## Photonic synthesis of high fidelity microwave arbitrary waveforms using near field frequency to time mapping |

Optics Express, Vol. 21, Issue 19, pp. 22974-22987 (2013)

http://dx.doi.org/10.1364/OE.21.022974

Acrobat PDF (1824 KB)

### Abstract

Photonic radio-frequency (RF) arbitrary waveform generation (AWG) based on spectral shaping and frequency-to-time mapping has received substantial attention. This technique, however, is critically constrained by the far-field condition which imposes strict limits on the complexity of the generated waveforms. The time bandwidth product (TBWP) decreases as the inverse of the RF bandwidth which limits one from exploiting the full TBWP available from modern pulse shapers. Here we introduce a new RF-AWG technique which we call near-field frequency-to-time mapping. This approach overcomes the previous restrictions by predistorting the amplitude and phase of the spectrally shaped optical signal to achieve high fidelity waveforms with radically increased TBWP in the near field region.

© 2013 OSA

## 1. Introduction

1. M. Z. Win and R. A. Scholtz, “Ultra-wide bandwidth time-hopping spread-spectrum impulse radio for wireless multiple-access communications,” IEEE Trans. Commun. **48**(4), 679–689 (2000). [CrossRef]

3. A. Dezfooliyan and A. M. Weiner, “Evaluation of time domain propagation measurements of UWB systems using spread spectrum channel sounding,” IEEE Trans. Antenn. Propag. **60**(10), 4855–4865 (2012). [CrossRef]

4. J. D. McKinney, D. E. Leaird, and A. M. Weiner, “Millimeter-wave arbitrary waveform generation with a direct space-to-time pulse shaper,” Opt. Lett. **27**(15), 1345–1347 (2002). [CrossRef] [PubMed]

10. M. H. Khan, H. Shen, Y. Xuan, L. Zhao, S. Xiao, D. E. Leaird, A. M. Weiner, and M. Qi, “Ultrabroad-bandwidth arbitrary radiofrequency waveform generation with a silicon photonic chip-based spectral shaper,” Nat. Photonics **4**(2), 117–122 (2010). [CrossRef]

11. J. Yao, “Photonics for ultrawideband communications,” IEEE Microw. Mag. **10**(4), 82–95 (2009). [CrossRef]

5. J. Chou, Y. Han, and B. Jalali, “Adaptive RF-photonic arbitrary waveform generator,” IEEE Photon. Technol. Lett. **15**(4), 581–583 (2003). [CrossRef]

10. M. H. Khan, H. Shen, Y. Xuan, L. Zhao, S. Xiao, D. E. Leaird, A. M. Weiner, and M. Qi, “Ultrabroad-bandwidth arbitrary radiofrequency waveform generation with a silicon photonic chip-based spectral shaper,” Nat. Photonics **4**(2), 117–122 (2010). [CrossRef]

12. A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. **71**(5), 1929–1960 (2000). [CrossRef]

5. J. Chou, Y. Han, and B. Jalali, “Adaptive RF-photonic arbitrary waveform generator,” IEEE Photon. Technol. Lett. **15**(4), 581–583 (2003). [CrossRef]

10. M. H. Khan, H. Shen, Y. Xuan, L. Zhao, S. Xiao, D. E. Leaird, A. M. Weiner, and M. Qi, “Ultrabroad-bandwidth arbitrary radiofrequency waveform generation with a silicon photonic chip-based spectral shaper,” Nat. Photonics **4**(2), 117–122 (2010). [CrossRef]

13. J. Azana and M. A. Muriel, “Real-time optical spectrum analysis based on the time-space duality in chirped fiber gratings,” IEEE J. Quantum Electron. **36**(5), 517–526 (2000). [CrossRef]

14. V. Torres-Company, D. E. Leaird, and A. M. Weiner, “Dispersion requirements in coherent frequency-to-time mapping,” Opt. Express **19**(24), 24718–24729 (2011). [CrossRef] [PubMed]

*far-field*condition, which imposes a minimum amount of dispersion and limits the complexity of signals that can be handled. For RF-AWG, this condition imposes an onerous limit on the amount of information that can be placed on to the signal, severely restricting the TBWP. The TBWP decreases as the inverse of the RF bandwidth and becomes disappointingly small for RF bandwidth beyond those already available with electronic arbitrary waveform generators.

16. S. Shen and A. M. Weiner, “Complete dispersion compensation for 400-fs pulse transmission over 10-km fiber link using dispersion compensating fiber and spectral phase equalizer,” IEEE Photon. Technol. Lett. **11**(7), 827–829 (1999). [CrossRef]

## 2. Frequency-to-time mapping

12. A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. **71**(5), 1929–1960 (2000). [CrossRef]

*ψ*

_{2}

*) to meet the far-field criterion. As explained above, this condition corresponds to tolerating phase errors up to π/8 within the quadratic factor exp*

_{-min}*(−jt*

^{´}^{2}

*/*2

*ψ*

_{2}

*)*of the Fresnel integral [13

13. J. Azana and M. A. Muriel, “Real-time optical spectrum analysis based on the time-space duality in chirped fiber gratings,” IEEE J. Quantum Electron. **36**(5), 517–526 (2000). [CrossRef]

14. V. Torres-Company, D. E. Leaird, and A. M. Weiner, “Dispersion requirements in coherent frequency-to-time mapping,” Opt. Express **19**(24), 24718–24729 (2011). [CrossRef] [PubMed]

*δt*) are determined by frequency-to-time mapping of the finest optical spectral features (

_{RF}*δf*). Equation (4) shows that the maximum achievable RF bandwidth of the conventional FTM method is proportional to the optical spectral resolution. To synthesize an undistorted RF waveform with larger frequency content, a coarser spectral resolution is required which limits one from exploiting the full TBWP of modern pulse shapers.

*T*) can be expressed as:where

_{RF}*N = T/δt = B/δf*is the ratio of the time aperture and temporal resolution of the shaped optical signal prior to dispersive propagation, or equivalently the ratio of the optical bandwidth and finest spectral feature. In this regime, the TBWP is:

*TBWP*is inversely proportional to the required RF bandwidth and becomes disappointingly small for RF bandwidth beyond those already available with electronic arbitrary waveform generators. Although experiments reaching bandwidths beyond the limit presented in Eq. (6) have been reported [18

_{FTM}18. C. Wang and J. Yao, “Chirped microwave pulse generation based on optical spectral shaping and wavelength-to-time mapping using a sagnac loop mirror incorporating a chirped fiber bragg grating,” J. Lightwave Technol. **27**(16), 3336–3341 (2009). [CrossRef]

## 3. Near-field frequency-to-time mapping

*a*represented by:where as mentioned above,

_{NF-FTM}(t)*a*is defined in terms of the target RF waveform assuming frequency-to-time mapping strictly applies. Here

_{FTM}(t)*a*is multiplied by a new quadratic phase term that cancels out the phase factor exp

_{FTM}(t)*(−jt*

^{´}^{2}

*/*2

*ψ*

_{2}

*)*in Eq. (1). As a result the target waveform

*|a*

_{out}(t)|^{2}that appears in the frequency-to-time mapping expression, Eq. (2), is obtained exactly independent of the far-field condition. Experimentally we realize this condition simply by reprogramming the pulse shaper according to the Fourier transform of Eq. (7); no new physical device is needed.

19. B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. **30**(8), 1951–1963 (1994). [CrossRef]

20. M. T. Kauffman, A. A. Godil, B. A. Auld, W. C. Banyai, and D. M. Bloom, “Applications of time lens optical-systems,” Electron. Lett. **29**(3), 268–269 (1993). [CrossRef]

### 3.1 Theory of near-field frequency-to-time mapping

*(-jt*

^{´}^{2}

*/*2

*ψ*

_{2}

*)*in Eq. (1) to produce the desired RF waveforms. This makes our new NF-FTM method tolerant of small phase errors that may arise in experimental systems.

#### 3.1.1 Maximum RF bandwidth limit

*nth*temporal feature (

*ϕ*) of the quadratic factor applied in Eq. (7) can be written as:Where

_{n}*N,*as defined above, is the total number of resolvable features of the pulse shaper. The maximum phase change from one temporal feature to the next (

*δϕ*) which occurs at the edges of the quadratic phase is:

_{max}*ω = −t/ψ*

_{2}, we have:

*(δϕ*is proportional to the ratio of the generated RF bandwidth

_{max})*(B*to the optical bandwidth. Although the applied temporal quadratic phase shift is essential to get a faithful frequency to time mapping in the near field region, it remains small except at very high RF bandwidth such that

_{RF})*B*approaches the optical bandwidth. For example for ultrabroadband waveforms with bandwidth in the range of ~10 to ~100 GHz, the

_{RF}*B*is orders of magnitude smaller than the optical bandwidth (optical bandwidth of 5 THz is assumed), and the corresponding

_{RF}*δϕ*is limited to the range of only ~0.004π to ~0.04π.

_{max}*(Δf*[17]. This parameter is defined in terms of the time derivative of the applied temporal phase (

_{inst})*δϕ*) as:

*Δf*calculated in Eq. (11) is the shift we would get if we multiplied by a real quadratic phase. Since we cannot increase the optical bandwidth in our method (since we employ time invariant spectral shaping, which implements a virtual not a real time lens), we require the implied instantaneous frequency shift (or equivalently the RF bandwidth) to be much smaller than optical spectrum bandwidth. Otherwise the near-field frequency-to-time mapping process will be disturbed.

_{inst}*(B)*(e.g., for a pulse shaper with 5 THz spectral bandwidth, this number is roughly 625 GHz). This means that NF-FTM can be applied over a very wide microwave frequency range while maintaining waveform fidelity, which is quite distinct from conventional FTM. From another viewpoint, this condition is equivalent to limiting the maximum temporal phase change from one feature to the next to be smaller than π/4:

#### 3.1.2 Time aperture versus RF bandwidth

*TBWP*in near-field frequency-to-time mapping is only a function of pulse shaper’s characteristics. In particular, the available TBWP is equal to one half the number of spectrally resolved control elements within the optical bandwidth. As long as Eq. (12) is satisfied,

_{NF-FTM}*TBWP*is directly proportional to the optical bandwidth and is independent of the targeted RF bandwidth. This is in contrast to Eq. (6), for which the

_{NF-FTM}*TBWP*in conventional frequency-to-time mapping was inversely proportional to the required RF bandwidth.

_{FTM}## 4. Simulation results

21. A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert, “Programmable shaping of femtosecond optical pulses by use of 128-element liquid-crystal phase modulator,” IEEE J. Quantum Electron. **28**(4), 908–920 (1992). [CrossRef]

22. J. T. Willits, A. M. Weiner, and S. T. Cundiff, “Line-by-line pulse shaping with spectral resolution below 890 MHz,” Opt. Express **20**(3), 3110–3117 (2012). [CrossRef] [PubMed]

23. S. Xiao and A. M. Weiner, “Coherent photonic processing of microwave signals using spatial light modulators: programmable amplitude filters,” J. Lightwave Technol. **24**(7), 2523–2529 (2006). [CrossRef]

21. A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert, “Programmable shaping of femtosecond optical pulses by use of 128-element liquid-crystal phase modulator,” IEEE J. Quantum Electron. **28**(4), 908–920 (1992). [CrossRef]

^{5}π, while the maximum temporal phase required to implement the virtual time lens is only ~10π.

*a*, the optical power spectrum for NF-FTM, Fig. 3(d), is a scaled replica of the temporal distortion of Fig. 3(b). A simple derivation explaining this scaling relationship is presented as an appendix. Although Fig. 3(d) shows only the power spectrum, the corresponding field must have the spectral phase function as prescribed by NF-FTM. This is unlike FTM, where input spectral phase does not affect output power spectrum. When this pre-distorted signal propagates through the dispersive medium, a time domain RF waveform with undistorted chirp is obtained, Fig. 3(e), which is indistinguishable from the target waveform–refer to Fig. 3(a), appropriately scaled. The RF spectrum of this signal, Fig. 3(f), extends smoothly out to ~20 GHz with less than 6.5 dB roll-off in respect to the 1 GHz frequency component.

_{FTM}(t)## 5. Experiment

### 5.1 Experimental setup

^{2}. In the experiments we program the pulse shaper assuming only linear group delay dispersion. Higher order dispersion is taken into account in the simulations to most closely model the experiment. The RF signal is detected by a high-speed photodetector with bandwidth of ~50 GHz. A digital sampling oscilloscope and an RF spectrum analyzer with respective bandwidths of 60 GHz and 50 GHz are used to characterize the generated RF waveforms in time and frequency. The optical spectrum is also measured with an optical spectrum analyzer with spectral resolution of 0.01 nm.

### 5.2 Experimental result

### 5.3 Verification of the experiment

^{2}to yield a time aperture of ~6.8 ns (here the dispersion slope is included to allow meaningful comparison of experimental and simulation results). The effect of the dispersion slope with our experimental parameters is to increase the frequency span of the generated chirp function by <4% compared to the targeted frequency span.

## 6. Discussion

## 7. Conclusion

## Appendix

*A*, the Fourier transform of

_{FTM}(ω)*a*, is real.

_{FTM}(t)*a*has a real Fourier transform.

_{FTM}(t)## Acknowledgments

## References and links

1. | M. Z. Win and R. A. Scholtz, “Ultra-wide bandwidth time-hopping spread-spectrum impulse radio for wireless multiple-access communications,” IEEE Trans. Commun. |

2. | M.-G. Benedetto, T. Kaiser, A. F. Molisch, I. Oppermann, C. Politano, and D. Porcino, |

3. | A. Dezfooliyan and A. M. Weiner, “Evaluation of time domain propagation measurements of UWB systems using spread spectrum channel sounding,” IEEE Trans. Antenn. Propag. |

4. | J. D. McKinney, D. E. Leaird, and A. M. Weiner, “Millimeter-wave arbitrary waveform generation with a direct space-to-time pulse shaper,” Opt. Lett. |

5. | J. Chou, Y. Han, and B. Jalali, “Adaptive RF-photonic arbitrary waveform generator,” IEEE Photon. Technol. Lett. |

6. | I. S. Lin, J. D. McKinney, and A. M. Weiner, “Photonic synthesis of broadband microwave arbitrary waveforms applicable to ultra-wideband communication,” IEEE Microw. Wirel. Compon. Lett. |

7. | V. Torres-Company, J. Lancis, and P. Andres, “Arbitrary waveform generator based on all-incoherent pulse shaping,” IEEE Photon. Technol. Lett. |

8. | J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics |

9. | C. Wang and J. Yao, “Photonic generation of chirped millimeter-wave pulses based on nonlinear frequency-to-time mapping in a nonlinearly chirped fiber bragg grating,” IEEE Trans. Microw. Theory Tech. |

10. | M. H. Khan, H. Shen, Y. Xuan, L. Zhao, S. Xiao, D. E. Leaird, A. M. Weiner, and M. Qi, “Ultrabroad-bandwidth arbitrary radiofrequency waveform generation with a silicon photonic chip-based spectral shaper,” Nat. Photonics |

11. | J. Yao, “Photonics for ultrawideband communications,” IEEE Microw. Mag. |

12. | A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. |

13. | J. Azana and M. A. Muriel, “Real-time optical spectrum analysis based on the time-space duality in chirped fiber gratings,” IEEE J. Quantum Electron. |

14. | V. Torres-Company, D. E. Leaird, and A. M. Weiner, “Dispersion requirements in coherent frequency-to-time mapping,” Opt. Express |

15. | A. Dezfooliyan and A. M. Weiner, “Temporal focusing of ultrabroadband wireless signals using photonic radio frequency arbitrary waveform generation,”Optical Fiber Commun. Conf. (OFC), pp. 1–3 (Anaheim, Calif., 2013). |

16. | S. Shen and A. M. Weiner, “Complete dispersion compensation for 400-fs pulse transmission over 10-km fiber link using dispersion compensating fiber and spectral phase equalizer,” IEEE Photon. Technol. Lett. |

17. | A. M. Weiner, |

18. | C. Wang and J. Yao, “Chirped microwave pulse generation based on optical spectral shaping and wavelength-to-time mapping using a sagnac loop mirror incorporating a chirped fiber bragg grating,” J. Lightwave Technol. |

19. | B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. |

20. | M. T. Kauffman, A. A. Godil, B. A. Auld, W. C. Banyai, and D. M. Bloom, “Applications of time lens optical-systems,” Electron. Lett. |

21. | A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert, “Programmable shaping of femtosecond optical pulses by use of 128-element liquid-crystal phase modulator,” IEEE J. Quantum Electron. |

22. | J. T. Willits, A. M. Weiner, and S. T. Cundiff, “Line-by-line pulse shaping with spectral resolution below 890 MHz,” Opt. Express |

23. | S. Xiao and A. M. Weiner, “Coherent photonic processing of microwave signals using spatial light modulators: programmable amplitude filters,” J. Lightwave Technol. |

24. | A. J. Metcalf, V. Torres-Company, V. R. Supradeepa, D. E. Leaird, and A. M. Weiner, “Fully programmable ultra-complex 2-D pulse shaping,” Conf. of Lasers and Electro-Optics (CLEO), pp. 1–2 (San Jose, Calif., 2012). |

**OCIS Codes**

(320.5540) Ultrafast optics : Pulse shaping

(350.4010) Other areas of optics : Microwaves

(060.5625) Fiber optics and optical communications : Radio frequency photonics

**ToC Category:**

Signal Generation and Processing

**History**

Original Manuscript: May 31, 2013

Revised Manuscript: August 15, 2013

Manuscript Accepted: August 19, 2013

Published: September 23, 2013

**Virtual Issues**

Microwave Photonics (2013) *Optics Express*

**Citation**

Amir Dezfooliyan and Andrew M. Weiner, "Photonic synthesis of high fidelity microwave arbitrary waveforms using near field frequency to time mapping," Opt. Express **21**, 22974-22987 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-19-22974

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### References

- M. Z. Win and R. A. Scholtz, “Ultra-wide bandwidth time-hopping spread-spectrum impulse radio for wireless multiple-access communications,” IEEE Trans. Commun.48(4), 679–689 (2000). [CrossRef]
- M.-G. Benedetto, T. Kaiser, A. F. Molisch, I. Oppermann, C. Politano, and D. Porcino, UWB communication systems A comprehensive overview (Hindawi Publishing Corporation, 2006).
- A. Dezfooliyan and A. M. Weiner, “Evaluation of time domain propagation measurements of UWB systems using spread spectrum channel sounding,” IEEE Trans. Antenn. Propag.60(10), 4855–4865 (2012). [CrossRef]
- J. D. McKinney, D. E. Leaird, and A. M. Weiner, “Millimeter-wave arbitrary waveform generation with a direct space-to-time pulse shaper,” Opt. Lett.27(15), 1345–1347 (2002). [CrossRef] [PubMed]
- J. Chou, Y. Han, and B. Jalali, “Adaptive RF-photonic arbitrary waveform generator,” IEEE Photon. Technol. Lett.15(4), 581–583 (2003). [CrossRef]
- I. S. Lin, J. D. McKinney, and A. M. Weiner, “Photonic synthesis of broadband microwave arbitrary waveforms applicable to ultra-wideband communication,” IEEE Microw. Wirel. Compon. Lett.15(4), 226–228 (2005). [CrossRef]
- V. Torres-Company, J. Lancis, and P. Andres, “Arbitrary waveform generator based on all-incoherent pulse shaping,” IEEE Photon. Technol. Lett.18(24), 2626–2628 (2006). [CrossRef]
- J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics1(6), 319–330 (2007). [CrossRef]
- C. Wang and J. Yao, “Photonic generation of chirped millimeter-wave pulses based on nonlinear frequency-to-time mapping in a nonlinearly chirped fiber bragg grating,” IEEE Trans. Microw. Theory Tech.56(2), 542–553 (2008). [CrossRef]
- M. H. Khan, H. Shen, Y. Xuan, L. Zhao, S. Xiao, D. E. Leaird, A. M. Weiner, and M. Qi, “Ultrabroad-bandwidth arbitrary radiofrequency waveform generation with a silicon photonic chip-based spectral shaper,” Nat. Photonics4(2), 117–122 (2010). [CrossRef]
- J. Yao, “Photonics for ultrawideband communications,” IEEE Microw. Mag.10(4), 82–95 (2009). [CrossRef]
- A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum.71(5), 1929–1960 (2000). [CrossRef]
- J. Azana and M. A. Muriel, “Real-time optical spectrum analysis based on the time-space duality in chirped fiber gratings,” IEEE J. Quantum Electron.36(5), 517–526 (2000). [CrossRef]
- V. Torres-Company, D. E. Leaird, and A. M. Weiner, “Dispersion requirements in coherent frequency-to-time mapping,” Opt. Express19(24), 24718–24729 (2011). [CrossRef] [PubMed]
- A. Dezfooliyan and A. M. Weiner, “Temporal focusing of ultrabroadband wireless signals using photonic radio frequency arbitrary waveform generation,”Optical Fiber Commun. Conf. (OFC), pp. 1–3 (Anaheim, Calif., 2013).
- S. Shen and A. M. Weiner, “Complete dispersion compensation for 400-fs pulse transmission over 10-km fiber link using dispersion compensating fiber and spectral phase equalizer,” IEEE Photon. Technol. Lett.11(7), 827–829 (1999). [CrossRef]
- A. M. Weiner, Ultrafast Optics (Wiley, 2009).
- C. Wang and J. Yao, “Chirped microwave pulse generation based on optical spectral shaping and wavelength-to-time mapping using a sagnac loop mirror incorporating a chirped fiber bragg grating,” J. Lightwave Technol.27(16), 3336–3341 (2009). [CrossRef]
- B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron.30(8), 1951–1963 (1994). [CrossRef]
- M. T. Kauffman, A. A. Godil, B. A. Auld, W. C. Banyai, and D. M. Bloom, “Applications of time lens optical-systems,” Electron. Lett.29(3), 268–269 (1993). [CrossRef]
- A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert, “Programmable shaping of femtosecond optical pulses by use of 128-element liquid-crystal phase modulator,” IEEE J. Quantum Electron.28(4), 908–920 (1992). [CrossRef]
- J. T. Willits, A. M. Weiner, and S. T. Cundiff, “Line-by-line pulse shaping with spectral resolution below 890 MHz,” Opt. Express20(3), 3110–3117 (2012). [CrossRef] [PubMed]
- S. Xiao and A. M. Weiner, “Coherent photonic processing of microwave signals using spatial light modulators: programmable amplitude filters,” J. Lightwave Technol.24(7), 2523–2529 (2006). [CrossRef]
- A. J. Metcalf, V. Torres-Company, V. R. Supradeepa, D. E. Leaird, and A. M. Weiner, “Fully programmable ultra-complex 2-D pulse shaping,” Conf. of Lasers and Electro-Optics (CLEO), pp. 1–2 (San Jose, Calif., 2012).

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