Figure 1

(Color online) Amplified dispersive Fourier transformation. $u_S\left(0\right)$ and $u_S\left(z\right)$ are the normalized amplitude of the input and output Stokes fields, respectively, $z$ is the propagation distance (equivalent to the dispersion length), ${\beta }_m$ are the GVD coefficients at the Stokes frequency, and $u_{p_j}$ is the normalized amplitude of pump field $j$. In the ADFT, the spectrum of each pulse is mapped into a time-domain waveform and simultaneously amplified by distributed Raman amplification.Reuse & Permissions

Figure 2

(Color online) The real (top) and imaginary (bottom) parts of the exponential in the integral in Eq. (17). $\varphi$ is defined to be $\varphi ={\beta }_{2}z{\left(\omega -{\Omega }_S\right)}^2/2=a{\varpi }^2$, where $a={\beta }_{2}z{\Omega }_S^2/2$ and $\varpi =\omega /{\Omega }_S-1$, which is the normalized frequency. $a=1$ is used for the figure. Both $\text{cos}\varphi$ and $\text{sin}\varphi$ oscillate rapidly as $\varpi$ varies from the origin. Only the frequencies in the central region of the two plots contribute to the temporal waveform at a particular time instant. Contributions from all the other frequencies oscillate rapidly between positive and negative values and cancel out when integrated over all frequencies.Reuse & Permissions

Figure 3

(Color online) Top: An input spectrum which is the sum of two Gaussian lines. Bottom: The ADFT waveform of the input spectrum that is mapped into the time domain for various amounts of GVD: (a) 1 km, (b) 3 km, and (c) 9 km. The values used for the dispersion parameter and dispersion slope are $-120\left(\text{ps}/\text{nm}\right)/\text{km}$ and $0.3\left(\text{ps}/{\text{nm}}^2\right)/\text{km}$, respectively. The center wavelength is 1550 nm. As the amount of GVD increases, the input pulse is transformed into a temporal waveform which resembles the input spectrum. It is important to note that the two Gaussian lines can only be resolved when sufficient GVD is used for ADFT.Reuse & Permissions

Figure 4

(Color online) Top: An input spectrum which is the sum of two Lorentzian-shaped absorption lines with a continuum background. Bottom: The ADFT waveform of the input spectrum that is mapped into the time domain for various dispersion lengths: (a) 1, (b) 3, (c) 9, and (d) 27 km. The values used for the dispersion parameter and dispersion slope are $-120\left(\text{ps}/\text{nm}\right)/\text{km}$ and $0.3\left(\text{ps}/{\text{nm}}^2\right)/\text{km}$, respectively. The center wavelength is 1550 nm. As the amount of GVD increases, the input pulse is transformed into a temporal waveform which resembles the input spectrum. It is important to note that the two absorption lines can only be resolved when sufficient GVD is used for ADFT. The ripples on top of the rectangular spectrum shape are less evident for larger GVD, indicating that the transformation is near completion. The ripples at the edges are due to the sharp edges of the input spectrum.Reuse & Permissions

Figure 5

(Color online) The effect of ${\beta }_3$ and the bandwidth ${\Omega }_S-{\omega }_S$ on the frequency-time relation given in Eq. (33) for various values of $dD/d\lambda {\mid }_{{\lambda }_S}$ (the relation between the GVD coefficients and dispersion parameters is given in Appendix ): (a) $-2\left(\text{ps}/{\text{nm}}^2\right)/\text{km}$, (b) $-1\left(\text{ps}/{\text{nm}}^2\right)/\text{km}$, (c) $0\left(\text{ps}/{\text{nm}}^2\right)/\text{km}$, (d) $1\left(\text{ps}/{\text{nm}}^2\right)/\text{km}$, and (e) $2\left(\text{ps}/{\text{nm}}^2\right)/\text{km}$. The dispersion length, the center wavelength, and the dispersion parameter $D\left({\lambda }_S\right)$ used for the figure are $z=50\text{km}$, 1550 nm, and $-120\left(\text{ps}/\text{nm}\right)/\text{km}$, respectively. The higher-order dispersion parameters are not considered here. It is important to note that in absence of the dispersion slope (or when the dispersion parameter is constant), the frequency-time relation is completely linear regardless of the bandwidth. Otherwise, the frequency-time relation is nonlinear and hence needs to be recalibrated. The effect of ${\beta }_3$ becomes evident with larger bandwidth.Reuse & Permissions

Figure 6

(Color online) The spectral resolution given in Eq. (34) as the dispersion length varies for various dispersion parameter values: (a) $-50\left(\text{ps}/\text{nm}\right)/\text{km}$, (b) $-100\left(\text{ps}/\text{nm}\right)/\text{km}$, (c) $-200\left(\text{ps}/\text{nm}\right)/\text{km}$, and (d) $-400\left(\text{ps}/\text{nm}\right)/\text{km}$. The dispersion slope, the center wavelength, and the bandwidth are $-0.3\left(\text{ps}/{\text{nm}}^2\right)/\text{km}$, 1550 nm, and 3 THz, respectively. The higher-order dispersion parameters are not considered here. The spectral resolution improves with larger GVD, as expected.Reuse & Permissions

Figure 7

(Color online) The effect of ${\beta }_3$ and the bandwidth ${\Omega }_S-{\omega }_S$ on the spectral resolution given in Eq. (34) for various values of $dD/d\lambda {\mid }_{{\lambda }_S}$ (the relation between the GVD coefficients and dispersion parameters is given in Appendix ): (a) $-1\left(\text{ps}/{\text{nm}}^2\right)/\text{km}$, (b) $-0.5\left(\text{ps}/{\text{nm}}^2\right)/\text{km}$, (c) $0\left(\text{ps}/{\text{nm}}^2\right)/\text{km}$, (d) $0.5\left(\text{ps}/{\text{nm}}^2\right)/\text{km}$, and (e) $1\left(\text{ps}/{\text{nm}}^2\right)/\text{km}$. The dispersion length, the center wavelength, and the dispersion parameter $D\left({\lambda }_S\right)$ used for the figure are $z=50\text{km}$, 1550 nm, and $-120\left(\text{ps}/\text{nm}\right)/\text{km}$, respectively. The higher-order dispersion parameters are not considered here. The effect of ${\beta }_3$ becomes evident with larger bandwidth.Reuse & Permissions

Figure 8

(Color online) The required temporal resolution of the digitizer or oscilloscope ([(a)–(c)] and the temporal resolution imposed by ADFT [(d)–(e)] for various values of the digitizer bandwidth $f_{\text{dig}}$ and dispersion parameters $\left[D\left({\lambda }_S\right),dD/d\lambda {\mid }_{{\lambda }_S}\right]$: (a) 1 GHz, (b) 3 GHz, (c) 9 GHz, (d) $-60\left(\text{ps}/\text{nm}\right)/\text{km}$ and $0.3\left(\text{ps}/{\text{nm}}^2\right)/\text{km}$, and (e) $-120\left(\text{ps}/\text{nm}\right)/\text{km}$ and $0.3\left(\text{ps}/{\text{nm}}^2\right)/\text{km}$. The center wavelength and bandwidth are 1550 nm and 1 THz, respectively. It is ideal to have an ADFT system whose temporal resolution is limited by the available dispersion. For the system with (a) and (d), a dispersion length of about 125 km is required for the dispersive fiber, whereas for the system with (c) and (e), it is an entirely dispersion-limited system, not influenced by the bandwidth of the digitizer for a dispersion length of 1–1000 km.Reuse & Permissions

Figure 9

(Color online) The digitizer bandwidth required for the given dispersion length for various values of the dispersion parameters $\left[D\left({\lambda }_S\right),dD/d\lambda {\mid }_{{\lambda }_S}\right]$ (the relation between the GVD coefficients and dispersion parameters is given in Appendix ): (a) $-40\left(\text{ps}/\text{nm}\right)/\text{km}$ and $0.3\left(\text{ps}/{\text{nm}}^2\right)/\text{km}$, (b) $-120\left(\text{ps}/\text{nm}\right)/\text{km}$ and $0.3\left(\text{ps}/{\text{nm}}^2\right)/\text{km}$, and (c) $-360\left(\text{ps}/\text{nm}\right)/\text{km}$ and $0.3\left(\text{ps}/{\text{nm}}^2\right)/\text{km}$. The center wavelength and bandwidth are 1550 nm and 1 THz, respectively. As the dispersion increases, the requirement for the bandwidth of the digitizer is relaxed.Reuse & Permissions