**4. Problem statement**

At present, in large-dimension models of laser OEO (**Figure 1**) with the fiberoptical delay line the low phase noise level of 157 dB/Hz [5, 6] is achieved on the 10 GHz generation frequency at 1 kHz offset from a carrier.

Experimental and theoretical investigations of the power spectral density of the laser oscillator phase noise described in [16], show that reduction of the phase noise level of OEO in many respects depends on the laser phase noise level. At oscillation frequency 8 … 10 GHz at standard offsets from 1 to 10 kHz, the power spectral density of the phase noise is 120 dB/Hz … -140 dB/Hz.

Appearance on the commercial market of nano-dimension optical fibers with low losses (down to 0.001 dB per one bend, at small bend radii up to 2 … 5 mm) becomes the stimulus for improvement of OEO radiofrequency generation methods. This allows implementation of comparably small (by geometric linear maximal dimensions) fiber-optical 5*μ*s delay lines of 10 … 30 mm.

In spite of the growth of publications devoted to OEO experimental investigations, the theoretical analysis and systematization of main mechanisms of the phase noise suppression in the low-noise laser OEO was not yet described in known literature. The laser phase noise influence on the OEO radiofrequency phase noise was not researched yet.

The purpose of the presentation is to consider the main features of OEO as a low-noise generator. This includes consideration based on the study of differential equations, the study of transients in OEO, and the calculation of phase noise. It is shown that different types of fibers with low losses at small bending radii can be used as a FOLD in OEO.

Following to an approach described in **[**16], for OEO noise analysis, we consider the system in **Figure 1**, in which two different oscillation processes are developed: laser oscillations with the generation frequency of approximately 200 THz and 10- GHz oscillations in the radiofrequency network closed into a loop. At that, the frequency multiplicity is approximately 20,000.

### **5. Laser in OEO**

We will assume that the laser in OEO has high coherence and the spectral line width is much smaller than the average generation frequency, and the laser

oscillations can be considered close to sinusoidal, and with a phase component of noise with normalized amplitude noises *mLm*ð Þ*t* and the phase noise component *φLm*ð Þ*t* :

$$E\_L(t) = \left[E\_{0L} + m\_{Lm}(t)\right] \cos\left[2\pi\nu\_{0L}t + \varphi\_{0L} + \varphi\_{Lm}(t)\right].\tag{1}$$

Here *EL*ð Þ*t* , *E*0*L*, *mLm*ð Þ*t* are normalized non-dimensional quantities, respectively: the instantaneous intensity, the EMF intensity amplitude, and the EMF amplitude noise, *ν*0*<sup>L</sup>* is the average laser oscillation frequency, *φ*0*<sup>L</sup>* is the initial constant phase shift, *t* is the current time.

In the opto-electronic oscillator system, under fulfillment of excitation conditions in the electronic part of such an oscillator, the radiofrequency oscillations *u* ¼ *ug*ð Þ*t* give rise. At that, the radiofrequency signal passes to the electric MZ input from the output of a nonlinear amplifier through the C coupler during oscillation generation. The instantaneous voltage of this signal is

$$u\_{\mathfrak{g}}(t) = [U\_{10MZ} + m\_{em}(t)] \cos\left[2\pi\hat{\mathfrak{f}}t + \phi\_{0\epsilon} + \varphi\_{em}(t)\right],\tag{2}$$

where *U*0*:*1*MZ* ¼ *U*01*<sup>C</sup>* is the amplitude of fundamental oscillation at the electric input of the MZ modulator or at the C output, *f* is the oscillation radiofrequency, *ϕ*0*<sup>e</sup>* is the constant phase shift, *φem*ð Þ*t* are electronic phase fluctuations, *mem*ð Þ*t* are electronic amplitude fluctuations.

The low-noise single-mode and single-frequency quantum-dimension laser diodes or the fiber optical lasers are used as the light sources in OEO.

The laser included in the OEO structure (**Figure 1**) is formed by (closed in the loop) the nonlinear OA, the narrowband optical filter (OF), and the optical delay line. The optical oscillation frequency *ν*0*<sup>L</sup>*, which is generated by the quantumdimension laser diodes in the autonomous steady-state, can be found (under excitation condition fulfillment) on the basis of the phase balance equations solution for the steady-state optical intensity oscillations in the optical resonator and in the laser active element.

To reveal the main mechanisms of the laser noise influence on the OEO radiofrequency noise, the laser can be described by a system of semi-classical equation with the Langevin's sources of the white noise (*ξE*,*ξP*,*ξN*), relatively, for the EMF intensity *EL*, a polarization of the laser active material *Pn*, a population difference *N*. We studied the laser equation system under its operation in the single-frequency single-mode regime. At that, oscillation are linear-polarized. The main assumption for utilization of semi-classical equations is that the carrier life time on the upper operation level and the time constant *T*0*<sup>F</sup>* of the laser optical filter (OF) are much larger than the relaxation time of polarization *T*2. At that, the equation system with the Langevin's sources for the laser can be written as:

$$\begin{cases} \frac{d^2 E\_L}{dt^2} + \frac{1}{T\_{0F}} \frac{dE\_L}{dt^2} + (2\pi\nu\_{0F})^2 E\_L = \frac{2\nu^2 P\_n}{\varepsilon\_n} + \xi\_E; \\\\ \frac{d^2 P\_n}{dt^2} + \frac{1}{T\_2} \frac{dP\_n}{dt} + (2\pi\nu\_{12})^2 P\_n = \frac{p\_\varepsilon^2}{h} N E\_L + \xi\_P; \\\\ \frac{dN}{dt} = a\_{N0} \cdot f\_{0N} - \frac{N}{T\_1} - \frac{1}{h} P\_n E\_L + \xi\_N; \end{cases} \tag{3}$$

In (3) *T*<sup>2</sup> is the polarization time constant, the excited particles at the upper energy level, *T*<sup>1</sup> is the lifetime of the excited particles at the upper energy level, *Laser Opto-Electronic Oscillator and the Modulation of a Laser Emission DOI: http://dx.doi.org/10.5772/intechopen.98924*

*T*0*<sup>F</sup>* is the time constant of the optical resonator, *pe* is the combined dipole moment, *h* is the Planck constant, *ν*0*<sup>F</sup>* is the natural frequency of the optical resonator on the specific n-th longitudinal mode, *ν*<sup>12</sup> is the optical frequency of the transition, *J*<sup>0</sup> is the constant pump current, *αN*<sup>0</sup> � *J*<sup>0</sup> ¼ ð Þ *N*<sup>02</sup> � *N*<sup>01</sup> *=*ð Þ *N*02*T*<sup>1</sup> is the constant pump, *ε<sup>n</sup>* is the permittivity, *ν*0*<sup>F</sup>* is the intrinsic optical frequency of the resonator, *Pn* is the polarization of the active material, *N* ¼ ð Þ *N*<sup>02</sup> � *N*<sup>01</sup> is the population difference between the excited and unexcited levels produced by the pumping.

It should be noted that Eqs. (3) are similar to well-studied equations in the oscillator theory for the double-circuit autonomous oscillator with the inertial auto-bias chain with fluctuations.

#### **6. Compact fiber optic delay line in OEO**

At first, we would like to note that in RF FODL with geometric length of the optical fiber of 1 … 5 km, the useful volume (in which emission propagates in the regime of one transverse mode) is not more that one cubic centimeter.

The extremely small geometric dimensions and dimensions of the FODL OEO are important for its use in on-board systems of flying unmanned vehicles, since it is possible to implement effective systems for suppressing force vibrations and accelerations and to make high-precision thermal stabilization systems.

It is comparable in size to other commercial low-noise sources of microwave oscillation. **Figure 3**, and represents a diagram of the maximum sizes for various oscillators that operate in the 10 GHz frequency range: 1 - the quartz resonator (QR), 2 – the disk dielectric resonator from ceramic alloys (DR), 3 – the disk dielectric resonator from leuco-sapphire (DDLS), 4 - OEO the fiber-optical delay line (OEO RF FODL) (delay time is 10–50), 5 - the optical disk microresonator (ODR).

**Figure 3**, and shows that the smallest dimensions of the resonator have ODR. The dimensions of modern microresonators, taking into account optical input and output devices, lie in the range of about 10...100 cubic microns. **Figure 3b** shows a

#### **Figure 3.**

*Maximal dimensions of resonators and delay lines used in modern high-stable OEOs and microwave oscillators (а). Dependence of the resonator size in years (delay time is 50 μs). 1 – QR – The quartz resonator, 2 – DR – The disk dielectric resonator from ceramic alloys, 3 – DDLS – The disk dielectric resonator from leuco-sapphire, 4 - FODL – The fiber-optical delay line (delay time is 10–50 μs), 5 - ODR – The optical disk resonator. The plot of maximal overall dimensions' variations of the fiber reels in years(b).*

graph of the geometric dimensions of the FODL coil of a single-mode optical fiber (the geometric length of the optical fiber is 5...10 km).

The development of microstructural optical fiber technologies with low bending losses suggests that in a few years the maximum geometric dimensions of RF FODL will be 10...50 mm. This becomes possible because microstructured nanofibers have a minimum loss of 0.001 dB per bend at a bending radius of 2...3 mm. It becomes possible to reduce the thickness of the optical shell and reduce the required volume. In this case, it is possible to apply the technology developed by the author [16] for creating fibers by the plasma method when heating the quartz fiber support tube in the temperature range from 1000° C to 19500° C.

The improvement of the technology of heating blanks of nitrogen-doped quartz glass with the help of microwave generators, the automatic movement of plasma columns along the support tubes will lead to the creation of small-sized low-noise OEO with overall dimensions of 0.5 cm3 with a delay of optical oscillation in it of 50 microseconds.

**Figure 4** shows images of various RF FODL with the optical fiber length of 10 km used in OEO.

We note (**Figures 3** and **4**) that FODL geometric dimensions for the length of 10 km with the delay 50 *μs* is about 100х100х20 (mm<sup>3</sup> ), and dimensions of the optical disk resonator are 100х100х<sup>100</sup> ð Þ *<sup>μ</sup><sup>m</sup>* <sup>3</sup> . The record small dimensions of FODL and optical disk resonators allow manufacturing of microwave and mmwave oscillators in the miniature implementation with relatively high characteristics in noises and frequency tuning.

At the present stage, the geometric dimensions of the FORD AO are approximately equal to the resonator made of leucosapphyre. If we talk about using them in oscillators when generating oscillations with a frequency of 10 GHz.

Note the advantage of the linear topology of the fiber optic delay line FODL in contrast to the leuco sapphire crystal. The optical fiber in the FODL is less susceptible to extreme forces, which results in higher mechanical strength. These technical characteristics are very important, since on-board systems are subject to destructive shock effects and accelerations of several g.

#### **Figure 4.**

*Views of Fiber optic delay line (FODL) with the optical fiber length of 10 km with dimensions 100х100х20 mm<sup>3</sup> (a) . View of FODL with optical fiber length of 0.2 km with dimensions 20х 20х100 mm<sup>3</sup> (b).*

#### *Laser Opto-Electronic Oscillator and the Modulation of a Laser Emission DOI: http://dx.doi.org/10.5772/intechopen.98924*

Note that the FODL volume consists of only 10% of optical fiber wound on a quartz cylinder. Therefore, by reducing the critical bend radius when winding the optical fiber and the cladding diameter, it is possible to potentially significantly reduce the maximum dimensions of the FODL OEO.

Moreover, in contrast to the disk microcavity, which is used in synthesizers, the nonlinear optical Kerr effect, FODL in OEO operates in a linear mode. This means that when light passes through an optical fiber, nonlinear effects and additional optical harmonics do not appear. The width of the spectral line of the laser after passing through the optical fiber in the FOLD does not change.

**Figure 5a** and **b** shows profiles of commercial "perforated" optical fibers with the nano-dimension structure of the light-guiding thread.

Note that microstructured fibers with extremely low optical bending losses (**Figure 5**) are used in photonic devices to generate the second optical harmonic. But for this purpose, higher power (more than 20 MW) is used at the input to the single-mode fiber. As a rule, an optical amplifier is placed after the laser or modulator.

Plots of optical losses for different types of optical fibers are shown in **Figure 5a**. From this plot, we can conclude that fibers with HALF type perforation are promising for the development of small-sized delay lines.

**Figure 5a** shows the dependences of optical losses for different types of optical fibers, and **Figure 5b** shows the cross-sectional profile of a microstructured optical fiber with extremely small losses at small bending radii. Analysis of the research results and optical fibers, gives the right to declare. That the HALF type optical fiber is promising for creating compact FODL [16]. Application of special or nanodimension optical fibers (**Figure 5**) with low losses at small bend radii (1..3 mm) (0.001 dB/one bend) allows creation of miniature delay lines (1...50 *μs*) with overall sizes from 10 to 30 mm [17, 18].

Thus, when using microstructured optical fibers in OEO, it is possible to significantly reduce the dimensions of the fiber-optic delay line of the FODL.

#### **Figure 5.**

*a) 1)the plot of bending radius dependencies for standart optical fiber (OF) single-mode fiber G. 652 type with core diameter about 10 microns SMF-28e (corning). 2)the plot of bending radius dependencies for special microstructured optical fiber or hole-assisted light guide fiber (HALF). b)profile of commercial microstructured "perforated" optical fiber with the nano-dimension structure of the light-guiding core.*

#### **7. OEO differential equations**

To make the differential equations of a closed OEO circuit, it is necessary to keep in mind the following. The positive feedback circuit includes FOS (fiber optic system (FOS), which contains optical filters OF (**Figure 1**) optical amplifier (OA), modulators, photodetector (PD), electronic amplifier(A), electronic filter (F) and couple (C).

Taking into account the remark made, for the transfer function of the "feedback loop" *KFB*, it is possible to write for the case of OEO DM (**Figure 1a**):

*KDL* <sup>¼</sup> *KFB* <sup>¼</sup> *<sup>i</sup>*1*<sup>L</sup> E*2 *n* <sup>¼</sup> *<sup>J</sup>*1*<sup>L</sup> E*2 *n* , where *En* ¼ *EL* is the normalized strength in the QWLD output, which is equal to the value of the FOS input, *i*1*<sup>L</sup>* ¼ *im* is a component of the AC input voltage MZ in the OEM MZ structure and is simultaneously a component of the AC input QWLD in the OEO DM structure. We obtain the following symbolic equation for the variable component of the current:

$$J\_{1L} = \frac{\left[\cos\left(\Delta\phi\_{OF}\right)\right] \left|E\_{\pi}\right|^2 \left(1/T\_{EF}\right) \mathbf{K}\_{OF} \mathbf{K}\_{PD} p \exp\left(-p\,T\_{DL}\right) \mathbf{S}\_{NV}(f\_{1L})}{\left[p^2 + \left(1/T\_{EF}\right)p + \left(2\pi f\_{0\epsilon}\right)^2\right]}.\tag{4}$$

Taking into consideration the circuit of positive FB, we transfer to equation system in the time domain for OEO DM [16]:

$$\begin{cases} d\mathcal{E}\_{0L}^{2}/dt = \mathbf{G}\_{0} \cdot \mathcal{E}\_{0L}^{2} N\_{L} - \mathcal{E}\_{0L}^{2}/T\_{0F} \\\\ d\mathcal{N}\_{L}/dt = a\_{N00} \cdot J\_{0L} + a\_{N01} \cdot J\_{1L} - \frac{\mathcal{N}\_{0L}}{T\_{1}} - \mathcal{G}\_{0} N\_{L} \mathcal{E}\_{0L}^{2}, \\\ d\boldsymbol{\varrho}/dt = 2\pi\nu\_{0P}(\mathcal{N}\_{L}) - 2\pi\nu\_{0} + \sigma\_{0L} + \rho\_{0L}\mathcal{E}\_{0L}^{2}, \\\ \frac{d^{2}f\_{1L}}{dt^{2}} + \frac{1}{T\_{F}}\frac{dJ\_{1L}}{dt} + \left(2\pi f\_{eF\_{0}}\right)^{2}J\_{1L} = \mathcal{S}\_{\rm NY} \left[\mathcal{E}\_{0L}^{2}\mathcal{K}\_{\rm FOS}\mathcal{K}\_{\rm PD}, J\_{1L}(t - T\_{\rm DL})\right] \frac{dJ\_{1L}(t - T\_{\rm DL})}{dt^{2}}, \end{cases} \tag{5}$$

where the transfer function *KDL* ¼ *KFOSKPD*, *KFOS* is the transfer function of FOS, which contains the optical fiber of two fibers of different length, *KPD* is the transfer function of the photo-detector, which were defined in Chapter 2 in book [16].

Now we present for comparison the similar to (4.18) system from four timeequation for OEO MZ with QWLD [16]:

$$\begin{cases} d\mathcal{E}\_{0\mathcal{L}}^{2}/dt = \mathcal{G}\_{0}\mathcal{E}\_{0\mathcal{L}}^{2}N\_{L} - \mathcal{E}\_{0L}^{2}/T\_{0\mathcal{F}},\\ d\mathcal{N}\_{L}/dt = a\_{\rm N00} \cdot J\_{0\mathcal{L}} - \frac{N\_{L}}{T\_{\mathcal{L}}} - \mathcal{G}\_{0}N\_{L}\mathcal{E}\_{0\mathcal{L}}^{2},\\ d\boldsymbol{\rho}/dt = 2\pi\nu\_{\rm OP}(N\_{L}) - 2\pi\nu\_{0} + \sigma\_{0\mathcal{L}} + \rho\_{0L}\mathcal{E}\_{0\mathcal{L}}^{2},\\ \frac{d^{2}U}{dt^{2}} + \frac{1}{T\_{F}}\frac{dU}{dt} + \left(2\pi f\_{\rm eF\_{0}}\right)^{2}U = \mathcal{S}\_{\rm NY}\left[\mathcal{E}\_{0L}^{2}K\_{\rm MZ}K\_{\rm PO}\mathcal{K}\_{\rm PD} \cdot U(t - T\_{\rm FOLD})\right]\frac{dU(t - T\_{\rm FOLD})}{dt^{2}}.\end{cases} \tag{6}$$

#### **8. Dynamics of transients in OEO DM**

Let us consider the transient process of the exit to the steady-state mode of the free generation of OEO DM at representation of the oscillator in **Figure** 1**a**. As it had *Laser Opto-Electronic Oscillator and the Modulation of a Laser Emission DOI: http://dx.doi.org/10.5772/intechopen.98924*

been mentioned earlier, such a structure is described by the system of differential Eqs. (5). Let us describe in more detail the results of the study of the system of differential Eqs. (5) for OOO DM (**Figure 1a**).

On the base of mentioned OEO differential Eqs. (5), the analog model of OEO was constructed presented in **Figure 1a**.

**Figure 5** presents the obtained solutions of system (5) and shows plots of the square of the intensity, population, and pump current, as well as phase portraits in the transient mode under the influence of a constant pump current in the form of a step pulse.

The one of difficulties at solution of (4) finding at the analog modeling is the determination of the nonlinearity of the RF nonlinear amplifier (A) in order to "compensate of the multiplicative QWLD nonlinearity".

At the same time, in the analog models, the following laser parameters were taken in solution (5) (the same as in Chapter 3 [16]): for the mesa-strip laser with the thickness of the dielectric film *<sup>d</sup>* = 1.2 *<sup>μ</sup>m*: *<sup>g</sup>*<sup>0</sup> = 10<sup>3</sup> , *<sup>τ</sup>D*=7,2�10�<sup>12</sup> s, *<sup>I</sup>*thr = 12 mA, *εsh* = 0. The values of parameters of QWLD are: the life time of carriers *T*<sup>1</sup> ¼ *τ<sup>n</sup>*<sup>1</sup> ¼ 0.5�10�<sup>9</sup> s, the threshold level population difference is 10<sup>18</sup> 1/cm<sup>3</sup> , the life time of photons or the time constant of the optical resonator *<sup>T</sup>*0*<sup>F</sup>* <sup>¼</sup> *<sup>τ</sup>ph* <sup>¼</sup> 1,2�10�<sup>12</sup> s, the volume of the QWLD active zone is 10�<sup>11</sup> cm�<sup>3</sup> .

The modes with and without delay in the OEO feedback ring were investigated. (**Figure 6**).

The pulsations of the square of the intensity and population of the laser in the simulation of the transition process OOO DM are established.These dependences are shown in **Figure 6**. The nonlinear distortions are related to the multiplicative nonlinearity of the laser. And their level depends on the value of the DC pump current of the laser.

The period of laser pulsations in transients, which is approximately 0.4 ns, depends on the level of the pump current and is determined by the carrier lifetime.

#### **Figure 6.**

*The transient process in OEO and in the laser. The constant pumping level* J*<sup>0</sup> = 30 mA. Activation of pumping occurs in the time moment* t *= 0. Time-functions of normalized values: а) the population difference* N *(10181/ cm<sup>3</sup> ), b) the normalized square of strength (*E*)2 = (*E*0*L*)2 , (1.0 point = 1 mW), c) AC component of pumping current, d) the normalized square of strength (*E*)2 = (*E*0*L*)2 , (1 point = 1 mW) at initial part [0,50]. The transient process is presented with the exit to the limit cycle on the time diagram* E*0*L*,* N*0. The scale on the time axis* t*: 5 points = 0,1 ns. The setting time for the laser oscillations is 40 points (or 0,8 ns). The setting time of OEO RF oscillations is on the time axis* t *50 … 250 points (or 40 ns).*

The setting time of the laser oscillations is 0.8 ns (or from 0 to 40 points in **Figure 6a** and **b**). The time of setting the RF oscillations of the OEO on the time axis t is 40 ns (or from 50 to 100 points in **Figure 6**). The oscillation frequency in the steady-state mode is close to the natural frequency of the electronic filter (F) and was approximately 10 GHz.

#### **9. OEO DM system of the laser emission**

As follows from the theory of oscillations, in a transient process, a special or critical point can be a stable node, with real and negative roots p1 and p2 of the characteristic equation of the system of differential Eqs. (5). When the feedback coefficient in the OEO ring increases, the special point A becomes unstable. In this case, the characteristic roots p1 and p2 must be positive.

As shown in **Figure 7**, there is a stable limit cycle around the unstable point A. The generation is impossible if the isoclinic lines *F*1ð Þ *NL* and *F*2ð Þ *EL* are not intercepted.

The process of establishing the laser radiation oscillation ends, and then, due to the positive feedback in the OEO DM ring, there are increasing oscillations of the laser charge current, which also modulate the inverse population of the laser. This leads to subsequent oscillations of the square of the electromagnetic field strength of the laser.

If there is a single singular point A in the upper half-plane (**Figure 7**) the condition of self-excitation of the laser is fulfilled. Therefore, the excitation of the OEO DM occurs in a gentle way. This is also true for the case of an arbitrary odd numbers of nontrivial singular points. OEO DM generation may not be possible if the isoclinic lines and are not intersect (**Figure 7**). If the number of singular points is even (if there are two singular points points), the condition of self-excitation of the OEO may not be met.

Laser generation can only be excited in a "hard" way, which is initiated by a pulse from an external source.

If we consider the case of an unstable point A, the oscillatory system develops a process of oscillation growth. The nonlinearity of the electronic amplifier limits the growth of oscillations and the conditions for the existence of a limit or closed stable cycle are met (**Figure** 7**b**).

#### **Figure 7.**

*Transition mode scenario in OEO DM. A phase portrait of the normalized square force is presented. a) the phase portrait of the normalized square strength (*E*)2 = (*E*0*L*)<sup>2</sup> and the population difference* N *and . Y-axis – The normalized square strength (or the intensity), Х-axis – The population. The scale on the Y axis is 1.0 = 1 (V/m)2 . The scale on the Х axis is 1.0 = 1 mW.* N *(1.0 point on the scale is 1018 1/cm<sup>3</sup> ). b) the enlarged image of the phase portrait is shown and the transient development with the exit to the limit cycle on the time diagram* E*0*L*,* N*0. The scale on the time axis* t *is 5 points = 0,1 ns.*

*Laser Opto-Electronic Oscillator and the Modulation of a Laser Emission DOI: http://dx.doi.org/10.5772/intechopen.98924*

If we consider the question the stability of a given oscillation cycle, then its existence is determined by the sign of the partial derivatives of the right-hand sides with respect to one variable when analyzing the characteristic equation obtained in [16], Chapter 4.

The limit cycle is stable only if the corresponding expressions for the coefficients are greater than zero [16].

When considering the hard-excited OEO DM mode, it is necessary to note more complex dynamics, and the picture of the phase plane in the transition mode becomes diverse. The number of singular points (intersection points of isoclinic lines) becomes even. Therefore, long-term generation of OEO DM oscillations in hard mode is possible only when an external generator is operating.

The transients of the oscillation tuning in the OEO DM with no lag in the positive feedback loop are shown in **Figure 8**.

Modeling has shown that strong nonlinear distortions caused by the multiplicative nonlinearity of the laser occur at a large oscillation amplitude [16] and their level is determined by the choice of the operating point or the direct pump current of the laser.

For example, a level of 1 ... 10% of the maximum possible values is performed when a constant bias current is selected at a level of 1.5 to 5.0 exceeding the threshold laser pump current. It is established that the nature of the transient process is determined by the type of non-linearity of the electronic amplifier, the

**Figure 8.**

*Plots of function of the dependences of the square of the electromagnetic field strength of the laser in the optical channel, the population of carriers and the pump current. In abscissa axis – The normalized time, in ordinate axis – <sup>а</sup>)* <sup>N</sup> *– The inversed population; b) (*E*)2 = (*E*0*L*)<sup>2</sup> - intensity of laser; c), <sup>О</sup>scillations of OEO the electrical current of laser pumping.*

selection of the natural frequency of the electronic filter, and the delay value in the FODL. At the same time, the duration of the OEO DM oscillation transition process changes significantly.

Positive feedback is included in the DE system, taking into account the photodetection of optical radiation, selectivity in the radio frequency, and nonlinear gain on a nonlinear amplifier.

What is new in the analysis of the OEO DM operation is that the Lotka-Volterra laser differential equations for the optical field intensity, inverted population, and optical phase with positive selective feedback with a delayed argument can be reduced to a single van der Pol differential equation for the pump electric current.

From our studies of differential Eqs. (5), it follows that in OOO DM, singlefrequency and two-frequency modes of relaxation oscillations are possible.

For a stable single-frequency mode of OOO DM generation, the following conditions must be met: a twofold excess of the electron filter time constant (F) over the electron relaxation time constant in the active layer of the laser.

#### **10. Laser phase noise and OEO phase noise**

Expressions for SSB PSD of the laser phase noise do not reflect the important property of the laser oscillating system: a presence of the relaxation resonance on the frequency *ν*00*<sup>L</sup>* at the offset from a carrier *ν*0*<sup>L</sup>*, i.e., at *F*00*<sup>L</sup>* ¼ 2*π ν*ð Þ <sup>00</sup>*<sup>L</sup>* � *v*0*<sup>L</sup>* . We can take this "resonance peak" into account at linearization of system [16] with account of the population equation. At that, the expression for SSB PSD of the laser phase noise take a form:

$$\text{S}\_{\text{PL}}/P\_{0L} \approx \frac{\text{S}\_{\text{SL Im}}}{\left(F T\_{0F} \right)^2} + \frac{\text{S}\_{\text{LE}} D\_{11}^{\ \ 2} + \text{S}\_{LN} D\_{22}^{\ \ 2}}{T\_1^4 \left(\left(F^2 - F\_{00L}^2\right)^2 + \left(F a\_{00l} \right)^2\right)^2} \tag{7}$$

where *<sup>F</sup>*00*<sup>L</sup>* <sup>¼</sup> ð Þ <sup>1</sup>*=T*<sup>1</sup> ð Þ ð Þ *<sup>T</sup>*0*<sup>F</sup>=T*<sup>1</sup> *<sup>α</sup>*<sup>0</sup> � <sup>1</sup> <sup>1</sup>*=*<sup>2</sup> , *α*<sup>0</sup> is an excess of DC laser pumping over its threshold value, *α*00*<sup>l</sup>* is a damping decrement, *T*<sup>1</sup> is the lifetime of the excited particles at the upper energy level, *D*<sup>11</sup> and *D*<sup>22</sup> are the constant coefficients, and *SLE*,*SLN* are relatively, spectral densities of impacts in [15, 16] the Langevinian noise of the laser *ξE*, *ξN*, relatively. Here *ξ<sup>E</sup>* is the noise of the EMF intensity *EL*, *ξ<sup>N</sup>* is the noise of a population difference *N*. **Figure 9** shows the curve 1 of SSB PSD of the

**Figure 9.** *Laser phase noise SSB PSD (curve 1), and OEO phase noise SSB PSD (curve 2).*

laser phase noise calculated by formula (7) for *SSL* Im ≈*SLED*<sup>11</sup> 2 *SLND*<sup>22</sup> <sup>2</sup> <sup>≈</sup> � 105dB/ Hz, *<sup>F</sup>*00*<sup>L</sup>* <sup>≈</sup>14 kHz, the time constant of the laser resonator *TOF* <sup>¼</sup> <sup>10</sup>�<sup>7</sup> s.

### **11. OEO as the EMF correlator**

We studied OEO as a correlator of two random variables *ξ*1, *ξ*<sup>2</sup> with probability density at the input of the correlator *p*<sup>1</sup> *ξ*1, *ξ*<sup>2</sup> ð Þ. The random variables in the extraction of two optical harmonics [16] are phase noise *ξ*<sup>1</sup> ¼ *φ*10*Lm*ð Þ*t* and *ξ*<sup>2</sup> ¼ *φ*20*Lm*ð Þ*t* corresponding harmonics with the amplitudes *A*1*E*0*<sup>L</sup>* and *A*2*E*0*L*. The resulting phase noise of the current in the load of the PD photodetector is the result of statistical averaging.The distribution probability *p*2ð Þ*η* determines the appropriate correlation function of the output process. Where *f*ð Þ*η* is the nonlinear characteristic of the photo-detector, *ητ* ¼ *η*ð Þ *t* � *τ* of the *η*ð Þ*t* process in the correlator output. At that, *p*<sup>1</sup> *ξ*1, *ξ*<sup>2</sup> ð Þ defines the probability density *p*2ð Þ*η* of the statistical process in the correlator output (**Figure 2**) of the OEO MZ (**Figure 1b**) at the *closed loop* of OEO MZ for *τ* >*TFOS*.

The spectral density of radio frequency OEO oscillations *SRFL*ð Þ *F* is determined by the formula:

$$\begin{aligned} S\_{\rm RFL}(F) &= \\ \frac{E\_{0L}^4}{2} \frac{U\_{10\text{MZ}}^2 \text{S}\_L}{2} K\_{\Gamma \text{PN}}^2 \cdot \left[ \mathbf{1} - \frac{A\_2}{A\_1} \exp\left( -\frac{2(\Delta T\_M + T\_{\rm FOS})}{T\_c} \right) \right], & \end{aligned} \tag{8}$$

where *SPL*- SSB PSD of the laser phase noise (7), *K*<sup>2</sup> <sup>2</sup>Γ*PN* is the coefficient of the noise suppression, which depends upon *TFOS*, the laser optical power *E*<sup>2</sup> <sup>0</sup>*<sup>L</sup>*, the transfer function of FODLj j *KFODL* , *<sup>U</sup>*<sup>2</sup> <sup>10</sup>*MZ* is the square of the AC amplitude in the MZ electrical input. When considering OEO as a correlator of two random variables *ξ*1, *ξ*<sup>2</sup> with a probability distribution density at its input *p*<sup>1</sup> *ξ*1, *ξ*<sup>2</sup> ð Þ, we can conclude from (8) that the SSB PSD of the laser phase noise is significantly determined by the ratio of the delay time in FOS and the laser coherence time, and it significantly depends on the ratio harmonics amplitudes *A*2*=A*1.

#### **12. OEO phase noise**

From Eqs. (7) with account for nonlinear characteristic of the amplifier A (**Figure 1a**) as a cubic polynomial *iA*ð Þ¼ *<sup>u</sup> <sup>α</sup><sup>e</sup>*0*<sup>u</sup>* � *<sup>β</sup><sup>e</sup>*0*u*<sup>3</sup> (where *<sup>u</sup>* is the instantaneous voltage at the amplifier input, and the average slope of this characteristics is *σ<sup>U</sup>* ¼ *σ<sup>e</sup>*<sup>00</sup> � ð Þ 3*=*4 *β<sup>e</sup>*00*P*0*<sup>G</sup>*) and we can obtain through laser and delay line parameters the power of the Opto-Electronic oscillator radiofrequency generation *P*0*<sup>G</sup>*:

$$P\_{0G} = \frac{a\_{\epsilon00}}{\beta\_{\epsilon00}} \left( 1 - \frac{1}{P\_{0L} |K\_{F0LD}| a\_{\epsilon00} \beta\_{\epsilon00}} \right) . \tag{9}$$

We introduce the designation: *Y*00*=P*0*<sup>L</sup>* ¼ *yM*½ � 1 þ *FTEF =*j j *KFODL* ,

where *yM* is the input normalized conductivity of the MZ modulator. Similarly to (7) for laser PSD, we obtain from the general symbolic Equations [16] the equation for SSB PSD *S*<sup>Ψ</sup> of the OEO phase noise.

SSB PSD *S*<sup>Ψ</sup> reduced to the radiofrequency oscillation power *P*0*<sup>G</sup>* is determined by expression derived in [14–16] according to the Evtianov-Kuleshov approach.

The *K*<sup>2</sup> <sup>Γ</sup>*PN*<sup>2</sup> coefficient depends on the delay time in the optical fiber and on the laser optical power and it is equal

$$\begin{split} K\_{\text{IPN}}^2 &= \\ & \quad \left\{ \left( Y\_{00} / P\_{0L} \right) \left[ \sqrt{2} \sin \left( \pi / 4 - F T\_{\text{FOS}} \right) \right] - \sigma\_U \right\}^2 \\ & \quad \left\{ \left[ \left( Y\_{00} / P\_{0L} \right) \right]^2 - \left( Y\_{00} / P\_{0L} \right) \cdot \left( 1 + \sigma\_U \right) \cos \left[ F T\_{F \text{OS}} \right] + \sigma\_U \right\}^2 . \end{split} \tag{10}$$

where *yM* is the input normalized conductivity of the MZ modulator. Then the function of OEO phase noise PSD can be represented [14–16] as

$$S(F) = \frac{S\_{\Psi}}{P\_{0G}} = \frac{K\_{\Gamma PN}^2 C\_A h v N\_{sp}}{P\_{0G}},\tag{11}$$

where *CA* - the constant coefficient, *Nsp* is a number of spontaneous photons received by PD.

Plots of (10) are shown in **Figure 9**. which are limited functions of OEO of the phase noise PSD with account of small noises of PD and the A amplifier, at laser phase noise for the offset frequency 1 kHz equaled to about �120 dB/Hz, at laser power 30 mW, the delay of *TBC* <sup>¼</sup> <sup>5</sup> � <sup>10</sup>�6s (the OF length is 1000 m), *<sup>σ</sup><sup>U</sup>* <sup>¼</sup> 1. We see that the first peak is defined by the laser phase noise PSD, and average suppression of the phase noise for 50 kHz offset is more, that �10 dB/Hz.

It should be noted that at the optical fiber length of 2 km the uniform suppression of the laser phase noise is achieved in the offset range 1 … 50 kHz.

Calculation of the phase noise suppression factor *K*2ð Þ *F* suppression factor according to (**9**) is presented in **Figure 10** *σ<sup>U</sup>* ¼ 1*σ<sup>U</sup>* ¼ 1 : *TFOS=TF* ¼ 1, *P*0*<sup>L</sup>*∣*KFODL*∣ ¼ 2 (curve 1); *TFOS=TF* ¼ 10, *P*0*<sup>L</sup>*∣*KFODL*∣ ¼ 2 (curve 2); *TFOS=TF* ¼ 10, *P*0*<sup>L</sup>*∣*KFODL*∣ ¼ 4 (3 curve). It can be seen that increase of delay time from *TFOS=TF* ¼ 1 (curve 1) to *TFOS=TF* ¼ 10 (curve 2) results in reduction of *K*<sup>2</sup> factor more than 10 times in the rated offset frequency *F* � *TF* range 0.05 … 0.5.

It is shown that at OF length, the further reduction of the OEO phase noise is possible using the PLL (phase-locked loop) system. Calculation results are wellagreed with experimental dependences of OEO phase noise PSD, which can be found in [14–16]. Here, we should remind that first publications on research of frequency stability in OEO with the help of FOLD were fulfilled in 1987–1989 at

**Figure 10.**

*The phase noise suppression factor K*<sup>2</sup> <sup>¼</sup> *<sup>K</sup>*<sup>2</sup> <sup>Γ</sup>*PN (9) versus the rated offset frequencyF* � *TF: TFOS=TF* ¼ 1, *P*0*<sup>L</sup>*∣*KFODL*∣ ¼ 2*, (curve 1); TFOS=TF* ¼ 10, *P*0*<sup>L</sup>*∣*KFODL*∣ ¼ 2 *(curve 2); TFOS=TF* ¼ 10, *P*0*<sup>L</sup>*∣*KFODL*∣ ¼ 4 *(3 curve).*

Radio Transmitter Dept. of Moscow Power Engineering Institute (now NRU MPEI) while the OEO circuit was offered in [12, 13].
