Abstract

A new kind of 3n feed-forward programmable optical fiber true delay line was proposed. Theoretical analysis was presented on its delay performance and expandability. Experimental demonstration was given to show the implementation of such delay lines using SOAs and Farady rotation mirrors. Delay step as small as 0.5 ps with precision of about 0.03 ps was achieved. Measurement was performed to verify the feasibility and results are given.

© 2007 Optical Society of America

1. Introduction

Programmable optical fiber delay line (OFDL) has attractive applications, such as antenna beam forming in phase array radar system [1] and microwave signal processing[2,3], due to various advantages in comparison with the traditional methods. It can provide true delay of microwave signal, very high time-bandwidth product, immunization to electromagnetic interference (EMI) and the possibility of spatial and wavelength parallelism using WDM techniques. So far, two types of optical fiber delay lines have been proposed as buffers for optical communication networks: re-circulating structure and traveling-wave structures [4]. Re-circulating buffers are compact and require fewer components to implement. However, the fiber loop length is fixed and hence it is not suitable for applications in phase array radar system and microwave signal processing. Traveling-wave buffer has the potential for the above applications. It can be cascaded to provide desired delay using, e.g., 2×2 optical switches or broadcast-and-select configuration [4]. Of course, high-speed (≤1ns) and low-loss (≤1dB) optical switches are required [1,5].

Two typical high speed devices, i.e. LiNbO3 switch [6] and semiconductor optical amplifier (SOA) ON–OFF gate [4], can be considered. For LiNbO3 2×2 switch, the large insertion loss, polarization dependence and crosstalk need to be addressed before they can be used in phase array radar system and microwave signal processing. SOA ON–OFF gate has nanosecond switching speed, large on-off extinction ratio and wide optical bandwidth. It seems more suitable for traveling-wave type of fiber delay lines[7].

In this paper, we proposed a novel 3n feed-forward OFDL using SOAs as switching components. Theoretical analysis and experimental verification are presented. Important issues, such as the attenuation [8], polarization [9,10], delay precision, and etc., are addressed in detail.

2. Design of kn feed-forward optical fiber true delay line

Traditionally, the programmable traveling-wave fiber delay line is implemented by using 1×N optical switch, or equally by using 1×N coupler plus on-off gate (e.g., SOA) in each arm, as shown in Fig. 1(a) [11]. In this configuration, the maximum number of possible delays, y, is equal to the mount of SOAs, x. It is apparently not suitable to be scaled for large delays. The delay performance can be enhanced by configuring it in an alternative way with 2×2 switches(or 2×2 couplers plus ON-OFF SOA gates) and same number of SOAs as shown in Fig. 1(b). In this configuration, the total number of delays can be expressed as follows:

y=2x2

where the number of SOAs, x, is supposed to even. We define such structure as 2n structure, where, n=x/2, is the number of stages.

Similarly, we can use x SOAs to build kJ structure as shown in Fig. 1(c), where k is the dimension of each stage and J is the number of stages(J=x/k is an integer). It can be derived the total number of delays, y, is given by:

y=kxk

Equation (2) can be rewritten as:

log(y)x=log(k)k

Equation (3) shows that for the same x, there is an optimal k that gives maximum y. We use L(k) to express the right side of Equation(3):

L(k)=log(k)k

The peak value of L(k) can be obtained mathematically by differential operation on L(k) as follows:

dL(k)dk=0

So, the k can be obtained by solving Equations (4) and (5):

k=e=2.718
 

Fig. 1. The architectures of traveling-wave fiber delay lines (y=x for (a), y=2x/2 for (b), and y=kx/k for (c), where y is the maximum number of possible delays, and x is the number of SOAs)

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However, as shown in Fig. 1, the physics meaning of k is the number of delay lines in each stage, so, k should be integer. Figure 2 shows the relationship between L(k) and k. It can be easily found that the optimal value k is 3. It can also be proved that 4n structure is exactly the same as 2n structure:

y(x)=kxkk=4y(x)=4x4=(22)x4=22x4=2x2k=2

When the number of SOAs used is small, the difference between the maximum numbers of delays available for k=2 and k=3, respectively, are also small. However, as more SOAs are used, the difference becomes significant, as shown in Fig. 3. From the figure, we can see that fast tunable true delay line with large number of delays can be obtained by employing 3n structure and certain number of SOAs.

 

Fig. 2. L(k) as a function of k

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Fig. 3. Dependence of total delay number on the delay units per stage (x: the number of SOAs)

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3. Implementation and performance measurement

According to the above analysis, 3n feed-forward optical fiber true delay line based on SOAs can provides large amount of delays with fast tunability and easier scalability. For practical applications, problems such as insertion loss and polarization dependence should be solved. The former can be overcome by SOA itself for it provides gain in addition to ON-OFF gating. In fact, this is also one of advantages of employing SOAs. We use Farady rotation mirrors (FRMs) to alleviate the polarization dependence in optical fiber [12] and in SOAs. The SOAs we used are bi-directional SOA without inside optical isolators at input and output. Due to the low backward reflection of the SOA and the polarization rotation by FRM, no oscillation was observed. Figure 4 shows the structure for practical implementation and we built a two-stage (the first two stages) feed-forward optical fiber true delay line for experimental verification. For each stage, the light is fed from the input port of the optical circulator. It experiences delay whose amount is determined by turning on one of the SOAs and is then forwarded to the next stage via the output port.

The main insertion loss is caused by 1×3 coupler. The light beam passes the coupler twice and experience about 12 dB loss (excluding the coupler’s excess loss). By taking the losses resulted from the FRM and the circulator into consideration, the total loss of each stage is around 13 dB. It should be noted that the beam also passes SOA twice. Usually, an SOA can provide with 5–8 dB gain for small optical signals, e.g. -10dBm, under moderate drive current (~80mA). Therefore, by careful design of the drive current of each SOA, it is possible to obtain a delay stage with no insertion loss. This feature is particularly important for practical applications. It enables the proposed structure to be cascaded without limitation on the stage number.

 

Fig. 4. A practical structure of 3n feed-forward optical fiber true delay line

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The resolution is another key issue to be addressed. The delay introduced by a light path distance of 0.15mm, or fiber length of 0.103mm (group refractive index of 1.46 is supposed), is about 0.5 ps. It corresponds to a phase control precision of about 1.8° in X-band (λ~3cm). In order to reach the above precision level, we used hydrogen flame to precisely adjust the fiber length, with the help of the high precision reflectometer (HP8504)[12]. This equipment is used to precisely measure the lightpath difference between the measured arm and reference arm of Michelson interferometer. The measurement span (equivalent distance in air) is 1–400 mm with resolution of 20 µm. Figure 5 shows the result when we set all six SOAs ON. It can be clearly seen that there are 9 peaks, which represent the all possible delays to be obtained, and each delay can be programmablely switched by setting of the relative one SOA per stage. The measured relative light path distances in three arms of the first stage are 20.756mm, 20.914mm, 21.056mm, respectively. The correspondent fiber lengths are 14.216 mm, 14.325mm, 14.422mm, respectively. These data indicate that τ as shown in Fig. 3 is 0.5 ps and the precision is kept within 0.03ps. One may note that there are insertion losses in stages 2 and 3. This is because of the limited drive current of the circuit we used, resulting in small SOA gain incapable to compensate the inherent insertion loss mentioned above.

Figure 6 shows the measured result of tuning speed. The rise time is 18.6 ns and the fall time id 39.4 ns. Such speed can enhance the scanning performance of microwave significantly. The scanning frequency may be improved from tens of kHz up to 10 MHz. The drive circuit for the SOAs was implemented with conventional electronic devices at hand. There is large potential to improve the tuning speed, especially the switching-off performance.

One may note that there is extra delay in switching-off process, which is mainly resulted in from the large junction capacitance in the device.

We also measured the polarization dependence loss (PDL) using polarization controller. The measured result is 0.18 dB. In comparison with the PDL of SOA (~0.3dB), there is improvement that is mainly due to the employment of FRM.

The relatively low noise figure of the SOA may bring a certain limitation to the proposed structure for some applications, especially for those requiring high transmission SNR. This indicates that for expansion purpose, noise performance should be taken into consideration, which is a topic worthy of intensive study and, however, beyond the scope of the paper.

 

Fig. 5. Measured result of relative fiber lengths x-axis: light path difference(0.2 mm/div), y-axis: relative intensity of the output(dB)

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Fig. 6. Photograph of tuning speed measurement (lower trace: drive pulse, upper trace: optical output)

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4. Conclusions and discussion

In this paper, we proposed a new kind of 3n feed-forward optical fiber delay lines. Theoretical analysis shows that it can provide large delays and is easy to be expanded. Experimental demonstration to implement such delay lines using SOAs and Farady rotation mirrors is presented. Delay step as small as 0.5 ps with precision of about 0.03 ps was realized. Measured results verify the feasibility of the proposed method. Its excellent expandability, low insertion loss, low polarization dependence loss and high tuning speed are suitable for application in phase array radar system and microwave signal processing.

Acknowledgments

The work was partially supported by 863 Project (ID: 2006AA01z242, 2007AA01Z275), Dawn Program for Excellent Scholars by Shanghai Municipal Education Commission, New Century Program for Excellent Scholars by Education of Ministry of China, National Spaceflight Innovation Fund of China (2007 Key Project), Shanghai Municipal Science and Technology Commission Project (05DZ22312), and partially supported by the MIC(Ministry of information and Communication), Korea, Under the ITFSIP(IT Foreign Specialist Inviting Program) supervised by the IITA (Institute of Information Technology Advancement).

References and links

1. A. J. Seeds and K. J. Williams, “Microwave Photonics,” J. Lightwave Technol. 24, 4628–4641(2006) [CrossRef]  

2. J. Capmany, B. Ortega, D. Pastor, and S. Sales, “Discrete-time optical Processing of microwave signals,” J. Lightwave Technol. 23, 702–723(2005) [CrossRef]  

3. K. Wilner and A. P. Van Den Heuvel, “Fiber-optic delay lines for microwave signal processing,” Proc. IEEE 64, 805–807(1976) [CrossRef]  

4. G. K. Chang, J. J Yu, Y. K. Yeo, A. Chowdhury, and Z. S. Jia, “Enabling technologies for next-generation optical packet-switching networks,” Proc. IEEE 94, 892–910(2006) [CrossRef]  

5. A. J. Seeds, “Microwave photonics,” IEEE Transactions on Microwave Theory and Techniques 50, 877–887(2002) [CrossRef]  

6. E. J. Murphy, T. O. Murphy, A. F. Ambrose, R. W. Irvin, B. H. Lee, P. Peng, G. W. Richards, and A. Yorinks, “16×16 strictly nonblocking guided-wave optical switching system,” J. Lightwave Technol. 14, 352–358(1996) [CrossRef]  

7. Y.K. Yeo, J. J. Yu, and G. K. Chang, “A dynamically reconfigurable folded-path time delay buffer for optical packet switching,” IEEE Photonics Technol. Lett. 16, 2559–2561(2004) [CrossRef]  

8. B. Moslehi, “Fibre-optic filters employing optical amplifiers to provide design flexibility,” Electron. Lett. 28, 226–228(1992) [CrossRef]  

9. J. Capmany, B. Ortega, and D. Pastor, “A tutorial on microwave photonic filters,” J. Lightwave Technol. 24, 201–229(2006) [CrossRef]  

10. A. Ho-Quoc, S. Tedjini, and A. Hilt, “Optical polarization effect in discrete time fiber-optic structures for microwave signal processing,” IEEE MTT-S International Microwave Symposium Digest(1996),. pp.907–910

11. D.J. Blumenthal, P. R. Prucnal, and J. R. Sauer, “Photonic packet switches: architectures and experimental implementations,” Proc. IEEE 82, 1650–1667(1994) [CrossRef]  

12. C.H. Shi, J.P. Chen, G.L. Wu, and X.W. Li, “Stable dynamic detection scheme for magnetostrictive fiber-optic interferometric sensors,” Opt. Express 14: 5098–5102(2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-12-5098 [CrossRef]   [PubMed]  

References

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  1. A. J. Seeds and K. J. Williams, “Microwave Photonics,” J. Lightwave Technol. 24, 4628–4641(2006)
    [CrossRef]
  2. J. Capmany, B. Ortega, D. Pastor, and S. Sales, “Discrete-time optical Processing of microwave signals,” J. Lightwave Technol. 23, 702–723(2005)
    [CrossRef]
  3. K. Wilner and A. P. Van Den Heuvel, “Fiber-optic delay lines for microwave signal processing,” Proc. IEEE 64, 805–807(1976)
    [CrossRef]
  4. G. K. Chang, J. J Yu, Y. K. Yeo, A. Chowdhury, and Z. S. Jia, “Enabling technologies for next-generation optical packet-switching networks,” Proc. IEEE 94, 892–910(2006)
    [CrossRef]
  5. A. J. Seeds, “Microwave photonics,” IEEE Transactions on Microwave Theory and Techniques 50, 877–887(2002)
    [CrossRef]
  6. E. J. Murphy, T. O. Murphy, A. F. Ambrose, R. W. Irvin, B. H. Lee, P. Peng, G. W. Richards, and A. Yorinks, “16×16 strictly nonblocking guided-wave optical switching system,” J. Lightwave Technol. 14, 352–358(1996)
    [CrossRef]
  7. Y.K. Yeo, J. J. Yu, and G. K. Chang, “A dynamically reconfigurable folded-path time delay buffer for optical packet switching,” IEEE Photonics Technol. Lett. 16, 2559–2561(2004)
    [CrossRef]
  8. B. Moslehi, “Fibre-optic filters employing optical amplifiers to provide design flexibility,” Electron. Lett. 28, 226–228(1992)
    [CrossRef]
  9. J. Capmany, B. Ortega, and D. Pastor, “A tutorial on microwave photonic filters,” J. Lightwave Technol. 24, 201–229(2006)
    [CrossRef]
  10. A. Ho-Quoc, S. Tedjini, and A. Hilt, “Optical polarization effect in discrete time fiber-optic structures for microwave signal processing,” IEEE MTT-S International Microwave Symposium Digest(1996),. pp.907–910
  11. D.J. Blumenthal, P. R. Prucnal, and J. R. Sauer, “Photonic packet switches: architectures and experimental implementations,” Proc. IEEE 82, 1650–1667(1994)
    [CrossRef]
  12. C.H. Shi, J.P. Chen, G.L. Wu, and X.W. Li, “Stable dynamic detection scheme for magnetostrictive fiber-optic interferometric sensors,” Opt. Express 14: 5098–5102(2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-12-5098
    [CrossRef] [PubMed]

2006 (4)

2005 (1)

2004 (1)

Y.K. Yeo, J. J. Yu, and G. K. Chang, “A dynamically reconfigurable folded-path time delay buffer for optical packet switching,” IEEE Photonics Technol. Lett. 16, 2559–2561(2004)
[CrossRef]

2002 (1)

A. J. Seeds, “Microwave photonics,” IEEE Transactions on Microwave Theory and Techniques 50, 877–887(2002)
[CrossRef]

1996 (1)

E. J. Murphy, T. O. Murphy, A. F. Ambrose, R. W. Irvin, B. H. Lee, P. Peng, G. W. Richards, and A. Yorinks, “16×16 strictly nonblocking guided-wave optical switching system,” J. Lightwave Technol. 14, 352–358(1996)
[CrossRef]

1994 (1)

D.J. Blumenthal, P. R. Prucnal, and J. R. Sauer, “Photonic packet switches: architectures and experimental implementations,” Proc. IEEE 82, 1650–1667(1994)
[CrossRef]

1992 (1)

B. Moslehi, “Fibre-optic filters employing optical amplifiers to provide design flexibility,” Electron. Lett. 28, 226–228(1992)
[CrossRef]

1976 (1)

K. Wilner and A. P. Van Den Heuvel, “Fiber-optic delay lines for microwave signal processing,” Proc. IEEE 64, 805–807(1976)
[CrossRef]

Ambrose, A. F.

E. J. Murphy, T. O. Murphy, A. F. Ambrose, R. W. Irvin, B. H. Lee, P. Peng, G. W. Richards, and A. Yorinks, “16×16 strictly nonblocking guided-wave optical switching system,” J. Lightwave Technol. 14, 352–358(1996)
[CrossRef]

Blumenthal, D.J.

D.J. Blumenthal, P. R. Prucnal, and J. R. Sauer, “Photonic packet switches: architectures and experimental implementations,” Proc. IEEE 82, 1650–1667(1994)
[CrossRef]

Capmany, J.

Chang, G. K.

G. K. Chang, J. J Yu, Y. K. Yeo, A. Chowdhury, and Z. S. Jia, “Enabling technologies for next-generation optical packet-switching networks,” Proc. IEEE 94, 892–910(2006)
[CrossRef]

Y.K. Yeo, J. J. Yu, and G. K. Chang, “A dynamically reconfigurable folded-path time delay buffer for optical packet switching,” IEEE Photonics Technol. Lett. 16, 2559–2561(2004)
[CrossRef]

Chen, J.P.

Chowdhury, A.

G. K. Chang, J. J Yu, Y. K. Yeo, A. Chowdhury, and Z. S. Jia, “Enabling technologies for next-generation optical packet-switching networks,” Proc. IEEE 94, 892–910(2006)
[CrossRef]

Hilt, A.

A. Ho-Quoc, S. Tedjini, and A. Hilt, “Optical polarization effect in discrete time fiber-optic structures for microwave signal processing,” IEEE MTT-S International Microwave Symposium Digest(1996),. pp.907–910

Ho-Quoc, A.

A. Ho-Quoc, S. Tedjini, and A. Hilt, “Optical polarization effect in discrete time fiber-optic structures for microwave signal processing,” IEEE MTT-S International Microwave Symposium Digest(1996),. pp.907–910

Irvin, R. W.

E. J. Murphy, T. O. Murphy, A. F. Ambrose, R. W. Irvin, B. H. Lee, P. Peng, G. W. Richards, and A. Yorinks, “16×16 strictly nonblocking guided-wave optical switching system,” J. Lightwave Technol. 14, 352–358(1996)
[CrossRef]

Jia, Z. S.

G. K. Chang, J. J Yu, Y. K. Yeo, A. Chowdhury, and Z. S. Jia, “Enabling technologies for next-generation optical packet-switching networks,” Proc. IEEE 94, 892–910(2006)
[CrossRef]

Lee, B. H.

E. J. Murphy, T. O. Murphy, A. F. Ambrose, R. W. Irvin, B. H. Lee, P. Peng, G. W. Richards, and A. Yorinks, “16×16 strictly nonblocking guided-wave optical switching system,” J. Lightwave Technol. 14, 352–358(1996)
[CrossRef]

Li, X.W.

Moslehi, B.

B. Moslehi, “Fibre-optic filters employing optical amplifiers to provide design flexibility,” Electron. Lett. 28, 226–228(1992)
[CrossRef]

Murphy, E. J.

E. J. Murphy, T. O. Murphy, A. F. Ambrose, R. W. Irvin, B. H. Lee, P. Peng, G. W. Richards, and A. Yorinks, “16×16 strictly nonblocking guided-wave optical switching system,” J. Lightwave Technol. 14, 352–358(1996)
[CrossRef]

Murphy, T. O.

E. J. Murphy, T. O. Murphy, A. F. Ambrose, R. W. Irvin, B. H. Lee, P. Peng, G. W. Richards, and A. Yorinks, “16×16 strictly nonblocking guided-wave optical switching system,” J. Lightwave Technol. 14, 352–358(1996)
[CrossRef]

Ortega, B.

Pastor, D.

Peng, P.

E. J. Murphy, T. O. Murphy, A. F. Ambrose, R. W. Irvin, B. H. Lee, P. Peng, G. W. Richards, and A. Yorinks, “16×16 strictly nonblocking guided-wave optical switching system,” J. Lightwave Technol. 14, 352–358(1996)
[CrossRef]

Prucnal, P. R.

D.J. Blumenthal, P. R. Prucnal, and J. R. Sauer, “Photonic packet switches: architectures and experimental implementations,” Proc. IEEE 82, 1650–1667(1994)
[CrossRef]

Richards, G. W.

E. J. Murphy, T. O. Murphy, A. F. Ambrose, R. W. Irvin, B. H. Lee, P. Peng, G. W. Richards, and A. Yorinks, “16×16 strictly nonblocking guided-wave optical switching system,” J. Lightwave Technol. 14, 352–358(1996)
[CrossRef]

Sales, S.

Sauer, J. R.

D.J. Blumenthal, P. R. Prucnal, and J. R. Sauer, “Photonic packet switches: architectures and experimental implementations,” Proc. IEEE 82, 1650–1667(1994)
[CrossRef]

Seeds, A. J.

A. J. Seeds and K. J. Williams, “Microwave Photonics,” J. Lightwave Technol. 24, 4628–4641(2006)
[CrossRef]

A. J. Seeds, “Microwave photonics,” IEEE Transactions on Microwave Theory and Techniques 50, 877–887(2002)
[CrossRef]

Shi, C.H.

Tedjini, S.

A. Ho-Quoc, S. Tedjini, and A. Hilt, “Optical polarization effect in discrete time fiber-optic structures for microwave signal processing,” IEEE MTT-S International Microwave Symposium Digest(1996),. pp.907–910

Van Den Heuvel, A. P.

K. Wilner and A. P. Van Den Heuvel, “Fiber-optic delay lines for microwave signal processing,” Proc. IEEE 64, 805–807(1976)
[CrossRef]

Williams, K. J.

Wilner, K.

K. Wilner and A. P. Van Den Heuvel, “Fiber-optic delay lines for microwave signal processing,” Proc. IEEE 64, 805–807(1976)
[CrossRef]

Wu, G.L.

Yeo, Y. K.

G. K. Chang, J. J Yu, Y. K. Yeo, A. Chowdhury, and Z. S. Jia, “Enabling technologies for next-generation optical packet-switching networks,” Proc. IEEE 94, 892–910(2006)
[CrossRef]

Yeo, Y.K.

Y.K. Yeo, J. J. Yu, and G. K. Chang, “A dynamically reconfigurable folded-path time delay buffer for optical packet switching,” IEEE Photonics Technol. Lett. 16, 2559–2561(2004)
[CrossRef]

Yorinks, A.

E. J. Murphy, T. O. Murphy, A. F. Ambrose, R. W. Irvin, B. H. Lee, P. Peng, G. W. Richards, and A. Yorinks, “16×16 strictly nonblocking guided-wave optical switching system,” J. Lightwave Technol. 14, 352–358(1996)
[CrossRef]

Yu, J. J

G. K. Chang, J. J Yu, Y. K. Yeo, A. Chowdhury, and Z. S. Jia, “Enabling technologies for next-generation optical packet-switching networks,” Proc. IEEE 94, 892–910(2006)
[CrossRef]

Yu, J. J.

Y.K. Yeo, J. J. Yu, and G. K. Chang, “A dynamically reconfigurable folded-path time delay buffer for optical packet switching,” IEEE Photonics Technol. Lett. 16, 2559–2561(2004)
[CrossRef]

Electron. Lett. (1)

B. Moslehi, “Fibre-optic filters employing optical amplifiers to provide design flexibility,” Electron. Lett. 28, 226–228(1992)
[CrossRef]

IEEE Photonics Technol. Lett. (1)

Y.K. Yeo, J. J. Yu, and G. K. Chang, “A dynamically reconfigurable folded-path time delay buffer for optical packet switching,” IEEE Photonics Technol. Lett. 16, 2559–2561(2004)
[CrossRef]

IEEE Transactions on Microwave Theory and Techniques (1)

A. J. Seeds, “Microwave photonics,” IEEE Transactions on Microwave Theory and Techniques 50, 877–887(2002)
[CrossRef]

J. Lightwave Technol. (4)

Opt. Express (1)

Proc. IEEE (3)

D.J. Blumenthal, P. R. Prucnal, and J. R. Sauer, “Photonic packet switches: architectures and experimental implementations,” Proc. IEEE 82, 1650–1667(1994)
[CrossRef]

K. Wilner and A. P. Van Den Heuvel, “Fiber-optic delay lines for microwave signal processing,” Proc. IEEE 64, 805–807(1976)
[CrossRef]

G. K. Chang, J. J Yu, Y. K. Yeo, A. Chowdhury, and Z. S. Jia, “Enabling technologies for next-generation optical packet-switching networks,” Proc. IEEE 94, 892–910(2006)
[CrossRef]

Other (1)

A. Ho-Quoc, S. Tedjini, and A. Hilt, “Optical polarization effect in discrete time fiber-optic structures for microwave signal processing,” IEEE MTT-S International Microwave Symposium Digest(1996),. pp.907–910

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Figures (6)

Fig. 1.
Fig. 1.

The architectures of traveling-wave fiber delay lines (y=x for (a), y=2x/2 for (b), and y=kx/k for (c), where y is the maximum number of possible delays, and x is the number of SOAs)

Fig. 2.
Fig. 2.

L(k) as a function of k

Fig. 3.
Fig. 3.

Dependence of total delay number on the delay units per stage (x: the number of SOAs)

Fig. 4.
Fig. 4.

A practical structure of 3 n feed-forward optical fiber true delay line

Fig. 5.
Fig. 5.

Measured result of relative fiber lengths x-axis: light path difference(0.2 mm/div), y-axis: relative intensity of the output(dB)

Fig. 6.
Fig. 6.

Photograph of tuning speed measurement (lower trace: drive pulse, upper trace: optical output)

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

y = 2 x 2
y = k x k
log ( y ) x = log ( k ) k
L ( k ) = log ( k ) k
dL ( k ) dk = 0
k = e = 2.718
y ( x ) = k x k k = 4 y ( x ) = 4 x 4 = ( 2 2 ) x 4 = 2 2 x 4 = 2 x 2 k = 2

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