## Abstract

Optical fiber ring circuits constructed with frequency shifters and EDFAs are applicable to pulsed lightwave frequency sweepers, wavelength converters, and optical packet buffers. The salient criterion for those applications is how many times the optical pulse can circle the ring. Optical band-pass filters in the ring can serve an important role for pulse circulation because the filter determines the gain bandwidth at every circulation under the condition of signal wavelength shift. This paper clarifies the effects of optical filter response on pulse circulation in the ring through numerical simulation of the EDFA dynamic model, considering the gain spectrum.

©2006 Optical Society of America

## 1. Introduction

Optical ring circuits with frequency shifters and Erbium-doped fiber amplifiers (EDFAs) are useful for many purposes for optical sensing systems and optical communications. For example, pulsed lightwave frequency sweepers [1, 2], optical tunable delay lines for optical packet networks [3, 4], and wavelength converters [5] employing rings have all been proposed. A typical ring configuration is shown in Fig. 1. The rectangular optical pulse is incident into the ring; subsequently, the pulse can make round trips. The ring output is an optical pulse sequence constructed from replicas of the incident pulse. The frequency shift from every circulation of the ring is useful for both optical frequency shifters and wavelength converters. The sweeper was used to measure chromatic dispersion of fiber-optic components [6] and distributed strain/loss along fibers combined with optical time-domain reflectometry techniques [7]. Optical tunable delay line applications in packet network nodes take advantage of both the delay time in a round trip and many circulations without unexpected oscillation.

An important issue for these applications is the maximum circulation of the optical pulse in the ring, i.e., how long an optical pulse can remain in the ring. Maximum circulation can determine specifications of a sweeping frequency range for sweeper applications, the wavelength channel number for wavelength conversion applications, and buffering time to store the packet for tunable packet delay line applications. K. Shimizu and K. Aida explained qualitatively [2, 8], using a simple EDFA model with constant EDFA output power, that the pulse circulation is limited by accumulated amplified spontaneous emission (ASE) noise from EDFA, which is located in the ring for compensation of the optical loss. The growth of ASE results from the combination of EDFA gain saturation with transmittance of optical filters, which are employed to suppress ASE noise and a residual pump light in the ring. However, no studies have clarified optimum bandwidth and detuning of optical filters for a long circulation of the optical pulse. Intuitively, the pulse circulation can continue as long as the signal, of which the wavelength is shifted by a frequency shifter, can maintain the unity loop gain. The gain spectrum of a ring configuration such as that shown in Fig. 1, however, is determined mostly by optical filters. Broad-bandwidth filters, then, have a large gain spectrum, but ASE accumulation occurs more quickly. This paper presents a quantitative evaluation of the pulse circulation by changing optical filter characteristics using EDFA numerical simulation, while taking account of the gain spectrum and gain dynamics. Results show that a sufficient filter bandwidth is about 10-20 nm, and that filter detuning can raise the number of pulse circulations. Moreover, the center frequency of the filter has an optimum value for maximizing pulse circulation. This study provides useful knowledge for selecting filters for ring design.

## 2. Fiber ring with frequency shifter and EDFA

#### 2.1 Configurations of the ring circuit

Figure 1 shows the optical ring configuration and a timing chart for operation. The optical ring circuit is constructed with an optical frequency shifter, an EDFA, an optical band-pass filter (BPF), and a fiber delay line. For an optical frequency shifter, most applications, except for a wavelength converter in dense wavelength-division-multiplexing (WDM) fiber communications, are more adaptable to acousto-optic frequency shifters (AOS) because of their features, which include stability of frequency shift, no harmonic generation, adequate frequency shift values, and so on. The ring requires a master laser and an optical switch with a high extinction ratio, such as an acousto-optic modulator (AOM), when the ring is used for an optical frequency sweeper. The optical switch can convert the continuous wave (CW) signal light of the master laser to an optical pulse and the pulse is incident into the ring. The pulse experiences a round trip in the ring and both a frequency shift and a time delay occur every round trip. The ring output is an optical pulse sequence constructed from replicas of the incident optical pulse. When the oscillation frequency of the master laser is stabilized, then the output pulse train is also frequency-stabilized; in addition, its frequency can be swept discretely. That is the key function of an optical frequency sweeper. An EDFA is employed to compensate the optical loss of the ring. The following BPF rejects ASE noise and residual pump light from the EDFA.

#### 2.2 Numerical simulation model

Optical signal power at the output port of the ring after the *n*-th circulation is derived from that of the (*n* - 1)th circulation using the following equation:

where *P* is the optical power, *v* is the optical frequency, *Δv*_{FS}
is the frequency shift of AOS, and *n* is the ring-circulation number. Subscript *SIG* indicates the signal component. Superscript (*n*) indicates the *n*-th circulation, *G* is the EDFA gain, *F* is BPF transmittance, *α*_{ring}
is the optical loss of the ring except for both EDFA and BPF, and 0< *t* <*T*_{ring}
.

The ASE component is also derived similarly as

where the subscript of *ASE* indicates the ASE component and *S*
^{(n)} is the additional ASE power from the EDFA at the *n*-th circulation.

The EDFA gain and ASE spectrum are calculated using the following EDFA dynamic model. The simulation is based on the model of a homogeneously broadened two-level system described by propagation equations and rate equations [9] incorporating wavelength dependence of the gain and ASE noise. The propagation equation concerning optical power *P* in EDF can be described as the following.

$$+\mathit{umh}\phantom{\rule{.2em}{0ex}}vg\left(v\right)\Delta v{N}_{2}\left(z,t\right)$$

In that equation, *u* indicates the propagation direction, i.e., *u* = 1 denotes the coincidence of propagation direction with signal light, and *u* = -1 denotes the opposite propagation direction with signal light. Also, *N*_{1}
and *N*_{2}
respectively denote the atomic population densities at ground level (level 1) and the excited level (level 2). In addition, *g* and *α* respectively indicate the gain and absorption coefficients of EDF; *l* denotes excess fiber loss. The number of transversal modes is expressed as *m*, which is normally 2 because of the two polarization states of the fundamental mode; *h* is Planck’s constant. In our numerical model, *Δv* represents the frequency resolution of calculation for spectra.

The rate equation for two-level models can be written as

where *A*_{21}
is spontaneous decay rate from level 2 to level 1. The stimulated emission and absorption rates between level *i* and level *j* are indicated as *W*_{ij}
and *W*_{ji}
(*i* > *j*), respectively. They are described through the following relations for two-level model as

where *k* is the integer number indicating the optical frequency to solve Eq. (4) numerically. In addition, *P*_{SAT}
is the saturation power parameter of EDF.

The EDFA dynamic model described above is valid for prediction of practical EDFA operation in the ring in which an optical pulse goes around because of the ability to calculate the wavelength dependency and the time-dependent response. The accuracy of this model for wavelength dependency was clarified previously [10], and agreement with experimental results about the dynamic response of EDFAs was reported in other works [9, 11]. For EDFA simulation in the fiber ring with a frequency shift, we have produced a special coding for wavelength resolution. Intuitively, the frequency resolution of EDFA simulation *Δv* is presumed to be identical to the frequency shift *Δv*_{FS}
. However, the EDFA gain bandwidth is broad, several terahertz, whereas the frequency shift of AOS is several hundreds of megahertz. Consequently, the intuitive condition of *Δv*= *Δv*_{FS}
must consume many computer resources and spend too much calculation time. To overcome that difficulty, our simulation code has frequency resolution *Δv* of 0.05 nm, which differs from *Δv*_{FS}
written in Table 1 and which moves to *Δv*_{FS}
every pulse circulation, incorporating an interpolation technique for frequency dependencies of both *α* and *g*.

To verify our simulation, we compared simulation results with the experimental ring output waveform. Figures 2(a) and 2(b) respectively show experimental input and output waveforms of the ring. The optical power was observed using an oscilloscope after detection by photodiodes. The accumulation of ASE noise from numerous circulations was observed because the input pulse width was adjusted to be slightly shorter than the ring round-trip time. The waveform is similar to the result described in Ref [1]. Figure 2(c) shows a simulation result assuming that the ring parameters were identical to those of the experiment. Simulation results showed good agreement with experimental results for power distribution and ASE accumulation. Our simulation is therefore demonstrably valid for analyses of the ring.

Table 1 shows simulation conditions using the following calculation. The wavelength dependence of the Er^{3+} absorption coefficient *α*, gain coefficient *g*, EDF excess loss *l*, and the saturation parameter *P*_{SAT}
take equal values of fiber type (2a), as shown in a previous study [12]. The respective wavelength dependences of *α* and *g* are shown in Fig. 3(a). The wavelength range of our simulation was 1460-1570 nm.

Under that condition, EDFA gain characteristics are shown in Fig. 3(b) as a function of EDFA input power at the wavelength of *λ*_{SIG}
, which is the WDM anchor wavelength. Small signal gain of EDFA *G*_{small}
is 14.2 dB, whereas the ring loss, except for EDFA and BPF *α*_{ring}
, is 13.2 dB. The absolute condition to maintain pulse circulation is that the whole ring gain is unity. Therefore, BPF transmittance loss at the signal frequency is necessarily smaller than the gain margin *G*_{margin}
defined as *G*_{small}
- *α*_{ring}
. The *G*_{margin}
is 1.0 dB in this simulation. The EDFA noise figure is 5.8 dB. For incident light to the ring, a quasi-CW condition (*T*_{p}
= *T*_{ring}
) is employed for simplicity and the peak power of the initial pulse at the EDFA input port ${P}_{\mathit{\text{SIG}}}^{\left(0\right)}$ is adjusted so that it is almost equal to that of the EDFA operating point at which the gain of ${v}_{\mathit{\text{SIG}}}^{\left(0\right)}$ is equal to the ring loss, to thereby suppress unexpected relaxation oscillations of optical power. The BPF transmittance frequency response is assumed as a Gaussian shape:

where *B* is the full-width at half-maximum (FWHM) of the filters and *V*_{0}
is the center frequency of filters. The polarization-dependent loss of the ring and the polarization mode dispersion are ignored in this simulation. In our simulation model of the homogeneously broadened two-level system, spectral hole burning effect of EDF was not taken into account because shallow gain saturation was assumed in our condition.

Employing the model above, we evaluate the pulse circulation limit caused by ASE accumulation with changing the filter parameter, i.e., filter bandwidth *B*, filter detuning *Δλ*_{d}
defined as *c/v*_{0}
- *λ*_{SIG}
, and filter shape.

## 3. Results and discussion

#### 3.1 Power distribution

Figure 4(a) shows the response envelope of the output pulse train in the case of *B* = 10.0 nm and *Δλ*_{d}
= 1.4 nm in terms of wavelength; Fig. 4(b) shows output spectra at circulations of 200, 400, 600, 800, and 900. The output power level is almost constant with a slight relaxation oscillation. Rapid growth of the ASE component is visible at about the 800th circulation of the ring. Here, the circulation limit *N* is defined as the circulation number at which the signal power is equal to the ASE power. This case achieves *N* of 846. It corresponds to the 211.5 GHz frequency translation range as a sweeper. The ASE level variation is categorizable into three stages. At the first stage, the ASE level tends to increase with circulation because the ASE at the wavelength of the BPF peak transmittance grows as a result of BPF detuning, as shown in Fig. 4(b). At the second stage, the ASE component is decreasing with circulation. The signal component increases with circulation because the signal wavelength has moved near the center wavelength of the BPF. At the circulation corresponding to the ASE peak wavelength, the ASE component is reduced through the EDFA saturation. For the case shown in Fig. 4, the minimum ASE power in the second stage is obtained at the 650th circulation corresponding to the 162.5 GHz (1.3 nm) signal frequency shift. It is almost identical to BPF detuning. At the third stage, the ASE is again increasing because of the low transmittance of BPF at the signal wavelength.

#### 3.2 Filter bandwidth and center wavelength

Figure 5(a) shows the circulation limit *N* as a function of filter detuning *Δλ*_{d}
(= *c/v*_{0}
- *λ*_{SIG}
) with changing filter bandwidth. Filter detuning has an optimum value for the circulation limit at each filter bandwidth. Consequently, the center wavelength of BPF should show a wavelength shift from the master laser wavelength along with the direction of the frequency shift of AOS. Using the case of *B* = 15.0 nm as an example, we can explain the variation of the circulation limit as a function of filter detuning. For subsequent discussions, the maximum circulation limit *MCL* is defined as a maximum value among the data of circulation limit *N* with the same bandwidth. Optimum detuning *Δλ*_{opt}
is defined as detuning by which the maximum circulation is achieved. For 0 < *Δλ*_{d}
< 1.2 nm, the circulation limit increases with filter detuning. With larger filter detuning, the circulation limit increases rapidly and saturates (1.2 nm < *Δλ*_{d}
< 2.2 nm). The maximum circulation limit is 897 with *Δλ*_{d}
= 2.2 nm under the condition of *B* = 15.0 nm. Subsequently, the circulation limit shows a sudden drop to lower circulation than 200. To elucidate the difference between the case of optimum detuning and the non-optimum case, we show the response envelope of the output pulse train with *B* = 15.0 nm, as shown in Fig. 5(b). The middle graph of Fig. 5(b) shows the case of optimum detuning (*Δλ*_{d}
= 2.2 nm); other graphs of (a) and (c) show a non-optimum case (*Δλ*_{d}
= 1.0, 2.8 nm). The case of optimum detuning shows that the signal power is emphasized temporarily and that ASE power is reduced briefly before reaching a signal-to-noise ratio of unity. The phenomenon is described in the previous section as the second stage. For other cases, the phenomenon does not occur. For *Δλ*_{d}
= 1.0 nm, the case of smaller detuning from the optimum, the signal power tends to decrease monotonously with circulation, whereas the ASE power is inversely increasing. The case has only the first stage described in the previous section, which is caused by the decreased BPF transmittance, resulting from the signal wavelength increase with signal circulation of the ring. For *Δλ*_{d}
= 2.8 nm, which is the case of larger detuning from the optimum, more rapid growth of ASE is apparent than in the other two cases after several tens of circulations. The signal at less than several tens of circulations incurs an overly large optical loss from BPF, which results from strong detuning to maintain the signal power. It induces the rapid growth of ASE power.

The explanation above shows that filter detuning is useful to increase the circulation limit of the ring. Reviewing Fig. 5(a), no cases of *B* = 25.0 nm exhibit a dramatically increasing circulation limit attributable to the detuning effect. In this case, with detuning smaller than the optimum detuning, the filter bandwidth is so large that the wavelength dependence of the loop gain is determined dominantly by the EDFA gain spectrum. The peak gain wavelength, then, becomes ca. 1552.0 nm, which is a shorter wavelength than *λ*_{SIG}
. The peak wavelength of accumulated ASE is shorter than that of the signal, as shown in Fig. 6.

In Fig. 7, the circulation limit is described using color contrast with changing filter bandwidth and detuning. The magenta dashed line is the absolute upper limit of filter detuning calculated from *G*_{margin}
of 1.0 dB resulting from the unity ring gain (*F*(*λ*_{SIG}
) = 1/*G*_{margin}
). Pulse circulation is obtained with the condition of lower detuning than the upper limit detuning. Note that maximum circulation limit, which is defined as the maximum value among the data of circulation limit with the same bandwidth, forms a nearly straight line in Fig. 7.

Figure 8 depicts the *MCL* (solid circles) and the optimum detuning *Δv*_{opt}
(white circles) as a function of filter bandwidth. For *B* < 12 nm, the *MCL* increases linearly with increasing filter bandwidth *B*. For 12 < *B* < 20 nm, the *MCL* has an almost constant value of 900. These values of *B* are optimal for many pulse circulations. For *B* > 20 nm, the *MCL* drops suddenly. This drop occurs because the gain peak wavelength has shorter wavelength than *λ*_{SIG}
, whereas the signal wavelength is shifted to a longer wavelength through every round trip. The saturation and the sudden drop of *MCL* with increasing *B* were not predicted in one previous study [8] because the investigation ignored the EDFA gain spectrum and treated only small bandwidth filters.

The white circle plot in Fig. 8 shows that optimum detuning, which causes the maximum circulation at a certain bandwidth, is proportional to the filter bandwidth. The dashed line is an approximation of simulation data produced using least-squares fitting. In this case, the dashed line is *Δλ*_{d}
= 0.159 × *B*, where *λ*_{d}
and *B* are expressed in units of nanometers. The factor of the equation corresponds to the constant transmittance at initial pulse wavelength. Solving Eq. (8) for filter detuning *Δv*_{d}
with *v* = ${v}_{\mathit{\text{SIG}}}^{\left(0\right)}$, the relationship between detuning and the filter bandwidth can be expressed as

With reference to Eq. (9), the factor of 0.159 means the *F*(*λ*_{SIG}
) = 0.304 dB. Let us consider why the BPF transmittance for an initial signal of 0.304 dB induces optimum detuning. Figure 5 shows that filter detuning can expand the pulse circulation. However, overly large detuning induces rapid growth of accumulated ASE at several tens of circulations, as shown in Fig. 5(b) with *Δv*_{d}
of 2.8 nm, even if *F*(*λ*_{SIG}
) is larger than 1/*G*_{margin}
. The maximum circulation limit with optimum detuning is, therefore, the critical state with many circulations because of detuning effects. To facilitate careful observation of the rapid growth of accumulated ASE at early circulation, Fig. 9(a) depicts a different plot of the power envelope response at early circulation of Fig. 5(b) in terms of decibels. Figure 9(b) shows the dynamic gain response of EDFA in the case of a filter bandwidth of 15.0 nm. The signal component in the case of *Δv*_{d}
= 2.8 nm suffers a slightly larger (about 0.4 dB) ring loss *α*_{ring}
from filter detuning than the other two cases. It causes a slightly larger EDFA gain. Additional ASE power from every circulation is proportional to the EDFA gain [13]. Different ASE power is emphasized by the signal circulation in the ring. The ASE components in the case of *Δv*_{d}
= 2.8 nm, then, are clearly distinguishable from those of the other cases after the 30th circulation. On the analogy of ASE power accumulation of long-haul fiber transmission with concatenated EDFAs and with identical span loss, accumulation can be determined by the noise figure of EDFA.

The constant transmittance of initial pulse wavelength for optimum detuning is useful knowledge for filter design. Once designers select an EDFA for the ring and test optimum filter detuning with a certain filter bandwidth, the optimum detuning for other bandwidths are readily calculable from the results. If the operation point of EDFA is moved to a 1 dB more saturated condition but with little NF degradation by longer loop length, we have obtained almost identical results for both ASE accumulation and the constant transmittance of initial pulse wavelength for optimum detuning. Moreover, when we applied our simulation in other case with longer EDF (15 m) and lower pump power (25 mW), the input power of EDFA at the operation point became 5.5 dB smaller than the case of Fig. 3(b) and NF of EDFA was 6.0 dB. That situation readily generates more ASE components every circulation. However, a linear relation between filter bandwidth and optimum detuning is also observable. Therefore, the change of inversion condition does not degrade the linear relation. Of course, the factor has a lower value of 0.102, which means that *F*(*λ*_{SIG}
) = 0.125 dB.

#### 3.3 Filter shape dependency

In the previous section, the Gaussian function was employed for BPF transmittance shape. Recent filter fabrication techniques can provide flat-top transmittance. In this section, then, we evaluate the impact of flat-top filter response on the ring circulation of the optical pulse.

A super-Gaussian function was assumed for flat-top filter response in the following analysis, as

the curve is drawn in Fig. 10.

The maximum circulation limit is shown in Fig. 11(a) at optimum detuning as a function of the filter bandwidth. The simulation assumptions are the same as those shown in Table 1. Comparison with Fig. 8 shows that the maximum circulation limit of each case between 10 < *B* < 20 is almost identical to 900. The *MCL* increase rate at *B*< 10 is faster with larger Gaussian order *m*. The flat-top filter shape, therefore, presents the advantage of increasing *MCL* with the same bandwidth, or the expansion of filter bandwidth for maintaining the peak *MCL* value, 900 in this case. Figure 11(b) shows optimum detuning with changing filter bandwidth. The optimum detuning value tends to be linearly proportional to the BPF bandwidth in each Gaussian-order case. Dashed lines show approximations of simulation data using least-squares fitting. As mentioned in the previous section, linearity means the constant transmittance for the initial signal. Table 2 shows the factor of fitted lines and the transmittance of the initial signal, which are calculated using the equation expanded for a super-Gaussian function from Eq. (9), for each Gaussian order case. Optimum detuning for the maximum circulation limit engenders constant filter transmittance for the initial circulation signal, even if the flat-top filter is used. It would change when the EDFA noise figure changes.

## 4. Conclusion

The impact of optical filter response on the pulse circulation limit in a fiber ring with a frequency shifter and EDFA were evaluated using the EDFA dynamic model incorporating the gain spectrum.

The signal pulse circulation vanishes because of the growth of ASE power accumulation. The maximum circulation limit can be extended dramatically through the use of filter detuning, which is defined as the frequency difference between the BPF center frequency and optical frequency of the initial signal. Over-detuning, however, induces a situation in which the loop gain peak wavelength is determined dominantly by the EDFA gain spectrum, not by BPF. In such a situation, the expansion of maximum circulation limit because of filter detuning cannot be observed because the rapid accumulation of ASE can be induced at early signal circulation.

The optimum filter detuning for the maximum circulation limit is, in rough terms, linearly proportional to the BPF bandwidth. Consequently, filter detuning should be set so that the transmittance at the initial pulse frequency is a certain value. In the case of our EDFA condition, the transmittance is 0.3 dB. The value depends on the noise figure of EDFA. This feature of the linearity between the optimum detuning and the BPF bandwidth can be obtained when the pumping condition is changed. Then, qualitatively identical results are obtainable if other EDF parameters are used. The filter bandwidth has optimum area for long pulse circulation. Pulse circulation of 900 is achievable at the optimum bandwidth area of 12 < *B* < 20 nm, implying that the optical packet can be delayed 900 times longer than a packet duration for optical packet buffer applications or that the frequency translation range for pulsed sweepers can be achieved at 225 GHz. The optimum bandwidth range for long circulation can be expanded slightly using the flat-top optical filter in the ring.

## Acknowledgments

The authors would like to thank Makoto Ueda, Hironobu Sato, and Yukiko Suzuki for their assistance with simulations. The authors would like to thank the reviewers for their valuable comments.

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