}}}$ decrease, the magnitude of forced oscillations increases; the stationary steepness decreases, and consequently the amplitude of self-oscillations decreases too. The forced oscillations suppress the self-oscillations and at the mismatch $\left| {{f_{\text{c}}} - {f_{\text{g}}}} \right|<\Delta {f_{{\text{lock}}}}$ the self-oscillations magnitude reduces to zero. In this case, only oscillations with ${f_{\text{g}}}$ frequency exist in the system and the frequency locking effect takes place. Herewith the beatings magnitude ΔU equals zero. The more is the external force magnitude the wider is the frequency locking band $\Delta {f_{{\text{lock}}}}$.

Figure 3 represents the experimentally observed feedback circuit RF signal oscillograms and corresponding spectra for the mismatches near the frequency locking band in the case when both ${f_{\text{c}}}$ and ${f_{\text{g}}}$ frequencies exist.

Fig. 3

The observed shape (a, b) and spectra (c, d) of the oscillations in the system feedback circuit near the frequency locking band in the case of high RF generator signal and low feedback gain. a, c RF signal magnitude 5 V, gain 640, ${f_{\text{c}}} - {f_{\text{g}}}=8.55$ kHz; b, d RF signal magnitude 5 V, gain 640, ${f_{\text{c}}} - {f_{\text{g}}}=4.39$ kHz

The presence of two harmonic signals with ${f_{\text{g}}}$ and ${f_{\text{c}}}$ frequencies in the AO cell leads to the appearance of the beatings with the difference frequency outside the locking band. The AO system signal modulation depth and beatings shape depend on the RF and AO generators magnitudes and frequencies and feedback circuit gain. If RF generator magnitude is high (5 V) and gain is comparatively small ($\kappa =640$), the shape of the beatings is very close to the harmonic in the case of high mismatches (Fig. 3a). This statement is proved by the corresponding spectra of the oscillations presented at Fig. 3c. It consists of only three components that is typical for the harmonic signal with amplitude modulation. Figure 3b was observed right near the locking band border. The modulation depth increases till 100% with the mismatch decrease and the beatings shape slightly distorts (Fig. 3b). This distortion results in the enrichment of oscillations spectrum (Fig. 3d). Instead of three spectral components we observe nine.

Another variant of the frequency locking is realized when the external driving force has comparatively small amplitude (2 V). Higher feedback gain is required in this case ($\kappa =1450$). The locking effect is observed at narrow band $\Delta {f_{{\text{lock}}}}$ and the averaged steepness of the ${I_{\text{d}}}\left( \Gamma \right)$ dependence turns out not to be constant in time but varying slowly.

The variation in the average steepness value causes a change of the self-oscillations amplitude. The shape of the beatings distorts as a result of this self-consistent influence on the self-oscillations magnitude. The change of the resulting oscillation amplitude in time ceases to be harmonic.

This situation is illustrated by Fig. 4 where the oscillograms and spectra of the electric signal oscillations observed in the feedback circuit of the examined optoelectronic system are presented. They were obtained for the case of small RF generator signal, high gain coefficient κ, and various mismatches.

Fig. 4

The observed shape (a, b) and spectra (c, d) of the oscillations in the system feedback circuit near the frequency locking band in the case of low RF generator signal and high feedback gain. a, c RF signal magnitude 2 V, gain 1450, ${f_{\text{c}}} - {f_{\text{g}}}=3.25$ kHz; b, d RF signal magnitude 2 , gain 1450, ${f_{\text{c}}} - {f_{\text{g}}}=877$ Hz

The RF generator signal magnitude equals 2 V and feedback gain $\kappa =1450$ for both cases presented at Fig. 4. The gain growth leads to the increase AO generator self-oscillations amplitude and the decrease of the signal modulation depth. The higher gain causes also the system nonlinearity growth. Thus, the shape of the oscillations is not harmonic (Fig. 4a,b) and the frequency locking band turns out to be narrower than in the case of driving force with high magnitude. The signal spectrum contains more components (Fig. 4c,d) due to the nonlinearity. Figure 4b was observed right near the locking band border.

The entire spectrum merges into single line when the driving force frequency approaches the boundary of the locking band, since the beatings frequency tends to zero.

5 Frequency locking band

Let us now consider in detail the mechanisms that determine the locking band width in the examined system.

It is important to define the dependence of the signal amplitude in the feedback circuit to understand the processes occurring in this system. The feedback signal is determined by the photodetector signal that depends on the amplitudes of the locking (RF generator) and locked (AO generator) oscillators. The photodetector signal is being set by the AO diffraction efficiency in the collinear AO cell and depends on the amplitude of the acoustic wave aroused in it. This dependence was already mentioned above (Eqs. 2 and 10), is nonlinear, and determines the shape and magnitude of the oscillations established in the feedback circuit.

We have measured the AO interaction efficiency dependences on the amplitude of the RF generator signal and the feedback gain at first. The results of these measurements are shown in Fig. 5. The AO diffraction efficiency magnitude is normalized to the incident light intensity.

Fig. 5

Experimental AO diffraction efficiency dependences in the collinear AO cell for ${I_1}$ light component; a on the RF generator signal magnitude without feedback circuit; b on the feedback gain κ value for 2 V RF generator signal amplitude

Figure 5a represents the experimentally measured dependence of AO interaction efficiency for ${I_1}$ output light intensity component (Eq. 1) on the RF generator signal magnitude for the case when the feedback is absent. This dependence is described by the second term of Eq. 2 obtained at first time in papers [27, 28]. The experimental dependence presented in Fig. 5a coincides with the theoretical one [27]. The dependence is linear on the RF magnitude range between 0 and 6 V. The maximal diffraction efficiency is achieved near 16 V and equals 0.92.

The experimentally obtained dependence of AO diffraction efficiency on the feedback gain κ for RF generator signal magnitude 2 V is shown in Fig. 5b. This dependence differs much from one presented in Fig. 5a. The diffraction efficiency changes with gain linearly if κ < 1200 and κ > 2500.

The next step is to consider the experimentally obtained dependences of the frequency locking band on the high-frequency generator signal magnitude and gain value. They are presented in Fig. 6. The AO generator frequency locking area measured for a fixed gain value that equals 1120 and varying amplitude of the generator signal is shown in Fig. 6a. Along the vertical axis, the RF generator signal amplitude is plotted, along the horizontal axis—RF generator frequency. The locking region has a triangular shape in full accordance with the classic theory of the oscillator frequency locking [30]. The area gradually narrows with RF signal magnitude decreasing and lies between two straight lines indicating the boundaries of the capture region. Points mark the values measured experimentally; straight lines represent the data linear approximation.

Fig. 6

The experimental dependence of the AO generator frequency locking band on: a RF generator magnitude, $\kappa =1120$; b feedback circuit gain κ, 1 V RF generator magnitude

The locking band alternation for fixed amplitude of the generator 1 V and varying feedback loop gain is shown in Fig. 6b. The gain magnitude is shown in the vertical axis of the graph. The locking band changes linearly in the gain variation ranges $800<\kappa <1200$ and $2300<\kappa <4000$; the higher is the gain the narrower is the band width. The nonlinear region of the dependence is located in the range of amplification factors from 1200 to 2300. Such shape of the presented dependence may be explained by the form of the AO diffraction efficiency dependence on the feedback circuit gain shown in Fig. 5b. In fact, the shape of the curves limiting the locking band borders presented in Fig. 6b repeats the shape of the curve in Fig. 5b taking into account that gain values in Fig. 5b are plotted on horizontal axis and in Fig. 6b—on the vertical.

The AO interaction characteristic change average steepness variation that appears in the case of low RF generator signal has another effect except the beatings shape distortion—it retards the resulting oscillation amplitude alternation. As a result, the frequency of the beatings decreases faster than linearly when ${f_{\text{g}}}$ frequency is close to the ${f_{\text{c}}}$ frequency of self-oscillations. The frequency of self-oscillations is “pulled” to the frequency of the external driving force.

This effect is illustrated by the experimental dependences presented in Fig. 7. The curves show the dependence of beatings frequency on RF generator signal frequency (driving force) obtained for $\kappa =1450~$ and generator signal magnitudes 1 V (curve 1) and 2 V (curve 2).

Fig. 7

The experimental dependences of beatings frequency on the driving force frequency for two RF generator signal magnitudes (1 V—curve 1, 2 V—curve 2)

The beatings frequency depends linearly on the RF generator frequency at high mismatches ${f_{\text{c}}} - {f_{\text{g}}}$ and the dependence is transformed into nonlinear for the small mismatches. The zero beatings frequency corresponds to the AO generator frequency locking effect. This situation is in full agreement with known locking effect theory [30]. The locking band increases with driving force magnitude growth; it equals 0.67 kHz for 1 V magnitude and 1.65 kHz for 2 V RF generator signal.

Figure 8 summarizes the experimental results as shown in Figs. 6 and 7. Here, the dependences of frequency locking band (curve 1) and AO generator frequency pulling effect magnitude near the locking border (curve 2) on RF generator signal magnitude are represented for the same incident light intensity and gain values (in Fig. 8a both dependences use the same vertical scale).

Fig. 8

The experimental dependences of AO system frequency locking band (curve 1) and frequency pulling magnitude (curve 2) on the RF generator signal magnitude (a) and feedback gain (b)

The experimental measurements have shown that the AO system frequency locking band increases from 0.5 kHz at the RF generator signal amplitude 0.05 V to 8.7 kHz at the amplitude 2.5 V (more than in 17 times). The bandwidth increases linearly. The linear dependence takes place as the range of the RF signal amplitude variation corresponds to the linear part of the AO interaction efficiency dependence presented in Fig. 5a. The AO system self-oscillations frequency pulling magnitude also increases with the RF generator signal amplitude linearly and varies from 0.1 to 1.88 kHz.

The similar dependences for the case of RF generator signal constant amplitude and the varying feedback circuit gain are shown in Fig. 8b. In such variant, the frequency pulling near the locking band is much smaller; it does not exceed 0.8 kHz as the external driving force has less influence on the system. The locking effect bandwidth decreases from 10.5 to 0.5 kHz with gain increasing from 1000 to 4500 times. The decrease of the bandwidth with the increase of the gain is explained by the concomitant decrease in the RF generator influence.

The next moment that was examined in our experimental study is the RF generator signal magnitude effect on the signal modulation depth arising at the beatings near the locking threshold frequency. Figure 9 represents the AO system signal modulation depth dependence on the (curve 1) and the frequency locking bandwidth (curve 2) on the RF generator when $\kappa =1450$.

Fig. 9

The experimental dependences of AO generator signal magnitude modulation depth (curve 1) and frequency locking band (curve 2) on RF generator magnitude for the constant gain $\kappa =1450$

The frequency locking bandwidth dependence on RF generator magnitude has nonlinear shape in this case. This situation may be explained firstly by a twice larger range of RF signal amplitudes and secondly by the slightly higher feedback amplification coefficient that is also the reason of system nonlinearity increment.

6 The phase shift influence

The phase shifter is an important element of the feedback system. It is applied to fulfill the phase matching condition Eq. (3). Let us examine the phase shift influence on the frequency locking effect in this part of the paper. The electric signal phase is controlled by changing the DC voltage applied to the phase shifter Mini-Circuits JSPHS-51+ in our experimental setup. The voltage variation from 0 to 15 V allows to rotate the RF signal phase in the range from 0° to 264°.

The experimental study of the phase shift influence on the generation effect in the AO system has shown that the self-oscillations are maintained in the system for the phase range between 0° and 87°. The phase variation causes the slight change in the oscillations frequency. The broad possible phase variation limits may be explained by the specific shape of system transmission function that has the flat top of the main maximum [23, 24, 28, 29].

The phase shifter has such characteristics that the phase tuning voltage variation affects the transmission coefficient of the phase shifter and, consequently, the feedback circuit gain. The feedback gain dependence on the phase shift evaluated from the Mini-Circuits JSPHS-51+ datasheet is presented in Fig. 10.

Fig. 10

The feedback circuit gain dependence on the phase shift

Fig. 11

The experimental dependences of AO generator self-oscillations frequency, locking band, and frequency pulling magnitude on phase rotation magnitude

The presented dependence shows that the increase of the phase shift leads to the decrease of the gain. The phase variation from 0° to 86° causes 10% gain reduction.

The dependences of AO system self-oscillations frequency, frequency locking band, and pulling magnitude on the introduced phase shift were measured. The RF generator magnitude was set to be 2 V during the measurements (see Fig. 11).

The frequency locking band and pulling magnitude increase with the phase shift. This effect may be explained by the concomitant decrease of feedback gain. The change in the AO system self-oscillations frequency caused by the RF signal phase rotation does not exceed 8 kHz and can be explained by the change in the AO interaction phase-matching conditions and by the shape of the collinear AO filter transmission function with chosen geometry of polarizer and analyzer mutual orientation [23, 24, 28, 29].

7 Frequency locking effect and optical spectrum

The locked AO generator, like any other locked generators, behaves like some adjustable passband circuit with bandwidth equal to the locking band. This feature can be applied for the signal filtering. Such studies are known in optoelectronics. For example, it was proposed to use the frequency locking phenomenon to improve the characteristics of radiation sources [33].

In this section of the paper, we will consider the possibility of the observed locking phenomenon similar application as in [33] where the feedback was applied to enhance the laser radiation spectrum. Firstly, the self-excited AO system electric signal spectrum in the feedback circuit was examined in three cases. The first case is realized when the optical radiation source is a single-mode laser (a He–Ne laser was used). Second case takes place when the radiation source is the multimode cheap semiconductor laser. The third case is realized when the radiation source is the same semiconductor laser, but the frequency of the AO generator oscillations is locked by the RF generator. The spectra of the electric signal observed in these cases are represented in Fig. 12a-c, respectively.

Fig. 12

The spectra of the electric signal in the feedback circuit of the examined AO system. a Single mode He–Ne laser, b multimode semiconductor laser, c multimode semiconductor laser, AO system frequency is locked

The RF signal spectrum analyzer span band in all three cases is 20 kHz; the spectral resolution is 100 Hz. The electric signal spectrum contains only one component in the case of single-mode laser. The bandwidth of this component is defined by the spectrum analyzer transfer function. The spectrum has some finite width defined by the spectral composition of optical radiation source and phase and amplitude balance conditions in the case of multimode laser (semiconductor laser) or source with continuous spectrum. The frequency band of the spectrum presented in Fig. 12b is 8 kHz.

The spectrum of AO system self-oscillations presented in Fig. 12b is transformed into the spectrum presented in Fig. 12c when the frequency locking occurs. The only one vivid spectral component is observed. The width of this component is also defined by the spectrum analyzer transfer function. The only difference between spectra presented in Fig. 12a, c is the noise level, defined by the laser noise.

A part of the multimode laser optical radiation was extracted with a beam splitter mounted after the analyzer from the system and fed through an optical fiber to the input of the optical spectrum analyzer Agilent 86142B with 0.06 nm spectral resolution in order to determine the influence of the frequency locking effect and the electric signal spectrum in the feedback circuit on the spectral composition of the optical radiation. The spectrum of the transmitted laser radiation was measured for three cases. The first case takes place when there is no acoustic power in the AO cell. There is no AO interaction and it is possible to measure semiconductor laser radiation spectrum in the raw. The second case occurs when AO system is excited, but its frequency is not locked. The third case is realized when the AO generator is excited and its frequency is locked. The results of the measurements are shown in Fig. 13. Figure 13 shows the spectrograms obtained on the screen of the optical spectrum analyzer; curve 1 (black) represents the case of the AO generator with locked frequency; curve 2 (green) is the optical spectrum in the case of AO generator excitation, and curve 3 (red) is the spectrum of a semiconductor laser.

Fig. 13

The optical radiation spectra on the output of the AO system in three cases. 1—AO system frequency is locked; 2—AO system is exited but not locked; 3—AO diffraction is absent

The experimental results show that the intensity of the central component increases, while the magnitude of the side components near the main spectral component decreases in the case of frequency locking. The curve 1 may be considered as ${I_0}$ in Eq. (1). The influence of this component may be reduced by the increase of AO diffraction efficiency. The magnitude of the spectra central component increases due to the AO interaction.

Here, we should notice that the passband of the collinear AO filter used in the experiment without feedback is 1.3 nm [23, 24, 28, 29]. So the presented results show that the passband of self-exited or locked AO system with feedback is much narrower.

The detailed analysis of the experimentally obtained optical spectra is presented in Fig. 14. Here, the curves representing the change of spectral components intensity for the examined cases are shown. The curve 1 is the residual of optical spectra in the frequency locking case (curve 1 in Fig. 13) and spectra of light when AO diffraction is absent (curve 3 in Fig. 13). It is possible to notice that the highest increase of light intensity for curve 1 occurs in the narrow band near the optical wavelength that corresponds to the locked frequency value. The bandwidth of the maxima is 0.06 nm that equals to the spectrum analyzer maximal spectral resolution. Thus, in our experiment, the spectral band of curve 1 is defined not by AO system characteristics but by the spectrum analyzer transmission function. Curve 2 shows the residual of optical spectra in the cases of self-exited AO system without locking (curve 2 in Fig. 13) and the absence of AO interaction (curve 3 in Fig. 13); this dependence is wider than curve 1 and has lower intensity. It means that the AO interaction occurs in wider spectral region and the energy of AO system self-oscillations is not concentrated at the single component, as shown in Fig. 12b. Curve 3 represents the residual of optical radiation spectra in the cases of locked AO system frequency (curve 1 in Fig. 13) and self-exited AO system without locking (curve 3 in Fig. 13). The negative values of the residual near the main maximum indicate that the radio frequency locking really allows suppressing the optical spectrum lateral components.

Fig. 14

The variations of output optical radiation spectrum. Curve 1—the residual of optical spectrum in the frequency locking mode and spectrum when AO diffraction is absent; Curve 2—residual of optical spectra in the cases of self-exited AO system without locking and absence of the AO interaction; Curve 3—the residual of optical radiation spectra of locked AO system frequency and self-exited AO system without locking

Thus, it can be concluded that AO generator frequency locking effect actually makes it possible to improve the spectral composition of the laser radiation by suppressing the lateral components and selecting the component with maximal intensity. The optical wavelength of this component corresponds to the locked frequency of the electric signal in the feedback circuit.

8 Conclusions

We have examined the operation of the optoelectronic system consisting of the collinear AO cell and electronic feedback circuit. It was found that such system has at least three operation modes. The regeneration mode, when it operates like conventional AO filter but with the transmission function that can be tuned changing the gain and RF generator signal magnitude, self-excitation and generation modes when self-oscillations with damped or constant magnitude are excited in the system. The boundaries between the modes depend mainly on the incident light intensity and feedback circuit gain.

The new phenomenon for acousto-optics that consists in the AO system self-oscillations frequency locking was discovered and examined experimentally. This phenomenon takes place in the AO system when it operates above the self-excitation threshold. In this operation mode, the system may be treated as the acousto-optic generator [13, 14]. The frequency of the AO generator may be locked by the external RF generator that is a device with higher frequency stability. The parameters of locking effect were observed experimentally for the wide range of the examined system parameters. It was found out that the frequency band of the AO system frequency locking depends on the amplitude of the RF generator signal and the gain of the feedback circuit included in the AO system. The observed effect is in full correspondence with the general theory of frequency locking effect developed in the theory of oscillations for RF devices.

It was also found out that the increase of the RF generator magnitude at a constant gain leads to the growth of the locking band width. The increase of the gain at constant RF generator signal magnitude contrary narrows the locking band.

It was experimentally shown that the spectral composition of the RF signal in the self-excited AO system feedback circuit depends on the incident optical radiation spectrum and may be controlled by frequency locking effect. It was discovered that the AO system frequency locking effect may be used for the multimode semiconductor laser longitudinal modes selection and consequently for the improvement of the laser optical radiation spectral composition. The laser optical modes selection possibility appearance for the locked AO system indicates that it has extremely narrow passband that could not be achieved in AO devices without feedback.