Fulltext

CHINESE JOURNAL OF PHYSICS

JUNE 1998

VOL. 36, NO. 3

Nonlinear Response of Optogalvanic F’r actal and Its Application to Signal Processing

Jyh-Long ChernI, Yuh-Fung Huang2, Herng-Yih Ueng2, and Tsu-Chiang Yen3 ‘ N onlinear Science Group, Department of Physics, National Cheng Kung University, Tainan, Taiwan 701, R.O.C. 2Department of Electrical Engineering, National Sun Yat-Sen University, Kaohsiung, Taiwan 804, R. 0. C. 3Department of Physics, National Sun Yat-Sen University. Kaohsiung, Taiwan 804, R. 0. C.

(Received February 9, 1998)

For a frequency locking system with the devil’s staircase structure, between the two frequency locking states there embeds an almost infinite number of states whose periodicities could be specified by the Farey tree. Since the range between two specific states is usually small, such a fractal system may be utilized as a generating source of signals with specific periodicity without greatly modifying the system parameters. On the other hand, due to its nonlinear response to external perturbations, a fractal system may be tuned to act as a detector for a variety of weak periodic signals. These two potential applications are investigated and experimentally demonstrated in a He-Ne laser optogalvanic system.

PACS. 05.45.+b - Theory and models of chaotic systems. PACS. 64.60.Ak - Renormalization-group, fractal, and percolation studies of phase transitions. PACS. 82.80.Kq - Energy-conversion spectra-analytical methods. PACS. 42.79.Hp - Optical processors, correlators, and modulators.

I. Introduction Recently the application of nonlinear dynamics, particularly the ideas of mastering chaos, has attracted many attentions in a variety of fields [l]. In another front, fractal is utilized in the field of image compression [2]. The essential key of the success is the selfsimilarity in the geometry of image, i.e., the image can be decomposed into some basic units and thus the capacity requirement in transmission can be greatly reduced. After transmission, the original image can be reconstructed quickly based on the rule of self-similarity [a]. Further extension of self-similarity to different applications is of great interest. However, an extension in the time domain requires a point of view vastly different from geometrical one. The self-similarity implies that an infinite number of states, which repeatedly follow some unique rules, are embeded within a narrow (fin&) region of parameter. When one hopes to drive (or switch) a specific system among many different states without greatly modifying the system, the idea behind the self-similarity described above should be attractive. 501

@ 1998 THE PHYSICAL SOCIETY OF THE REPUBLIC OF CHINA

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NONLINEAR RESPONSE OF OPTOGALVANIC FRACTAL AND

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This suggests that fractal may be useful in signal generation and detection. On the other hand, in terms of time variation, the dynamical response of fractal remains less addressed. Therefore, this issue is not only of practical potential application but also interesting in its academic aspect. TO illustrate this point further, let us consider a system with a devil’s staircase whose self-similarity can be described by the Farey tree [3]. The states within the Farey tree correspond to the mode-locking (frequency locking) states. More specifically, the Farey tree means that between the two frequency locking states, p/q and r/s, there exists a state with frequency (p+r)/(q+s). 1 n p rinciple, the sequence can repeat itself infinitely and thus there embeds an infinite number of frequency locking states within the region. Thus a system with such a devil’s staircase provides a range of parameter where an infinite number of periodic orbits embedded. Therefore this type of system may be utilized as a generating source of periodic signal with desired periodicity provided that the signal can be well controlled. On the other hand, due to the nonlinear resonance, the frequency-locking states could be rather sensitive when it is exposed to small external signals, particularly with some resonant frequencies. In such a case, this system may be utilized as a detector. However, due to the external noise, such an ideal infinite sequence would be terrninated even in the classical, non-quantum level. Therefore, in order to utilize the self-similarity for signal generation and detection, one would need to clarify the response of the fractal structure to external weak periodic signals under the infiuence of noise. As an experimental demonstration we address this interesting issue in a He-Ne laser system, where an optogalvanic fractal can be identified. We first clarify the terminology. Optogalvanic fractal simply means a special class of fractal, period adding, occurred to an optogalvanic system. Period-adding is a feature in devil’s staircase structure. In the phase diagram (or bifurcation diagram) between two “macroscopic” regions, when the period of output time series is locked at nT and (ntl)T, there lies a narrower region over which the system oscillates with a period (2n+l)T. Here n > 1 and T is the fundamental period. Furthermore, between nT and (2n+l)T, one can find a region for which the period is (3n+l)T, and between (2ntl)T and (n+l)T, a region with period (3nt2)T. In practice, due to the influence of noise it is not easy to observe high periods, which are also within narrower parameter regions. Nevertheless, period-adding has been experimentally confirmed in nonlinear electric circuits [4], and recently in an optogalvanic system which is consisting of a neon bulb shone by a modulated green-light He-Ne laser [5]. Optogalvanic effect means that, say in the neon bulb, its electric circuit parameters may vary when the discharged plasma absorbs optical radiation [6]. Interest in this effect and its applications to many fields, such as element analysis and frequency stabilization of lasers [7], have grown since Green et. al. demonstrated its use in laser spectroscopy [8]. In this work, we adopt some optogalvanic fractal found in Ref. [5] but focus on its possible application to signal generation and detection. II. Experimental setup

To create optogalvanic fractal, as in Ref. [5] we use a commercially available neon bulb, a RSD-01 neon bulb made by Rainbow Machinery Inc., Taiwan, which contains two

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JYH-LONG CHERN, YUH-FUNG HUANG, HERNG-YIH UENG, .

503

green-light He-Ne laser

synthesizer

FIG.

1. Schematic diagram of experimental setup

straight-wire electrodes, 4 mm long and separated by a gap of about 1 mm. The bulb’s diameter is 5.1 mm and its length is about 11 mm. In experiment, the neon bulb is shone by a green-light He-Ne laser (Melles Griot, model number 05 LGR 193). Before shinning on the neon bulb, the intensity of laser light is modulated by an acoustic-optic modulator (AOM) (Crystal Technology Inc., model number 3080-14) which is driven by a multifunctional synthesizer (HP 8904A). We use it in a squarewave function operation mode and label this modulation frequency as fm. As shown in Fig. l(a), the neon lamp is connected in parallel with a capacitor C (3.3_nF) and charged through a resistor R, (390kR) [5]. In this paper, we add a functional generator (HP 33120A) which is in series with the neon bulb and a resistor R,(lkR). This functional generator is operated in sinusoidal mode at the frequency as fe. We measure the voltage across the capacitor with an oscilloscope and the voltage across the resistor R, with a data acquisition board (NI DAQ MIO-16-F5, sampling rate 500 KHz and resolution 2.4 mV). Sustained relaxation oscillation occurs as the dc applied voltage V, exceeds the threshold voltage Vth M 78.7 V. We label the output oscillation frequency as fout. The experiment has been performed in two ways based on the type of perturbation, i.e. electric or optical. III. Experimental result - perturbation of electrical signal In the case of electric perturbation, we first set V, = Vth + 0.4 V and use the AOM to modulate the green-light He-Ne laser (averaged power 0.34 mW). The discharged frequency fout changed as we varied the modulation frequency of AOM, fm. As shown in Fig. 2, period-adding exists. This is only one frequency fm, thus a fundamental period T = l/fm can be identified. Next we turn on the functional generator and set the signal amplitude at 260 mV,_, (peak-to-peak, i.e, oscillated between 130 mV and -130 mV). To ensure this electric perturbation is weak enough, we test the system within the region of fm/fout = 1 and fm/fout = 2 by adjusting fm. It is found that this electric perturbation does not affect these two ratios at all. Then we set f,,, within the region of period-adding. The ratio fm/fout could not fix often since the output time series is irregular as suggested by the typical return map in Fig. 2. This irregularity may be

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IA,

,

..

I___ ,_. .___-T

1 60 160 1 6 5

OL 0.0

170

,~.

175 180

600.0 300.0 450.0 modulation frequency 1, (Hz)

FIG. 2. Frequency-locking regions (devil’s staircase structure) for V, = Vth + 0.4 V. The number in parentheses shows the ratio fm/fout. In the left-top part, a typical return map is presented _ to show the chaos.

caused by the external noise. It is suspected that due to the external noise, the system randomly switches among too many frequency-locking state embedded in a very small region. Please note that this return map is made by using the successive maxima, i.e., (n+l)th, peak versus nth. peak. However, if we tune the frequency fe carefully, some regions with fixed ratios can be found, e.g. for fe = 200 Hz, fm/foUt = 2 persists. (Note fm = 400 Hz.). S ince there are multiple frequencies, it becomes difficult to ascertain the fundamental period. Nevertheless, we obtain a simple rule for the discharged periodic signal, i.e., fozlt = LLfel,

(I)

where [fm, fel means the largest common factor between fm and fe. This new rule established completely due to the influence of weak electric perturbation. We list part of the results in Table I and present typical time series in Fig. 3(a). It seems that the appearance of Eq. (1) rm ’ pl ies a common tours structure [3]. However, we notice that: It is not necessary to follow Eq. (1) exactly. In other words, fe(fm) is not necessary PI to be exactly a fraction of fm(fe) as required by Eq. (1). A slight difference in fe( fm) of [fm, fe] results in an increase of broad-band background and shifts foul a little bit in the same direction. In such cases, stable waveforms still can be seen. PI This rule depends on the perturbation intensity. Indeed, Eq. (1) works only when the amplitude of electric signal is within the range of [260,440] mV,_,, for the case described above.

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[3] Some frequency-locking states do not follow Eq. (1) at all as shown by the case 3 of Table 1. Equation (1) indicates the onset of some extremely amazing nonlinear resonance but the mechanism is still unclear. Later, we will address some connections established based on mapping dynamics.

,,.,1,,,,,,.,,,,~,,,,,,~,

0

10

20

30

i A1=20.00

1

0

10

20

.,.I,,.,.,,.II,...III.I.

40

50

60

70

80

90

100

40

50

60

70

10

YU

IUO

111s 1

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time (Ins) FIG. 3. (a) Th e waveform (1) shows the discharged output time series with measured frequency fout = 50 Hz. The waveforms (2) and (3) are the AOM driven signal (frequency f,,, = 400 Hz) and the electric signal (frequency fe = 250 Hz) respectively. The ground levels are shown by the right side of the figure and the units are 10, 5 and 0.1 V for the waveforms (l), rsc ar (2) and (3) respectively. (b) D’ h ge doutput waveform with measure frequency fout = 50 Hz when the driven electric modulation frequency fe = 400 Hz and a weak optical signal with modulation frequency fm = 250 Hz.

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NONLINEAR RESPONSE OF OPTOGALVANIC FRACTAL AND . . .

TABLE I. Creating output waveform with specific periodicity (foUt) by weak periodic electric perturbation (amplitude: 260 mV,_,; frequency f,J. The laser light is modulated at fm = 400.0 Hz and all items are in the unit of Hz.

Case No.

range of fe

1 2 3 4 5 6 7

100.0 f 2.0 133.3 f 0.7 114.0(113.5-115.0) 120.0 f 0.3 125.0 f 0.3 150.0 f 1.0 200.0 f 3.0

[fm, fe] fDUt 100 133.3 40 25 50 200

100.0 133.3 57.0 40.0 25.0 50.0 200.0

Case

No. 8 9 10 11 12 13 14

range 250.0 300.0 400.0 560.0 600.0 700.0 800.0

of f f f f f f f

1.0 3.0 4.0 2.0 6.0 2.0 8.0

fe [fm, fe] foUt 50 100 400 40 200 100 400

50.0 100.0 400.0 40.0 200.0 100.0 400.0

IV. Experimental result - perturbation of optical signal

To explore the applicability to optical signal processing, we turn to use optical signal as the weak perturbation. When the applied voltage is set at V, = 78.7 V, the sustained relaxation oscillation appears. The electric signal, A sin (2~f&), serves the main modulation. In experiment, we set A = 2.0 V. Similar period-adding can also be found by tuning the frequency fe, and the fundamental period is T = l/fe. Again, we have to test the system to ensure the optical perturbation is weak enough. During the experiment, the averaged power of laser light is 0.43 mW. We found the locking regions, fe/foUt = 1 and fe/foUt = 2, are not affected. Next we set fe within the region of the period-adding. Exactly the same features as that of electric perturbation occurs. The rule for the discharged periodic waveform is still fout = [fe, fm]. Again we can see that the fundamental period becomes blurred and a new rule takes place. For comparison, we present the time series of [fe,fm] = 50 Hz in Fig. 3(b) and one can see that it is rather stable. It should be noted that to realize a high fe/[fe,fm] state is difficult. Nevertheless, creating waveform with specific periodicity in a wide parameter regime is possible for both optical and electric perturbation cases. V. Connection to mapping dynamics

Although the experimental results suggest some unusual features in an optogalvanic system, a simple picture related to Eq. (1) can be derived based on mapping dynamics. Let us consider a dynamical system which could be described by a mapping X(n+ 1) = f(X(n)), where n is the time step and the nonlinearity is characterized by the function f. Since in the experiments, there are two modulation sources, we consider the case with two additional periodically driving terms. Therefore, the dynamics may follow

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X(n + 1) = f(X(n)) + Y(n) + z(n),

507

(2)

where Y(n) and Z(n ) are periodic driving terms. As a simple illustration, let us set Y(n) to be of period-2 with the values Ya and Yb, and Z(n) of period-3 with the value Za, Zb, and Zc. Considering the time evolution of Eq. (2), we may have X(n + 1) = f(X(n)) +

ya + 2%

X(n + 2) = f(X(n t 1)) t Yb t Zb, X(n + 3) = f(X(n t 2)) t Ya t zc, x(n + 4) =

f(X(n t 3)) t Yb t Zo,

x(n + 5) = f(x(n t 4)) t Yo t Zb, X(n + 6) = f(X(n t 5)) t Yb t Zc, x(n t 7) = f(X(n + 6)) + Yo + Zo, ... One of the permitted perioidicities suggested in the time evolution, say from X(n) to X(n + 7), is X(n) = X(n t 6). Thus it becomes obvious that the allowable periodicity for the X(n) is 6k where k = 1,2,. . . . Here the number 6 is the least common multiple of the periodicities of two driving terms. By the same argument, one can deduce the periodicity of X(n) is Pz, which follows Px = {Py, Pz}k,

(3)

where Py is the periodicity of Y(n), P, is the periodicity of Z(n), {,} means the least common multiple and k = 1,2,. . . Eq. (3) has exactly the same meaning as in Eq. (1). Another example is that Y(n) is of period-2 and Z(n ) is of period-4, then the allowable periodicity is 4k where k = 1,2,. . . . Let us take the well-known logistic mapping as a numerical example. The modified logistic mapping follows

X(n t 1) = rX(n)(l - X(n)) - au(n) - bZ(n),

(4)

Y(n + 1) = 3.2Y(n)(l - Y(n)),

(5)

Z(n + 1) = 3.832+)(1 - Z(n)).

(6)

Here we set a = 0.01 and b = 0.02 in Eq. (4). Di fferent choices of a and b show the same feature. It can be verified that the output time series of Y(n) and Z(n) are of period-2 and period-3, respectively. As shown in Fig. 4, all the periodic orbits are of period-6k where k = 1,2,..., though some periodic orbits requires higher resolution for identification. We provide the return map at T = 3.2 as an example. As a note, we have replaced Eq. (6) as Z(n + 1) = 3.52+)(1 - Z(n)). It can be verified that the output time series of this new Z(n) is of period-4. This modified Logistic mapping does show different periodicity. All

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Return Map at r =3.2

0.80 1

:.

0.70. F‘ + c xv

3.00

3.20

3.40

3.60

3.60

Control Parameter r

4.00

1 0.60 (

0.50 ~.}

---7---T

0.50

-‘-[-T7

0.70

0.60

0.60

x(n)

FIG. 4. Bifurcation diagram of the modified Logistic mapping with two periodic terms where the external periodic sources are of period-2 and period-3 and the return map at T = 3.2. the periodic orbits are of period-4k where k = 1,2,. . . . An extension of the modified Logistic

mapping to the case with additional noise also has been carried out. We can simply replace Eq. (4) as X(n + 1) = rX(n)(l - X(n)) - au(n) - bZ(n) + q(n) where q(n) is uniform noise. The general feature of periodicity are the same, though some periodic orbits are hardly recognized to be a period-6k (or period-4k) orbit due to the influence of noise. Although the mapping approach closely resembles the salient feature of the optogalvanic system as described in Eq. (l), we emphasize that there are some fundamental differences between the mapping dynamics and the dynamics of the optogalvanic system. The comments listed in [l] - [3], particularly the anomaly [3], support our point. VI. Conclusions and discussions

In conclusion, we have shown that one can create the waveform of some specific periodicity without greatly modifying the system’s parameter in an optogalvanic system. This also suggest a possibility of a novel frequency demultiplexing. Without such an optogalvanic fractal, these could not be achieved. As shown above, this optogalvanic fractal is very sensitive to a weak periodic signal, either optical or electric. This means that it can serve as a detector. It is worthwhile to note such a detector could also serve as an intensity filter. For the technical interests, it is necessary to seek for the possible extension to highspeed optical system. Let us outline the physical conditions to explore the possibility. As shown in Fig. 5, the characteristic curve of neon bulb behaves like a negative resistance element. Optogalvanic effect shifts the threshold downward as shown in Fig. 5 where the characteristic portion of the VI curve changes from the arrow with solid line to that with dashed line as the laser light is on. Notice that the characteristic of negative resistance remains. An application of optical (electric) modulation causes a periodic change of

JYH-LONG CHERN, YUH-FUNG HUANG, HERNG-YIH UENG,

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z 8.0 E .. .z 4.0 zi 0.0

: : : 56.81 60.81

:

64.81

I I

:

:

68.81 72.81 (unit: V)

:

:

76.81

:

80.1 1

FIG. 5. Characteristic VI curve of neon bulb measured by a curve tracer (Good Will GW

GCT-1212A). (see text.) characteristic curve. This period will interplay with the sustained relaxation oscillation inherent in the system resulting in phase-locking, tours, period-adding and so on. This suggests an integrated circuit consisted of negative resistance element and optosensitive device could be a potential system of optogalvanic fractal, and hence a new device possesses theTunctionalities as reported here. With the current semiconductor technology assembled using commercial fast-response semiconductor laser diodes (or silicon alvananche photodiodes), such a device with frequency response up to several GHz should be possible. Admittedly, there are still many works needed to be done, e.g., the dependence on intensity. However, by tuning the main driven frequency, a weak periodic signal can be detected and identified in a large range. Acknowledgments This work was partially supported by the National Science Council, ROC under Contract No.s: NSC83-0117-C-168-003-M, NSC84-2112-MllO-004, and NSC 86-2112-MllO016. JLC thanks R.-R. Hsu and H.-T. Su for discussions. References

[II

[21 [31 [41

E. Ott, T. Sauer, and J. A. Yorke, eds, Coping with chaos, (Wiley, New York, 1994), and references therein. For an illustrated discussion, see Y. Fisher, in H.-O. Peitgen, H. Jurgens, and P. Saupe, Chaos and Fractal, (Springer-Verlag, New York, 1992), p. 903. H. G. Schuster, Deterministic Chaos, (VCH, Weinheim, 1989). M. P. Kennedy and L. 0. Chua, IEEE Trans. Circuits Syst., CAS-33, 974 (1986); L.-Q. Pei, F. Guo, S.-X. Wu, and L. 0. Chua, IEEE Trans. Circuits Syst. CAS-33, 438 (1986). Theoretical works should refer to K. Kaneko, Prog. Theor. Phys. 68, 669 (1982); ibid. 69, 403 (1983).

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[ 5 ] Y.-F. Huang, T.-C. Yen, and J.-L. Chern, Phys. Lett. A199, 70 (1995). [ 6 ] F. M. Penning, Physica 8, 137 (1928). [ 7 ] R. B. Green, R. A. Keller, G. G. Luther, R. K. Schenck, and J. C. Travis, Appl. Phys. Lett. 29, 727 (1976). [ 8 ] R. S. Stewart and J. E. Lawler, eds., O p t o g a l v a n i c s p e c t r o s c o p y , Proc. 2nd. Int. Meet. of Optogalvanic Spectroscopy and Allied Topics, (IOP, 1991), and references therein.