## Proposed flat-topped pulses bursts generation using all-pass multi-cavity structures

Optics Express, Vol. 17, Issue 16, pp. 13875-13880 (2009)

http://dx.doi.org/10.1364/OE.17.013875

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### Abstract

We propose a simple lossless method for the generation of flat-topped intensity pulses bursts from a single utrashort pulse. We have found optimum solutions corresponding to different numbers of cavities and burst pulses, showing that the proposed all-pass structures of optical cavities, properly designed, can generate close to flat-topped pulse busts.

© 2009 Optical Society of America

## 1. Introduction

1. J. Caraquitena, Z. Jiang, D. E. Leaird, and A. M. Weiner, “Tunable pulse repetition-rate multiplication using phase-only line-by-line pulse shaping,” Opt. Lett. **32**, 716–718 (2007). [CrossRef] [PubMed]

7. J. Azaña and M. A. Muriel, “Temporal Talbot effect in fiber gratings and its applications,” Appl. Opt. **38**, 6700–6704 (1999). [CrossRef]

8. A. M. Weiner and D. E. Leaird, “Generation of terahertz-rate trains of femtosecond pulses by phase-only filtering,” Opt. Lett. **15**, 51–53 (1990) [CrossRef] [PubMed]

8. A. M. Weiner and D. E. Leaird, “Generation of terahertz-rate trains of femtosecond pulses by phase-only filtering,” Opt. Lett. **15**, 51–53 (1990) [CrossRef] [PubMed]

9. J. Azaña, R. Slavík, P. Kockaert, L. R. Chen, and S. LaRochelle, “Generation of Customized Ultrahigh Repetition Rate Pulse Sequences Using Superimposed Fiber Bragg Gratings,” J. Lightwave Technol. **21**, 1490- (2003) [CrossRef]

## 2. Finding optimum solutions

*N*parameters, where

_{C}*N*is the number of cavities. In the case of mirrors based cavities, we can use the two sequences of

_{C}*N*elements, {

_{C}*r*

_{i}} and {

*ϕ*

_{i}}, where ri denotes the reflection coefficients corresponding to the

*i*-th mirror, and

*ϕ*

_{i}denotes the round-trip phase corresponding to the

*i*-th cavity. It is well known the equivalency of this structure with CROWs [12,13

13. J. Capmany, P. Muñoz, J.D. Domenech, and M. A. Muriel, ”Apodized coupled resonator waveguides,” Opt. Express **15**, 10196–10206 (2007). [CrossRef] [PubMed]

*K*is the

_{i}*i*-th coupling factor of the CROW. The impulse response of these optical structures,

*h*(

*t*), can be obtained from inverse Fourier transform of the spectral response in reflection,

*H*(

*ω*), which can be calculated by applying the transfer matrix model method [12–14]. Since all the cavities have similar FSRs, we obtain a spectrally-periodic

*H*(

*ω*) with this same FSR too. Thus, we have a discrete-time function

*h*(t) that can be expressed as:

*T*=FSR

^{-1}is the period of the sequence, and

*c*are complex coefficients which are function of the 2×

_{n}*N*cavities parameters, i.e.,

_{C}*c*=

_{n}*f*({

*r*

_{i}},{

*ϕ*

_{i}},

*n*). Moreover, we are interested in generating a sequence of

*N*pulses of similar intensity from a single pulse. Let us define a causal discrete rectangular function of length

_{P}*N*:

_{P}*c*∣

_{n}^{2}define the envelope of the output burst intensity, we are interested in having a sequence {∣

*c*∣

_{n}^{2}} similar to rect

_{Np}[

*n*] in order to have a flat-topped envelope. Let us define a figure of merit (

*FM*) based on the normalized cross-correlation coefficient [15] to measure this similarity:

*AJ*), and extinction ratio (

*ER*) of the burst, defined as:

_{10}stands for the base 10 logarithm,

*i*∈[1,

*N*],

_{P}*j*∈[1,

*N*], and

_{P}*k*∈[

*N*+1, ∞). Obviously, we are interested in solutions with low

_{P}*A*and high

_{J}*E*values. It is worth noting that a different

_{R}*FM*definition to Eq. (4) can be used, in other to get solutions with a different AJ and ER trade off. However, we have to take into account that the optimization algorithm convergence is also affected by the concrete

*FM*definition.

## 3. Examples

*N*=4 and

_{C}*N*=9). Since

_{P}*AJ*=0.71 dB and

*ER*=15 dB (see Table 1), a good trade-off between burst accuracy and structure complexity (only four cavities) is obtained. The structure parameters of this solution are {

*r*

_{i}}={0.339, 0.386, 0.483, 0.822} and {

*ϕ*

_{i}}={0, 0.139, 0.023, -0.037}. For a CROW implementation, from Eq. (1) we can obtain {

*K*}={0.885, 0.851, 0.767, 0.325}. Let us assume an input Gaussian pulse with a full width at half maximum (FWHM) of 200 fs. In order to avoid pulses burst interference, we must impose

_{i}*FSR*

_{0}

^{-1}>200 fs, where

*FSR*

_{0}stands for the FSR of the structure. We choose

*FSR*

_{0}

^{-1}=1 ps, i.e.,

*FSR*

_{0}=1 THz (a length of cavity in the order of 100 μm), which, from Eq. (2), lead to an output pulse burst with a repetition rate of 1 THz. The different values of {

*ϕ*

_{i}} are obtained by very slight variations of the cavities

*FSR*

^{4,5}, i.e.,

*FSR*

_{i}=

*FSR*

_{0}·(1+

*ϵ*

_{i}), where

*FSR*is the exact value of the FSR of the i-th structure cavity, and

_{i}*ϵ*

_{i}is the relative variation of

*FSR*regarding

_{i}*FSR*

_{0}. Assuming a central frequency of (

*ω*

_{0}/2π)=193 THz for the input signal, we can calculate

^{4,5}{

*ϵ*

_{i}}= 10

^{-3}×{0, -0.114, -0.019, 0.031}.

*ω*

_{0}’/2π)=195 THz. The output pulses burst obtained from a numerical simulation of the designed structure for both input signals are represented in Fig. 3, where practically identical outputs can be observed.

*FM*,

*AJ*,

*ER*, and energy efficiency (

*EE*), considering several values of round-trip power loss (

*RTPL*) of each cavity, assuming same

*RTPL*for all the cavities of the structure. As expected, the energy efficiency and the pulse train uniformity decrease with the cavity losses. Moreover, the CROW dispersion may be significant for a 200 fs pulse, which affect as a increasing of the pulse width and a distortion of the pulse shape.

## 4. Conclusion

## Acknowledgements

## References and links

1. | J. Caraquitena, Z. Jiang, D. E. Leaird, and A. M. Weiner, “Tunable pulse repetition-rate multiplication using phase-only line-by-line pulse shaping,” Opt. Lett. |

2. | C. -B. Huang and Y. Lai, “Loss-less pulse intensity repetition-rate multiplication using optical all-pass filtering,” IEEE Photon. Technol. Lett. |

3. | J. Azaña, “Pulse repetition rate multiplication using phase-only filtering,” Electron. Lett. |

4. | M. A. Preciado and M. A. Muriel, “Repetition-rate multiplication using a single all-pass optical cavity,” Opt. Lett. |

5. | M. A. Preciado and M. A. Muriel, “All-pass optical structures for repetition rate multiplication,” Opt. Express |

6. | M. A. Preciado and M. A. Muriel, “Repetition Rate Multiplication Using All-Pass Optical Structures,” Optics & Photonics News |

7. | J. Azaña and M. A. Muriel, “Temporal Talbot effect in fiber gratings and its applications,” Appl. Opt. |

8. | A. M. Weiner and D. E. Leaird, “Generation of terahertz-rate trains of femtosecond pulses by phase-only filtering,” Opt. Lett. |

9. | J. Azaña, R. Slavík, P. Kockaert, L. R. Chen, and S. LaRochelle, “Generation of Customized Ultrahigh Repetition Rate Pulse Sequences Using Superimposed Fiber Bragg Gratings,” J. Lightwave Technol. |

10. | B. Muralidharan, V. Balakrishnan, and A. M. Weiner, “Design of Double-Passed Arrayed-Waveguide
Gratings for the Generation of Flat-Topped Femtosecond Pulse Trains,” J. Lightwave Technol. |

11. | V. García-Muñoz, M. A. Preciado, and M. A. Muriel, “Simultaneous ultrafast optical pulse train bursts
generation and shaping based on Fourier series developments using superimposed fiber Bragg gratings,” Opt. Express |

12. | A. Yariv and P. Yeh, “Wave propagation in periodic media,” in |

13. | J. Capmany, P. Muñoz, J.D. Domenech, and M. A. Muriel, ”Apodized coupled resonator waveguides,” Opt. Express |

14. | J. Capmany and M. A. Muriel, “A new transfer matrix formalism for the analysis of fiber ring resonators: Compound coupled structures for FDMA,” J. Lightwave Technol. |

15. | A. Papoulis, |

**OCIS Codes**

(140.4780) Lasers and laser optics : Optical resonators

(230.1150) Optical devices : All-optical devices

(320.0320) Ultrafast optics : Ultrafast optics

(350.4600) Other areas of optics : Optical engineering

(140.3538) Lasers and laser optics : Lasers, pulsed

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: May 26, 2009

Revised Manuscript: July 8, 2009

Manuscript Accepted: July 10, 2009

Published: July 27, 2009

**Citation**

Miguel A. Preciado and Miguel A. Muriel, "Proposed flat-topped pulses bursts generation using all-pass multi-cavity structures," Opt. Express **17**, 13875-13880 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-16-13875

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### References

- J. Caraquitena, Z. Jiang, D. E. Leaird, and A. M. Weiner, "Tunable pulse repetition-rate multiplication using phase-only line-by-line pulse shaping," Opt. Lett. 32, 716-718 (2007). [CrossRef] [PubMed]
- C. -B. Huang and Y. Lai, "Loss-less pulse intensity repetition-rate multiplication using optical all-pass filtering," IEEE Photon. Technol. Lett. 12, 167-169 (2000). [CrossRef]
- J. Azaña, "Pulse repetition rate multiplication using phase-only filtering," Electron. Lett. 40, 449-451 (2004). [CrossRef]
- M. A. Preciado and M. A. Muriel, "Repetition-rate multiplication using a single all-pass optical cavity," Opt. Lett. 33, 962-964 (2008). [CrossRef] [PubMed]
- M. A. Preciado and M. A. Muriel, "All-pass optical structures for repetition rate multiplication," Opt. Express 16, 11162-11168 (2008). [CrossRef] [PubMed]
- M. A. Preciado and M. A. Muriel, "Repetition Rate Multiplication Using All-Pass Optical Structures," Optics & Photonics News 19, 37-37 (2008). [CrossRef]
- J. Azaña and M. A. Muriel, "Temporal Talbot effect in fiber gratings and its applications," Appl. Opt. 38, 6700-6704 (1999). [CrossRef]
- A. M. Weiner and D. E. Leaird, "Generation of terahertz-rate trains of femtosecond pulses by phase-only filtering," Opt. Lett. 15, 51-53 (1990) [CrossRef] [PubMed]
- J. Azaña, R. Slavík, P. Kockaert, L. R. Chen, and S. LaRochelle, "Generation of Customized Ultrahigh Repetition Rate Pulse Sequences Using Superimposed Fiber Bragg Gratings," J. Lightwave Technol. 21, 1490- (2003) [CrossRef]
- B. Muralidharan, V. Balakrishnan, and A. M. Weiner, "Design of Double-Passed Arrayed-Waveguide Gratings for the Generation of Flat-Topped Femtosecond Pulse Trains," J. Lightwave Technol. 24, 586- (2006) [CrossRef]
- V. García-Muñoz, M. A. Preciado, and M. A. Muriel, "Simultaneous ultrafast optical pulse train bursts generation and shaping based on Fourier series developments using superimposed fiber Bragg gratings," Opt. Express 15, 10878-10889 (2007) [CrossRef] [PubMed]
- A. Yariv and P. Yeh, "Wave propagation in periodic media," in Photonics: Optical electronics in modern communications (Oxford University Press, 2007).
- J. Capmany, P. Muñoz, J.D. Domenech, and M. A. Muriel, "Apodized coupled resonator waveguides," Opt. Express 15, 10196-10206 (2007). [CrossRef] [PubMed]
- J. Capmany and M. A. Muriel, "A new transfer matrix formalism for the analysis of fiber ring resonators: Compound coupled structures for FDMA," J. Lightwave Technol. 8,1904-1919 (1990). [CrossRef]
- A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, New York, 1962).

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