Extinction ratio improvement by an all-optical signal regenerator with a semiconductor optical amplifier and a Sagnac loop

Optics Communications 266 (2006) 80–87 www.elsevier.com/locate/optcom

Extinction ratio improvement by an all-optical signal regenerator with a semiconductor optical amplifier and a Sagnac loop Er’el Granot a,*, Shalva Ben-Ezra a, Reuven Zaibel a, Haim Chayet a, Niv Narkiss a, Nir Shahar a, Shmuel Sternklar a, Sagie Tsadka a, Paul R. Prucnal b b

a KaiLight Photonics, 2b Bergman, Rehovot 76124, Israel Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA

Received 22 December 2005; received in revised form 27 March 2006; accepted 18 April 2006

Abstract In this paper, we present the operation of an all-optical signal regenerator (AOSR). The AOSR is based on a semiconductor optical amplifier (SOA), which is incorporated in an asymmetric Sagnac loop (ASL). We show experimentally that the AOSR is capable of improving the input signal extinction ratio (ER) considerably. We present a theoretical model, which explains this improvement, and we illustrate a qualitatively similar behavior with a simulation. Ó 2006 Elsevier B.V. All rights reserved. OCIS: 060.2340; 060.4370; 230.1150

1. Introduction The transmission of high-speed optical signals through very long fibers requires transponders or regenerators. These devices are distributed along a communication line and should overcome noise. The noise, which accumulates in the signal, is mainly due to the multiple amplifiers in the communication line. Part of the noise essentially increases the zero level, and thus reduces the signal’s extinction ratio (ER). Therefore, ER improvement is a required property in optical regenerators. Moreover, in many cases the modulator operates in the weak modulation regime (as opposed to deep modulation), either due to the relative ease of operation (and affordable cost) to operate the modulator in this regime, or because of changes or drift in operation conditions. In these cases, an affordable regenerator, which improves ER, is very useful. It has been shown [1] that all-optical regenerators have the potential of improving

*

Corresponding author. Tel.: +972 89470770x9221; fax: +972 89470771. E-mail address: [email protected] (E. Granot). 0030-4018/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2006.04.021

the signal’s BER as well as the signal’s SNR. Many successful demonstrations of optical signal reshaping (i.e., a component of regeneration) have been demonstrated with semiconductor optical amplifiers (SOA) in interferometric devices based on cross-gain (XG) and cross-phase (XP) modulation schemes (e.g. [1–3]). The general idea to improve ER is relatively simple. The device is essentially an interferometer, in which the ER is improved by employing the nonlinear response function to squeeze and truncate the zeros-level and ones-level. Additionally, the noise near the truncation level is also truncated and reduced. The interference conditions are determined by the relative phase of the interfering signals. This is achieved by passing the input signal through an SOA and interfering it with a reference arm. In general, ER-improving interferometric devices have two main deficiencies: their speed is limited by the relatively slow carrier recovery time (CRT) of the SOA (e.g. Ref. [1]), and due to the interference, they are vulnerable to thermal and mechanical noise [2,3]. In the past, most all-optical regenerators were designed to work in a configuration where the data transcription is done within a relatively fast SOA, and the final DC

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reduction and extinction ratio improvement are carried out using a Mach–Zehnder (MZ) interferometer [2–8]. The SOA’s fast response was essential for the signal’s reconstruction. The use of a slow SOA resulted in intrasymbol-interference (ISI) problems. However, this configuration has two drawbacks. Currently, it is almost impossible to obtain, even from a fast SOA, high performance at very high bit-rates (P40 Gbs), and MZ interferometer configurations are highly unstable. Both of these drawbacks introduce noise to the system’s operation. This problem was partially overcome for the return-to-zero (RZ) format using a delay MZ interferometer [6–8], Sagnac loop [9] and bandpass filter [10]. However, in most communication lines the ruling format is nonreturn-to-zero (NRZ). Other techniques, like Brillouin filters, which were implemented with the NRZ format, suffered from ISI [11,12]. In this paper, we present a device based on an asymmetric Sagnac loop (ASL) which does not suffer from these problems. In contrast to the MZ, which is highly unstable in the presence of environmental changes, a Sagnac interferometer is very robust against external fluctuations. Evidently, integrated MZI’s are considerably more robust, however, their performances should be compared to integrated Sagnac loops. For a given technology, Sagnac loops are always more stable than MZI. Furthermore, instead of working in the strong modulation, short CRT (fast) regime, where the SOA’s output signal bears a resemblance to the data signal, our working point is in the relatively weak modulation, long CRT (slow) regime, where the SOA’s output signal resembles the integration of the incident signal over some portion of the bit-period. The Sagnac loop was already incorporated in other signal-manipulating devices such as optical switches, optical logical gates and wavelength converters, for RZ signals [13–15] and NRZ signals [16,17] and even in 40 Gb/s [18]. Moreover, a Sagnac loop in a reflection configuration was also used as a waveform restoration amplifier for NRZ signal [19]. In this paper, we demonstrate a signal regenerator (ER improver) based on ASL in a transmission configuration for NRZ signals. We expect this device will work equally well for the RZ format. We also elaborate on the theoretical basis of the ASL operation. 2. Theory In order to regenerate a signal, the zero level must be reset to a sufficiently small value. By resetting the zero level, three goals are achieved: the extinction ratio improves (and shot-noise is reduced); the noise at this level decreases; and susceptibility to dispersion is reduced. Obviously, a destructive interference with a reference arm can accomplish this task, but as was explained in Section 1, interference with an external reference is usually noisy. In the proposed regeneration scheme, zero-level nullification is done with a Sagnac interferometer, as depicted in Fig. 1.

81

ΔT SOA

PCW , λ i

C2

φR

φL PC2

Pin , λ s

PC1

C1

F

PC3

Pout , λ s P

Fig. 1. System schematic. Pin is the incident signal with the carrier wavelength ks, PCW is the CW beam with an idler wavelength ki, the PC’s are polarization controllers, C1 and C2 are couplers, F is a spectral filter, P is an optical polarizer and /R and /L are the phases of the clockwise and counter-clockwise propagating beams, respectively. The dotted elements: the idler wavelength coupler C2, the filter F, PC3 and the polarizer P are only auxiliary components and are not part of the basic operation.

In order to understand the device dynamics and to simplify the analysis, we assume the SOA is very slow (long CRT) and that it operates in the weak modulation regime. We will see later that the device will work properly even when these requirements are not fully kept. In this regime of operation, the SOA’s output phase is proportional to a running average (i.e., an integral) over the input power (see Appendix A). The SOA is incorporated into a Sagnac interferometer, whose role is to differentiate the SOA’s output and to restore the original shape of the incident signal, except for the DC part. Operating in this mode provides two advantages: The SOA can operate in a relatively slow mode, and since the output is proportional to the integral over the input signal, the high-frequency noise is washed out. Moreover, a Sagnac interferometer is considerably more stable than a Mach–Zehnder or Michelson interferometer, since the interference occurs between the co- and counter-propagating beams, which traverse the same optical path. The regenerator scheme is shown in Fig. 1. A coupler (C1) splits the incident signal beam (Pin) into two counter-propagating beams. Both the co- and counter-propagating signal beams experience a nonlinear phase shift induced in the SOA by changes in the signal’s intensity. Since the SOA’s gain is proportional to the integral over the input signal (see Appendix A), its phase can be written approximately as Z s 1 /out ðsÞ ¼ /0 þ aC pAC ðnÞ dn; ð1Þ 2 1 where we separated Pin(s) = pDC + pAC(s) into its DC and AC components, and a, C and /0 are constants. Due to this integral behavior, we take the derivative of the SOA’s output to recover the shape of the input signal.

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This is done by subtracting the two beams (co- and counter-propagating) from each other at the coupler C1, and by carefully placing the SOA with a slight offset (the time delay DT in Fig. 1) from the midpoint of the Sagnac loop. As a consequence, although the two counter-propagating beams arrive at the coupler with perfect synchronization, their phases do not, and the coupler’s output resembles a temporal derivative of the SOA’s output phase. At the same time, amplitude modulation occurs, but its influence is negligible in comparison to the initial amplitude changes, since in most SOA’s the linewidth enhancement factor is larger than 1. Therefore, amplitude modulation is ignored in this theoretical treatment. R Mathematically, if ER SOA ðtÞ ¼ eSOA ðtÞ exp½iu ðtÞ and L L ESOA ðtÞ ¼ eSOA ðtÞ exp½iu ðtÞ are the electric fields that emerge from the SOA in the clockwise (‘‘R’’) and counter-clockwise (‘‘L’’) directions, respectively, then the two phases can be described using the SOA’s output phase u0(t) [where u0(t) ffi uout(t)], according to uR(t) = u0(t + DT/2) and uL(t) = u0(t  DT/2). Hence, the field at the exit of C1 is given by L ER SOA ðtÞ  ESOA ðtÞ

Fig. 2. An ideal representation of the phases in the system. The upper panel stands for the input signal’s power. The central panel represents the phases of the beams that exist the SOA. DT is the time lag between the two beams and b is the relative phase between them (in this case b is negative). L In the lower panel the two phases /R CW and /CW are represented. Hereinafter we plot only 10 Gb/s NRZ signals.

¼ eSOA ðtÞfexp½iuR ðtÞ  exp½iuL ðtÞg ¼ eSOA ðtÞfexp½iu0 ðt þ DT =2Þ  exp½iu0 ðt  DT =2Þg ffi iDT eSOA ðtÞ exp½iu0 ðtÞ du0 ðtÞ=dt:

ð2Þ

In principle, this derivative operation completes the reconstruction of the input signal, since the derivative du0(t)/dt is proportional (according to Eq. (1)) to the AC component of the input signal. However, since this analysis reconstructs only the AC part of the input signal, the DC component still has to be calibrated (i.e., resetting the zero level). In order to add an artificial DC shift to the resultant signal, we put a device inside the loop, which distinguishes between co- and counter-propagating trajectories. One such device is the polarization controller (PC), which we use in our scheme to add a phase shift (b) between the two propagation directions (see Fig. 2 and Appendix B). Because the interferometer’s output intensity is approximately proportional to the square of the phase difference times the input signal’s power, when this phase shift (b) is exactly equal to the maximum absolute phase difference between the two beams (see Fig. 2), the interference that takes place at the coupler forms an improved regenerated signal. Thus, we can summarize the mathematical concepts behind the device’s operation as follows. Since the recovery time of the SOA is long, the exiting phase is proportional to an integral over the incident signal (pin(t) = pDC + pAC(t)). Similar to Eq. (1), the phase of the left-propagating beam can be written as Z t L pAC ðnÞ dn; ð3Þ /CW ðtÞ ¼ /0 þ /1 1

where /1  12aC is a constant and the superscript ‘‘L’’ stands for the counter-clockwise (left) propagating beam.

The clockwise (right) propagating beam gains, without loss of generality, an additive phase b relative to the left propagating one (with the PC2, according to the explanation given in Appendix B), and due to the SOA’s offset (DT) the right-propagating beam gains a time delay with respect to the left-propagating beam (see Fig. 2), i.e., Z tþDT ðtÞ ¼ / þ / pAC ðnÞ dn þ b: ð4Þ /R 0 1 CW 1

It should be stressed that the amplitude of the two beams, besides small deformations, are almost identical, since the two beams travel exactly the same distance, i.e., PR(t) ffi PL(t). At the exits of coupler C1 the counter-clockwise propagating beam (‘‘L’’) field is subtracted from the clockwise one (‘‘R’’). Due to the time lag between the two beams (DT), the original signal’s shape is restored by the phase difference. As was stressed above, the changes in the amplitudes are negligible. Thus, at the exit of the coupler, the intensity can be written as 2 L P out ðtÞ / P in ðtÞj exp½i/R CW ðtÞ  exp½i/CW ðtÞj  Z tþDT  b 2 /1 ¼ 4P in ðtÞ sin pAC ðnÞ dn þ : 2 2 t

ð5Þ

Again, in the case of a small DT (considerably smaller than the bit-period) P out ðtÞ / 4P in ðtÞ sin2 ½ð/1 DTpAC ðtÞ þ bÞ=2:

ð6Þ

By carefully adjusting the relative phase b (by controlling the phases h and l of PC2) to b ¼ /1 DTpmin AC

E. Granot et al. / Optics Communications 266 (2006) 80–87

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(where pmin AC is the minimum value of pAC(t)) the argument of the sine will vanish at the minima of Pin(t). Thus, the ER is improved (see Fig. 3). When DT increases, the power is raised (along with the SNR) but so does the integration time, which distorts the signal and washes out high frequency noise. Since the zero level of the output power Pout is proportional to the square of the input signal P 2in (note that this statement is correct even when DT is not small), then the noise in the zero level can be reduced, provided the SOA and the interferometer do not introduce additional noise. Another important feature of this device is that it can invert the incident signal provided the input signal has a low modulation depth. By changing the phase difference between the propagating beams to min b ¼ /1 DTpmax AC ffi /1 DTp AC

[where pmax AC is the maximum value of pAC(t)] instead of b ¼ /1 DTpmin AC the output of the interferometer will vanish whenever Pin reaches its maximum value, and the signal is inverted (see Fig. 4). As was explained above, the two primary mechanisms that help in the regeneration process are:

Fig. 4. Illustration of the inversion algorithm. By changing the relative phase between the two propagating beams from b ¼ /1 DTpmin AC to b ¼ /1 DTpmin AC an inverted pattern appears. This inversion is possible provided the input signal has a low modulation depth.

(1) The extinction ratio is improved, the DC is reduced and therefore the shot noise of the detector is reduced. (2) The noise of the zero-level bits is reduced with the interferometric process. The effects of the two mechanisms are illustrated in Figs. 5 and 6 for the ideal case. When DT is relatively small (compared to the bit-period), the bit pattern of the input signal remains almost intact, but the SNR is not increased considerably (see Fig. 5). On the other hand, when DT is Fig. 5. Calculation results of the ASL as a regenerator for a time delay of 12 ps. The SNR increases slightly but the bit shape remains almost intact.

Fig. 3. A schematic presentation of the coupler operation. In the upper L panel the two phases /R CW and /CW are presented again for comparison purposes (as in Fig. 2). In the central panel the phase difference L /R CW  /CW is plotted, while the lower panel represents the power of the coupler exist.

Fig. 6. Same as Fig. 5 but for the delay time of 40 ps. In this case the SNR increases significantly but in expense of a small pattern deformation.

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raised, the SNR increases, while the bit pattern is a bit distorted (see Fig. 6). In both cases the noise of the zeros level was reduced substantially. In practice, because the SOA and the fibers introduce additional noise in the system, the experimental results are not as dramatic as the theory predicts in terms of noise reduction. In practice, to maintain the system operation in the weak modulation regime, even in cases where the input signal is not weakly modulated, it is possible to introduce a CW beam at a different idler (ki) wavelength (it is denoted by PCW in Fig. 1). This beam keeps the SOA working near the saturation point at the weak modulation depth. However, it requires an additional filter to block the CW idler beam (denoted by ‘‘F’’ in Fig. 1). 3. Simulation of ASL with fast SOA As can be understood from Appendix A, Eq. (1) is merely an approximation, which allows us to understand the main principle of the ASL operation. In general, however, the SOA’s relaxation time se is not considerably larger than the bit-period. In fact, in many cases we prefer to choose a relatively small se so that the SOA’s output intensity will not be impaired (when the SOA is slow its output peak-to-peak value is small). In these cases the signal regeneration is more complicated, and the dynamics takes advantage of the fact that the SOA does not amplify the two beams (‘‘R’’ and ‘‘L’’) evenly. In fact, since the two beams traverse the SOA with different polarizations, the SOA’s polarization sensitivity amplifies them to different extent. We will show in this section by an accurate simulation that this additional feature of the SOA is responsible for ER improvement even for a fast SOA (It should be emphasized that the discussion in the previous section still applies to the transition regime. Therefore, even in the relatively fast SOA regime, the theoretical discussion explains why the rise-time in the device’s output is shorter than the SOA’s rise-time). In this case, to compensate for the different gains of the two beams (which unlike the different phases cannot be compensated by PC2) we use a tunable coupler C1 instead of a fixed 50:50 one. In what follows we show that the tunable coupler can be replaced with an additional PC followed by a polarizer. Unlike the previous section, where the calculations were made on the ideal equation (1), in this section we perform the simulation on the more realistic differential equations (Eqs. (A.2)–(A.4)), and the two PC’s with the time delay were fully simulated. We chose to simulate the ASL for the following arbitrary, but realistic, parameters sc = 0.2 ns, Es/sc = 3 mW, g0L = 3, a = 2 rad and aint = 0 (see Appendix A for definitions). It is also assumed that the amplification in the ‘‘R’’ direction is 50% larger than the amplification in the ‘‘L’’ direction due to different polarizations, i.e., hR = 1.5hL, and the tunable coupler was tuned to 40:60 coupling ratio. In Fig. 7 the solution of the simulation is plotted. In the presented scenario the

Fig. 7. Simulation results. The upper panel stands for the incident signal; the central panel stands for the cross-gain modulation signal that exits the SOA; and the lower panel is the simulated ASL’s output. The input signals bit-rate is 10 Gb/s.

time delay and the PC’s matrices that give these 1 0results ¼ and are DT ffi 5% of the bit-period, M 1 0 1   0 e2:79i M2 ¼ , respectively, which correspond e2:79i 0 to h1 = 0, d1 = 0 (l1 is arbitrary), h2 = p/2, and l2 = 2.79 rad (d2 is arbitrary). The upper panel of Fig. 7 shows the input signal with ER  10. The middle panel represents the ‘‘L’’ propagating beam that emanates from the SOA. Due to the SOA’s cross-gain process (XG) it is a bit distorted. In the lower panel we show the regenerated signal, where the ER was drastically improved (to about 100, but since the minima can be reduced to zero the ER can be arbitrarily large). Although the time delay was only 5% of the bit-period, it helps in canceling the deformation in the XG signal. It can be seen from Fig. 8, that although the phases (/Lcw and /R cw Þ are not ideal piecewise linear functions, they

L Fig. 8. The phases /R CW and /CW that correspond to the simulation presented in Fig. 7.

E. Granot et al. / Optics Communications 266 (2006) 80–87

almost exactly cancel each other at the new ‘‘zeros’’ (former ‘‘ones’’). To show that the phase difference is almost a constant at the locations of the regenerated ‘‘ones’’ (former ‘‘zeros’’) we put two identical arrows in a zoom-in image of the figure (see Fig. 9). Since in practice there is a weak cross-polarization at the SOA’s exit, an additional PC (PC3) at the device’s exit followed by an optical polarizer can improve the signal (see Fig. 1). In our experiment we used these two elements to improve the extinction ratio. Moreover, in many systems it is cumbersome to use a tunable coupler. One of the benefits of controlling the polarization with PC3 is that the tunability of the coupler is not required. Let us assume, for simplicity, that the clockwise and counter-clockwise beams arrive to couplerC1 orthogonal  with approximately  polarizations, i.e. ER ðtÞ 10 and EL ðtÞ 01 . Then, after passing through PC3 described by the Jones  id3

85

is to block the idler CW beam. In cases where there is no idler beam the filter is not required). The input (and output) signal’s wavelength is 1550 nm at 10 Gb/s, and the idler wavelength is 1545 nm. For technical reasons, the time delay was set to 20 ps (and not shorter), which is preferred in this configuration. The SOA was designed to operate at 40% modulation depth, with an operation current of 200 mA, and with almost no polarization sensitivity. An ordinary SMF28 fiber was used, both couplers were 50:50 ± 5% (C1 and C2), and Pin ffi 14.5 mW, Pcw ffi 2.5 mW. In Fig. 10, the input and regenerated output measurements of the ASL are plotted. As can be seen from this figure, the signal’s ER was improved substantially from 3.2 to 13.6 dB without impairment of the SNR. Indeed, the SNR effectively increases, since the noise level decreases to half

il3

sin h3 e matrix M 3 ¼ cossinhh3 eeil3 cos and the polarizer h3 eid3 3 1 0 P ¼ 0 0 (where we assumed polarization in the arbitrary x-direction), the field at the device output will be polarized in the x-direction with the amplitude   1 Eout ¼ ½cos h3 eid3 ER ðtÞ  sin h3 eil3 EL ðtÞ: 0

Clearly, changes in h3 are equivalent to changes in the coupling constant of the tunable coupler, and the phases d3 and l3 can be regarded as an additional factor in the phase tuning. Obviously, exchanging of the tunable coupler with the PC3 and the polarizer causes additional intensity loss. However, in many cases a tunable coupler is more expensive and the alternative (polarizer) is preferred. 4. Experimental setup and results We employ the setup shown in Fig. 1, using an Alcatel SOA, DFB lasers (5 MHz spectral width) and a 0.8 nm spectral width Dicon filter (the only purpose of this filter

Fig. 9. A zoom-in of Fig. 8 around the 7th bit. To show that the difference between them is almost a constant we added the two identical arrows.

Fig. 10. The experimental results of the input (upper plot) and output (lower plot) signals of the ASL.

Fig. 11. Eye-pattern of the input (lower eye) and output (upper eye) signals. The arrows indicate the zero-levels of the eye-patterns. The units of the x-axis are 20 ps/div.

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E. Granot et al. / Optics Communications 266 (2006) 80–87

its initial value, and the signal (the difference between ‘‘1’’ and ‘‘0’’) is increased by a factor of 2 (note that the power scale in Fig. 10 is different for the two plots). Moreover, the ASL re-inverted (after inversion caused by the SOA) the signal. In Fig. 11, we show the corresponding eye-patterns of the input and output signals. In these plots the ER improvement is clearly seen, as is the opening of the eyepattern.

SOA’s length, / is the phase a is the linewidth enhancement factor and sc is the spontaneous carrier lifetime. The solution to the set of Eqs. (A.1) of the (self phased) amplified beam is:

5. Summary and conclusions

dh g0 L  h P in ðsÞ þ P 0 ¼  ½expðhÞ  1: ds sc Es

We have demonstrated an all-optical regenerator device for NRZ signals. The device is based on an SOA in an ASL, and has achieved excellent results in the signal’s DC reduction (ER increment), in a relatively simple and affordable device. We also have presented a theory, which explains the general properties of the AOSR, mainly its ability to improve a signal’s ER, and showed the qualitative behavior in a simulation. The excellent performance and robust nature of the ASL makes the ASL a promising component for realizing alloptical regeneration for ER improvement in optical communication lines.

P out ðsÞ ¼ P in ðsÞ exp½hðsÞ; ðA:2Þ 1 ðA:3Þ /out ðsÞ ¼ /in ðsÞ  ahðsÞ; 2 RL where hðsÞ ¼ 0 dzgðz; sÞ solves the ordinary differential equation

Assume that the SOA is short enough (shorter than a single bit length) and P(s) = P0 + Pin(s) is the instantaneous power inside the SOA where P0 is the idler CW power and Pin(s) is input signal’s power. When the SOA operates in the weak modulation regime, the DC component of P(s) is considerably larger than its AC component, i.e., pAC(s)  pDC. Note that the AC component of Pin(s) is also pAC(s). Also, assume h(s) = h0 + dh(s) where dh(s)  h0 for every s. Therefore, h0 solves the transcendental equation g0L  h0 = U[exp(h0)  1] where U  PDCsc/Es, and dh satisfies ddh dh þ ¼ pAC ðsÞC; ds se

Appendix A The general configuration of a SOA in self-phase modulated operation is presented in Fig. A1. The incident signal modulated on the carrier wavelength ks and an idler beam at wavelength ki (not essential) are inserted into the SOA from opposite directions (see Fig. A1). The SOA dynamics can be presented by the three equations (see, for example, Refs. [17,20]): the spatial derivative (with respect to the longitudinal coordinate z) of the optical power (P), its phase (/) and the temporal derivative of the gain coefficient (g) oP ¼ ðg  aint ÞP ; oz o/ 1 ¼  ag; oz 2 og g0  g gP ¼  ; os sc Es

ðA:4Þ

ðA:5Þ

where s1e  s1c þ PEDCs expðh0 Þ is the reciprocal of the effective relaxation time, and C  [exp(h0)  1]/Es is a constant that depends on the working point parameters. The solution to Eq. (A.5) is straightforward Z s dh ¼ C expðs=se Þ pAC ðnÞ expðn=se Þ dn: ðA:6Þ 1

For relatively long relaxation time this integral can be simplified Z s dh ffi C pAC ðnÞ dn: ðA:7Þ 1

ðA:1Þ

where Es is the saturation energy, aint is the distributed loss (will be ignored in the following discussion, but this conjecture is not a restrictive one), P is the light power according to P(z = 0, s) = Pin(s), and P(z = L, s) = Pout(s), L is the

P0 (τ) ( λ i )

Pin ( λ s)

SOA Pout ( λ s ) Fig. A1. A schematic representation of the SOA in a cross-gain/phase operation.

Thus, in the first approximation the phase of the SOA’s output is proportional to integration over the AC part of the incident signal (pAC(s)): Z s 1 pAC ðnÞ dn; ðA:8Þ /out ðsÞ ¼ /0 þ aC 2 1 where /0 is a constant phase. Appendix B The PC is a device that modifies the polarization state, where no polarization dependent losses are involved. Thus, it can be described as a Jones matrix   cos heid sin heil M¼ ; ðB:1Þ  sin heil cos heid where h, d and l are tunable phases.

E. Granot et al. / Optics Communications 266 (2006) 80–87



PC

PC

M¼

0

eil

eil

0

87

 ðB:5Þ

and W

V

FV

FWe iβ

T 

A  ðM Þ FM F ¼

Fig. B1. The two figures show the Jones representation of the propagating beam’s polarization.

If the incident polarization Jones vector is V and the transmitted one (that passes through PC2) is W, then MV ¼ W:

ðB:2aÞ

On the other hand, when the beam propagates in the opposite direction its initial polarization state is FV and after PC2 it is MTFV, therefore, in order to obtain a phase shift b between the two counter-propagating beams the vectors V and W should maintain (see Fig. B1). FM T F V ¼ W expðibÞ; ðB:2bÞ 1 0 where F  0 1 is a flipping matrix. It should be noted that the addition of the left F matrix is equivalent to adding p/2 to both d and l, and therefore has no qualitative effect on the working point of the device. It can easily be shown that for both equations, V must be an eigenvectors of the matrix A, defined by: ! 2 2 2il id h þ sin he i sin 2h sin le cos  A  ðM T Þ FM T F ¼ i sin 2h sin leid cos2 h þ sin2 he2il ðB:3Þ with the eigenvalue exp(ib) (the asterisk stands for complex conjugate), where cos b ¼ cos2 h þ sin2 h cos 2l:

T

ðB:4Þ

Therefore, by carefully adjusting h and l (by tuning PC2, see Fig. 1) one can control the relative phase between the counter-propagating beams. However, one must change, accordingly, the incident vector V keeping it an eigenvalue of A. This is done by tuning PC1. In particular, when PC2 is tuned so that d = 0 and h = p/2 then

Fig. B2. A physical realization of a device (by a PC) that adds a different phase for the two beams (co and counter).



e2il 0

 : 2il 0

e

ðB:6Þ

In this case the eigenvalues (K1,2) and eigenvectors (V1,2) are   1 K1 ¼ e2il ; V1 ¼ 0   0 and K2 ¼ e2il ; V2 ¼ accordingly: ðB:7Þ 1 Physically, matrix (B.5) describes a 90° rotator followed by a birefringence plate (see Fig. B2). Clearly, in this case the phase shift is b = 2l. References [1] J.Y. Emery, M. Picq, F. Poingt, F. Gaborit, R. Brenot, M. Renaud, B. Lavigne, A. Dupas, in: Technical Digest OFC’ 01, Paper Mb4-1, 2001. [2] W. Idler, K. Daub, G. Laube, M. Schilling, P. Wiedemann, K. Dutting, M. Klenk, E. Lach, K. Wunstel, IEEE Photon. Technol. Lett. 8 (1996) 1163. [3] M.L. Masˇanovic´, V. Lal, J.A. Summers, J.S. Barton, E.J. Skogen, L.A. Coldren, D.J. Blumenthal, IEEE Photon. Technol. Lett. 16 (2004) 2299. [4] M.L. Masˇanovic´, V. Lal, J.S. Barton, E.J. Skogen, L.A. Coldren, D.J. Blumenthal, IEEE Photon. Technol. Lett. 15 (2003) 1117. [5] J.S. Barton, M.L. Masˇanovic´, M.N. Sysak, J.M. Hutchinson, J. Skogen, D.J. Blumenthal, L.A. Coldren, IEEE Photon. Technol. Lett. 16 (2004) 1531. [6] J. Leuthold, C.H. Joyner, B. Mikkelsen, G. Raybon, J.L. Pleumeekers, B.I. Miller, K. Dreyer, C.A. Burrus, Electron. Lett. 36 (2000) 1129. [7] J. Leuthold, G. Raybon, Y. Su, R. Essiambre, S. Cabot, J. Jaques, M. Kauer, Electron. Lett. 38 (2002) 890. [8] P. Bernasconi, W. Yang, L. Zhang, N. Sauer, L. Buhl, I. Kang, S. Chandrasekhar, D.T. Neilson, OFC 2005, PDP16. [9] M.L. Nielsen, B.-E. Olsson, D.J. Blumenthal, IEEE Photon. Technol. Lett. 14 (2002) 245. [10] B.-E. Olsson, P. Andrekson, IEEE Photon. Technol. Lett. 7 (1995) 120. [11] Er’el Granot, Shmuel Sternklar, Haim Chayet, Shalva Ben-Ezra, Niv Narkiss, Nir Shahar, Arieh Sher, Sagie Tsadka, Appl. Opt. 44 (2005) 4959. [12] Er’el Granot, Shmuel Sternklar, Shalva Ben-Ezra, Haim Chayet, Nir Shahar, Sagie Tsadka, J. Opt. Soc. Am. B (in press). [13] K.L. Deng, I. Glesk, K.I. Kang, P.R. Prucnal, IEEE Photon. Technol. Lett. 9 (1997) 830. [14] B.C. Wang, L. Xu, V. Baby, I. Glesk, P.R. Prucnal, IEEE Photon. Technol. Lett. 14 (2002) 989. [15] K.I. Kang, T.G. Chang, I. Glesk, P.R. Prucnal, Appl. Opt. 35 (1996) 417. [16] L. Xu, I. Glesk, V. Baby, P.R. Prucnal, IEEE Photon. Technol. Lett. 16 (2004) 539. [17] E. Granot, R. Zaibel, N. Narkiss, S. Ben-Ezra, H. Chayet, N. Shahar, S. Sternklar, S. Tsadka, J. Opt. Soc. Am. B 22 (2005) 2534. [18] S.L. Jansen, H. Chayet, E. Granot, S. Ben-Ezra, D. den van Borne, P.M. Krummrich, D. Chen, G.D. Khoe, H. De Waardt, IEEE Photon. Technol. Lett. 17 (2005) 2137. [19] K. Chan, C.-K. Chan, W. Hung, F. Tong, L.K. Chen, IEEE Photon. Technol. Lett. 14 (2002) 995. [20] G.P. Agrawal, N.A. Olsson, IEEE J. Quantum Electron. 25 (1989) 2297.