Nonlinear Dynamics of the Phonon Stimulated Emission in Microwave
  Solid-State Resonator of the Nonautonomous Phaser Generator
Nonlinear Dynamics of Phonon Stimulated Emission
Nonlinear Dynamics of the Phonon Stimulated Emission in Microwave Solid-
State Resonator of the Nonautonomous Phaser Generator* 
D.N. Makovetskii 
A. Usikov Institute of Radio Physics and Electronics, National Academy of Sciences of Ukraine 
12, Academician Proskura St., Kharkov 61085, Ukraine 
The microwave phonon stimulated emission (SE) has been investigated in a solid-
state resonator of a non-autonomous acoustic quantum generator (phaser). 
Branching and long-time refractority (absence of the reaction on the external pulses) 
for deterministic chaotic and regular processes of SE were observed in the 
experiments with weak and strong electromagnetic pumping. The clearly depined 
increase of the number of the independently co-existing SE states as the pumping 
level rise has been observed both in physical and computer experiments. This 
confirms the analytical estimations of the branching density in the phase space. The 
nature of the SE pulses refractority is closely connected with such branching and 
reflects the crises of strange attractors, i.e. their collisions with unstable periodic 
components of the higher branches. 
The stimulated emission (SE) of microwave phonons has been first observed in experiments on 
quantum hypersound amplification, when in 1960-ies it was reported for dielectric single crystals doped 
with paramagnetic ions of the iron group [1] - [3]. The hypersound amplification is due to inversed 
population of the ion spin levels, transitions between which are allowed for the spin-phonon interaction, 
and is, in fact, an acoustic analog of the linear maser amplification of electromagnetic fields [4], [5].  
At the same time, the nature of this effect – quantum generation of microwave phonons – has 
remained not fully understood for a long time. There was some inertia of thinking, with analogies 
proposed [6] between the photon quantum generator (phaser) and electromagnetic maser generator in the 
same way as it had been done for amplifiers.  
Experimental studies of phonon SE in Ni2+:Al2O3 and Cr3+:Al2O3 crystals ([7] - [10]) have shown 
that phaser generation mechanism is actually much closer to the processes in optical lasers than those in 
maser generators.  Really, due to very low velocity of hypersound (as compared with the velocity of 
light) the wavelength of acoustic SE in a microwave solid-state resonator is about 1-3 µm, i.e., it 
corresponds to the near-infrared range of electromagnetic radiation. The resonator Q-factor  CAVQ  is, 
like in typical lasers, ≈CAVQ 10
6, i.e., again it is by several orders of magnitude higher than in 
electromagnetic masers.  As a result, observed SE power spectra, transient processes, stability of 
stationary modes and other properties of a phaser are very much similar to these characteristics for solid-
state lasers at SCAVS TTT 21 >>>> . Here SS TT 21 ,  are, respectively, axial and transversal relaxation times 
of active (inverted) centers, CAVCAVCAV QT Ω= , where CAVΩ  is the operation mode frequency of the 
resonator with active medium, and SESCAV Ω≈Ω≈Ω , SΩ  is the quantum transition frequency for 
inverse population difference, SEΩ  - SE carrier frequency. 
However, there is also a fundamental difference between phaser and optical lasers, which is 
related to the intensity of intrinsic quantum noises (intrinsic, or spontaneous, emission) spontJ . As for a 
phaser  =ΩS 3 - 10 GHz, i.e., five orders of magnitude less than for a laser, the relative level of the 
intrinsic component in the first generator is by ~15 orders lower than in the second one (due to 
spont SJ ∞ Ω ). In fact, this allows us to consider a phaser as a deterministic dynamic system practically in 
all practically available ranges of SE intensity. 
 Investigation of complex (including irregular) determined movements in dynamic systems is 
now steadily proceeding.  This is related not only to the fundamental character of this problem, but also 
to prospects of practical application of the methods involved and results obtained in studies on the 
mechanisms of associative memory [11], [12], for development of computer image and symbol 
recognition [13], [14] etc. The multiplicative noises (which also include intrinsic radiation) affect the 
behavior of dynamical systems in a very non-trivial manner [15], e.g., leading to roughing of phase space 
topology [16] and substantial reduction of the associative memory volume. Such considerations 
substantially increase our interest in phaser generator-type systems, where, as distinct from optical lasers, 
internal noise level is low for practically all feasible combinations of controlling parameters (CP). 
 In the present paper, we summarize the data of our experimental studies of a non-autonomic 
phaser generator, which were partially reported in our separate presentations [17, 18]. We also report 
numerical modeling of SE dynamics on the basis of a deterministic generation model that is an acoustic 
analog of the laser model [19]. The experiments were carried out using a ruby phaser as described earlier 
[7] - [10] under modulation of various CP (pumping, static magnetic field H , intensity of hypersound 
injJ injected into the resonator, etc.) in the frequency range =mω  30 - 3000 Hz.  
Fig. 1. Energy level diagram for Cr3+:Al2O3 active crystal. 
The Cr3+:Al2O3 spin system is formed upon splitting of the principal energy level of Cr3+ ion 
with orbital quantum number L = 0 and spin S = 3/2 in a trigonal crystalline field and static magnetic 
field H (Fig.1). Electromagnetic pumping of the Cr3+:Al2O3 spin system was carried out with a klystron 
of power =pumpP 12 mW at frequency PΩ = 0P PΩ + ∆Ω  using a cylindrical cavity on the 011H  mode. 
The cavity was tuned to frequency 0PΩ = 23 GHz. This corresponds to the resonance magnetic field 
0H = 3,9 kOe at the angle =ϑ 54°44' between H  and the third order crystallographic axis C  of ruby 
(when conditions for symmetrical pumping are realized – see Fig.1). Detuning of the pumping source 
P∆Ω  varied within several tens of MHz. A solid-state microwave Fabry-Perot acoustic resonator 
(FPAR) of Q-factor ≈CAVQ 5⋅10
5 ( ≈CAVT 10
-5 s) was placed inside the pumping cavity along its axis. 
FPAR is a cylinder of synthetic rose ruby with optically planar and parallel end sides – acoustic mirrors. 
Upon one of the FPAR acoustic mirrors, deposition a thin textured ZnO film with Al sublayer 
was applied by vacuum evaporation, this film being the main element of the hypersound converter. 
Using this film, detection was carried out of SE signals having carrier frequency =ΩSE  9,16 GHz and 
intensity ( )tJ . The SE signal is formed on 2 3E E↔  transition of Cr3+ ions under electromagnetic 
pumping of 1 3E E↔  и 2 4E E↔  transitions (diagram of energy levels for Cr
3+:Al2O3 is presented in 
Fig.1, spin levels iE  and respective wave functions iψ  are numbered in the order of increasing 
energy). By means of the same film, external longitudinal hypersound of frequency ≈Ω≈Ω Sinj  9 GHz 
and pulse intensity ≤injJ 300 mW/cm
2 was injected into the ruby crystal. All experiments were carried 
out at 1.8 K. The axial relaxation time on the active transition was ≈ST1 0,1 s, the transverse relaxation 
time ≈ST2 10 ns. 
The third-order axis C of ruby was coincided with FPAR geometrical axis O , and conical 
refraction was observed for the transverse component of the phonon SE[7]; thus, the contribution of the 
longitudinal component to SE intensity J  was overwhelming. As the texture axis of ZnO film was also 
parallel to axes C  and O , the injected hypersound was also purely longitudinal. It should be stressed 
that the momentary value ( )tJ  is determined by SE processes in FPAR and is (alongside with the inverse 
population difference n ) a dynamic variable, while injJ  is among CP. The time-averaged SE intensity 
value J  can be different from zero only if ( )Gpumppump PP > . Here 
pumpP   is the value of electromagnetic 
pumping power at frequency =ΩP 23 GHz, where phaser generation begins. The pumping parameter A  
(the ratio of inverted population difference at frequency ≈ΩS 9 GHz to its threshold value) is equal to 
unity. 
At low levels of pumping ( A -1 ≈ 1/30) and small periodic perturbation of the system, ( mk << 1, 
where mk  is modulation coefficient of one of CP, e.g., pumpP  , H  or injJ ) regular modulation of the 
recorded SE signal J  with period mT  is always observed in experiments all over the modulation 
frequency range mω = 30 - 3000 Hz. This corresponds to a soft birth (emergence) of a limiting cycle of 
period mT  on the main (zeroth) generation branch 0B . When mk  is increased, the SE output signal 
modulation coefficient smoothly rises up to values normally not higher than 60 - 70 %. Then in a certain 
critical point ( )0m mk k=  the value of SER  increases stepwise to nearly 100 %, not acquiring, however, a 
pulse character and preserving the same SE period mT . If we change the sign of mdk d t , hysteresis will 
be observed of the ( )SE mR k  dependence – the jump of  SER  back (decreasing) is at 
( ) ( )1 0
m mk k< , indicating 
the existence of an above-located hard branch 1B  of IR. 
Thus, even in the range of mk  where SE is not yet of pulse character, a bistability of periodic 
movements is observed. Co-existence of the soft 0B  and hard 1B  branches with one and the same 
modulation period of integral intensity of the generated photon flux is realized. If now, being on the 
branch 1B , to increase (and not to decrease) mk , at ≈−1A 1/30 consequent doubling of the SE period 
will be observed according to Feigenbaum’s scenario, ending in the transition to the chaotic pulse 
modulation mode ( )tJ . Varying the system detuning over magnetic field 0H H H∆ = −  within several 
Oersteds, we succeeded in finding two qualitatively different types of irregular (chaotic) modulation with 
pseudoperiods mT  and 2 mT  (pseudoperiod is the maximum interval between the neighboring pulses 
rounded to the nearest integer value [20]). 
As for the soft branch 0B , at ≈−1A 1/30 it has strictly regular character with period mT  in the 
all range of ( )( )00 ,m mk k∈ , cutting itself off at the transition point to the hard branch 1B  independently of 
the H∆  detuning (though specific values of  ( )0mk  naturally change upon variation of H∆ ).  
As it was shown by experiments with modulation of each of the selected CP ( pumpP , H , injJ ) 
the further increase in mk  at low pumping level ( A -1 ≈ 1/30) leads to appearance of new hard branches 
2 3, , …B B , which begin by an appropriate periodic (which is now pulse) SE mode with period mMT , 
where, respectively, =M  2, 3, ... , and most often end by the chaotic mode after the series of 
Feigenbaum’s doublings. It should be noted that coexistence region of the SE branches was typically 
narrow for small 1−A , and at <∆H 4 Oe no bistability of periodical movements with 1>M  was 
observed at all. 
The picture becomes qualitatively different when pumping parameter A  is increased up to 
values of ≈A 1.5 – 1.8. Here strong overlapping of periodical regions occurs on branches 2 3, , …B B , 
accompanied by a hysteresis under slow scanning of any of non-modulated CP (including mk , p∆Ω , 
etc.). Fig.2 shows a typical picture observed for coexistence of the regular SE modes.  
a) b) 
Fig. 2. Regular SE of microwave phonons (experiment): coexistence of the modes 
with doubled (a) and tripled (b) sequence periods of generation pulses. 
The SE modes with doubled (Fig.2a) and tripled (Fig.3b) sequence periods of generated phonon 
pulses are coexisted for common modulation period mT  and a fixed set of other CP. Transitions 
2 3m mT T↔  are of hysteretic character when p∆Ω is scanned within about 1 MHz. The dashed line 
indicated the pumping modulation period mT = 1/120 s. The amplification factor of the recording 
equipment in Fig.2a is four times larger than in Fig.2b. 
It should be underlined that the observed movement with period 2 mT  (Fig.2a) is not an 
intermediate stage of the Feigenbaum’s scenario on the first hard branch, but is a primary SE mode for 
the second hard branch – just as the primary mode with period 3 mT  for the third hard branch (Fig.2b). 
This follows from the fact that at 0mdk /dt <  both for 2B  and 3B   SE frustration (suppression) points 
mk ↓  and 
mk ↓  were found experimentally, where after sign inversion of mdk /dt  (but with other CP 
unchanged) the disrupted movements were not restored. It was possible to restore the modes 2 mT  
(Fig.2a) and 3 mT  (Fig.2b) only by stepwise variation of other CP – the most convenient was to apply a 
hypersound monopulse injJ  up to 10 mW/сm
2 at ( ) ( ) ( )2,3 2,3 2,3mm mk k k↑ ↓= + ∆ . It appeared that restoration 
occurred even at ( ) ( )2,3 2,3210m mk k
∆ ≈ , with further increase in injJ  not leading to new modes for such low 
overcriticities.  
It follows from these experimental data that at ( )2m mk k ↑≈  the second hard branch, which is shown 
in Fig.2a, is also the toppest, i.e., located above the others. Similarly, at ( )3m mk k ↑≈  it is the third hard 
branch that is the toppest (Fig.2b). From the other side, the fact that ( ) ( )2,3 2,3m mk k↓ ↑≈  directly indicates that 
the periods 2 mT  and 3 mT  for these branches are the initial ones. In other words, SE refractority (i.e., 
absence of SE pulse during time R mT KT= , where 1K > ) for hard branches 2B  and 3B  appears already 
at the moment of their birth. The presence of refractority is the main qualitative difference between the 
mode with period 2 mT  on branch 2B  from the mode with the same period 2 mT  (after Feigenbaum’s 
doubling) on branch 1B . Really, for the latter mode doubling of the period ( 2SE mT T= ) is accompanied 
only by changes in the amplitude ratio of the neighboring SE pulses that follow with a unit pseudoperiod 
2R SE mT T T= =  (no refractority). At the same time, for branch 2B  we have 2R SE mT T T= = . 
a) b) 
c) d) 
Fig. 3. Chaotic SE of microwave phonons (experiment): one realization with 
constant refractority (a) and three realizations with variable refractority (b) 
Another type of refractority is observed at large P∆Ω , when, in particular, the effective pumping 
parameter is also decreasing. Fig.3 shows SE oscillograms recorded at =∆ΩH 1 Oe, mω = 120 Hz in the 
cases of detuning ≈∆Ω p 4 МHz (oscillogram a) and ≈∆Ω p 30 МHz (oscillograms b-d). A transition 
takes place from the so-called helical chaotic mode with pseudoperiod 1R mT T=  (when each period of 
pumping modulation mT  is matched by one SE pulse) to the chaotic mode with variable refractority, 
where pseudoperiods R mT KT=  (1 ≤≤ K 5) alternate irregularly. Amplification factor of the recording 
equipment in Fig.3 is the same for all oscillograms a-d. Besides large refractority intervals that appear on 
oscillograms b-d, sharp increase in the amplitude of pulses (as compared with oscillogram a) is observed, 
which is an evidence of substantial expansion of the attraction range of the corresponding attractor in the 
phase space of the system. 
Subsequent measurements have shown that refractority of variable duration  like shown in Fig.3 
(oscillogrammes b-d) are observed for chaotic SE pulse modes even more often that purely helical 
chaotic movements with the unit half-period shown in Fig.3a. It is essential that the observed transitions 
between SE chaotic modes with different maximum half-periods be of non-hysteretic character, as 
distinct from the above-described hysteretic transitions between independent regular branches with 
different periods of movement (like those shown in Fig.2). To clear up specific mechanisms of branching 
of SE modes and appearance of variable refractority in chaotic realizations of pulse phonon generation, 
we should address the deterministic SE model in a high Q-factor resonator with active centers, for which 
the condition 1S CAVT T>>  is also met (this model is an acoustic analog of the deterministic model for 
class “B” lasers [19]). In this paper, we will limit ourselves in our numerical modeling by a relatively 
simple case of a three-level active system. Although we used experimentally a more complex four-level 
system (Fig.1), which allowed us to substantially enhance the inversion. 
 Let us consider a three-level spin system 3 2 1E E E> > , where signal transition 1 2E E↔  (S-
transition) is allowed for interaction with a coherent microwave acoustic field (hypersound) of specified 
direction and polarization, and the pumping transition 1 3E E↔  is allowed for magneto-dipole 
interactions with the corresponding pumping field. The third transition  (F-transition) is an idle (no-load) 
one, and its frequency ( )3 2 /F E EΩ ≡ −  is, by definition, not equal to the frequency of S-transition 
( )2 1 /S E EΩ ≡ −  or its multiple values. In addition, FΩ  is not an integer divisor of the P-transition 
frequency ( )3 1 /P E EΩ ≡ − . 
 In the impurity paramagnetic Cr3+:Al2O3, where the ground state (orbital quantum number L = 0, 
spin S = 3/2 – see Fig.1) is weakly bonded with the lattice (just by spin-orbital interaction), longitudinal 
relaxation times 1 1 1, ,S P FT T T  for all the above-mentioned transitions at low temperatures are many orders 
of magnitude higher than the respective latitudinal relaxation times 2 2 2, ,S P FT T T . Consequently, it is 
possible to choose the amplitude of microwave electromagmetic pumping field 1PH  such as to make 
populations 1n  и 3n  of spin levels 1E   equal, but with broadening of these levels still the same as at 
1 0PH = . In other words, we assume that the following two inequalities hold simultaneously: 
               ( )2 11; 1P P P PZ T / T Z>> <<       (1) 
or, in a somewhat different form 
                 1 21P P P PY T Y T>> >> ,         (2) 
 where PZ   is saturation factor of  P-transition 
                 2 21 2 1
4P P P P P
Z T T г H= ,       (3) 
and PY  is interaction probability of the pumping field with the P-transition. Here Pγ  is effective 
gyromagnetic ratio for this spin transition (accounting for direction and polarization of vector 1PH ).  
 Besides this, phonon life time CAVT  in an FPAR of high Q-factor does normally meet the 
requirement 
     1 2S CAV ST T T>> >> .       (4) 
Similar inequalities are true for photon life times in a microwave electromagnetic pumping resonator. In 
addition, it is usually assumed that the pumping resonator has no intrinsic frequencies in the vicinity of 
FΩ . 
 In this case, to calculate the difference in populations 2 1N n n∆ ≡ −  on S-transition, one could 
use the balance approximation equations, which, accounting for equality 3 1n n=  , can be presented in 
the following form: 
12 1 32 1 2 2
21 2 31 1 1 1
W n W n W n Y N
W n W n W n Y N
= + − − ∆
 = + − + ∆
,      (5) 
 where 2 21 23W W W≡ + ; 1 12 13W W W≡ + ; SY  is the probability of interaction of the hypersound field with 
the spin S-transition; ijW  are probabilities of longitudinal spin relaxation. Following [1], [7], we obtain 
       22 2 22 12 1S S u S SY T q U F / Z / T= ≡ ,      (6) 
 where /u S uq V= Ω ; U and uV  are, respectively, the hypersound amplitude and phase velocity; 12F  is 
spin-phonon interaction factor for S-transition at specified values of  the hypersound polarization and 
propagation direction. General expressions for  12F  can be found in [1]. For the case of latitudinal 
hypersound propagating along the crystallographic axis O  of the third or higher order (as it is in the ruby 
phaser at 9 Ghz [7]), we have from [1] 
           12 1 2Hsu
F ψ ψ
.        (7) 
Here zzε  is a component of elastic deformation tensor; 1ψ  и 2ψ  are wave functions corresponding to 
spin levels 1E  and 2E ; Hsu  is the spin-phonon interaction Hamiltonian. 
 Using the approximation 3P Bk θΩ <<  (where  Bk  is the Boltzmann constant; θ  - temperatute 
of the thermostat), we find from (5) 
     
( ) ( )
2 S P
N Nd N
∆ − ∆∆
= − ∆ + ,       (8) 
 where 
( ) ( ) -10 4 2 2S F P E cN s W f W p W W Nθ θ θ∆ = − + − ; 
S ET / W= ;   1 2 3cN n n n= + + ; 
/ 3S Bs kθ θ= Ω ;   
12 21
/ 3P Bp kθ θ= Ω ;   
13 31
/ 3F Bf kθ θ= Ω ;   
23 32
( ) ( )6 2 3 2E S F PW s W f W p Wθ θ θ= + + − + . 
 The effective longitudinal relaxation time (relaxation time of active centers for the phaser signal 
channel) ( )1
ST  that is referred to in (8) is not the conventional spin-lattice relaxation time 
1ST  used in 
studies of passive systems, because pumping ("hidden" in (5) due to 1 3n n= ) leads to renormalization of 
the longitudinal relaxation time [26]. E.g., at ,F S PW W W>>  we find 
( ) ( )0
S F ST W T≈ << . Further we 
will omit the upper index for ( )1
ST . 
 Let us introduce a dynamical variable M , which is proportional to the average SE intensity in 
the solid-state resonator of a phaser generator 
     ( )2 12 2 /S SM U Y Bρ −′ ′= Ω ≡ ,      (9) 
 where 2 22 12S S uB T F Vρ′ ′= Ω ; ρ′  is the crystal density, and the line above denotes averaging over the 
FPAR volume. On the basis of the wave equation for hypersound in active paramagnetic medium [26], 
using approximations of works  [4], [5], we obtain an equation for the first derivative of phaser SE 
intensity  
                  
B M N
′= − ;        (10) 
 where ( )N N= ∆ . 
 Averaging (8) as well, we obtain the second equation of our system 
          0
N NdN
B M N
′= − + ;      (11) 
 The system (10) - (11) is a system of reduced movement equations for an autonomous phaser. 
This system, due to reduction of the pumping equations, is isomorphic to the movement equations for a 
two-level autonomous laser of class "В" [19]. Introducing 1St Tτ = , we will further work with a 
dimensionless form of these equations. For convenience, let us use the following dynamic variables: 
    ( ) 1 2S SJ B T M Zτ ′= ≡ ;   ( ) trn N Nτ =       (12) 
and the following CP : 
   0 trA N N= ;   1S CAVB T T= ,        (13) 
where trN  is the threshold value of the inverted population difference of spin levels corresponding to 
self-excitation of the phonon SE: 
CAV CAV S S
B T T T F
.      (14) 
Let us now introduce a periodic perturbation of amplitude mk  and frequency mω  into the 
equation system (10), (11). The concrete form of the perturbation naturally depends on the choice of the 
modulated CP. As it has been shown theoretically [19] and follows from our experiments, the transition 
from modulation of one parameter to the modulation of another primarily affects the quantitative 
movement characteristics of the system, while on the qualitative level, behavior of the modulated system 
remains essentially unchanged. For convenience, we introduce the modulation into the right-hand side of 
equation (10) (in this case, according to [19], the smallest values of mk  are needed to reach the first 
critical points). Then, finally, our equations for a non-linear phaser system containing a microwave solid-
state resonator with inverted paramagnetic centers acquire the form 
( ) ( ), cosJ m m
J n BJk
= Φ − ,      (15) 
( ),n
= Φ ,        (16) 
 where 1St Tτ = ; 1m m STω ω= ; 1S CAVB T T= ; JΦ  and nΦ   are components of the unperturbed vector 
field Φ  of our system having the following form: ( )1 ;J BJ nΦ = −  ( 1)n A n JΦ = − + . 
 The equation system (15), (16) at 0mk =  has two singular points: saddle [st1]: 
( )0,J n A= = ,where [st1]1Re 0Λ > ; [st1]2Re 0Λ < ; [st1]1,2Im 0Λ =  ( 1,2Λ  - Lyapunov’s indexes) and attractor 
[st2]: ( )1, 1J A n= − = , where: 
[ ] ( )st21,2 2AΛ = − ± ( ) ( )
2 22 1A A B − −
.     (17) 
Taking into attention that 1B >> , at ( ) 14 1 4B A B− << − <<  we find that [st2] is a focus with so-
called relaxation frequency: [st2]1,2 1 1Imrel S rel ST Tω ω= Λ ≡ , where we introduced, for convenience, the 
dimensionless frequency ( )1rel A Bω = − . 
As it is easy to find for our oscillator (15), (16), at small mk  the branch 0B  does softly emerge 
from this focus as a limiting cycle of period 1 mT , what has been actually observed in the experiments. 
The oscillator non-linearity (15), (16) begins to really manifest itself as mk  increases and  mT  comes 
close to the resonance: 2m rel relT T π ω≈ ≡ . Here numerical methods are needed. Let us make some 
necessary estimates by introducing parameters 1ε , 2ε  in the following way: 
1 1 2
1 ;A B
ε ε ε
= + = .        (18) 
 At ( )1 1A O− =  we find that relationships 1 21 2 1Bε ε −≈ ≈ << ; relT ≡ 2 relπ ω ≈ 12 S CAVT Tπ  are 
fulfilled. At small 1A− ≈ 24 Bπ  we obtain 1ε ≈ 1 2π ; 2 2 Bε π≈ ; 1rel ST T≈ . Let us now evaluate the 
value of ppN  , which corresponds to the minimum number of required steps with time stepT  for the 
modulation period, i.e., ( ) 1/ 2pp m step m stepN T T Tπ ω
= = . Accounting for the fact that the field subsystem 
in our case is fast with respect to the atomic subsystem, for m relω ω≈  we obtain 
κ π κ
> ≡ ≡
,      (19) 
 where 2 1κ > . E.g., for A - 1=1/30 and B = 3.7⋅10
3 , from (19) we find that ppN > 2⋅10
The full evolution time of the system wholeT  is another important parameter in looking for 
solutions of our equation system. In all cases, we carried out our trial calculations for such number of 
modulation periods /p whole mN T T≡  as to surpass the value of 1 /S mT T  ratio. This means that there should 
be ( )1 12pN κ πε≈ , where 1κ > 1. In the vicinity of some special points (e.g., period doubling points) 
effect of critical deceleration is observed – like at phase transition points in conservative systems. Here 
pN  values were adapted to the real duration of the transient process by the trial-and-error method. The 
total number of points for calculations calc p ppN N N≡ ⋅  outside the critical deceleration regions does not 
depend on A and has a simple form p ppN N⋅ > 1 2Bκ κ . However, it should be taken into account that the 
problem with 1 1A− <<  requires larger ppN  because of inverse root dependence of mT  upon the overrun 
of the generation threshold ( pN  is considered to be small while 1m rel ST T T≈ → ).  
This circumstance is related to the A –dependence of the normalized dissipativity  nD   
introduced in [20] as Re ImnD ≡ Λ Λ , where Lyapunov’s indexes are taken at the stationary point in 
the absence of modulation. As shown in [20], the value of nD  can be used for preliminary prediction of 
the degree of phase space stratification – when nD  decreases, the number of existing attractors, 
generally speaking, tends to be increased at the given CP set [20].  
For the attracting point [2] of our system in the autonomous mode, using (17), we obtain 
( ) ( )
11 222 1 2nD A A B A
  =− − −   
,       (20) 
Formula  (20) is defined at ( ) ( )NF NFA A A
− +< < , where saddle-focus bifurcation points  ( )±NFA  are 
( ) ( ){ }1/ 212 1 1NFA B B± −= ± − . For the phaser system, 1B >>  (typical values are B ≈ 103-105), so 
( ) 4NFA B
+ ≈ ; ( ) ( ) 11 4NFA B
−− ≈ + . Then for 1 2A A A< < , where ( )
1 1 4A B
− >> ; 2 4A B<< , we obtain 
( )2 1
.        (21) 
As distinct from the unperturbed vector field divergence ( )0 0mkΦ ≡Φ =  characterizing the 
contraction (i.e., the rate of phase volume contraction) 
( ) ( )[st2] [st2]0div 1 1n B J AΦ = − − + = −        (22) 
the value of nD  has a non-monotonous dependence on А: 
d A ( )
.        (23) 
Instead of linear increasing, which is characteristic for 0div Φ , the normalized dissipativity module is 
rapidly decreasing when А is changed from 1 to 2, and only then  (at 2A > ) begins to slowly increase 
with the asymptotics nD ∞ A . Consequently, there are three qualitatively different regions of the 
pumping parameter А (both from physical and calculation point of view) that require separate 
approaches:  
  1. Pumping parameter values are small: 1 1A− << , accounting for the aforementioned limitation 
1A− >> ( )1 4B . In this case, slight stratification (layering) of the phase space is predicted (due to large 
nD ) in combination with weak contraction (as  0div Φ  is small); 
 2. Vicinity of the nD  minimum: 2A ≈ . Here contraction is moderate, but maximum phase 
space stratification is predicted; 
 3. The region of high pumping parameter values, 2A >>  (accounting to be limitation 4A B<< ). 
With increased А, a certain decrease in stratification degree is predicted (as compared with the previous 
case) on the background of enhanced contraction.  
Condition 3 is much less typical in real experiments (it is difficult to obtain such high inversion 
of the spin system) than conditions 1, 2, but we consider all three cases in order to obtain a full picture. 
 Main results of our numerical modeling of SE dynamics in a solid-state resonator of the phaser 
generator can be summarized as follows. The total number of the observed coexisting attractors, 
including limiting cycles with periods mMT  (where М  is the external force undertone number), invariant 
toroids and strange attractors, is first increased with increased А (upon transition from condition 1 to 
condition 2), and is then decreased upon further monotonous increase of A  (transition from condition 2 
to condition 3) – according to formula (21). From the other side, the generated SE pulse amplitude is 
monotonously increasing upon increase of A  - in a complete accordance with formula (23).  This 
supports interpretation of nD  and div Φ  as correct characteristics of different behavior aspects of one 
and the same dissipative system [20]. 
Let us go now to concrete numerical results. Phase space stratification of our system begins 
under the scenario of primary saddle-knot bifurcations, which had been already known (e.g., for 
biochemical systems) in the beginning of 1970-ies [21], and which adequately describes the global 
properties of Class “B” modulated lasers [19]. Pairs of stable and unstable limiting cycles with periods 
mMT  ( 1, 2, 3,M = … ), i.e., natural undertones  mω , evolutionate (with, say, increasing  mk ) from phase 
space singularities in all the studied range low highA A A< <  ( 1 1 30lowA − = ; 30highA = ), but the  
"density" of the coexisting limiting cycles changes non-monotonously in this range. 
 The simples phase space structure was observed in region 3. The calculations were carried out at 
A = 30; B = 3700; mω = 290. The primary hard branch  1B  is developed into a series of Feigenbaum’s 
period doublings, becoming chaotic. This chaotic attractor at mk ≈ 0,18 has maximum refractority time 
2 mT≈ , it coexists with the higher regular branch 2B  with initial period 2 mT , and at  mk ≈ 0.19 the 
chaotic attractor belonging to branch 1B  is destroyed. As itself, the limiting cycle on branch 2B  is not a 
result of the first step of the doubling series (though its secondary evolution proceeds according to the 
Feigenbaum’s scenario) – this cycle is a primary one, i.e., it is born just with the period 2 mT  and 
refractority that is also equal to 2 mT . The nature of this primary undertone (as all the subsequent primary 
undertones – see below) is one and the same – continuous formation of independent subharmonic orbits 
in a dissipative system [21]. This becomes clear with further increase in the modulation amplitude: at 
mk = 0.44, the only attractor of the system is the limiting cycle with period 3 mT  and refractority that is 
also equal to 3 mT  (for the Feigenbaum’s series, the period of 4 mT  should emerge at once, and 
refractority should remain equal to 1 mT ).  At the same time, for this third hard branch, when mk  is 
increased from 0.44 to 0.62, a secondary Feigenbaum’s evolution is observed, with the tripled initial 
period: 3 mT → 6 mT → 12 mT →  24 mT →…  and the unchanged refractority time 3R mT T= . For even 
higher values of mk  , chaotic SE mode is realized. 
 The bifurcation sequences in the region 1 for small mk  are similar to those realized in the 
region 3.  However, upon  mk  increase in the region 1 we also observed much more complex dynamic 
properties. A very small contraction leads to the to excitation of chaotic oscillations with very large 
refractority time. At A -1 = 1/30; B = (1/27)⋅105 ≈ 3704; mk  = 2.4⋅10
-2; mω  = 10; ( )0J  = 10-10; 
( )0n  = 1.0 we obtain maximum values of RT  , up to 5 mT  (Fig.4). Even higher values - RT 12 mT=  - are 
noted when the modulation coefficient becomes as high as mk  = 0.1 (at the same CP as in Fig.4). 
Fig. 4. Calculated time dependence of SE intensity. 
Refractority, as in the experiment (see 
Fig. 3, oscillograms b-d), varies within the 
limits of one to five modulation periods. 
Fig. 5. Calculated time dependence of inverse 
population difference at the same 
controlling parameters and initial 
conditions as in Fig. 4. 
 Comparing the experimentally obtained dependences ( )J t  for the mode with prolonged 
variable refractority (oscillograms b-d in Fig.3) with results on ( )J t  obtained by calculations using our 
SE model in the region 1 (Fig.4), we can see their good agreement: both in experimental and calculated 
SE pulse sequences all refractority times are present, without exceptions - from mT  to 5 mT , including the 
last value. Moreover, comparing Fig.4 and Fig.5, one can clearly see the qualitative difference in 
behavior of dynamic variables J and n, related to substantial difference in relaxation times for the field 
and the atomic subsystems.  
As a matter of fact, the above-described refractority effects are in one or another way related to 
this substantial difference. As 1B >>  (т. е. 1S CAVT T>> ), the slow relaxation atomic subsistem 
(variable n) has not enough time “to be tuned" to the fast phonon processes (variable J). Therefore, when 
there is a periodic perturbation imposed from outside, the population difference n for certain ranges of 
CP does not reach over-critical values (ensuring phonon generation) in each of the modulation periods. 
Correspondingly, it is clearly seen from both experimental (Fig.3, oscillograms 6d) and calculated 
(Fig.4) pulse sequences ( )J t  that larger intensity of “irradiated” SE pulse is accompanied by a longer 
subsequent state of refractority – the phaser is not responding to several modulation pulses. Let us 
underline once more that in our experiments (where B ≈ 104) such modes are more typical than SE 
without refractority. 
An essential point is that each increase in the maximum refractority time observed in our 
computer experiments occurred stepwise., was accompanied by an increase in maximum SE pulse 
amplitude and showed no hysteresis, which was also in full agreement with our real physical 
experiments.  
It is generally assumed [22] that such effects are caused by a qualitative rearrangement of the 
stratified phase space as a result of so-called external crises (or crises of second kind [22]), when the 
attraction region of a strange attractor is suddenly expanded, accompanied by an increase in the average 
refractority time. Each subsequent crisis gives rise to lower and lower pseudoperiods as a result of 
collisions of the strange attractor with unstable regular manifolds, specifically – with saddle components 
emerging (at primary saddle-knot bifurcations) together with branches MB  [22]. Therefore, the strange 
attractor crises can be realized only in these regions of CP space where branching of the above-lying 
regular attractors has already taken place. We have noted such behavior for all SE modes with 
pseudoperiods up to 5 mT  (physical experiment) and 12 mT  (computer modeling).  
As further calculations have shown, when pumping power is increased, and values of A ≈ 2 are 
reached, the number of coexisting attractors becomes higher, enhancing stratification of the phase space, 
which is also in good agreement with experimental data.. The bifurcation diagrams acquire now a very 
complex indented structure with multiple interweaving in the CP space. As a result, the system 
sensitivity becomes even higher both to CP perturbations and variation of initial conditions.  
As a conclusion, let us stress once more that all the above-described SE processes are adequately 
described within the deterministic non-linear oscillator model (15), (16) under the double inequality 
1 2S CAV ST T T>> >>  that is typical for phaser generators. For electromagnetic maser generators, which are 
also characterized by very low level of intrinsic radiation, condition 2CAV ST T<≈  is normally fulfilled 
[24], [25]; therefore, for this case a separate consideration is needed, which will be done in our 
forthcoming papers. 
 The author is grateful to E.D. Makovetsky, who kindly performed a large part of work on the 
computer modeling of phaser generation, and to S.D. Makovetskiy, who has written DLL-modules with 
non-standard library functions for computer experiments. 
References 
1. Такер Э. Парамагнитное спин-фононное взаимодействие в кристаллах  (Paramagnetic spin-
phonon interaction in crystals.) // Физическая акустика. - М.: Мир, 1969. – T. 4А. - С.63-138. 
2. Shiren N.S. Ultrasonic maser amplification by stimulated phonon-photon double quantum emission in 
MgO // Appl. Phys. Lett. - 1965. – Vol. 7, No.5. - P.142-144. 
3. Peterson P.D., Jacobsen E.H. Maser amplification of 9.5-Gigahertz elastic waves in sapphire doped 
with Ni2+ // Science (USA). - 1969. - 164, No.3883. - P.1065-1067. 
4. Маковецкий Д.Н., Ворсуль К.В. Полевые зависимости коэффициента инверсии в 
парамагнитных мазерах и фазерах с нелинейными (бистабильными) резонаторами накачки 
(Field dependences of inversion coefficient in paramagnetic masers and phasers with non-linear 
(bistable) pumping resonators.) // Журн. техн. физики. - 1991. – T. 61, № 1. - С.86-90.  
5. Makovetskii D.N., Vorsul K.V. Field dependence of the inversion factor in paramagnetic masers and 
phasers with nonlinear (bistable) pump cavities // Tech.Phys. - 1991. – Vol. 36, No.1. - P.50-52. 
6. Fain B. Phonon enhancement and saturation in a three-level system // Phys Rev. B. - 1982. – Vol. 26, 
No.10. - P.5932-5940. 
7. Маковецкий Д.Н. Исследование усиления и генерации гиперзвука парамагнитными центами в 
корунде при инверсии населенностей спиновых уровней: (Studies of hypersound amplification 
and generation by paramagnetic centers in corundum at inverted spin level population.) Автореф. 
дис. ... канд. физ.-мат. наук. - Харьков: ФТИНТ АН УССР, 1984. - 22 с. 
8. Makovetskii D.N. Microwave power spectra of stimulated phonon emission and space-time structures 
in a spin-phonon system of ruby phaser // Telecommun. & Radioeng. - 1997. – Vol. 51, No.6-7.  
- P.133-139. 
9. Маковецький Д.М. Динамічне звуження спектрів індукованого випромінювання фононів у 
нелінійному акустичному резонаторі (Dynamic narrowing of phonon induced radiation spectra in a 
non-linear acoustic resonator.) // Укр. фіз. журн. - 1998. – T. 43, № 5. - С.537-539. 
10. Маковецкий Д.Н. Нестационарная тонкая структура микроволновых спектров мощности 
индуцированного излучения фононов при инверсии населенностей спиновых уровней в 
твердотельном резонаторе (Non-stationary fine structure of microwave power spectra of phonon 
induced radiation at inverted spin level population in a solid-state resonator.) // Радиофизика и 
электроника. - Харьков: Ин-т радиофизики и электрон. НАН Украины, 1999. – T. 4, № 2.  
- С.91-98. 
11. Лоскутов А.Ю., Михайлов А.С. Введение в синергетику. (Introduction to synergetics.) - М.: 
Наука, 1990. - 272 с. 
12. Hopfield J.J. Neural networks and physical systems with emergent collective computation abilities // 
Proc. Natl. Acad. Sci. USA. - 1982. – Vol. 79. -P.2554-2558. 
13. Анисимов Б.В., Курганов В.Д., Злобин В.К. Распознавание и цифровая обработка 
изображений. (Recognition and digital processing of images.) - М.: Высшая школа, 1983. - 295 с. 
14. Павлидис Т. Алгоритмы машинной графики и обработки изображений. (Algorithms of computer 
graphics and image processing.) - М.: Радио и связь, 1986. - 400 с. 
15. Хорстхемке В., Лефевр Р. Индуцированные шумом переходы. (Noise-induced transitions.) - М.: 
Мир, 1987. - 400 с. 
16. Pecora L.M., Carroll T.L. Pseudoperiodic driving: Eliminating multiple domains of attraction using 
chaos // Phys. Rev. Letters. - 1991. – Vol. 67, No.8. - P.945-948. 
17. Маковецкий Д.Н. Критические явления в примесном парамагнетике Cr3+: Al2O3 при 
насыщении электронного парамагнитного резонанса (Critical phenomena in doped paramagnetic 
Cr3+:Al2O3 at saturation of electron paramagnetic resonance.) // XIX Всесоюз. конф. по физике 
магн. явлений (24-27 сент. 1991): Тез.докл. - Ташкент, 1991. - С.102. 
18. Маковецкий Д.Н. Критические явления и нелинейные резонансы в неравновесном примесном 
парамагнетике (Critical phenomena and non-linear resonances in a non-equilibrium doped 
paramagnetic.) // Тр. III Междунар. конф. “Физические явления в твердых телах” (21-23 янв. 
1997). - Харьков, 1997. - С.105. 
19. Tredicce J.R. e.a. On chaos in lasers with modulated parameters: A comparative analysis // Opt. 
Commun. - 1985. – Vol. 55, No.2. - P.131-134. 
20. Meucci R., Poggi A., Arecchi F.T., Tredicce J.R. Dissipativity of an optical chaotic system 
characterized via generalized multistability // Opt. Commun. - 1988. – Vol. 65, No.2. - P.151-156. 
21. Николис Г., Пригожин И. Самоорганизация в неравновесных системах. (Self-organization in 
non-equilibrium systems.) - М.: Мир, 1979. - 512 с. 
22. Grebogi C., Ott E., Yorke J.A. Crises, sudden changes in chaotic attractors, and transient chaos // 
Physica D. - 1983. – Vol. 7. - P.181-200. 
23. Дмитриев А.С., Кислов В.Я. Стохастические колебания в радиофизике и электронике. 
(Stochastic vibrations in radiophysics and electronics.) - М.: Наука, 1989. - 280 с.  
24. Маковецкий Д.Н., Лавринович А.А., Черпак Н.Т. Ветвление стационарных инверсионных 
состояний в квантовом парамагнитном усилителе с резонаторной накачкой (Branching of 
stationary inversion states in a cavity-pumped paramagnetic quantum amplifier.) // Журн. техн. 
физики. - 1999. – T. 69, № 5. - С.101-105.  
25. Makovetskii D.N., Lavrinovich A.A., Cherpack N.T. Branching of stationary inversion states in a 
cavity-pumped paramagnetic maser amplifier // Tech. Phys. - 1999. – Vol. 44, No.5. - P.570-574. 
26. Маковецкий Д.Н. Нелинейное спин-фононное взаимодействие и усиление гиперзвука в 
активной парамагнитной среде (Non-linear spin-phonon interaction and hypersound amplification 
in an active paramagnetic medium.) // Укр. физ. журн. - 1985. – T. 30, № 11. - С.1737-1740.
