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Abstract
The development of the optical frequency comb (OFC) generation and practical application methods is one of the most important and rapidly developing areas of the modern optical electronics. One of the comb types is acousto-optical (AO) OFCs. This type of OFCs is obtained by the multiple passage of an optical signal through a closed loop containing an acousto-optic frequency shifter (AOFS). Despite the fact that AO OFCs have been studied quite intensively lately, the published papers did not focus on the influence of the main element, the AO cell used as AOFS, parameters on the characteristics of the obtained optical comb, primarily on the comb spectral width, number of spectral components and its envelope shape. In this paper, we perform a theoretical analysis of all possiblities in paratellurite crystal wide-angle AO diffraction geometries in order to determine the most suitable for the application as AOFS in a frequency shifting loop.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Optical frequency combs (OFCs) represent an optical signal with spectrum containing a number of lines with clearly defined spectral intervals between them [1]. Such combs have become key tools in many practical applications such as optical precision measurements [2], molecular spectroscopy [3], dual-comb spectroscopy [4] and others [5–8].
First OFC sources were developed for the near-infrared (NIR) spectral range, while frequency combs in the visible (VIS), ultraviolet (UV) or mid-infrared (MIR) are in great demand [7–13]. Many techniques for optical combs generation depending on the particular application were invented in recent years [5,14–23].
One of the widely used techniques for OFC synthesis is based on the frequency shifting of a seed CW laser with electro-optic (EO), acousto-optic (AO), or single side band (SSB) modulators inserted in the optical feedback loop [24–32], so-called, frequency shifting loop (FSL). AO cell may also play the role of SSB modulator, in this case it is called AO frequency shifter (AOFS). The frequency shift appears in the process of AO diffraction due to the Doppler effect and may be positive or negative depending on the use of minus or plus first AO diffraction order.
OFC generated applying EO modulator are called an electro-optic combs. The OFC is called acousto-optic comb if the AOFS is used. Both methods pave the way to tunable comb generator with both tunable central frequency and the comb spacing. The AOFS application also allows to tune the OFC envelope [33]. The spacing between OFC spectral components is being set by the radio-frequency (rf) function generator driving the modulators and may vary from MHz to tens of GHz for EO OFCs and from MHz to GHz for AO OFCs (the upper frequency in the AO devices is limited by the ultrasound attenuation in the AO cell that is proportional to the ultrasound frequency in the second power). The comb position and the span are defined by the seed laser and the gain range of the optical loop.
Both EO combs and AO are perfectly suited for dual-comb spectroscopy since coherence of two combs may be achieved by a single laser pumping two optical loops with slightly different characteristics [34–36]. Electro-optic dual-comb technique measuring broadband optical waveforms at ultrahigh single-interferogram refresh rates has been demonstrated recently [37].
While OFCs based on electro-optical modulators are widely applied, acousto-optic devices based OFCs are faintly studied. However, AO OFCs have some advantages comparing with EO combs. AOFSs are relatively simpler as the EO cell for effective operation requires the application of higher voltages compared to the AO cell. Moreover the EO frequency shifters produce an inevitable third harmonic, resulting in cross-talk effects in the generated comb lines [38]. It was also shown recently that AO OFCs may be applied for multiheterodyne spectroscopy with high spectral resolution [39].
Application of AO devices in the FSL may allow obtaining optical combs with central frequencies in the range from UV to medium IR (this spectral range is confidently mastered by modern AO devices). Due to the AO interaction peculiarities AO OFCs may be generated effectively in the UV and VIS ranges of the optical spectrum and this may be treated as one of the most important features of AO combs.
By installing several AOFS in series, which shift the frequency of the optical signal in opposite directions, one may obtain combs with extremely tight spacing between spectral components [39]. The new concepts for photonic generation of broadband optical combs, consisting of relatively flat spectra with $>1000$ modes and providing tunable linear rf chirp rate, were proposed [40].
Typical AO comb generation circuit contains an AO modulator, an optical amplifier and a band-pass optical filter [24]. One of the variants of this scheme is shown in Fig. 1.
An AO modulator may be used as AOFS, then, in the process of AO interaction, spectral components will appear with both longer and shorter wavelengths than pump laser has. One can also apply the AO cell in the AO interaction geometry that corresponds to tunable AO filter to obtain the SSB modulation.
The tunable bandpass filter may be inserted into the loop to limit the spectral band of the amplified spontaneous emission and to control the number of spectral lines of the OFC if needed. The control of OFC spectral lines amount may be also realized by changing the optical amplifier gain or the parameters of AO interaction in the AO cell.
The half-wave plate (HWP) is another important element of the system. It is applied to change the polarization of diffracted light. Since the AO interaction is accompanied not only by a shift in the light frequency due to the Doppler effect, but also by a change in the type of optical mode (anisotropic AO diffraction). The HWP rotates the polarization of the diffracted optical radiation so that it coincides with the incident light polarization.
The optical radiation passes through the optical loop many times and each time, in the process of AO diffraction, acquires a frequency shift that equals to the ultrasound frequency aroused in the AO cell.
A significant drawback of AO devices, when used in OFCs generation systems, is that there is a dependence of the diffracted light wave propagation direction on the wavelength of the incident light radiation, the so-called scanning effect. Thus, different spectral components will be inserted into the optical feedback circuit with different losses. Of course, for optical combs with a narrow spectral width, this effect will not be of great importance, but as the width of the combs increases, it will become more and more significant. This effect can be reduced by choosing the orientation of the output optical face of the AO cell, but, as was shown in [41], it cannot be fully compensated. Thus, diffracted light beam scanning will limit the spectral width of the optical comb.
The OFC generation scheme has been proposed recently applying an AO cell of collinear AO interaction geometry [33]. In such cells, the incident and diffracted light beams, as well as the acoustic wave, propagate along the same direction in crystal and the scanning effect is absent. In the same paper it was shown also that the collinear AO cell allows realizing the AO OFCs generation without using an rf generator by introducing not only an optical feedback circuit, but also an electrical feedback circuit. In this case, the signal in the electric feedback circuit is aroused due to the optical heterodyning effect upon interference of first (or minus first) and zero diffraction order light beams as they are of different frequencies. The frequency difference equals to the AO phase matching frequency.
Unfortunately, collinear AO cells are manufactured from materials that have comparatively low AO figure of merit and require the application of high rf signal power for efficient operation.
We should note that in the existing papers on the generation of AO OFCs, pay very little attention to the operation of AOFS and does not examine the AO diffraction characteristics influence on the generated OFCs parameters. It is stated that the AO modulator is only responsible for the interval between the spectral components of the comb [42]. Of course, this is correct if the passband of the AO device is much wider than the passband of the applied bandpass filter or the spectral width of the comb [42], but in the general case this is not quite correct, and AO cell will have the main impact on the shape and width of the comb. This was demonstrated in [33].
This paper is devoted to the analysis of the characteristics of wide-angle AO cells [43–46] fabricated on the base of a paratellurite crystal in order to determine the AO interaction geometries optimal for the generation of AO OFCs.
Paratellurite crystal [47,48] was chosen as it is the most important AO material with very high AO figure of merit. The higher is the AO figure of merit, the less acoustic power is required for the effective operation of the device. On the one hand, this simplifies the electrical part of the system, and on the other, it increases the thermal stability of the system (the frequency of AO phase matching varies with the temperature of the AO crystal due to a change in the propagation velocity of the acoustic wave [49,50], this causes the appearance of AO mismatch that influence the OFC envelope shape [33]).
The wide-angle geometry of AO interaction was chosen as it is less sensitive to the angle of incidence of light on the AO cell, which means that it simplifies the optical alignment of the system and reduces its sensitivity to chromatic aberrations.
In this paper we were focused on the on the possibilities offered by the application of AO cells for the OFCs generation. We do not take into account the characteristics of real optical amplifiers, first of all, their operation band, assuming that the optical feedback loop simply has some gain and it is constant. We also do not take into account the AO cells electrical characteristics variation in a wide range of ultrasound frequencies, also assuming that they are constant.
2. Basic relations
2.1 Possible AO OFCs generation schemes
2.1.1 Conventional FSL scheme description
Acousto-optic diffraction, as it was shown in [33], allows to implement two fundamentally different optical combs generation schemes. The first scheme, shown in Fig. 1 is the typical FSL scheme with the possible introduction of additional elements, such as bandpass filter. In this embodiment, the AO cell piezoelectric transducer is being fed by the rf generator signal. The frequency of this signal $f_{g}$ is defined by the AO interaction phase matching condition:
where $\vert \overrightarrow {k_{i}}\vert =2\pi n_{i}/ \lambda$, $\vert \overrightarrow {k_{d}}\vert =2\pi n_{d}/ \lambda$ and $\vert \overrightarrow {K}\vert =2\pi f_{g}/V$ are the wave vectors of incident and diffracted light and ultrasound wave correspondingly. Here $V$ is the ultrasound velocity, $\lambda$ - optical radiation wavelength, $n_{i}$ and $n_{d}$ are the refraction indices of the paratellurite for incident and diffracted optical beams.In the general case the ultrasound wave vector direction does not coincide with the direction of the Poynting vector $\overrightarrow {S}$ (the presence acoustic energy walk-off).
Using Eq. (1) it is possible to obtain the relation for the AO phase matching frequency $f_{c}$:
The RF generator frequency determines the spectral interval between the components of the comb. So if the seed laser radiation has $f_{l}$ frequency, the frequencies of the other comb spectral components will be $f_{n}=f_{l}\pm nf_{g}$ where $n$ is the number of light passes through the FSL and the sign depends on the applied AO diffraction order.
The selected seed laser wavelength $f_{l}$ corresponds to the single ultrasound frequency $f_{c}$ defined by Eq. (2) at which the AO diffraction efficiency will be maximal. For other ultrasound frequencies $f_{g}$, AO phase matching condition will be violated and the AO mismatch will appear. In this case the wave vector equation will look like:
where $\overrightarrow {\eta }$ is the mismatch vector that is usually directed perpendicularly to the AO interaction area borders ($\overrightarrow {\eta }\perp \overrightarrow {S}$). It was shown in [51,52] that the AO mismatch in acoustically anisotropic media is defined by the following equation:It was shown in [33] that the AO mismatch presence helps to increase the number of spectral components in the OFC approximately in two times, so varying ultrasound frequency one is able to tune the spectral width and shape of the OFC.
For the first scheme of the AO OFCs generation it is possible to write the following equation describing the OFC spectrum after $N$ passes of light through the system [33]:
AO cell transmission follows the well known equation:
where $\Gamma$ is the Raman-Nath parameter, proportional to the acoustic wave power and $A_{m}=\sqrt {\Gamma ^2 + R_{m}^2}$. Here $R_{m}=\eta _{m}l$ is the dimensionless AO mismatch and $l$ is the AO interaction length. AO cell transmission $T_{1}=1$ if $f_{g}=f_{c}$ and $\Gamma =\pi$.The process of OFC generation in the scheme with FSL is illustrated in Fig. 2. At the first passage through the closed-loop system light from seed laser with wavelength $\lambda$ diffracts by the acoustic wave with $f_{g}$ frequency with dimensionless mismatch $R_{1}$ (for definiteness, it is assumed that $-1$-st diffraction order is applied) and obtains wavelength shift $\Delta \lambda$ that corresponds to $f_{g}$ frequency. At the next passage optical wave with shifted wavelength which appeared during the previous passage through the loop diffracts again by the same acoustic wave but with another mismatch $R_{2}>R_{1}$. This process is repeated further until the $N$-th passage.
Thus Eqs. (5) and (6) describe the influence of the AO cell on the OFC spectral components magnitudes. So, in the FSL scheme applying AOFS the generated OFC spectral band will be the wider, the higher is the AO cell passband, and one does not need an AO cell with high spectral selectivity for broad AO OFC generation.
2.1.2 Description of the FSL scheme with optoelectronic feedback
Applying AOFS it is also possible to realize another OFC generation scheme [33] represented in Fig. 3. In this case, the external rf generator is replaced by the second - electric feedback circuit [53–56]. The electric feedback circuit contains photodetector, rf amplifier and phase shifter. The input signal in the electric feedback circuit is formed due to the heterodyning effect that occurs during the interference of zero and first orders of AO diffraction, since the frequency of light at AO diffraction changes by the frequency of the ultrasonic wave due to the Doppler effect. The HWP is mounted just at the AO cell output to rotate the diffracted light polarization and make AO interference possible [53,54].
Amplitude modulation of light intensity with an ultrasound frequency is detected by a photodetector displaced in the area of transmitted and diffracted light beams overlapping. The optical heterodyning may be also realized with the beam splitter directing the part of diffracted beam intensity to the input of optical adjustment scheme which brings diffracted and zero optical beams together in the photodetector plane. Of course, the magnitude of the heterodyne signal will depend on the diffracted beam scanning effect, but it is possible to compensate this influence by tuning the electric feedback gain.
The zero diffraction order is suppressed in the optical feedback loop as zero and first diffraction orders propagate in different directions, so when the optical input of the loop is optimally adjusted for first order, the zero will inserted into the optical fibre with large losses.
The signal from the photodetector is fed to the input of an electric feedback. The output of the electric feedback circuit is connected to the piezoelectric transducer of the AO cell. It was shown in [55–60] that with a sufficient value of the feedback gain, the system turns into an AO generator [53–56,61], with a self-oscillation frequency determined by the seed laser wavelength and the geometry of AO interaction.
The passage of light through the AO cell and further along the optical feedback gives rise to new components of the optical spectrum. The appearance of which leads to the excitation of new rf signal components. Thus, in the AO cell, with a sufficient gain of the electric feedback circuit, there will be as many acoustic frequencies as the optical radiation spectrum contain. It was shown in [33] that the following equation describing the OFC spectrum after $N$ passes of light through the system takes place:
It is possible to notice that the AO cell transmission function $T$ is excluded in this case, as the AO phase matching condition if fulfilled for each spectral component of the OFC in the generation mode. It was also shown [33] that the comb generated in such a system will be chirped, the magnitude of the chirp will be determined by the birefringence of the AO cell material and the AO interaction geometry.
The process of OFC generation in the scheme with two feedback loops is illustrated by Fig. 4. The system operates in the AO generator mode. At the first passage through the closed-loop system light from seed laser with wavelength $\lambda$ diffracts by the acoustic wave with $f_{c1}$ - the AO phase matching frequency for this optical wavelength (for definiteness, it is assumed again that $-1$-st diffraction order is applied) and obtains wavelength shift $\Delta \lambda _{1}$ that corresponds to $f_{c1}$ frequency. At the next step the component with $\lambda +\Delta \lambda _{1}$ arouses acoustic wave with corresponding frequency $f_{c2}$ and obtains wavelength shift $\Delta \lambda _{2}>\Delta \lambda _{1}$. This process is repeated further until the $N$-th passage. Thus, the spectral interval between the OFC components increases with $N$ growth and OFC turns out to be chirped.
Since two schemes of AO OFCs are considered, the analysis and optimization of wide-angle AO diffraction geometry is carried out for application in both of them. In the common FSL scheme, the main criteria is the AO cell passband. The wider it is, the higher the spectral width of the comb may be achieved. An important parameter is also AO figure of merit, as the higher it is the lower rf signal power is required for efficient system operation.
In the second scheme, the AO cell bandwidth does not play a significant role, since for each optical signal spectral component its own acoustic wave exists with a frequency corresponding to the phase matching condition. Nevertheless, the narrow passband is preferred to prevent the optical spectrum components diffraction by the adjacent ultrasonic frequencies. AO figure of merit should be high in order to reduce the total acoustic power circulating in the system.
An analysis of the AO phase matching frequency dependence on the AO interaction geometry is performed to determine the possible values of the spectral interval between the OFC components.
2.2 Wide-angle AO diffraction
Wide-angle AO diffraction is a specific type of AO interaction commonly applied for the spectral filtering of divergent optical beams [62]. The wave vector diagram corresponding to such type of AO interaction is shown Fig. 5. The peculiar feature of such diffraction geometry is that the tangents to the interaction medium refractive indices surfaces, drawn at the points of their intersection with the wave vectors of the incident and diffracted light waves, are parallel.
Here $\alpha$ is the crystal cut angle that sets the $\overrightarrow {K}$ orientation relative to the $\left [ 110 \right ]$ axis, $\varphi _{i}$ and $\varphi _{d}$ define the propagation directions of incident and diffracted optical waves.
In mathematical terms, the wide-angle AO diffraction corresponds to the case when the derivative of Eq. (2) $\delta f_{c} / \delta \Theta _{B}$ equals zero.
Calculations of the Bragg angles frequency dependencies for various paratellurite crystal cut angles in the range from $0^{\circ }$ to $20^{\circ }$ and optical wavelength 0.63 $\mu$m fulfilled with Eq. (2) are presented in Fig. 6.
Fig. 6. Frequency dependence of Bragg angles in the case of anisotropic diffraction in paratellurite crystal for 0.63 $\mu$m optical wavelength.
The presented dependencies have two features. Firstly, for each cut angle there are two points at which the condition $\delta f_{c} / \delta \Theta _{B}$ is fulfilled [63–65]. These points are marked in figure by circles. The first point corresponds to relatively low frequencies of ultrasound and small Bragg angles, so we will call it low-frequency (LFP). The second is observed at high frequencies and Bragg angles; therefore, we will call it high-frequency (HFP). The second feature is that the wide-angle AO diffraction geometry is observed only in a certain range of the AO crystal cut angles $\alpha$. For paratellurite, this range lies between 0 and 18.9 degrees slightly depending on optical radiation wavelength.
The dependencies illustrating the influence of paratellurite cut angle $\alpha$ on the wide-angle AO diffraction phase matching frequency are shown in Fig. 7. These curves define the spectral spacing between the OFC components. The simulations were done for three optical wavelengths 0.45 $\mu$m, 0.63 $\mu$m and 1.50 $\mu$m. Here and further solid lines correspond to low-frequency (LFP) wide-angle AO diffraction and dotted lines to the high-frequency wide-angle point (HFP).
Fig. 7. The dependencies of low-frequency (LFP) and high-frequency (HFP) wide-angle AO diffraction phase matching frequency on the paratellurite crystal cut angle.
The phase matching frequency in both cases increases with angle $\alpha$. The dependencies for LFP are linear unlike the HFP curves. The ultrasound frequency increases with optical wavelength decreasing. The acoustic wave frequency varies in approximately 17.5 times for LFP geometry and in 1.7 times for HFP geometry.
The general view of the wide-angle AO cell is represented in Fig. 8.
An acoustic wave is excited by a piezoelectric transducer of $d$ length and propagates in the paratellurite crystal with acoustic energy walk-off defined by angle $\psi$ depending on the cut angle $\alpha$. AO cell length is $L$ and width $D$. The direction of light beam propagation is determined by the Bragg angle $\Theta _{B}$ and the angle of incidence of the light on the acoustic wave $\phi _{i}=\theta _{i}+\psi$. In the examined case the walk-off angle $\psi$ takes negative values as it is being counted from $\left [110\right ]$ axis in the opposite direction compared with the $\Theta _{B}$ and $\phi _{i}$ angles.
It was already mentioned that it is possible to apply several methods to obtain the input signal of the feedback electric circuit. Firstly, the HWP could be placed at the AO cell output in the area of optical beams overlapping and a photodetector behind it. Then the HWP will rotate the polarization of diffracted light and interference between zero and diffracted beams will be observed in the photodetector plane. Thus, it is possible to obtain a signal with amplitude modulation at the frequency of ultrasound aroused in the AO cell. Secondly, it is possible to mount a beam splitter in order to separate a part of the diffracted light beam and the optical adjustment system to make the zero and diffracted light beams interfere in the photodetector plane. Beams overlapping will determine the effectiveness of the heterodyning. At the same time, the overlapping does not affect the system operation principle and its decrease only leads to a decrease in the photodetector signal magnitude at the ultrasound frequency and, consequently, to an increase in the threshold gain of the feedback electric circuit.
The dependencies of Bragg angles $\Theta _{B}$ and angles of light beam incidence $\phi _{i}$ on $\alpha$ angle for both LFP and HFP are shown in Fig. 9. This figure also represents the well known dependence of acoustic energy walk-off angle on the direction in the tellurium dioxide $\left ( 1\overline {1}0 \right )$ plane.
The $\Theta _{B}$ decreases from $87^{\circ }$ ($\Theta _{B}=90^{\circ }$ corresponds to the case of collinear AO diffraction) to $35^{\circ }$ for HFP case when $\alpha$ increases from $1.3^{\circ }$ to $18.9^{\circ }$. We do not consider the cut angle values lower than $1.3^{\circ }$ as in this case the AO cell dimensions will be extremely large for HFP case. For the LFP AO diffraction the opposite situation takes place - $\Theta _{B}$ increases from approximately $1.3^{\circ }$ to $35^{\circ }$.
The walk-off angle increases also with angle $\alpha$. The maximal angle $\psi =57.25^{\circ }$ is observed for $\alpha =16.5^{\circ }$.
Taking into account the dependencies shown in Fig. 9 and applying simple geometric considerations it is possible to estimate the minimal AO cell dimensions needed for the practical realization of all possible LFP and HFP AO diffraction geometries. It was assumed also that the light beam should completely intersect the ultrasound beam that is 2 cm width.
Both AO cell length and width increase with angle $\alpha$ if we consider low-frequency diffraction. The extremely low minimal length $L$ (less than 0.1 cm) needed for LFP realization for $\alpha <3^{\circ }$ is explained by low $\Theta _{B}$ values that take place in this case. This mean that the AO diffraction geometry close to the orthogonal. The AO crystal width in this case is defined mainly by the piezoelectric transducer length $d$.
The $L$ growth with increasing $\alpha$ for LFP diffraction is explained by an increase of Bragg angle and light beam angle of incidence shown in Fig. 9. The $\phi _{i}$ growth means that the area of optic and ultrasound beams intersection increases also.
Fig. 9. Dependencies of HFP and LFP AO diffraction Bragg angles $\Theta _{B}$, light beam angles of incidence $\phi _{i}$ and acoustic energy walk-off $\psi$ on the paratellurite crystal cut angle $\alpha$
The opposite situation takes place for the HFP diffraction. It was already mentioned that when $\alpha =0^{\circ }$ and $\Theta _{B}=90^{\circ }$ the collinear AO diffraction takes place. When $\alpha =0^{\circ }$ the acoustic walk-off is absent so the area of intersection is infinite. With the cut angle growth the walk-off appears and increases more and more, so the intersection area dimensions reduce. Thus, both AO cell length and width reduce with $\alpha$ increasing. We should also note that the simulation results presented in Fig. 10 take place for $d=2$ cm. If the transducer length variation causes the AO cell dimensions changing in the same way.
Fig. 10. Wide-angle TeO$_{2}$ AO cell parameters $L$-length and $D$-width needed for LFP and HFP diffraction realization.
3. Results of simulations
Thus, we have defined the ultrasound frequencies corresponding to wide-angle AO diffraction for various crystal cut angles $\alpha$. We have also estimated the AO cell dimensions needed. In this section of the paper we will examine the AO wide-angle diffraction characteristics that are of most importance for the AO OFCs generation. These are the AO figure of merit $M$ and AO cell passband $\delta \lambda$. These characteristics define the acoustic power needed for the effective AO cell operation, the higher is $M$ the lower power is needed to achieve maximal diffraction efficiency, and the number of OFC spectral components that can be obtained with this cell.
It is known that AO figure of merit is defined by the following equation:
where $V$ is ultrasound wave velocity, $\rho$ AO cell material density and $p_{eff}$ is the the effective photoelastic constant.The effective photoelastic constant for tellurium dioxide $\left ( 1\overline {1}0 \right )$ plane is defined as [66]:
The transmission function passband $\delta \lambda$ is defined as the transmission function width at the 0.5 level. The transmission function shape is evaluated with Eqs. (4) and (6) for the given optical wavelength $\lambda$. The simulations were carried for $\lambda = 0.45$ $\mu$m, 0.63 $\mu$m and 1.50 $\mu$m. The results obtained are presented in Fig. 11.
The presented dependencies have the same shape but differ in numerical values. It is possible to notice that both $\delta \lambda$ and $M$ reduce with increasing $\alpha$ for the case of LFP diffraction. Passband variation equals 174 times, while $M$ decreases in 7 times
The opposite situation is observed for HFP case. Both $M$ and $\delta \lambda$ increase with $\alpha$. The $\delta \lambda$ increases in 9 times and $M$ - in 123 times.
AO figure of merit reduces also with optical wavelength, passband, on the contrary, increases. Both these effects are typical for AO diffraction.
Thereby, if one needs AO cell with broad transmission band the LFP case with small cut angles should be chosen. If one needs extremely high spectral selectivity the HFP geometry also with small cut angles should be used, but in this case $M$ values will be comparatively small. Nevertheless, it is possible to find AO diffraction geometries, among the examined, that will provide much higher spectral resolution and $M$ values than it is achieved in conventional collinear AO filters fabricated from calcium molybdate or lithium niobate.
We should also note that the results presented take place for the piezoelectric transducer length $d=2$ cm and $\delta \lambda$ is proportional to it.
With results presented in Fig. 7 and Fig. 11 it is possible to estimate the number of spectral components in the OFCs generated with the examined AO cells. We introduce the following method to compare various wide-angle AO cells applicability for OFCs generation. It was shown in [33] that OFC spectral width is defined mainly by the AO cell transmission band, so the wider is the $\delta \lambda$ the wider OFC will be generated. The OFC spacing $\Delta \lambda$ is defined by ultrasound frequency that equals $f_{c}$ in the absence of mismatch. So the number of spectral lines in the case when AO phase matching condition is fulfilled will be defined as:
where and $c$ is the light wave velocity.If $f_{g}\neq f_{c}$ the AO mismatch appears that causes the increase of OFC spectral band. It was shown in [33] that mismatch appearance may increase the OFC width in approximately 2 times.
The results of OFC spectral lines amount $N$ estimations carried with Eqs. (11) and (12) are shown in Fig. 12.
One may notice that for the LFP case $N$ decreases with $\alpha$ and on the contrary for the HFP case. The highest $N$ values in LFP case exceed $10^{5}$ lines in the examined spectral region. Analysing the results presented in Fig. 7 and Figs. 10, 11, 12 it is possible to conclude that the maximal $\delta \lambda$ and $N$ will be observed for the cut angles range between $1.5^{\circ }$ and $4^{\circ }$. For transducer length $d=2$ cm it is possible to achieve as much as $10^{5}-10^{6}$ OFC spectral components. The OFC spectral band varies between 1-10 nm for $\lambda =0.45$ $\mu$m, 10-50 nm for $\lambda =0.63$ $\mu$m and 50-100 nm for $\lambda =1.50$ $\mu$m. This range of crystal cut angles correspond also to AO figure of merit values about $1000\cdot 10^{-18}$ s$^{3}$/g that are extremely high, so low rf power is needed for the system operation. The only disadvantage of this cut angles range is low ultrasound frequencies that means small spacing between OFC spectral components.
Fig. 11. AO figure of merit $M$ and transmission functions passband $\delta \lambda$ dependencies on cut angle $\alpha$ for LFP and HFP AO diffraction.
Fig. 12. The dependence of OFC spectral lines amount in the AO cell passband $\delta \lambda$ on cut angle $\alpha$ for LFP and HFP AO diffraction.
If one needs higher spectral spacing the larger cut angles should be selected. As it is shown in Fig. 7 by choosing $\alpha$ the spacing for LFP may be tuned in more than 17 times. One is also able to increase the spectral width of the OFC and number of lines reducing transducer length and introducing AO phase mismatch by choosing $f_{g}\neq f_{c}$. Thus, if we set $d=0.4$ cm the $\delta \lambda$ and $N$ will increase in 5 times. It was shown in [33] that the presence of mismatch may increase the spectral width and $N$ in about 2 times. So it is possible to obtain as many ans $10^{6}-10^{7}$ lines for the OFC with spectral width varying from 100 nm for $\lambda =0.45$ $\mu$m to 1000 nm for $\lambda =1.5$ $\mu$m. Of course, in real OFCs generation systems it will be difficult to achieve such OFC spectral band values primarily due to the characteristics of optical amplifiers.
The HFP AO diffraction geometry was proposed for the application in the OFC generation scheme with both optical and optoelectronic feedback circuits. It was mentioned that in this case the less spectral lines are in the AO cell passband the better it is for the system operation. We should also keep in mind that the $M$ values should be high enough to lower the system self-excitation threshold.
It turns out that for the HFP geometry it is better to use the cut angles between $1.5^{\circ }$ and $4^{\circ }$ too. In this case $N$ varies from a few dozens to several hundreds depending on the seed laser wavelength. Taking into account the system transmission band narrowing observed when it operates in the generation mode [56] it should be enough for correct operation.
The AO figure of merit values for these case of wide-angle AO diffraction are not high, for example for $\lambda =0.63$ $\mu$m $M\approx 10\cdot 10^{-18}$ but this is almost 5 times higher then calcium molybdate crystal has for collinear diffraction.
4. Conclusion
In this paper we have examined all possible variants of wide-angle AO interaction in tellurium dioxide crystal in order to determine the most suitable ones for the application as the AOFS in the FSL optical frequency combs generation scheme.
Application of these AO cells in two OFC generation schemes was considered. The first scheme is conventional FSL scheme with AO cell operating as the SSB modulator. The second scheme applies two feedback loops - the optical feedback and optoelectronic feedback which replaces the external rf generator. It is shown that the AO cell transmission function defines the OFC spectral width in the conventional scheme. The double feedback scheme allows to eliminate the AO cell transmission function influence.
The spectral spacing between OFC components is being set by the rf generator frequency in the first scheme and by the AO phase matching frequency in the second case.
The characteristics of AO wide-angle diffraction were simulated for the seed laser optical radiation wavelengths varying from 0.45 $\mu$m to 1.5 $\mu$m and 2 cm AO cell transducer length with taking into consideration paratellurite acoustic anisotropy and optical rotatory power. Both high- and low-frequency wide-angle AO interaction geometries (HFP and LFP correspondingly) were examined.
The simulation results showed that by choosing the crystal cut angle one is able to tune the OFC spectral interval in 17.5 times for LFP geometry and in 1.7 times for HFP case.
The highest number of OFC spectral lines is achieved in the LFP geometry for small cut angles. It is possible to obtain OFC containing as much as $10^{5}-10^{6}$ spectral lines for the 2 cm transducer and to increase these values in about 10 times by reducing the transducer length and introducing AO mismatch.
The HFP AO interaction geometry gives the possibility to obtain extremely narrow spectral transmission that is suitable for the double feedback OFC generation scheme. In this case, as well as for LFP geometry, the crystal cut angles that do not exceed 4$^{\circ }$ should be chosen.
The AO figure of merit values observed for HFP geometry are much lower than for LPF case but still significantly higher then those that can be achieved in calcium molybdate of lithium niobate crystals.
Thus, wide-angle AO diffraction in TeO$_{2}$ crystal gives excellent opportunities for the application in AO OFCs generation systems.
Funding
Russian Science Foundation (18-72-00036).
Disclosures
The authors declare no conflicts of interest.
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