## Complete family of periodic Talbot filters for pulse repetition rate multiplication

Optics Express, Vol. 14, Issue 10, pp. 4270-4279 (2006)

http://dx.doi.org/10.1364/OE.14.004270

Acrobat PDF (239 KB)

### Abstract

We introduce a new family of equivalent periodic phase-only filtering configurations that can be used for implementing the Talbot-based pulse rate multiplication technique. The introduced family of periodic Talbot filters allows one to design a desired pulse repetition rate multiplier with an unprecedented degree of freedom and flexibility. Moreover, these filters can be implemented using all-fiber technologies, and in particular (superimposed) linearly chirped fiber Bragg gratings. The design specifications and associated constraints of this new class of Talbot filters are discussed.

© 2006 Optical Society of America

## 1. Introduction

1. K. Tai, A. Tomita, J. L. Jewell, and A. Hasegawa, “Generation of subpicosecond solitonlike optical pulses at 0.3 THz repetition rate by modulation instability,” Appl. Phys. Lett. **49**, 236–238 (1986). [CrossRef]

7. J. Magné, J. Bolger, M. Rochette, S. LaRochelle, L. R. Chen, B. J. Eggleton, and J. Azaña, “4×100 GHz pulse train generation from a single wavelength 10 GHz mode-locked laser using superimposed fiber gratings and nonlinear conversion,” J. Lightwave Technol.24, 2006 (in press). [CrossRef]

3. J. Azaña and M. A. Muriel, “Technique for multiplying the repetition rates of periodic pulse trains of pulses by means of a temporal self-imaging effect in chirped fiber gratings,” Opt. Lett. **24**, 1672–1674 (1999). [CrossRef]

5. S. Atkins and B. Fischer, “All-optical pulse rate multiplication using fractional Talbot effect and field-to-intensity conversion with cross-gain modulation,” IEEE Photon. Technol. Lett. **15**, 132–134 (2003). [CrossRef]

6. J. Azaña, P. Kockaert, R. Slavík, L.R. Chen, and S. LaRochelle, “Generation of a 100-GHz optical pulse train by pulse repetition-rate multiplication using superimposed fiber Bragg gratings,” IEEE Photon. Technol. Lett. **15**, 413–415 (2003). [CrossRef]

7. J. Magné, J. Bolger, M. Rochette, S. LaRochelle, L. R. Chen, B. J. Eggleton, and J. Azaña, “4×100 GHz pulse train generation from a single wavelength 10 GHz mode-locked laser using superimposed fiber gratings and nonlinear conversion,” J. Lightwave Technol.24, 2006 (in press). [CrossRef]

3. J. Azaña and M. A. Muriel, “Technique for multiplying the repetition rates of periodic pulse trains of pulses by means of a temporal self-imaging effect in chirped fiber gratings,” Opt. Lett. **24**, 1672–1674 (1999). [CrossRef]

4. J. Azaña and M. A. Muriel, “Temporal self-imaging effects: Theory and application for multiplying pulse repetition rates,” IEEE J. Sel. Top. Quantum Electron. **7**, 728–744 (2001). [CrossRef]

6. J. Azaña, P. Kockaert, R. Slavík, L.R. Chen, and S. LaRochelle, “Generation of a 100-GHz optical pulse train by pulse repetition-rate multiplication using superimposed fiber Bragg gratings,” IEEE Photon. Technol. Lett. **15**, 413–415 (2003). [CrossRef]

7. J. Magné, J. Bolger, M. Rochette, S. LaRochelle, L. R. Chen, B. J. Eggleton, and J. Azaña, “4×100 GHz pulse train generation from a single wavelength 10 GHz mode-locked laser using superimposed fiber gratings and nonlinear conversion,” J. Lightwave Technol.24, 2006 (in press). [CrossRef]

8. N. K. Berger, B. Levit, A. Bekker, and B. Fischer, “Compression of periodic optical pulses using temporal fractional Talbot effect,” IEEE Photon. Technol. Lett. **16**, 1855–1857 (2004). [CrossRef]

3. J. Azaña and M. A. Muriel, “Technique for multiplying the repetition rates of periodic pulse trains of pulses by means of a temporal self-imaging effect in chirped fiber gratings,” Opt. Lett. **24**, 1672–1674 (1999). [CrossRef]

5. S. Atkins and B. Fischer, “All-optical pulse rate multiplication using fractional Talbot effect and field-to-intensity conversion with cross-gain modulation,” IEEE Photon. Technol. Lett. **15**, 132–134 (2003). [CrossRef]

6. J. Azaña, P. Kockaert, R. Slavík, L.R. Chen, and S. LaRochelle, “Generation of a 100-GHz optical pulse train by pulse repetition-rate multiplication using superimposed fiber Bragg gratings,” IEEE Photon. Technol. Lett. **15**, 413–415 (2003). [CrossRef]

**24**, 1672–1674 (1999). [CrossRef]

5. S. Atkins and B. Fischer, “All-optical pulse rate multiplication using fractional Talbot effect and field-to-intensity conversion with cross-gain modulation,” IEEE Photon. Technol. Lett. **15**, 132–134 (2003). [CrossRef]

**15**, 413–415 (2003). [CrossRef]

## 2. Temporal Talbot effect as a periodic phase-only filtering

*a*

_{1}(

*t*) which is centered at the optical frequency

*ω*

_{0}and has a repetition period

*T*

_{r}. This pulse train is propagated through a first-order dispersive medium in the linear regime. A first-order dispersive medium is a phase-only filter characterized by a spectral transfer function of the form

*H*(

*ω*)∝exp[

*j*Φ(

*ω*)], where the spectral phase function can be expressed as Φ(

*ω*)=Φ

_{0}+

_{0})+(

_{0})

^{2}, with Φ

_{0}=Φ(ω=ω

_{0}) and

*k*=1, 2. It is well known that the factors Φ

_{0}and

_{0})(constant phase factor and average group delay introduced by the medium, respectively) are not responsible for the distortion experienced by the optical signal after propagation through the dispersive medium and can be consequently ignored in our following analysis. The only factor responsible for the optical signal distortion is the quadratic phase factor, characterized by the term

_{0})

^{2}.

4. J. Azaña and M. A. Muriel, “Temporal self-imaging effects: Theory and application for multiplying pulse repetition rates,” IEEE J. Sel. Top. Quantum Electron. **7**, 728–744 (2001). [CrossRef]

*q*=1, 2, 3,…, and

*m*=2, 3, 4, … such that (

*q/m*) is a non-integer and irreducible rational number. This is the so-called fractional temporal Talbot condition. Under this condition, the original temporal sequence reappears at the output of the dispersive medium (the shape and duration of the individual pulses are not affected by the dispersive process) but with a repetition rate

*m*-times that of the input one. This is true as long as the new repetition period is still longer than the time width of the individual pulses (to avoid interference among them). Assuming nearly transform-limited optical pulses, this additional condition can be mathematically stated as Δω

_{in}>2Δω

_{r,out}, where Δω

_{in}is the total input pulse bandwidth and Δω

_{r,out}=

*mω*

_{r}is the multiplied repetition rate of the generated pulse train.

*ω*

_{r}=2π/

*T*

_{r}. After propagation through the linear dispersive medium, each of the signal’s Fourier components is affected by an additional phase factor. Specifically, the quadratic phase distortion introduced by the dispersive medium on the signal’s discrete Fourier component of order

*p*is Φ

_{p}=Φ′(ω=ω

_{0}+

*pω*

_{r})=(

*p*

^{2}

*p*=0, ±1, ±2, …). If the medium’s dispersion coefficient satisfies the Talbot condition for repetition rate multiplication in Eq. (1), the dispersion-induced phase shifts fulfill the following equality:

*j*|Φ

_{p}|), with |Φp| given by Eq. (2), is a periodic function of

*p*with a period given by the repetition rate multiplication factor

*m*. In particular, |Φ

_{p+m}|=(

*p+m*)

^{2}(

*q/m*)π=|Φ

_{p}|+(

*q.m*)π. The function exp(

*j*|Φ

_{p}| is obviously periodic (with

*m*period) when the product (

*q.m*) is an even number. In the case when (

*q.m*) is an odd number, the phase filter can be also made periodic by assuming a concatenated linear phase filtering operation of the form

*Tr/2m*. The reader can easily prove that in this latter case, exp(

*j*||Φ

_{p+m}+φ

_{p+m}|)=exp(

*j*||Φ

_{p}+φ

_{p}|) for any p. Thus, in the Talbot approach, the dispersive medium distorts the phase of the input discrete spectrum in a periodic fashion, with a period given by the desired output repetition rate, i.e. (

*mω*

_{r}).

*mωr*), in which the phase (or group delay) variation in each period is identical to that in the Talbot dispersive medium, i.e. linear group delay with a slope (dispersion coefficient) given by the Talbot condition in Eq. (1). A schematic of this equivalence is presented in Fig. 1, which shows the spectral phase and group delay responses of two equivalent filtering configurations over 5 spectral periods, assuming a repetition rate multiplication by

*m*=3 and with the q parameter fixed to

*q*=1.

9. K. Miyamoto, “The phase Fresnel lens,” J. Opt. Soc. Am. **51**, 17–20 (1961). [CrossRef]

*L*≈(

*c/2n*

_{eff})Δτ, where c is the speed of light in vacuum and

*n*

_{eff}is the effective refractive index of the mode propagating through the optical fiber. In the case of the conventional Talbot implementation based on a single dispersive medium (e.g. single LCFBG), the total group delay span Δ

*τ*

_{0}must be calculated over the entire input pulse bandwidth (Δ

*ω*

_{in}), see Fig. 1 (top plot); in particular,

*N*=Δ

*ω*

_{in}/ω

_{r,out}>2 is the inverse of the duty-cycle of the generated optical pulse train (assuming transform-limited pulses). In contrast, in the case of a periodic phase-only filtering device, the total group delay span Δ

*τ*

_{1}must be calculated over a single spectral period, see Fig. 1 (bottom plot),

_{r,out}=

*mω*

_{r}. A very relevant feature of a periodic phase-only filter (of period mωr) is that its total group-delay span is shorter than in the equivalent Talbot filter based on a single dispersive medium. This translates into shorter required devices. The difference is more significant for smaller output duty-cycles (i.e. for a larger

*N*).

*q*=1 (case represented in Fig. 1), the total group delay span in the periodic phase-only filter is equal to the input repetition period, Δ

*τ*

_{1}=

*T*

_{r}. In this particular case, the design specifications of the periodic Talbot filter coincide exactly with those previously used for implementing pulse repetition rate multipliers based on SI-LCFBGs [6

**15**, 413–415 (2003). [CrossRef]

## 3. Complete family of periodic Talbot filters

*m*. In principle, this filtering operation can be realized using either of the following solutions: (i) a single dispersive medium (e.g. LCFBG) providing a first-order dispersion that satisfies the Talbot condition in Eq. (1) [3

**24**, 1672–1674 (1999). [CrossRef]

**15**, 132–134 (2003). [CrossRef]

_{r,out}=

*mω*

_{r}, and providing a linear group delay in each period with a slope (dispersion) given by Eq. (1). It should be noted that as compared with the conventional solution based on periodic phase filtering [6

**15**, 413–415 (2003). [CrossRef]

*q*, which can be fixed to be any arbitrary integer as long as (

*q/m*) is a non-integer and irreducible rational number. The conventional periodic filtering solution is a particular case of the more general solution given here (for the case when

*q*=1).

_{r,out}=

*mω*

_{r}(when operating on the corresponding optical pulse train), implies that in general, the filter spectral response is periodic with a period given by

*any*integer multiple of its fundamental period. Based on this generalization, the same rate multiplication process as above can be achieved using a periodic phase-only filter with a spectral period given by

*any integer multiple of the desired output repetition rate*, i.e. with a spectral period given by

*p*ω

_{r,out}=

*p*(

*mωr*) where

*p*=1, 2, 3, …, and where the spectral phase variation in each period is that required to induce the corresponding Talbot effect [in each period, the filter must exhibit a linear group delay with a slope fixed to satisfy the Talbot condition in Eq. (1)]. These general design specifications define a complete family of different periodic phase filters that allow implementing the same Talbot-based rate multiplication process.

**15**, 413–415 (2003). [CrossRef]

*p*=1 in the general family of solutions. Similarly, the solution based on a single dispersive medium is the solution resulting from a sufficiently large value of

*p*so that the spectral period

*pω*

_{r,out}is larger than the input pulse bandwidth, Δω

_{in}<

*p*ω

_{r,out}; in this case, the general periodic filter is composed by a single spectral period [3

**24**, 1672–1674 (1999). [CrossRef]

**15**, 132–134 (2003). [CrossRef]

*p*=2 and

*q*=1). It is important to note that the fact that a periodic optical pulse train can be obtained at the output of a spectrally periodic filter having a spectral period larger than the repetition rate of the output train (by an integer factor

*p*) is a counter-intuitive fact, and this may explain in part why the complete family of periodic Talbot filters have not been found out before. The reader should be reminded that the introduced family of equivalences is only valid assuming a suitable periodic pulse train as the input signal to the filtering systems.

## 4. Design considerations

*p*=1). We also recall that the total group delay span in a Talbot filter based on a single dispersive medium is given by Eq. (3).

*ω*

_{in}using a general Talbot filter of parameter

*p*is

*R*{•} rounds the argument to the nearest integer towards infinity.

*p*), the longer the required filtering device will be but at the same time, the smaller the number of spectral periods in the filter will be, thus resulting in a simpler solution. It should be emphasized that a larger number of spectral periods in the phase filter translates into a higher degree of complexity in regards to its practical realization. For instance, if the filter is implemented using SI-LCFBGs, each different spectral period is realized by a different LCFBG, in such a way that the number of spectral periods fixes the required number of LCFBGs to be superimposed on the same section of optical fiber [6

**15**, 413–415 (2003). [CrossRef]

**24**, 1672–1674 (1999). [CrossRef]

4. J. Azaña and M. A. Muriel, “Temporal self-imaging effects: Theory and application for multiplying pulse repetition rates,” IEEE J. Sel. Top. Quantum Electron. **7**, 728–744 (2001). [CrossRef]

*p*=1) [6

**15**, 413–415 (2003). [CrossRef]

*N*

_{1}=

*R*{

*N*}=

*R*{Δω

_{in}/ω

_{r,out}}. In between the two mentioned extremes, we have now a complete collection of possible solutions where the specific tradeoff between device length and complexity of implementation can be chosen at the designer convenience.

## 5. Numerical example

*T*

_{r}=100ps) composed by 25 transform-limited Gaussian pulses, each with a FWHM time width of 3 ps. The associated total input pulse bandwidth is Δ

*ω*

_{in}/2π≈400 GHz. Three different but equivalent filtering configurations have been designed and numerically tested to multiply the input repetition rate by a factor of

*m*=4 to obtain an output pulse repetition rate of 40 GHz.

*Δτ*

_{0}≈1 ns. If the dispersive filter is implemented using a LCFBG, a grating length

*L*>10 cm is required.

*Δτ*

_{1}=

*T*

_{r}=100ps. If the filter is implemented using SI-LCFBGs, a minimum of 10 LCFBGs should be superimposed on the same section of optical fiber; however, each one of these gratings (i.e. the grating structure) would be as short as

*L*≈1 cm.

*p*=2). The new filter consists of 5 spectral periods and exhibits a linear group delay response in each period with the same slope (dispersion) as for the two previous filters [given by Eq. (1)]. This translates into a total group delay span of

*Δτ*

_{2}=200ps. In practice, this filter can be implemented using only 5 superimposed LCFBGs of length L≈2 cm.

10. L. Cohen, “Time-frequency distributions - A review,” Proc. IEEE **77**, 941–981 (1989) [CrossRef]

**15**, 132–134 (2003). [CrossRef]

*m*=4) on the same input pulse train as in the previous cases. This last filter simply added a π phase shift over one spectral mode every 4 modes of the original signal. The results corresponding to this repetition rate multiplication process are presented in Fig. 4, with the same definitions as for Fig. 3. Though the pulse repetition rate was increased (in intensity) in the same way as in the previous cases, the joint TF distribution reveals a different phase structure for the generated pulse train to that obtained with the Talbot filters. Thus, as expected, a different filtering process to that of the Talbot technique is now implemented.

## 6. Conclusions

**24**, 1672–1674 (1999). [CrossRef]

**15**, 132–134 (2003). [CrossRef]

**15**, 413–415 (2003). [CrossRef]

*T*

_{r}can be multiplied (by a factor

*m*) using a periodic phase-only filter with a spectral period given by any integer multiple of the desired output repetition rate, i.e. with a spectral period given by

*p*ω

_{r,out}=

*p*(

*m*ω

_{r}) where

*p*=1, 2, 3, …, and where the spectral phase variation in each period is that required to induce the corresponding Talbot effect, i.e. in each period, the filter must exhibit a linear group delay with a slope fixed to satisfy the fractional Talbot condition in Eq. (1).

## References and links

1. | K. Tai, A. Tomita, J. L. Jewell, and A. Hasegawa, “Generation of subpicosecond solitonlike optical pulses at 0.3 THz repetition rate by modulation instability,” Appl. Phys. Lett. |

2. | P. Petropoulos, M. Ibsen, M.N. Zervas, and D.J. Richardson, “Generation of a 40 GHz pulse stream by pulse multiplication with a sampled fiber Bragg grating,” Opt. Lett. |

3. | J. Azaña and M. A. Muriel, “Technique for multiplying the repetition rates of periodic pulse trains of pulses by means of a temporal self-imaging effect in chirped fiber gratings,” Opt. Lett. |

4. | J. Azaña and M. A. Muriel, “Temporal self-imaging effects: Theory and application for multiplying pulse repetition rates,” IEEE J. Sel. Top. Quantum Electron. |

5. | S. Atkins and B. Fischer, “All-optical pulse rate multiplication using fractional Talbot effect and field-to-intensity conversion with cross-gain modulation,” IEEE Photon. Technol. Lett. |

6. | J. Azaña, P. Kockaert, R. Slavík, L.R. Chen, and S. LaRochelle, “Generation of a 100-GHz optical pulse train by pulse repetition-rate multiplication using superimposed fiber Bragg gratings,” IEEE Photon. Technol. Lett. |

7. | J. Magné, J. Bolger, M. Rochette, S. LaRochelle, L. R. Chen, B. J. Eggleton, and J. Azaña, “4×100 GHz pulse train generation from a single wavelength 10 GHz mode-locked laser using superimposed fiber gratings and nonlinear conversion,” J. Lightwave Technol.24, 2006 (in press). [CrossRef] |

8. | N. K. Berger, B. Levit, A. Bekker, and B. Fischer, “Compression of periodic optical pulses using temporal fractional Talbot effect,” IEEE Photon. Technol. Lett. |

9. | K. Miyamoto, “The phase Fresnel lens,” J. Opt. Soc. Am. |

10. | L. Cohen, “Time-frequency distributions - A review,” Proc. IEEE |

11. | S. Gupta, P. F. Ndione, J. Azaña, and R. Morandotti, “A new insight into the problem of temporal Talbot phenomena in optical fibers,” Proc. SPIE |

**OCIS Codes**

(070.6760) Fourier optics and signal processing : Talbot and self-imaging effects

(120.2440) Instrumentation, measurement, and metrology : Filters

(320.5540) Ultrafast optics : Pulse shaping

(320.5550) Ultrafast optics : Pulses

**ToC Category:**

Fourier Optics and Optical Signal Processing

**History**

Original Manuscript: March 1, 2006

Revised Manuscript: May 4, 2006

Manuscript Accepted: May 5, 2006

Published: May 15, 2006

**Citation**

José Azaña and Shulabh Gupta, "Complete family of periodic Talbot filters for pulse repetition rate multiplication," Opt. Express **14**, 4270-4279 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-10-4270

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### References

- K. Tai, A. Tomita, J. L. Jewell, and A. Hasegawa, "Generation of subpicosecond solitonlike optical pulses at 0.3 THz repetition rate by modulation instability," Appl. Phys. Lett. 49, 236-238 (1986). [CrossRef]
- P. Petropoulos, M. Ibsen, M.N. Zervas and D.J. Richardson, "Generation of a 40 GHz pulse stream by pulse multiplication with a sampled fiber Bragg grating," Opt. Lett. 25, 521-523 (2000). [CrossRef]
- J. Azaña, and M. A. Muriel, "Technique for multiplying the repetition rates of periodic pulse trains of pulses by means of a temporal self-imaging effect in chirped fiber gratings," Opt. Lett. 24, 1672-1674 (1999). [CrossRef]
- J. Azaña and M. A. Muriel, "Temporal self-imaging effects: Theory and application for multiplying pulse repetition rates," IEEE J. Sel. Top. Quantum Electron. 7, 728-744 (2001). [CrossRef]
- S. Atkins and B. Fischer, "All-optical pulse rate multiplication using fractional Talbot effect and field-to-intensity conversion with cross-gain modulation," IEEE Photon. Technol. Lett. 15, 132-134 (2003). [CrossRef]
- J. Azaña, P. Kockaert, R. Slavík, L.R. Chen, and S. LaRochelle, "Generation of a 100-GHz optical pulse train by pulse repetition-rate multiplication using superimposed fiber Bragg gratings," IEEE Photon. Technol. Lett. 15, 413-415 (2003). [CrossRef]
- J. Magné, J. Bolger, M. Rochette, S. LaRochelle, L. R. Chen, B. J. Eggleton, J. Azaña, "4×100 GHz pulse train generation from a single wavelength 10 GHz mode-locked laser using superimposed fiber gratings and nonlinear conversion," J. Lightwave Technol. 24, 2006 (in press). [CrossRef]
- N. K. Berger, B. Levit, A. Bekker and B. Fischer, "Compression of periodic optical pulses using temporal fractional Talbot effect," IEEE Photon. Technol. Lett. 16, 1855-1857 (2004). [CrossRef]
- K. Miyamoto, "The phase Fresnel lens," J. Opt. Soc. Am. 51, 17-20 (1961). [CrossRef]
- L. Cohen, "Time-frequency distributions - A review," Proc. IEEE 77, 941-981 (1989) [CrossRef]
- S. Gupta, P. F. Ndione, J. Azaña, R. Morandotti, "A new insight into the problem of temporal Talbot phenomena in optical fibers," Proc. SPIE 5971, paper 59710O 1-12 (2005).

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