Coherence effect on nonlinear dynamics in fiber-optic ring resonator

1 August 2001

Optics Communications 195 (2001) 259±265

www.elsevier.com/locate/optcom

Coherence e€ect on nonlinear dynamics in ®ber-optic ring resonator Yoh Imai *, Tomoyuki Tamura Department of Computer Science and Electronics, Kyushu Institute of Technology, 680-4 Kawazu, Iizuka-shi, Fukuoka-ken 820-8502, Japan Received 13 February 2001; received in revised form 9 April 2001; accepted 5 May 2001

Abstract Dependence of nonlinear dynamics generated in an optical ®ber ring resonator on source coherence is investigated numerically. Chaotic and periodic behaviors are found to have reduced moduli with maintained patterns which become time independent as source spectrum width increases. Output bifurcation characteristics are independent of source coherence but are dependent on loss and other resonator parameters. Coherence length which is longer than resonator length is required for chaotic output. The critical spectrum width at which output becomes time independent is inversely proportional to ®ber ring resonator length. In addition, critical spectrum width decreases with increased ®ber ring resonator loss. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Optical chaos; Optical ®ber ring resonator; Optical coherence

1. Introduction An optical ®ber ring resonator (OFRR) is a fundamental and useful ®ber-optic device. Ikeda et al. found that chaotic dynamics are generated in OFRR [1,2]. Since then, great e€ort has been paid to analyze OFRR nonlinear dynamics experimentally and theoretically [3±6]. Recently, OFRR chaos has also been attractive for potential application for secure communications [7]. However, it has been assumed that input light is always monochromatic and coherent. Input light spectrum width is considered to a€ect output dynamics

*

Corresponding author. Fax: +81-948-29-7651. E-mail addresses: [email protected], imai@leo20cse. kyutech.ac.jp (Y. Imai).

signi®cantly because nonlinear dynamics result from interference among coupled input light and multiple circulated components in OFRR. Highly coherent input light is expected to be required for many components for interference in long cavity OFRR. It is important to know how coherence a€ects nonlinear dynamics from a viewpoint of understanding potential optical chaos applications' performance (such as in secure communication) as well as fundamental characteristics. This paper investigates numerically: the coherence effect, i.e., the input light spectrum width as well as input power, loss and the ®ber coupler coupling coecient; and the e€ect of other ®ber parameters, such as the nonlinear refractive index and ®ber ring resonator length, on OFRR output dynamics. Critical input light spectrum width required for chaotic output is presented.

0030-4018/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 1 ) 0 1 2 9 0 - 1

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2. Numerical model of optical ®ber ring resonator Consider the OFRR con®guration with an optical ®ber coupler as shown in Fig. 1. The resonator output ®eld, Eout …t†, consists of transmitted and circulated components and is expressed by the following iterative equation: p Eout …t† ˆ i j…1 q†Ein …t† p …1† ‡ …1 j†…1 q†Ep …t†; where Ein and Ep are input light ®eld and circulated (round trip) components in the resonator, j and q are the ®ber coupler coupling coecient and power loss. Here, input light is assumed to be monochromatic with constant amplitude and random phase modulation which results in temporal coherence degradation. Hence, the input light ®eld is expressed as Ein …t† ˆ E0 exp ‰i/…t†Š;

…2†

where E0 ˆ A exp‰i…xt ‡ /0 †Š (/0 : constant phase term). The circulated component is written as   aL Ep …t† ˆ Et …t s† exp 2  expf i‰/0 ‡ D/…t

s†Šg;

…3†

where

are the linear and nonlinear phase shifts, k is the wave number in a vacuum, L is the ®ber ring resonator length, and n2 is the ®ber core nonlinear refractive index. The random phase term /…t† is related to the input light temporal coherence function which is given by [8]. c…s† ˆ hexpfi‰/…t†

s†Šgi

ˆ exp… 2p Dms†:

…4†

Here, Dm stands for the input light spectrum width and h


i represents the time average. In derivation of the coherence function above, /…t† is assumed to obey the Gaussian random process. Then, the OFRR output power normalized by the input power is derived as follows: * + Eout …t† 2 Pout …t† ˆ E0 X p
n pn‡2 ˆ j…1 q† ‡ 2…1 j† 1 q j   exp

naL 2 j†4

‡ 2…1

n



NX …n 1† n‡2 D/k p‡ 2 kˆN

cos

N X jn 1 …1

!

q†n‡1 exp… naL†

nˆ1

p p Et …t† ˆ …1 j†…1 q†Ein …t† ‡ i j…1 q†Ep …t†;

‡ 2…1

and

 exp

j†4  "N

2

D/…t† ˆ kLn2 jEt …t†j ; where s is the delay time, a is the loss coecient of transmission power in the ®ber, /0 ˆ kLn0 and D/

/…t

 cos

N 1X N 1 X j2n‡m 2 …1 nˆ1 mˆ1

…2n ‡ m†aL 2 …n 1† X kˆN

N

D/k

q†2n‡m‡2

 exp… 2pDm ms† # …m‡n X 1† …5† D/l : lˆN

Here, the constant linear phase term is neglected. Eq. (5) is numerically analyzed under the conditions that the input light mean wavelength is set at k ˆ 1:55 lm, the e€ective core area is Aeff ˆ 30 lm2 , the core refractive index is n0 ˆ 1:45, and the ®ber loss coecient is a ˆ 0:4 dB=km for a single mode silica ®ber.

3. Numerical analysis Fig. 1. Model of an OFRR with a single ®ber coupler.

Typical time traces of output dynamics under the conditions that Dm ˆ 0 MHz (coherent), L ˆ

Y. Imai, T. Tamura / Optics Communications 195 (2001) 259±265

261

100 m, n2 ˆ 1  10 22 m2 =V2 , j ˆ 0:5, and q ˆ 0:1 are shown in Fig. 2. Here, the ordinate and abscissa stand for output power normalized by input power and time normalized by circulation (round trip) time in the resonator, T ˆ t=…n0 L=c†. The OFRR output power is time independent at low input power Pin ˆ 5 W after several tens s (s ˆ n0 L=c: circulation time in the resonator). Output power begins to ¯uctuate periodically and then becomes chaotic as input power increases to Pin ˆ 10 and 20 W. The period of the periodic pattern shown in the middle is 3s. Typical coherence e€ects on the chaotic state is shown in Fig. 3 in which input power is set at Pin ˆ 20 W and parameters except for spectrum width are the same as those in Fig. 2. Spectrum width is Dm ˆ 0:1, 1.0 and 3.0 MHz from top to bottom. The chaotic output reduces its ¯uctuation modulus while

Fig. 3. Time traces of OFRR output power under input power Pin ˆ 20 W. Input light spectrum width is set at Dm ˆ 0:1, 1.0, and 3.0 MHz from top to bottom. Other parameters match those in Fig. 2.

Fig. 2. Typical time traces of the OFRR output power under coherent (Dm ˆ 0 MHz) input light with input power Pin ˆ 5, 10, and 20 W from top to bottom. Other parameters are set at L ˆ 100 m, n2 ˆ 1  10 22 m2 =V2 , j ˆ 0:5, q ˆ 0:1.

maintaining the coherent pattern and then becomes time independent with an increase in spectrum width. Here, the time-independent state is de®ned such that the power ¯uctuation modulus becomes smaller than 0.1% of its average output power. The time-independent state generated at Pin ˆ 5 W in the coherent case does not change in dynamics ± independent of the coherence state. Fig. 4 shows the coherence e€ect on the bifurcation diagram in the input and output power relationship in which the spectrum width is set at Dm ˆ 0 MHz (coherent) in (a) and Dm ˆ 1:0 MHz (low coherent) in (b). The ¯uctuation modulus in the output power with Dm ˆ 1:0 MHz is much less than that in the coherent case. However, bifurcation characteristics such as the period doubling route to chaos are independent of source coherence and are almost identical between the two cases. Analysis of the diagram in detail con®rms

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Fig. 5. The Pin ±Dm state map of OFRR output with ®ber ring length L ˆ 10 m in (a) and L ˆ 100 m in (b). Other parameters match those in Fig. 2.

Fig. 4. Bifurcation diagram of the input and output power relation with spectrum width Dm ˆ 0 MHz (coherent) in (a) and Dm ˆ 1:0 MHz in (b). Other parameters match those in Fig. 2.

that the period changes from 2s to 3s, 6s, 12s, and then to chaos as input power increases from 6, 10, 13.8, 14.2 to 14.5 W in each case. Bifurcation characteristics are similar except for the power scale so long as the total loss of the resonator (aL) is constant. Dynamics of the OFRR output which are dependent on source coherence are arranged in the state map as shown in Fig. 5 where the time-independent, periodic, and chaotic states are designated by the symbols ( ), ( ), and ( ), respectively, and the smallest input power is chosen at 0.1 W. Fiber ring resonator length is set at L ˆ 10 m in (a) and 100 m in (b). Other OFRR

parameters are set at j ˆ 0:5, q ˆ 0:1, n2 ˆ 1  10 22 m2 =V2 . Critical spectrum width, at which the output becomes chaotic, decreases with increased resonator length. This observation supports the fact that OFRR output ¯uctuates as a result of interference among the coupled input light and circulated light components. The number of the circulated components which contribute coherently to output interference declines with decreased coherence length and increased resonator length. Both the chaotic and periodic states become time independent with a further increase in spectrum width. The critical input power, at which chaotic output appears, is inversely proportional to ring resonator length because sucient phase modulation due to the nonlinear refractive index is applied increasingly to circulated light as resonator length increases. The critical input power producing chaotic output is estimated as 300 W for L ˆ 5 m, 150 W for L ˆ 10 m, 15 W for L ˆ 100 m,

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1.7 W for L ˆ 1000 m, and 0.16 W for L ˆ 10 000 m. For a long OFRR, chaotic output appears in a practical range of input power. The critical input power directly implies that nonlinear phase shift imposed on the circulated component can be estimated roughly as p radian according to D/ ˆ kLn2 Pin . This fact implies that the chaotic state requires a nonlinear phase shift greater than p so that interference output ¯uctuates suciently, which agrees well with previous works [4,5]. The OFRR state map for input power and nonlinear refractive index variables (Pin n2 domain) is shown in Fig. 6 in which Dm ˆ 0 MHz (coherent), j ˆ 0:5, q ˆ 0:1, and L ˆ 100 m. The state map retains the same pattern with increasing Dm up to 3.0 MHz. In cases where Dm < 3:0 MHz, OFRR output becomes complicated as both Pin and n2 increase. On the other hand, OFRR output is always time independent, even in the high Pin and n2 region, when the spectrum width exceeds 3 MHz. This result re¯ects that the coupled input light and the many circulated components cannot interfere coherently but are added in intensity in the low coherent case (Dm > 3 MHz). The OFRR dynamics in the Pin ±j domain with the parameters Dm ˆ 0:1 MHz, q ˆ 0:1, and L ˆ 100 m are shown in Fig. 7. Chaotic output appears easily with a low input power around j ˆ 0:3. This is because amplitudes of the ®rst several circulated components are in a proximate level to each other when j ˆ 0:3. Consequently, interference occurs most e€ectively. The OFRR

Fig. 6. The Pin ±n2 state map of OFRR output with input spectrum width Dm ˆ 0 MHz (coherent). Other parameters match those in Fig. 2.

263

Fig. 7. The Pin ±j state map of OFRR output with input spectrum width Dm ˆ 0:1 MHz, coupler loss q ˆ 0:1, and ®ber ring resonator length L ˆ 100 m. Other parameters match those in Fig. 2.

state map in Pin ±q domain is shown in Fig. 8 in which parameters are set at Dm ˆ 0:1 MHz and j ˆ 0:5. Chaotic output appears as the loss becomes smaller with higher input power. The large loss causes a reduction in interference components which leads to the time-independent output. This e€ect is similar to that due to coherence reduction since the lower coherence brings about lower interference in the output. However, it is noted that coherence reduction results in a decrease in the modulus of the output ¯uctuation while keeping the coherent pattern and consequently leads to the time-independent state. On the other hand, the increased loss causes a state change from the

Fig. 8. The Pin ±q state map of OFRR output with input spectrum width Dm ˆ 0:1 MHz, coupling coecient j ˆ 0:4, and ®ber ring resonator length L ˆ 100 m. Other parameters match those in Fig. 2.

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Fig. 9. Critical spectrum width Dmc at which OFRR output changes from the chaotic to the convergent state as a function of ring resonator length L. Other parameters match those in Fig. 2.

chaotic to the periodic and then to the time independent. Fig. 9 shows the relationship between the critical spectrum width which produces the timeindependent state and the ®ber ring resonator length. Parameters are set at q ˆ 0:1 and j ˆ 0:5. Critical spectrum width is inversely proportional to resonator length. Critical coherence length is derived from the critical spectrum width as 10 m for 30 MHz, 30 m for 10 MHz, 300 m for 1 MHz. It should be noted that each critical coherence length corresponds to each resonator length. Namely, circulated components cannot interfere coherently with each other, followed by the timeindependent output, when coherence length is shorter than resonator length. Resonator length and ®ber loss a€ect OFRR dynamics in a compound manner, i.e., ®ber ring resonator loss is given by the product aL. Circulated components which contribute to interference output reduce with an increased ®ber loss, which is similar to the e€ect of coupler loss. Fig. 10 shows how coherence and total loss a€ect the dynamics for the four di€erent resonator lengths L ˆ 10, 50, 100, and 200 m. Each line represents the critical spectrum width at which dynamics change from the chaotic to the time-independent state. The region lower than the line in the graph indicates the chaotic, while the upper region is the time independent.

Fig. 10. Critical spectrum width Dmc at which OFRR output becomes time independent as a function of resonator loss at di€erent resonator lengths L ˆ 10, 50, 100 and 200 m. Other parameters match those in Fig. 2.

Critical spectrum width decreases as the total loss increases at each resonator length.

4. Conclusions The coherence e€ect on OFRR dynamics was investigated numerically. The OFRR output with coherent input light begins to ¯uctuate periodically and then, in a chaotic manner, which indicates the route to chaos, as input power increases. Chaotic and periodic behaviors calm down in the ¯uctuation modulus with an increase in input light spectrum width, i.e., with a decrease in coherence length, though patterns and bifurcation characteristics under coherent input light are maintained. Both ¯uctuation behaviors become time independent as coherence degrades further. The coherence length required for chaotic output is proportional to resonator length. This critical coherence length lengthens as the resonator loss increases. When input light coherence is reduced, many circulated components cannot contribute coherently to output interference, resulting in stable and timeindependent output. It should be noted that the coherence e€ect on output dynamics di€ers from that of ®ber loss, which changes the route to chaos characteristics. Chaotic dynamics are apt to generate in a higher nonlinear refractive index and a longer resonator region as well as lower ®ber

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coupler loss when input power is kept constant. This characteristic becomes hard to discern; consequently, OFRR output is always time independent as source coherence degrades.

Acknowledgement This research was partially supported by a Grant-In-Aid for Scienti®c Research from the Japanese Ministry of Education.

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