# Dispersive Fourier Transformation for Versatile Microwave Photonics Applications

## Abstract

**:**

## 1. Introduction

## 2. Dispersive Fourier Transformation Technique

**Figure 1.**Schematic diagram illustrating dispersive Fourier transformation process in a dispersive device.

#### 2.1. Theory of DFT

#### 2.1.1. Mathematical Description

#### 2.1.2. Impact of Higher-Order Dispersion

#### 2.1.3. Near-Field Condition

#### 2.2. Implementation of DFT

#### 2.2.1. Optical Source

#### 2.2.2. Dispersive Devices

#### 2.2.3. Operational Wavelength Bands of DFT

**Figure 2.**Group delay response of the linearly chirped fiber Bragg grating (LCFBG) operating at 800 nm. One-to-one wavelength-to-time mapping verifies the Dispersive Fourier transformation (DFT) at 800 nm band [77].

#### 2.2.4. Incoherent DFT

## 3. Microwave Photonics Applications of DFT

#### 3.1. DFT for Real-Time Spectroscopy

**Figure 4.**An alternative real-time spectral interferometry system for complete pulse characterization based on unbalanced temporal pulse shaping (UB-TPS). PUT: pulse under test, DE: dispersive element, PD: photodetector. [65].

#### 3.2. DFT for Microwave Arbitrary Waveform Generation

#### 3.2.1. General Concept

**Figure 5.**Schematic diagram to show microwave arbitrary waveform generation based on spectral shaping and DFT. PD: photodetector.

#### 3.2.2. All-Fiber DFT-Based Microwave AWG

**Figure 6.**An optical spectral shaper consisting of two superimposed chirped fiber Bragg gratings with different chirp rates and a small longitudinal offset. The spectral shaper has a chirped spectral response thus can be used to generate chirped microwave waveforms based on DFT technique. CFBG: chirped fiber Bragg grating.

#### 3.2.3. Nonlinear DFT for Photonic Microwave AWG

**Figure 7.**(

**a**) A nonlinearly chirped fiber Bragg grating (NL-CFBG) produced using strain-gradient beam tuning. (

**b**) Simulated group delay characteristics of the produced NL-CFBG at different beam displacements.

#### 3.2.4. “Two-in-One” Design

**Figure 8.**Illustration of the structure and fabrication of a spatially discrete chirped fiber Bragg grating (SD-CFBG) for simultaneous spectral slicing, frequency-to-time mapping and temporal shifting of the input optical pulse, leading to the generation of an optical pulse burst.

#### 3.2.5. Temporal Fourier Transform Pulse Shaping for Photonic Microwave AWG

**Figure 9.**Schematic an unbalanced TPS system for microwave arbitrary waveform generation based on frequency multiplication. MLL: mode-locked laser, LCFBG: linearly chirped fiber Bragg grating, EOM: electro-optic modulator, PD: photodetector.

**Table 1.**Time-bandwidth product (TBWP) of chirped microwave waveforms that are generated by various all-fiber DFT-based microwave AWG systems.

Systems for Generating Chirped Microwave Waveforms | TBWP | Chirp Rate |
---|---|---|

Based on superimposed chirped FBGs [103] | 37.5 | 23.8 GHz/ns |

Based on Sagnac loop mirror with a chirped FBG [104] | 44.8 | 22 GHz/ns |

Based on nonlinear DFT [56] | 8.4 | 74 GHz/ns |

Based on spatially-discrete chirped FBG [39] | 16.8 and 23.2 | 11.2–93.6 GHz/ns |

Based on temporal Fourier transform pulse shaping [114] | 1.8 | 0.715 GHz/ns |

#### 3.3. DFT for Microwave Spectrum Sensing

**Figure 10.**Schematic of real-time microwave spectrum sensing system based on DFT-enabled temporal channelization.

**Figure 11.**Compressive sensing improves the operational bandwidth of the DFT-based microwave spectrum sensing system. (

**a**) Schematic of compressive sensing module based on optical random mixing using a reflective spatial light modulator (R-SLM) with a pseudo-random bit sequence (PRBS) input. (

**b**) Illustration of sample-and-hold integration by spectral filtering in the optical channelizer.

#### 3.4. DFT for Photonic Analog-to-Digital Conversion (ADC)

**Figure 12.**Schematic diagram of DFT-enabled photonic time stretch analog-to-digital conversion (ADC).

## 4. Conclusions

## Acknowledgments

## Conflict of Interest

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**MDPI and ACS Style**

Wang, C. Dispersive Fourier Transformation for Versatile Microwave Photonics Applications. *Photonics* **2014**, *1*, 586-612.
https://doi.org/10.3390/photonics1040586

**AMA Style**

Wang C. Dispersive Fourier Transformation for Versatile Microwave Photonics Applications. *Photonics*. 2014; 1(4):586-612.
https://doi.org/10.3390/photonics1040586

**Chicago/Turabian Style**

Wang, Chao. 2014. "Dispersive Fourier Transformation for Versatile Microwave Photonics Applications" *Photonics* 1, no. 4: 586-612.
https://doi.org/10.3390/photonics1040586