224 IETE JOURNAL OF RESEARCH | VOL 57 | ISSUE 3 | MAY-JUN 2011

Phasor Notations-based Simple Analytical Method for an Intuitive Design of Wavelength Interleaver

with Flattop ResponseNaveen Kumar, M. R. Shenoy1 and Bishnu P. Pal1

Physics, Indian Institute of Information Technology Design and Manufacturing Kancheepuram, IIT Madras Campus, Chennai, 1Physics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, India

ABSTRACT

Mach–Zehnder interferometer (MZI)-based wavelength interleaver is an important device for wavelength multiplexing in dense wavelength division multiplexed optical communication systems. An MZI-based wavelength interleaver can be realized by concatenating 2×2 directional couplers through differential delay lines. The importance of the location of the second delay line in a 2-stage MZI-based wavelength interleaver, and its implication on the optimum value of the splitting ratios of the couplers, to achieve a flattop wavelength response, is highlighted. Here a novel method based on phasor notation to estimate the appropriate magnitude and location of the differential phase delay, and its dependence on the splitting ratios of the couplers is presented. The applicability of the method has been illustrated through analytical equations and simulated responses. The approach gives a better intuitive understanding of the working of interleavers, which is crucial in fabricating a practical wavelength interleaver with flattop response.

Keywords: Dense wavelength division multiplexed networks, Fiber couplers, Fiber slicer, Mach–Zehnder interferometer, Phasor notations, Wavelength interleaver.

1. INTRODUCTION

A wavelength interleaver is an important device that helps in easing out the tight tolerance requirements of multiplexing and demultiplexing of the signal channels in dense wavelength division multiplexed (DWDM) optical networks [1–8]. It is a device that combines 2, or more input streams of wavelength-channels having a constant spacing Dn in the frequency domain (eg, as per the International Telecommunication Union (ITU) grid), into a single dense stream of channels with separation Dn /2 at the output. A single-stage unbalanced Mach–Zehnder interferometer (MZI) can be realized by cascading 2 directional couplers such that one of the arms of the interferometer is longer than the other by an amount DL1 [Figure 1]. In order to achieve wavelength interleaving using this configuration, the splitting ratio of the 2 couplers should be 3 dB, and the differential length DL1 should satisfy the following relation [1–2]:

DD

Ln1

1 2

2=

l lleff

(1)

where Dl l l= -1 2 is the channel-spacing, l1 and l2being 2 adjacent channel wavelengths in the DWDM transmission window; neff is the effective index of the

guided mode at the channel wavelengths propagating through the fiber.

The important parameters that characterize an interleaver are its insertion loss, shape of the passband, and the level of cross-talk between the adjacent channels [4–5]. In addition to these features, a uniform flattop passband is very desirable to minimize/counter channel–power variations because the channel wavelength may drift due to variations in the environmental conditions in a real-world situation. Any initial imbalance in channel–power will have a cascading effect, as the signal is normally required to pass through multiple stages, which could affect the signal-to-noise performance of the system [8–11]. A 2-stage MZI configuration has been proposed to meet the above requirements [2–4]; this can be realized by concatenating three 2×2 directional couplers through 2 delay lines in the 2 stages, as shown in Figures 2 and 3. In order to achieve a uniform flattop response with minimum insertion loss, one has to choose optimum splitting ratios for the couplers K1, K2, and K3, as well as magnitudes of the delay linesDL1 , DL2 and D ¢L2.

There are reports of optimization of the splitting ratios of the coupler constituting a 2-stage MZI through different techniques, in order to achieve a flattop response [9–11]. In most of these reports, the splitting ratio of the first

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225IETE JOURNAL OF RESEARCH | VOL 57 | ISSUE 3 | MAY-JUN 2011

coupler was chosen to be 50:50, in order to get identical and symmetrical response at the 2 output ports 5 and 6. However, there is hardly any report wherein the precise recipe/logic is available with regard to implication of introducing the delay lines in specific arm(s) of a 2-stage MZI and its interdependence on the splitting ratios of the constituent couplers. In this article, we present a novel phasor approach to estimate the appropriate magnitude and location of the differential phase delay, to be introduced in the second stage of the MZI, in order to achieve a flattop response with minimum insertion loss and its dependence on the splitting ratios of the couplers. The phasor approach helps to deal with any situation arisen due to reversing the splitting ratios/delay lines, and gives more intuitive understanding of the working of interleavers based on multistage MZIs. To the best of our knowledge, such information has not been reported so far in the existing literature. The phasor approach [12] used to provide graphical representation of the supermode theory, has been applied in this study to analyze the effect of the splitting ratios of the couplers on the position and optimum magnitude of the second delay line of a 2-stage MZI.

2. MATHEMATICAL APPROACH

The matrix (M) for algebraic representation of the 2-stage MZI configuration, shown in Figure 2, is given as [7–9,11]:

M M M M M ML L2 2 1- =stageMZI coupler 3 coupler 2 coupler 1D D (2)

If the DWDM signal channels are input at Port 1, then the output fields at ports 5 and 6 can be expressed as:

EE

M15

162

10

ÊËÁ

ˆ¯̃

=ÊËÁ

ˆ¯̃-stageMZI (3)

EE

c jsjs c

e c jsjs c

j15

16

3 3

3 3

2 2

2 2

2 00 1

ÊËÁ

ˆ¯̃

=-

-ÊËÁ

ˆ¯̃

Ê

ËÁˆ

¯̃

--

ÊË

f

ÁÁˆ¯̃

Ê

ËÁˆ

¯̃

--

ÊËÁ

ˆ¯̃

ÊËÁ

ˆ¯̃

e c jsjs c

jf1 00 1

10

1 1

1 1

(4)

where c zi i= cos( )k , s zi i= sin( )k , i=1, 2, 3; k is the coupling coefficient and z is the interaction length of the couplers; f1 and f2 are the phase delays introduced by the first and the second differential delay lines, respectively. Thus the output fields emanating from ports 5 and 6 in the configuration, shown in Figure 2, can be derived as:

E c c c j L L s s c j Ls s c j

15 1 2 3 1 2 1 2 3 2

2 3 1

= +( ) --

exp ( ) exp( )exp(

b b bb

D D DDLL s s c1 1 3 2) - (5)

E j c c s j L s c c

s c c j L L s16 1 3 2 1 1 2 3

3 2 1 1 2 1

= - + +ÈÎ+( ) -

exp( )

exp ( )

b

b b

D

D D ss s j L3 2 2exp( )bD ˘̊ (6)

where b is the propagation constant of the guided mode at the channel-wavelength (l).

The fractional output powers emanating from ports 5 and 6, in this configuration [Figure 2], are given by the relations P E E15 15 15= * and P E E16 16 16= * , which are analytically expressible through

P c s c s c L L c s s s c

L L15 1 3 2

21 3 1 2 1 3 2

21 3

1 2

2 2= - + +

-

cos( )

cos( )

b b

b b

D D

D D ++ -( ) +

-( ) +

2

2

1 2 1 2 1 32

32

3 2 3 2 2 12

12

c c s s L s c

c c s s L s c

cos( )

cos( )

b

b

D

D cc c c s s c

c s s s c s

12

22

32

12

22

32

12

22

32

12

22

32 7

+ +

+ ( )

P c s c s c L L c s s s c

L L16 1 3 2

21 3 1 2 1 3 2

21 3

1 2

2 2= + -

- +

cos( )

cos( )

b b

b b

D D

D D 22

2

1 2 1 2 1 32

32

3 2 3 2 2 12

12

c c s s L c s

c c s s L c s c

cos( )

cos( )

b

b

D

D

-( ) +

-( ) + 112

22

32

12

22

32

12

22

32

12

22

32 8

c s s s s

c s c s c c

+ +

+ ( )

Figure 2: A 2-stage MZI configuration through concatenation of 3 couplers K1, K2, and K3, and 2 delay lines DL1 and DL2, introduced in the upper arms of the first and second stages, respectively.

Figure 3: A 2-stage MZI configuration through concatenation of 3 couplers K1, K2, and K3, and two delay lines DL1and D ¢L2, introduced in the upper and lower arms of the first and second stages, respectively.

Figure 1: Schematic of a single-stage all-fiber Mach–Zehnder interferometer-based interleaver, realized through concatenation of two 3-dB couplers; DL1 represents a delay line in the upper arm, corresponding to the path difference between the two arms of the MZIs.

Kumar N, et al.: Phasor Notations-based Simple Analytical Method for an Intuitive Design of Wavelength Interleaverwith Flattop Response

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226 IETE JOURNAL OF RESEARCH | VOL 57 | ISSUE 3 | MAY-JUN 2011

Similarly, the fractional output powers emanating from ports 5 and 6 (P E E15 15 15

' ' '*= and P E E16 16 16' ' '*= ) in the

configuration shown in Figure 3 can be obtained as [7–9]:

¢¢

ÊËÁ

ˆ¯̃

=¢ - ¢

- ¢ ¢ÊËÁ

ˆ¯̃

Ê

ËÁˆ

¯̃

¢ - ¢EE

c jsjs c e

c jsj

15

16

3 3

3 3

2 21 0

0 2f -- ¢ ¢ÊËÁ

ˆ¯̃

Ê

ËÁˆ

¯̃

¢ - ¢- ¢ ¢

ÊËÁ

ˆ¯̃

ÊËÁ

ˆ¯̃

js c

e c jsjs c

j

2 2

1 1

1 1

1 00 1

10

f(( )9

¢ = ¢ ¢ ¢ ¢ ¢ + ¢ - ¢ ¢ ¢ ¢ ¢P c s s s c L L c s c s c15 1 3 22

1 3 1 2 1 3 22

1 32 2cos( )

cos

b bD D

(( ) cos( )b b bD D DL L c c s s L s c

s c c

1 2 1 2 1 2 1 32

32

3 2 3

2

2

- ¢ + ¢ ¢ ¢ ¢ ¢ - ¢( )+¢ ¢ ¢ ¢¢ ¢ ¢ - ¢( ) + ¢ ¢ ¢ + ¢ ¢ ¢ +

¢ ¢

s L s c c c c s s c

c s

2 2 12

12

12

22

32

12

22

32

12

cos( )bD

222

32

12

22

32 10¢ + ¢ ¢ ¢s s c s ( )

¢ =- ¢ ¢ ¢ ¢ ¢ + ¢ + ¢ ¢ ¢ ¢ ¢P c s s s c L L c s c s c16 1 3 22

1 3 1 2 1 3 22

1 32 2cos( )

co

b bD D

ss( ) cos( )b b bD D DL L c c s s L s c

c c s

1 2 1 2 1 2 1 32

32

3 2

2

2

- ¢ - ¢ ¢ ¢ ¢ ¢ - ¢( ) -

¢ ¢ ¢33 2 2 12

12

12

22

32

12

22

32

12

¢ ¢ ¢ - ¢( )+ ¢ ¢ ¢ + ¢ ¢ ¢ +

¢ ¢

s L s c c c s s s s

c

cos( )bD

ss c s c c22

32

12

22

32 11¢ + ¢ ¢ ¢ ( )

where ¢ =c zi icos( )k , ¢ =si izsin( )k , i=1, 2, 3; the primes on “ci” and “si” are to distinguish these parameters from those of Equations (4)–(8). Equations (7), (8) and (10), (11) are similar to a finite Fourier series and might give flattop response [13] when

D DL L n2 12= ± p (12)

D D¢ = ±L L n2 12 p (13)

However, it is difficult to understand whether n=0 or 1, would lead to flattop response when the second delay line is inserted in the lower/upper arm or vice versa of the 2-stage MZI. It is also difficult to fully interpret, through these Equations (7)–(11), how the splitting ratios of the second and third couplers affect the position and magnitude of the second delay line, even and odd channels so as to achieve flattop response. In order to get more insight into the working of multistage MZI, and visualize the effects of the various terms in the Equations (7)–(11) on the spectrum of the wavelength interleavers, a phasor approach has been developed and described in the next section.

3. OPTIMIZATION OF THE DELAY LINES (DL1, DL2 AND DL¢2)

To get wavelength interleaving from a single-stage MZI, the magnitude of DL1 is optimized to satisfy Equation (1) [1–2], which is obtained using the basic concept of interference, as explained below through a

phasor diagram [Figure 4]. Here, the arrows denote the phasors assigned to the electric fields at wavelengths l1 and l2 . The length of the phasors represents the magnitude of the field associated with the wavelength channels, and the direction indicates the phase angle of the corresponding field. At the input end or at plane 1, the fields corresponding to the wavelengths l1 and l2 are assumed to be of equal magnitude and are in the same phase, as represented by the 2 phasors. The power at each of the 2 wavelengths is split into 2 equal parts after passing through the 3-dB coupler K1. At the throughput port of K1, signals at both the wavelengths are still in the same phase, whereas the phases of the signals at the coupled port on plane 2 are rotated by −90°. The differential path length (DL1) introduces an additional phase delay of ( )2 1n + p to the wavelength l1 (let us say odd channels) and 2np to the wavelength l2 (even channels). The phase change due to the propagation through equal lengths in the 2 arms has been neglected, and only the differential phase contribution by differential path length is considered. The phasors at the coupled port of K1 will maintain their phase relationship at plane 3. From the phasor orientation we can observe that the phase difference between the fields corresponding to wavelength l1 at plane 3 is 90° and that corresponding to wavelength l2 is −90°. These fields at plane 3 will interfere while passing through the second coupler and appear at plane 4. We have assumed that there is no cross-interference taking place between the signals at wavelengths l1 and l2 . The signals at wavelength l1 will interfere constructively at port 3 and destructively at port 4 on plane 4. Similarly, the signals at wavelength l2 will interfere constructively at port 4 and destructively at port 3. Thus the phase-matching condition for odd and even wavelengths that are to be interleaved, is satisfied when DL1 introduces a differential phase delay of ( )2 1n + p and 2np to odd and even wavelengths, respectively.

Figure 4: All-fiber wavelength interleaver based on single-stage MZI configuration with a schematic representation of phasors at various planes. Length of the arrows indicates the magnitude of the electric field and their orientations indicate the relative phase angles.

Kumar N, et al.: Phasor Notations-based Simple Analytical Method for an Intuitive Design of Wavelength Interleaverwith Flattop Response

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227IETE JOURNAL OF RESEARCH | VOL 57 | ISSUE 3 | MAY-JUN 2011

Similar phasor diagrams can be drawn for the 2-stage MZI configuration to derive the phase-matching condition for obtaining the optimum value of DL2 and D ¢L2 . The optimum value of DL2 (or D ¢L2 ) is the differential path length, which satisfies the condition of constructive interference and flattop response for odd and even wavelength channels, and separate out at ports 5 and 6 [Figures 5–8]. The position and magnitude of the differential delay line in the second stage depends on the situation whether the splitting ratio of the second coupler is greater or lesser than 50% (ie, s c or s c2

222

22

22≥ £ ).

There are 2 possible choices for a designer to realize a 2-stage MZI: The second delay line is introduced either in the upper arm or in the lower arm of the second stage [9–11]. Although these 2 optional configurations appear to be identical, we show below how they are dictated by the phase differences between the fields at ports 3 and 4, and also by the magnitude of the delay lines in the second stage. We consider these 2 cases separately and optimize the second delay line for each case using phasor notations.

Case 1: When the delay line is inserted in the upper arm of the second stage

The phasor diagram shown in Figure 4 is for the single-stage MZI configuration when both K1 and K2 are 3-dB

couplers. However, if one of them is not a 3-dB coupler, then some power at the odd wavelengths would appear at Port 4. Similarly, some power corresponding to the signals at even wavelengths would appear at Port 3. To obtain the optimum value of DL2 , consider the 2-stage MZI configuration with second delay line inserted in the upper arm of the second stage, along with phasor notations shown in Figures 5 and 6. In these configurations, the splitting ratio of the second coupler is assumed to be more than 50:50, that is, c s2 2< . The phase difference between the fields corresponding to wavelengths l1 is −90° (≡270°) at plane 4 [Figures 5 and 6]. In contrast, the phase difference between the fields corresponding to wavelength l2 is +90° at plane 4. In order to achieve a flattop response, the signals at wavelengths l1 and l2 (on plane 5) should interfere constructively at ports 5 and 6 after passing through the coupler K3. To satisfy these conditions, it requires that all the channels (even or odd), while traversing the differential delay lineDL2 , experience a differential phase delay of either (2n+1)p when splitting ratio of the third coupler K3 is less than 50:50 (c s3 3> ) or 2np when splitting ratio of the third coupler K3 is more than 50:50 (c s3 3< ). Mathematically, these conditions, after taking into account Equations (12) and (13), could be expressed as:

Figure 7: A 2-stage MZI-based all-fiber wavelength interleaver with a schematic representation of phasors at various planes corresponding to Case 2; even and odd channels emanating out of ports 6 and 5, respectively.

Figure 5: A 2-stage MZI-based all-fiber wavelength interleaver with a schematic representation of phasors at various planes, corresponding to Case 1; even and odd channels emanating out of ports 6 and 5, respectively.

Figure 6: A 2-stage MZI-based all-fiber wavelength interleaver with a schematic representation of phasors at various planes, corresponding to Case 1; even and odd channels emanating out of ports 5 and 6, respectively.

Figure 8: A 2-stage MZI-based all-fiber wavelength interleaver with a schematic representation of phasors at various planes corresponding to Case 2; even and odd channels emanating out of ports 5 and 6, respectively.

Kumar N, et al.: Phasor Notations-based Simple Analytical Method for an Intuitive Design of Wavelength Interleaverwith Flattop Response

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228 IETE JOURNAL OF RESEARCH | VOL 57 | ISSUE 3 | MAY-JUN 2011

b b pD DL L n2 12 2 1= ± -( ) , c s2 2< and s c3 3< (14)

b bD DL L2 12= , c s2 2< and s c3 3> (15)

Equation (14) corresponds to the situation, when the odd and even channels, with flattop response, would emerge from ports 5 and 6, respectively, as shown in Figure 5. Similarly, Equation (15) corresponds to the situastion, when the odd and even channels would emerge from ports 6 and 5, respectively, as shown in Figure 6. The differential delay line of magnitude 2DL1 and l/2neff will introduce differential phases of 2np and p, respectively, irrespective of even and odd wavelength channels. Equations (14) and (15) suggest P15=P16 if we substitute b b pD DL L n2 12 2 1= ± -( ) into Equation (8) and b bD DL L2 12= into Equation (7) and by doing so we get:

( ) ( )s c c s32

32

32

32- = - (16)

Since s c3 3π (see Equations (14) and (15)), the only other possibility, which satisfy Equation (16) and consistent with Equations (14) and (15) is that the splitting ratio of the third coupler be reversed, that is,

s c32

32Æ and c s3

232Æ (17)

Thus by introducing an additional phase delay of ( )2 1n ± p in the optimum delay lines in the upper arms of the second stage, one can achieve the flattop response provided the splitting ratio of the third coupler is reversed. However, by doing so, the even and odd channels emanating from the output ports would also be interchanged.

Case 2: When the delay line is inserted in the lower arm of the second stage

The corresponding MZI configuration with phasor notation is shown in Figures 7 and 8. In these configurations, the splitting ratio of the second coupler is assumed to be less than 50:50, that is, ¢ > ¢c s2 2 . The effect of the second coupler’s splitting ratio less than 50:50 (s c2 2< ) is visible on the phasor diagram at plane 4 [cf. Figures 5 and 7]. The phase difference between the fields at wavelength l1 at ports 3 and 4 on plane 4 is +90°, and that corresponding to wavelength l2 is −90° [Figure 7], which is different from that of Case 1 discussed above [Figure 5]. In order to achieve a flattop wavelength response with minimum loss at ports 5 and 6, all the channels, while traversing the differential delay line D ¢L2, should experience a differential phase delay of either 2np (when splitting ratio of the third coupler K3 is less than 50:50) or (2n+1)p (when splitting ratio of the third coupler K3 is more than 50:50). Mathematically, these conditions could be expressed as:

b bD D¢ =L L2 12 , ¢ > ¢c s2 2 and ¢ < ¢s c3 3 (18)

b b pD D¢ = ± +L L n2 12 2 1( ) , ¢ > ¢c s2 2 and ¢ > ¢s c3 3 (19)

Equation (18) corresponds to the situation, when the odd and even channels would emerge from ports 5 and 6, respectively [Figure 7]. Similarly, Equation (19) corresponds to the situation, when the odd and even channels would emerge from ports 6 and 5, respectively, as shown in Figure 8. Similar to Case 1, Equations (18) a n d ( 1 9 ) s u g g e s t P15

' = P16' i f w e s u b s t i t u t e

b b pD D¢ = ± +L L n2 12 2 1( ) into Equat ion (10) and b bD D¢ =L L2 12 into Equation (11) and by doing so we get:

( ) ( )¢ - ¢ = ¢ - ¢s c c s32

32

32

32 (20)

Since ¢ π ¢s c3 3 (see Equations (18) and (19)), the only other possibility that satisfies Equation (20) and consistent with Equations (18) and(19) is that the splitting ratio of the third coupler is reversed, that is,

¢ Æ ¢s c32

32 and ¢ Æ ¢c s3

232 (21)

Thus by introducing an additional phase delay of ( )2 1n ± p in the optimum delay lines in the lower arm of the second stage, one can achieve the flattop response provided the splitting ratio of the third coupler is reversed. As before, by doing so, the even and odd channels emanating from the output ports would also be interchanged. Just by looking at Equations (7), (8), (10), and (11), it is very difficult to estimate the exact magnitudes of optimum delay lines for the 2 cases in order to achieve a flattop response. However, the values of n in Equations (12) and (13), and hence the magnitude of the second delay line, can easily be known by the phasor approach described above. The effect of reversing the splitting ratios of the second and third couplers and the corresponding changes in the magnitudes of the optimum delay lines and odd and even channels can be qualitatively obtained through the phasor approach, which provides more intuitive understanding of the working of 2-stage MZI. For the sake of brevity and clarity, the summary of these 2 cases have been provided in Table 1.

4. RESULTS AND DISCUSSION

To know the optimum values of the splitting ratios of the couplers is the necessary condition for obtaining a flattop response from 2-stage MZI, but the sufficient condition is to have information on the optimum magnitude and location of the delay lines. In the literature, many methods have been suggested for calculating the optimum values of the splitting ratios for obtaining the flattop response. The empirical relations obtained from the best-fit curve for achieving optimum values of the splitting ratios of the couplers

Kumar N, et al.: Phasor Notations-based Simple Analytical Method for an Intuitive Design of Wavelength Interleaverwith Flattop Response

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229IETE JOURNAL OF RESEARCH | VOL 57 | ISSUE 3 | MAY-JUN 2011

has been described elsewhere [14] and these relations are mentioned as follows:

y= 693.043003–43.76664x+1.0398x2−0.01102x3+ 0.000041x4; s2>c2 (22)

where x s= ¥22 100 and y s= ¥3

2 100.

y x x x

x q

= - + - +

<

92 98277 8 09865 0 28183 0 00445

0 0000265201

2 3

42

. . . .

. ; pp2 (23)

where x q= ¥22 100 and y q= ¥3

2 100.

Equation (22) corresponds to the case when the second delay line is inserted in the upper arm such that the channel streams l l l1 3 5, , K and l l l2 4 6, , Kemanate from ports 5 and 6, respectively [Figure 5]. Similarly, Equation (23) corresponds to the case when the second delay line is inserted in the lower arm such that the channel streams l l l1 3 5, , K and l l l2 4 6, , Kemanate from ports 5 and 6, respectively [Figure 7].

In order to illustrate the importance of the location and magnitudes of the second delay line in achieving a flattop response, we show some examples through simulated responses. One of the combinations of the splitting ratios of the couplers to achieve a flattop response [8,14] is 50:50, 32:68, and 96:4, when the second delay line is introduced in the upper arm of the second stage of the MZI. Figure 9 shows the simulated flattop response of the 2-stage MZI when splitting ratios of the couplers are 50:50, 32:68, and 96:4, respectively and the second delay line of magnitude 2DL1± l/2neff is introduced in the upper arm. As discussed in Case 1 above, when the splitting ratio of the second coupler is more than 50:50, then the optimum magnitude of the second delay line in the upper arm should be chosen so as to create a differential phase delay of (2n+1)p for all the channel

wavelengths. This condition could also be satisfied if we chooseDL n2 2= l / eff ; however, it does not lead to flattop response, although it fulfills the condition for minimum loss as can be observed from Figure 10. It can also be seen from Figure 10 that the flat bandwidth of the spectrum is lesser compared with that shown in Figure 9 and the slope of the filter skirt is relatively small. Since the free spectral range (FSR) of an interleaver is inversely proportional to the differential path length (see Equation (1)), if we take a second delay line of magnitude D DL L n2 12 2= ± l / eff , the FSR would decrease, leading to a rapid fall of the skirt. Thus, both the conditions of rapid skirt-fall with the same channel spacing and minimum

Table 1: Summary of the major recommendations/observa-tions to achieve flattop wavelength response from a 2-stage MZI-based interleaverParameters For even and odd

channels to emanate from ports 6 and

respectively

For even and odd channels to emanate

from ports 5 and, respectively

When the optimum delay line is in the upper arm

Delay lines b b pD DL L2 12= ±n=1

b bD DL L2 12=n=1

Splitting ratios of the couplers

c s2 2< and s c3 3< c s2 2< and s c3 3>

When the optimum delay line is in the lower arm

Splitting delay lines b bD D¢ =L L2 12 n=1 b b pD D¢ = ±L L2 12 n=1Splitting ratios of the couplers

¢ > ¢c s2 2 and ¢ > ¢c s3 3 ¢ > ¢c s2 2 and ¢ < ¢c s3 3

Figure 10: Simulated wavelength response of a 2-stage MZI when the second differential delay line of magnitude l/2neff is inserted in the upper arm; the splitting ratios of the couplers K1, K2, and K3, are 50:50, 32:68, and 96:4, respectively.

Figure 9: Simulation result for the spectral response of a 2-stage MZI-based all-fiber wavelength interleaver with optimized parameters, that is, c1

2=0.5, and s12=0.5, c2

2=.32, s2

2=0.68, c32=0.96, and s3

2=0.04, when the optimum delay line DL2 =2DL n1 2± l / eff is introduced in the upper arm of the second stage.

Kumar N, et al.: Phasor Notations-based Simple Analytical Method for an Intuitive Design of Wavelength Interleaverwith Flattop Response

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loss are satisfied when D DL L n2 12 2= ± l / .eff If we introduce the differential delay of D DL L n2 12 2= ± l / eff in the lower arm of the second stage, while keeping the splitting ratios of all the couplers same, then the phase-matching conditions for constructive interference would not be satisfied for this case at the designated wavelength channels, but the region of flatness would shift to the wavelengths l l1 2 2, /± D . The corresponding simulated spectral response is shown in Figure 11. Although it may appear that the response of a 2-stage MZI would be same irrespective of the position of the second delay line, in practice it is not the case as could be observed by comparing Figures 9 and 11. Figure 12 shows the simulated response for the case when splitting ratios of the 3 couplers are 50:50, 32:68, and 4:96, respectively, and the second delay line of magnitude 2 1DL is introduced in the upper arm. It could be seen that the odd and even channels emerge from ports 6 and 5, respectively, confirming the Equations (14)–(17). Similarly, the findings of the approach can also be applied to the configuration, when second delay line inserted in the lower arm of the second stage using optimum values of the splitting ratios [14].

By employing the novel phasor approach, we can estimate the optimum magnitude of the second delay line, in order to achieve a flattop response from a 2-stage MZI. We also recognized and ascertained the relevance of the location of the second delay line and its dependence on the splitting ratios of the couplers through simulated responses [Figures 9–12]. However, if we employ the second coupler with splitting ratio 68:32 instead of 32:68 [Figure 9] keeping the second delay line of the same magnitude in the upper arm of MZI, the flattop response of the interleaver would be

degraded [15]. Thus, the identity of the throughput and coupled ports becomes extremely important, while forming interleavers through cascaded MZI [15]. Such ambiguity in identifying the coupled and throughput ports could lead to significant degradation in the flattop response of a wavelength interleaver [15]. The optimum value and position of the second delay line depends on the phase difference between the fields at ports 3 and 4, which in turn depends on the splitting ratio of the second coupler. For the sake of completeness, we would also like to mention the cases when we cannot get a flattop response even if the splitting ratios of the couplers are of optimum values. When the splitting ratio of the second coupler is more than 50:50 (s c'2 2> ¢ ), a delay line of whatever magnitude inserted in the lower arm of second stage would not yield flattop response. Similarly, when the splitting ratio of the second coupler is less than 50:50 ( ),s c2 2< a delay line of whatever magnitude inserted in the upper arm of second stage would not yield flattop response. The intuitive understanding gained through the novel phasor approach developed in Section 3 on the location of the second delay line and its dependence on the splitting ratios of the couplers is very crucial in realizing a wavelength interleaver with flattop response. The approach gives more freedom to the designer in designing/tailoring the response of the 2-stage MZI.

5. CONCLUSION

We have presented a novel phasor approach to estimate the optimum value of the second delay line in a 2-stage MZI configuration in order to achieve a flattop response. The applicability and implications of the approach is demonstrated and highlighted through analytical equations and simulated responses. The intuitive

Figure 11: Simulated wavelength response of a 2-stage MZI when the second delay line of magnitude D ¢L2 =2DL1± l/2neff is inserted in the lower arm of the MZI; the splitting ratios of the couplers K1, K2, and K3, are 50:50, 32:68, and 96:4, respectively.

Figure 12: Simulation result for the spectral response of a 2-stage MZI-based all-fiber wavelength interleaver with optimized parameters, that is, c1

2=0.5, and s12=0.5, c2

2=0.32, s2

2=0.68, c32=0.04, and s3

2=0.96, when the optimum delays line DL2=2DL1 is introduced in the upper arm of the second stage.

Kumar N, et al.: Phasor Notations-based Simple Analytical Method for an Intuitive Design of Wavelength Interleaverwith Flattop Response

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231IETE JOURNAL OF RESEARCH | VOL 57 | ISSUE 3 | MAY-JUN 2011

understanding developed through the approach is useful for the designer in tailoring the response of multistage MZI interleavers.

REFERENCES

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3. H Arai, T Chiba, H Uetsuka, H Okano, and L Mara, "Interleave filter with box-like spectral response and low chromatic dispersion", in Tech. Digest National Fiber Optic Engineers Conf. NFOEC, Denver, USA. Aug. 29-31, 2000.

4. C Huang, Y Li, J Chen, E Sidick, J Chon, and K G Sullivan, et al. "Low-Loss Flat-Top 50-GHz DWDM and Add/Drop Modules Using All-Fiber Fourier Filters", in Tech. Digest National Fiber Optic Engineers Conf. NFOEC-2000, Denver, USA, pp. 311-6, 2000.

5. J Chon, A Zeng, P Peters, B Jian, A Luo, and K Sullivan, "Integrated Interleaver Technology Enables High Performance in DWDM Systems", Tech. Digest National Fiber Optic Engineers Conf. NFOEC-2001, Baltimore, USA, pp. 1410-22, Jul. 8-12, 2001.

6. S Cao, J Chen, J N Damask, C R Doerr, L Guiziou, G Harvey, et al. "Interleaver technology: Comparisons and applications requirements", J. Lightwave Technol., Vol. 22, pp. 281, 2004.

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demultiplexer fabricated with phosphorous doped SiO2 waveguides on Si", J. Lightwave Technol., Vol. 6, pp. 1011-5, 1988.

8. N Kumar, M R Shenoy, and B P Pal, "All-fiber wavelength interleaver/slicer with flattop response", in Proc. Int. Conf. on Fiber Optics and Photonics- Photonics-2002, Mumbai, Paper FBR P 15, pp. 81, Dec. 14-8, 2002.

9. T. Lui, Y. C. Soh, Y. Zhang, and Z. Fang, “Parameter optimization of an all – fiber Fourier filter flat – top interleaver”, Opt. Eng., vol. 41(12), pp. 3217-3222, 2002.

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AUTHORSNaveen Kumar is currently an Assistant Professor of Physics at Indian Institute of Information Technology Design and Manufacturing Kancheepurm, working in the area of Fiber and Integrated optics and Solar/Green Energy based product development. He received his Ph. D. Degree in physics in 2006 from Indian Institute of Technology Delhi. Afterwards, he joined SPRERI as

senior scientific officer and worked as i/c head of solar energy division for nearly 2.5 years. He has over 12 research papers published in peer-reviewed journals and about the double number of presentations in national and international conferences. In January 2009, he joined IIITD&M Kancheepuram as a faculty and is working there since then. His current research is focused on components like fused fiber coupler, wavelength interleaver, EDFA gain-flattening filter, all-fiber interferometers, etc. for dense wavelength division multiplexed (DWDM) optical communication systems and optical fiber sensors and also solar energy/green energy based product development.

E-mail: vatsnaveen@yahoo.co.in

M. R. Shenoy received the M. Sc. in Physics in 1979 from Mysore University and the PhD in the field of Fiber and Integrated Optics from IIT Delhi in 1987. He joined the faculty of IIT Delhi in 1988, where he is currently Professor, working in the area of fiber and integrated optics. He was a Visiting Scientist with the Department of Electrical and Electronic Engineering, University of

Glasgow, Glasgow, U.K., in 1990 for 10 months, and on short-duration visits at the University of Nice – Sophia Antipolis, Nice, France, in 1992, 1997, 2006 and 2008 for collaborative research on integrated optical devices. He is an author/co-author of over 50 research papers in peer-reviewed journals, and is a co-editor of the book Fiber Optics Through Experiments (Viva Publications, New Delhi, 1994, 2008). He has supervised a number of student projects at the B.Tech/ M.Sc /M.Tech and Ph.D levels, and has been involved in several Sponsored R&D projects on the design and development of guided-wave optical components and devices. His current research interests are in the area of optical fiber components and nonlinear optical devices.

E-mail: mrshenoy@gmail.com

DOI: 10.4103/0377-2063.83643; Paper No JR 153_09; Copyright © 2011 by the IETE

Bishnu P. Pal, M. Sc. from Jadavpur University (Kolkata) and Ph.D. from IIT Delhi, is currently a Professor and the Chair of the Physics Department at IIT Delhi. He has worked as Visiting Professor/Scholar at Heriot-Watt University (Edinburgh, Scotland) as Erasmus Mundus Scholar of EU, Norwegian University of Science and Technology (Norway), University of Strathclyde in

Glasgow (UK), Optoelectronics Research Center at City University of Hong Kong, University of Nice (France) as Senior Foreign Scientist of CNRS, the Fraunhofer Institute IPM at Freiburg (Germany) as Alexander von Humboldt Fellow, and the National Institute of Standards and Technology at Boulder Colorado (USA) as a Fulbright Scholar for various periods. He has extensive teaching, research, sponsored R&D, and consulting experience on various aspects of components and devices for optical fiber communication and specialty optical fibers including microstructured optical fibers and has published books in these areas by publishers like Academic Press, Wiley, New Age, and Intech. He has been on the Editorial Advisory Boards of several research journals and has most recently joined the Advisory Boards of J. Optical Communication (Germany) and Nature Physics collection of the Nature Group. He has guest edited special topical issues of journals like Applied Optics of OSA, Proceedings IEE, and Journal of Optics. He has delivered over 30 plenary and invited talks at international conferences and has been chair/co-chair of several international conferences held in India and abroad. He is a Fellow of OSA The Optical Society (USA), Foreign Member by invitation of the Royal Norwegian Academy of Sciences and Letters (Norway), Senior Member of IEEE (USA), and Fellow of IETE (India). He has been a Distinguished Lecturer for 2005-07 of IEEE (USA), co-recipient of the First Fiber Optic Person of the Year award for 1997 of Lucent Technology in India, Homi Bhabha Award of UGC (India) for Applied Sciences, CEOT-94 award for 2010 from IETE (India) for Optoelectronic devices, OSI award of Optical Society of India in 2010 for life time contributions to growth and dissemination of Optics and Photonics, and Prof Y T Thathachari Prestigious Research Award in Physical Science of Bhramara Trust (Mysore) for 2010. Professor Pal is currently a Director of OSA The Optical Society (USA).

E-mail: bishnupal_in@yahoo.com

Kumar N, et al.: Phasor Notations-based Simple Analytical Method for an Intuitive Design of Wavelength Interleaverwith Flattop Response

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